User:Ageary/Chapter 11

Problems in Exploration Seismology and their Solutions

Problems-in-Exploration-Seismology-and-their-Solutions.jpg
Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

11.1 Salt lead time as a function of depth

11.1a The velocity of salt is nearly constant at 4.6 km/s. Calculate the amount of lead time per kilometer of salt diameter as a function of depth assuming the sediments have the Louisiana Gulf Coast velocity distribution shown in Figure 11.1a.

Background

Early seismic prospecting for salt domes involved locating geophones in different directions from the source at roughly the same distance from it. Rays that passed through salt arrived earlier than those that did not, the reduction in traveltime due to the high velocity in salt being the lead time.

Solution

The first two columns of Table 11.1a were obtained from the dashed curve in Figure 11.1a. The third column gives the lead time per kilometer of salt, that is, $ \Delta t=(1/V_{i}-1/4.6) $ s/km.

The lead time decreases rapidly with depth to the top of the dome because compaction causes the sediment velocity to increase.

11.1b Early refraction work searching for salt domes in the Gulf Coast considered a significant “lead” to be 0.25 s. Assuming a range of 5.6 km and normal sediment velocity at salt-dome depth of 2.7 km/s, how much salt would this indicate?

Solution

Let $ x $ be the path length in the salt. The lead time is the difference in traveltime for a salt path length of $ x $. Thus,

$ {\begin{aligned}0.25=x\left(1/2.7-1/4.6\right);\quad x=1.6\ {\rm {km}}.\end{aligned}} $

Figure 11.1a.  Gulf Coast interval velocity.

Table 11.1a. Calculation of lead time $ \Delta t $.

$ z $ (km)	$ V_{i} $ (km/s)	$ \Delta t $ (ms/km)
0.25	1.70	371
0.50	1.92	303
0.75	2.11	257
1.00	2.30	217
1.25	2.46	189
1.50	2.63	163
1.75	2.80	140
2.00	2.93	124

11.2 Effect of assumptions on refraction interpretation

The interpretation of refraction measurements necessarily involves a number of assumptions. How do these affect the interpretation?

Background

Refraction measurements are of apparent velocities (the inverses of the slopes) and intercept times observed from time-distance plots. Refraction events are generally defined by several points that approximately line up to define a straight line, which is drawn through the points. Refractor apparent velocity is then determined from the slope of the line and depth from the intercept with the time axis. If the refraction event from shooting in the opposite direction is also observed, the dip and refractor velocity can be determined. Events from shooting in opposite directions must be correlated correctly.

The basic refraction equations generally assume the following properties:

1. Homogeneous isotropic layers of constant velocity,
2. Each layer’s velocity is larger than that of any shallower layer,
3. Planar interfaces,
4. The profile is perpendicular to the strike.

Solution

Uncertainties in the data and correlations and differences between the real situation and the foregoing assumptions affect the interpretation. Where more than one head-wave event is present, the differences in slope must be large enough to distinguish them as separate events (see problem 11.3). Head waves where offsets are large often show shingling, an en-echelon pattern which may make traveltime picks a cycle late.

In the following we assume that the apparent velocities and intercepts are all measured correctly. Assumption (1) of homogeneous constant-velocity overburden is usually not valid and the velocity in the horizontal direction often exceeds that in the vertical direction. One result is errors in calculating refractor depths. Gradual changes in velocity with depth cause raypaths to be bent or curved, changing calculations as to where a critical raypath reaches the refractor (that is, changing the critical distance) and the distance the head wave travels in the refractor. This is generally the most serious violation of the assumptions. The values for velocity above a refractor generally should be obtained from independent data rather than from the refraction data alone.

Failure of assumption (2) that velocity increases monotonically creates depth errors (see problem 11.3). Layers that have smaller velocity than an overlying layer constitute one type of hidden-layer problem. Layers whose thicknesses are so small that their head waves do not become separate distinct events constitute another type of hidden-layer problem. Changes in overburden velocity in the horizontal direction create similar effects, and they also affect the critical angle at the refractor.

Assumption (3), that the refractor is planar, contrasts with the usual objective of mapping the relief on the refracting interface.

A refraction profile not perpendicular to the strike [assumption (4)] simply results in measuring only a component of the dip rather than the entire dip, and probably does not introduce a large error unless the dip is large.

As will be shown in subsequent problems, refraction mapping often uses more complicated methods than simply applying the refraction equations.

11.3 Effect of a hidden layer

Assume that you wish to map the 5.75 and 6.40 km/s formations in the Illinois basin. Given the velocity information shown in Figure 11.3a, what difficulties would you expect to encounter? The shale at 420–620 m and the lower velocity at 790–960 m form “hidden layers”; how much error will neglect of the hidden layers involve?

Solution

The velocity-depth data are summarized in Table 11.3a. Each of the three high-velocity layers will produce a head wave whose apparent velocity is that of the layers if the layering is all horizontal (which we assume, knowing that dips are generally gentle). We calculate the intercept times in order to plot the time-distance curves.

Because the layers are assumed to be horizontal, equation (3.1a) gives the angles of incidence for the ray that produces the head waves. For the 5150 m/s head wave,

$ {\begin{aligned}\sin \theta _{c1}/2650=1/5150,\qquad \theta _{c1}=31.0^{\circ }.\end{aligned}} $

Figure 11.3a.  Illinois Basin interval velocity.

Table 11.3a. Velocity-depth data.

Depth range	Velocity
0–300 m	2650 km/s
300–420	5150
420-620	3650
620-790	5750
790-960	5000
960-1200	5750
1200-1550	6400

We use equation (4.18d) to calculate the intercept time $ t_{i1} $:

$ {\begin{aligned}t_{i1}=2\times 300\cos 31.0^{\circ }/2650=0.194\ {\rm {s}}.\end{aligned}} $

For the 5750 m/s head wave we have

$ {\begin{aligned}\left(\sin \theta _{1}\right)/2650=\left(\sin \theta _{2}\right)/5150=\left(\sin \theta _{c2}\right)/3650=1/5750;\\\theta _{1}=27.4^{\circ },\quad \theta _{2}=63.6^{\circ },\quad \theta _{c2}=39.4^{\circ }.\end{aligned}} $

Its intercept time will be

$ {\begin{aligned}t_{i2}&=2\times \left(300\cos 27.4^{\circ }/2650+120\cos 63.6^{\circ }/5150+200\cos 39.4^{\circ }/3650\right)\\&=0.306\ {\rm {s}}.\end{aligned}} $

To complete the time-distance curve, we have for the 6400 m/s head wave, allowing for 170 m of 5000 m/s layer that interrupts the 5750 medium (note ray direction is the same in both parts at 5750 m/s),

$ {\begin{aligned}\left(\sin \theta _{1}\right)/2650&=\left(\sin \theta _{2}\right)/5150=\left(\sin \theta _{3}\right)/3650\\&=\left(\sin \theta _{4}\right)/5000=\left(\sin \theta _{c3}\right)/5750=1/6400;\\\theta _{1}=24.5^{\circ },\quad \theta _{2}&=53.6^{\circ },\quad \theta _{3}=34.8^{\circ },\quad \theta _{4}=51.4^{\circ },\quad \theta _{c3}=64.0^{\circ }.\end{aligned}} $

Its intercept time will be

$ {\begin{aligned}t_{i3}&=2\times [300\cos 24.5^{\circ }/2650+120\cos 53.6^{\circ }/5150+200\cos 34.8^{\circ }/3650\\&\qquad \qquad +170\cos 51.4^{\circ }/5000+\left(170+240\right)\cos 64.0^{\circ }/5750]\\&=2\times \left(0.103+0.014+0.044+0.021+0.031\right)=0.426\ {\rm {s}}.\end{aligned}} $

The crossover between the 5150 and 5750 m/s head waves is given by

$ {\begin{aligned}0.194+x/5150=0.306+x/5750,\quad {\rm {or}}\quad x=0.112/0.0203=5.52\ {\rm {km}};\end{aligned}} $

Figure 11.3b.  Time-distance plot.

and the crossover between the 5750 and 6400 m/s head waves is given by

$ {\begin{aligned}0.306+x/5.75=0.426+x/6.40,\quad {\rm {or}}\quad x=0.120/0.0176=6.78\ {\rm {km}}.\end{aligned}} $

The 5750 m/s curve will be responsible for first breaks for only 1.30 km.

The data are plotted in Figure 11.3b. Interpretation of this time-distance plot will be difficult because the slopes of the three head-wave curves are nearly the same. The ratios of the successive head-wave velocities in this situation are only 1.12 and 1.11; generally ratios should be 1.25 or larger to be interpreted unambiguously.

Failure to recognize a hidden layer means that the time spent in that layer will be interpreted as spent in a layer with higher velocity, which will make the depth appear too large. The shallow refraction event should be interpreted correctly because there are no hidden layers, but the depth calculated for the deeper interfaces will be too great because of the hidden layers.

If we recognize only the 5150 m/s and 6400 m/s head waves (the most probable situation unless additional information is available), that is, the 5750 m/s layer is a hidden layer, then we would calculate the thickness of the 5150 m/s layer $ h_{2} $ as

$ {\begin{aligned}0.426=2\left(300\cos 24.5^{\circ }/2650+h_{2}\cos 53.6^{\circ }/5150\right)=2[0.103+h_{2}(0.000115)].\end{aligned}} $

This gives $ h_{2}=960 $ m, which, when added to the 300 m thickness of the top layer, gives the depth of the 6400 layer as 1257 m. Comparing with the correct value of 1200 m, the error is 60 m or 5%.

If we should recognize the 5750 m/s head wave, but are not aware of the 3650 m/s layer, we would calculate the thickness of the 5150 m/s layer $ h_{2} $ as

$ {\begin{aligned}0.306=2\left(300\cos 27.4^{\circ }/2650+h_{2}\cos 63.6^{\circ }/5150\right).\end{aligned}} $

This gives $ h_{2}=600 $ m, which, when added to the 300-m thickness of the top layer, gives a depth of 900 m. Comparing with the correct value of 620 m, the error is 280 m or 45%.

The travel through the 170 m thick 5000 m/s layer, if it is not recognized, would probably be assumed to be at the velocity of 5750 m/s, producing a time error of only 4 ms:

$ {\begin{aligned}170\left(1/5000-1/5750\right)=170\left(0.000200-0.000174\right)=4\ {\rm {ms}}.\end{aligned}} $

The error is small because the difference in assumed velocities is small.

11.4 Proof of the ABC refraction equation

Prove the ABC refraction equation [equation (11.4a)].

Background

The ABC equation is often used to calculate the weathering thickness. Assuming reversed profiles as shown in Figure 11.4a and writing $ t_{AC} $, $ t_{BC} $ for the traveltimes from the sources to a geophone at $ C $ and $ t_{AB} $ for the traveltime from $ A $ to $ B $, the ABC equation gives the depth $ h_{C} $ as

$ {\begin{aligned}h_{C}={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)\left[V_{1}V_{2}/\left(V_{2}^{2}-V_{1}^{2}\right)^{1/2}\right].\end{aligned}} $

(11.4a)

Solution

Assuming that $ A $, $ B $, and $ C $ are coplanar and that elevation corrections have been made, we can write

$ {\begin{aligned}V_{1}\left(t_{AC}+t_{BC}-t_{AB}\right)&=V_{1}\left(t_{MC}+t_{NC}-t_{MN}\right)=2V_{1}t_{MC}-V_{1}t_{MN}\\&=2h_{C}/\cos \theta _{c}-MN\left(V_{1}/V_{2}\right)\\&=2h_{C}/\cos \theta _{c}-\left(2h_{C}\tan \theta _{c}\right)\sin \theta _{c}\\&=2h_{C}/\cos \theta _{c}-\left(2h_{C}\sin ^{2}\theta _{c}/\cos \theta _{c}\right)\\&=\left(2h_{C}/\cos \theta _{c}\right)(1-\sin ^{2}\theta _{c})=2h_{C}\cos \theta _{c}.\end{aligned}} $

Figure 11.4a.  The ABC method.

Thus,

$ {\begin{aligned}h_{C}&={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)V_{1}/\cos \theta _{c}\\&={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)V_{1}/[1-(V_{1}/V_{2})^{2}]^{1/2}\\&={\frac {1}{2}}\left(t_{AC}+t_{BC}-t_{AB}\right)V_{1}V_{2}/[V_{2}^{2}-V_{1}^{2}]^{1/2}.\end{aligned}} $

11.5 Adachi’s method

Given the data in Table 11.5a for a reversed refraction profile with sources $ A $ and $ B $, use Adachi’s method to find velocities, depths, and dips.

Background

Adachi (1954; see also Johnson, 1976) derived equations for reversed refraction profiles similar to equations (4.18b,d) but with two important differences: he used angles of incidence measured relative to the vertical ($ \alpha _{i} $ and $ \beta _{i} $ in Figure 11.5a) and vertical depths. The equations are valid for a series of refractors of different dips but with the same strike. Derivation of his equations is lengthy but not difficult (see Sheriff and Geldart, 1995, Section 11.3.2); we quote the final results without proof.

The notation is illustrated in Figure 11.5a where $ \alpha _{i} $ and $ \beta _{i} $ are angles of incidence relative to the vertical at the $ i^{\rm {th}} $ interface for the downgoing rays from sources $ A $ and $ B $, respectively (these are angles of approach at the surface for $ i=1 $), $ a_{i} $ and $ a_{i}^{'} $ are the angles of incidence and refraction for the downgoing ray at interface $ i $, $ b_{i} $ and $ b_{i}^{'} $ are the same for the upcoming ray, $ \xi _{i+1} $ is the dip of the $ i^{\rm {th}} $ interface, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{i} is the vertical thickness of the bed below this interface below the downdip source.

The traveltime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{n} for the refraction along the top of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n^{\rm th} layer is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{n} = \frac{x\sin \beta _{1} }{V_{1} } + \sum\limits_{i=1}^{n-1} \frac{h_{i} }{V_{i} } \left(\cos \alpha _{i} +\cos \beta _{i} \right). \end{align}

(11.5a)

If we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{n} becomes the intercept time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{in} at the downdip source; thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{in} = \sum\limits_{i=1}^{n-1} \frac{h_{i} }{V_{i} } \left(\cos \alpha _{i} +\cos \beta _{i} \right). \end{align}

(11.5b)

Table 11.5a. Reversed refraction times.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x\to	0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0	(km)
$ t_{A}\to $	0.00	0.25	0.50	0.74	0.98	1.24	1.50	1.70	1.81	1.91	2.02	(s)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_B\to	3.00	2.90	2.80	2.68	2.52	2.41	2.31	2.20	2.07	1.91	1.80	(s)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x\to	5.5	6.0	6.5	7.0	7.5	8.0	8.5	9.0	9.5	10.0		(km)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_A\to	2.16	2.28	2.38	2.44	2.56	2.64	2.72	2.80	2.89	3.00		(s)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_B\to	1.65	1.50	1.40	1.25	1.12	1.00	0.75	0.49	0.23	0.00		(s)

Figure 11.5a.  Notation used in Adachi’s equations.

The angles are related as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left.\begin{array}{l} \alpha _{i+1} =a_{i}^{'} +\xi _{i+1} =\alpha _{i+1} +\xi _{i+2},\\ \beta _{i+1} =b_{i}^{'} -\xi _{i+1} =b_{i+1} -\xi _{i+2}. \end{array} \right\} \end{align}

(11.5c)

Snell’s law [equation (3. 1a)] gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sin \alpha _{i}' =\left(V_{i+1} /V_{i} \right)\sin a_{i},\quad \sin b_{i}' =\left(V_{i+1} /V_{i} \right)\sin b_{i}. \end{align}

(11.5d)

For the refraction along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n^{\rm th} interface,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a_{n} =b_{n} =\theta _{cn} =\left(\alpha _{n} +\beta _{n} \right)/2,\qquad \xi _{n+1} =\left(\alpha _{n} -\beta _{n} \right)/2. \end{align}

(11.5e)

The initial interpretation stage is plotting the data and determining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} and the apparent velocities $ V_{un} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{dn} , and intercept times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{in} for each of the refraction events. The angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{1} are given by equation (4.2d). Next we use problem 4.24b to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{2} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{1} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{1} . The depth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{1} is now found using equation (11.5b).

For the next interface we find new values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{1} using the next pair of apparent velocities. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{2} is now known, we use equation (11.5c) to get new values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{1} , after which equation (11.5d) gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a_{1'} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{1'} and equation (5.11c) gives $ \alpha _{3} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{3} . We can now find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{3} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{2} .

Figure 11.5b.  Plot of the data in Table 11.5a.

Solution

Figure 11.5b shows the plotted data and the measured slopes and time intercepts. The average value of the near-surface velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} is 2.02 km/s. Two refraction events are observed with the apparent velocities and intercept times listed below.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{d2} =3.73\ {\rm km/s},\quad V_{u2} = 4.51\ {\rm km/s},\quad t_{i1} =0.92\ {\rm s};\\ V_{d3} =4.29\ {\rm km/s},\quad V_{u3} = 5.81\ {\rm km/s},\quad t_{i2} =1.28\ {\rm s}. \end{align}

First we calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{1} and $ \beta _{1} $:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \alpha _{1} =\sin ^{-1} \left(V_{1} /V_{d2} \right)=32.8^{\circ},\quad \beta _{1} =\sin ^{-1} \left(V_{1} /V_{u2} \right)=26.6^{\circ}. \end{align}

Equation (11.5c) gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{1} =a_{1} +\xi _{2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{1} =b_{1} -\xi _{2} . Since this interface is the refractor, equation (11.5e) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a_{1} &=b_{1} =\theta _{c1} =\left(\alpha _{1} +\beta _{1} \right)/2=29.7^{\circ}, \\ \xi _{2} &=\left(\alpha _{1} -\beta _{1} \right)/2=3.1^{\circ}, \\ V_{2} &=V_{1} /\sin \theta _{c1} =2.02/\sin 29.7^{\circ} =4.08\ {\rm km/s}. \end{align}

{Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{Checking:}\quad \quad V_{2} &=[(1/V_{d2} +1/V_{u3} )^{2} /2]^{-1} =4.08\ {\rm km/s}. \end{align} }

We find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{1} using equation (11.5b): so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h_{1} &= V_{1} t_{i1} /\left(\cos \alpha _{1} +\cos \beta _{1} \right)\\ &=2.02\times 0.92/\left(\cos 32.8^{\circ} +\cos 26.6^{\circ} \right)=1.07\ {\rm km}. \end{align}

For the second refractor, we calculate new angles of approach:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \alpha _{1} =\sin^{-1} \left(V_{1} /V_{d3} \right)=\sin^{-1} \left(2.02/4.29\right)=28.1^{\circ}, \\ \beta _{1} =\sin^{-1} \left(V_{1} /V_{u3} \right)=\sin^{-1} \left(2.02/5.81\right)=20.3^{\circ}. \end{align}

Then equation (11.5c) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a_{1} = \alpha_{1} - \xi _{2} = 28.1^{\circ} - 3.1^{\circ} = 25.0^{\circ}, \\ b_{1} = \beta_{1} + \xi _{2} = 20.3^{\circ} + 3.1^{\circ} = 23.4^{\circ}. \end{align}

Using equation (11.5d), we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a_{1^{'}} =\sin ^{-1} \left[\left(V_{2} /V_{1} \right)\sin a_{1} \right]=\sin ^{-1} [(4.08/2.02) \sin 25.0^{\circ} ]=58.6^{\circ}, \\ b_{1^{'}} =\sin ^{-1} \left[\left(V_{2} /V_{1} \right)\sin b_{1} \right]=\sin ^{-1} [(4.08/2.02) \sin 23.4^{\circ} ]=53.3^{\circ}. \end{align}

From equation (11.5c) we now get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \alpha _{2} =a_{1^{'}} + \xi _{2} =58.6^{\circ} +3.1^{\circ} =61.7^{\circ}, \\ \beta _{2} =b_{1^{'}} - \xi _{2} =53.3^{\circ} -3.1^{\circ} =50.2^{\circ}. \end{align}

From equation (5.11e) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a_{2} &= b_{2} =\theta _{c2} =\left(\alpha _{2} +\beta _{2} \right)/2=56.0^{\circ}, \\ \xi_{3} &=\left(\alpha _{2} -\beta _{2} \right)/2=5.8^{\circ}, \\ V_{3} &=V_{2} /\sin \theta _{c2} =4.08/\sin 56.0^{\circ} =4.92\ {\rm km/s}. \end{align}

{Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{Checking:}\quad \quad V_{3} &=[\left(1/V_{d3} +1/V_{u3} \right)/2]^{-1} =4.94\ {\rm km/s}. \end{align} }

Finally, we get the depth from equation (11.5b):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{i2} &=\left(h_{1} /V_{1} \right)\left(\cos \alpha _{1} +\cos \beta _{1} \right)+\left(h_{2} /V_{2} \right)\left(\cos \alpha _{2} +\cos \beta _{2} \right)\\ &=t_{u1} +\left(h_{2} /V_{2} \right)\left(\cos \alpha _{2} +\cos \beta _{2} \right), \\ h_{2} &=\left(1.28-0.92\right)\times 4.08/\left(\cos 61.7^{\circ} +\cos 50.2^{\circ} \right)=1.32\ {\rm km}. \end{align}

Total vertical depth at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A=h_{1} +h_{2} =1.07+1.32=2.39 km.

11.6 Refraction interpretation by stripping

11.6a Solve problem 11.5 by stripping off the shallow layer.

Background

Stripping is a method of interpreting refraction data by removing the effect of upper layers, the removal being accomplished by reducing the traveltimes and distances so that in effect the source and geophones are located on the interface at the base of the “stripped” layer. Stripping can be accomplished by calculation or graphically, or by a combination.

Solution

We wish to compare our results with those of problem 11.5, so we use the same measurements, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} =2.02 km/s and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{d2} =3.73\ {\rm km/s},\quad V_{u2} =4.51\ {\rm km/s},\quad t_{u1} =0.92\ {\rm s},\quad t_{d1} =0.46\ {\rm s}; \\ V_{d3} =4.29\ {\rm km/s},\quad V_{u3} =5.81\ {\rm km/s},\quad t_{u2} =1.28\ {\rm s},\quad t_{d2} =0.66\ {\rm s}. \end{align}

(To avoid triple subscripts, we denote intercept times at downdip and updip source locations by $ d $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u .)

We start by using equations (4.24f) to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 1/V_{2} =\left(1/V_{d2} +1/V_{u2} \right)/2,\quad V_{2} =4.08\ {\rm km/s}. \end{align}

Equations (4.24b,d) can be written

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sin \left(\theta _{c1} +\xi _{2} \right)=V_{1} /V_{d2},\qquad \sin (\theta_{c1} - \xi_2) = V_1/V_{u2}; \\ \hbox{so}\quad \quad \sin(\theta_{c1}+\xi_2) = 2.02/3.73,\qquad (\theta_{c1} + \xi_2) = 32.8^{\circ}; \\ \sin(\theta_{c1} - \xi_2) = 2.02/4.51,\qquad (\theta_{c1} - \xi_2) = 26.6^{\circ}; \end{align}

hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c1} =29.7^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{2} =3.1^{\circ} . These are the same as those in problem 11.5.

Next we calculate the distances perpendicular to the first refractor at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B (Figure 11.6a). We use equation (4.24b) to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{d} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{u} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h_{d} =\left(V_{1} t_{d1} \right)/2\cos \theta _{c1} =2.02\times 0.46/2\times \cos 29.7^{\circ} =0.53\ {\rm km}; \\ h_{u} =\left(V_{1} t_{u1} \right)/2\cos \theta _{c1} =2.02\times 0.92/2\times \cos 29.7^{\circ} =1.07\ {\rm km}. \end{align}

These results are identical with those in problem 11.5. We verify the dip using these depths:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \xi _{2} =\tan^{-1} \left[\left(1.07-0.53\right)/10.0\right]=3.1^{\circ}. \end{align}

The first step in stripping is to plot the shallow refractor; we do this by swinging arcs with centers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B and radii 1.07 and 0.53 km, the refractor being tangent to the two arcs. To get the “stripped” time values, we subtract the times down to and up from the first refractor, i.e., traveltimes along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): AA' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BB' for sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B . Although maximum accuracy would be achieved by stripping times for all geophones, the curves for the shallow refraction are so nearly linear that we calculate the stripped times only for each source and one intermediate point on each profile (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N ). We obtain the required distances by measuring the paths in Figure 11.6a. Calculation of the stripped times is given below. Path lengths: $ AA'\approx 1.30 $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN\approx 0.83 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BB'\approx 0.65 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ\approx 1.08 km.

Figure 11.6a.  Stripping for refraction interpretation. The numbers below the zero-time line are distances from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A' .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{Time along path:}\quad \quad \ AA'\approx 0.64, \ MN\approx 0.41, \ BB'\approx 0.32, \ PQ\approx 0.53\ {\rm s} \\ \hbox{Time along path:}\quad \quad \ AA'MN =2.50\ {\rm s} \\ \hbox{Stripped time for}\quad \quad \ AA'MN \approx 2.50-\left(0.64+0.41\right)\approx 1.45\ {\rm s}\ ({\rm point}\ E) \\ \hbox{Time along path}\quad \quad \ AA'B'B =3.00\ {\rm s} \\ \hbox{Stripped time for}\quad \quad \ AA'B'B \approx 3.00-\left(0.64+0.32\right)\approx 2.04\ {\rm s}\ ({\rm points}\ F, H) \\ \hbox{Time along path}\quad \quad \ BB'P Q=2.30\ {\rm s} \\ \hbox{Stripped time for}\quad \quad \ BB'PQ \approx 2.30-\left(0.32+0.53\right)\approx 1.45\ {\rm s}\ ({\rm point}\ G) \end{align}

Stripping off the first refractor in effect moves sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and $ B $ down to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B' and geophones at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N down to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M , so the stripped times are plotted above these shifted points, the new traveltimes curves being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): EF and $ GH $. Measurements on these stripped curves give the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{d3} =4.65\ {\rm km/s},\quad V_{u3} =5.13\ {\rm km/s},\quad t_{d3} =0.19\ {\rm s},\quad t_{u3} =0.30\ {\rm s}. \end{align}

Table 11.6a. Comparison of results of Adachi’s and stripping method.

Item	Problem 11.5	Problem 11.6	Difference
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{3} ({\hbox {km/s}})	4.92	4.88	0.8%
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 56.0^{\circ}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 56.7^{\circ}	1%
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5.8^{\circ}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5.7^{\circ}	2%
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{2}^{*} ({\hbox {km}})	1.32	$ 1.20^{*} $	10%

*Vertical depth measured at source A.

We now get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{3} &=[\left(1/4.65+1/5.13\right)/2]^{-1} =4.88\ {\rm km/s},\\ \theta _{c2} &=\sin ^{-1} \left(4.08/4.88\right)=56.7^{\circ}, \\ h_{u2} &=V_{2} t_{u3} /2\cos \theta _{c2} =4.08\times 0.30/2 \cos 56.7^{\circ} =1.11\ {\rm km}, \\ h_{d2} &=4.08\times 0.19/2 \cos 56.7^{\circ} =0.71\ {\rm km}, \\ \xi _{3} &=\tan^{-1} \left[\left(h_{u2} -h_{d2} \right)/A'B'\right]=\tan^{-1} \left[\left(1.11-0.71\right)/8.90\right]=2.6^{\circ}. \end{align}

This dip is relative to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A'B' , so the total dip is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{2} +\xi _{3} =\left(3.1^{\circ} +2.6^{\circ} \right)=5.7^{\circ}

11.6b Compare the solutions by stripping with those using Adachi’s method (problem 11.5).

Solution

To compare depths, we measured vertical depths below A. Results for the first layer are the same for both methods, those for the next layer are given in Table 11.6a.

11.6c What are some of the advantages and disadvantages of stripping?

Solution

Advantages of stripping:

Easy to understand

Straightforward in application

Can be used with beds of different dips if the strike is the same

As rapid as other methods when done graphically

Can be used to interpret irregular or curved surfaces

Disadvantages:

Very sensitive to velocity errors

Like most methods, assumes the same strike for all refractors

Difficult to apply when dips are steep

Figure 11.7a.  Refractors with the same strike but different dips.

11.7 Proof of a generalized reciprocal method relation

Prove equation (11.7a), assuming that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\xi _{i} -\xi _{j} \right)\approx 0 for all values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): j .

Background

The generalized reciprocal method (GRM) can be used with beds of different dips provided all have the same strike. Figure 11.7a shows a series of such beds. Depths normal to the beds are denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z_{Ai} and $ z_{Bi} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \beta _{i} are angles of incidence, those for the deepest interface being critical angles, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{i} is the dip of the interface at the top of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i^{\rm th} layer. To get the traveltime from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AB} , we consider a plane wavefront Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ that passes through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A at time $ t=0 $ in a direction such that it will be totally refracted at one of the interfaces, the third in the case of Figure 11.7a. The wavefront reaches Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AC} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AR} where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{AC} =\left(z_{A1} \cos \alpha _{1} /V_{1} \right),\quad t_{AR} = \sum\limits_{i=1}^{3} z_{Ai} \cos \alpha _{i} /V_{i}. \end{align}

The same wavefront will travel upward from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B in time

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{BT} = \sum\limits_{i=1}^{3} z_{Bi} \cos \beta _{i} /V_{i}. \end{align}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha _{3} is the critical angle,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{AB} = \sum\limits_{i=1}^{3} (z_{Ai} +z_{Bi} )/V_{i} +RV/V_{4}. \end{align}

Generalizing, we get for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n layers

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{AB} =\sum\limits_{i=1}^{n-1} (z_{Ai} +z_{Bi} )/V_{i} +RV/V_{n}. \end{align}

But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RV=YJ=EJ\cos \left(\xi _{3} -\xi _{2} \right)=AB\cos \xi _{1} \cos \left(\xi _{2} -\xi _{1} \right)\cos \left(\xi _{3} -\xi _{2} \right) ; for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n layers, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{AB} =\sum\limits_{i=1}^{n-1} (z_{Ai} +z_{Bi} )/V_{i} +AB\left(S_{n} /V_{n} \right), \end{align}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} S_{n} =\cos \xi _{1} \cos \left(\xi _{2} -\xi _{1} \right)\ldots \cos \left(\xi _{n-1} -\xi _{n-2} \right)\approx \cos \xi _{n-1}, \end{align}

(11.7a)

all differences in dip being small, that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\xi _{i} -\xi _{j} \right)\approx 0 . We shall not carry the derivation of the GRM formulas beyond this point; those who are interested should consult Sheriff and Geldart, 1995, Section 11.3.3, or Palmer (1980).

Solution

We are asked to prove that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \cos \xi _{1} \cos \left(\xi _{2} -\xi _{1} \right)\cos \left(\xi _{3} -\xi _{2} \right)\ldots \cos \left(\xi _{n-1} -\xi _{n-2} \right)\approx \cos \xi _{n-1}, \end{align}

where the differences in dip are all small, that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{j} -\xi _{j-1} \approx 0 . We start with the single cosine on the right-hand side of the equation and try to express it as a product of cosines. We write it as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \cos \left(\xi _{n-1} -\xi _{m} \right) and expand:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \cos \left(\xi _{n-1} -\xi _{m} \right)=\cos [\left(\xi _{n-1} -\xi _{n-2} \right)+(\xi _{n-2} -\xi _{m})]. \end{align}

Since all differences in dip are small, we expand the right-hand side and set the products of the sines equal to zero. Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \cos \left(\xi _{n-1} -\xi _{m} \right)\approx \cos \left(\xi _{n-1} -\xi _{n-2} \right)\cos \left(\xi _{n-2} -\xi _{m} \right). \end{align}

Next we treat the right-hand cosine in the same way, writing it as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \cos [\left(\xi _{n-2} -\xi _{n-3} \right)+(\xi _{n-3} -\xi _{m})] . We now expand the factor and drop the sine term. Continuing in this way we eventually arrive at the result

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \cos \left(\xi _{n-1} -\xi _{m} \right) &\approx \cos \left(\xi _{n-1} -\xi _{n-2} \right)\cos \left(\xi _{n-2} -\xi _{n-3} \right)\; \ldots \\ &\qquad \times \cos \left(\xi _{2} -\xi _{1} \right)\cos \left(\xi _{1} -\xi _{m} \right). \end{align}

We now take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi _{m} =0 and the result is equation (11.7a).

Figure 11.8a.  Illustrating delay time.

11.8 Delay time

Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): NQ in Figure 11.8a is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} NQ=V_{2} \delta_{NQ} \tan ^{2} \theta_{c}, \\ \hbox{i.e.,}\quad \quad \delta _{g} =\delta _{NQ} =NQ/V_{2} \tan ^{2} \theta_{c}. \end{align}

(11.8a)

Background

The concept of delay time has found wide application in refraction interpretation (see problems 11.9 and 11.11). We define the delay time associated with the refraction path SMNG in Figure 11.8a as the observed traveltime minus the time required to travel from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q at the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ is the projection of the path SMNG onto the refractor), Writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta for the total delay time, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta &=t_{SG} -PQ/V_{2} \end{align}

(11.8b)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &=\left(\frac{SM+NG}{V_{1} } +\frac{MN}{V_{2} } \right)-\frac{PQ}{V_{2} } =\left(\frac{SM}{V_{1} } -\frac{PM}{V_{2} } \right)+\left(\frac{NG}{V_{1} } -\frac{NQ}{V_{2} } \right) \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &=\delta_{s} + \delta_{g}, \end{align}

(11.8c)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{where}\quad \quad \delta_{s} = \hbox{source delay time}\ = \left(\frac{SM}{V_{1} } -\frac{PM}{V_{2} } \right) \end{align}

(11.8d)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{and}\quad \quad \delta_{g} = \hbox{geophone delay time}\ = \left(\frac{NG}{V_{1} } -\frac{NQ}{V_{2} } \right). \end{align}

(11.8e)

Solution

Referring to Figure 11.8a, we have, by definition,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta_{g} &=NG/V_{1} -NQ/V_{2} \\ &=NQ\left(\frac{1}{V_{1} \sin \theta _{c} } -\frac{1}{V_{2} } \right)=\frac{NQ}{V_{2} } \left(\frac{1}{\sin ^{2} \theta _{c} } -1\right)\\ &=NQ/V_{2} \tan ^{2} \theta _{c}. \end{align}

11.9 Barry’s delay-time refraction interpretation method

Source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B is 2 km east of source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A . The data in Table 11.9a were obtained with cables extending eastward from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x is the distance measured from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A ) with geophones at 200 m intervals. Interpret the data using Barry’s method (Barry, 1967); Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} = 2.50 km/s. Assume that the delay-time curve for the reverse profile is sufficiently parallel to yours that step (d) below can be omitted.

Background

Barry’s method requires that the total delay time be separated into source and geophone delay times. Two sources on the same side of the geophone are used to achieve this. In Figure 11.9a. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B are sources, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R are geophones, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BQ being the critical distance (problem 4.18) for source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B . We write $ \delta _{A} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{B} for the source delay times, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{Q} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{R} for the geophone delay times, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{AR} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{BR} , etc., for the total delay times. We get the source delay times from the intercept times if we assume zero dip [see equation (11.9a)]. The delay time at source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{B} , is due to travel along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BN , so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta_{B} &= BN/V_{1} - N'N/V_{2} =h_{N} /V_{1} \cos \theta _{c} - \left(h_{N} \tan \theta _{c} \right)/V_{2} \\ &=\left(h_{N} / V_{1} \cos \theta _{c} \right)\left(1-\sin ^{2} \theta _{c} \right)=\left(h_{N} \cos \theta _{c} \right)/V_{1} =\frac{1}{2} t_{iB}; \end{align}

(11.9a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{and so,}\quad \quad h_{N} =V_{1} \delta_{B} /\cos \theta _{c} =\frac{1}{2} V_{1} t_{iB} /\cos \theta _{c}. \end{align}

(11.9b)

Figure 11.9a.  Determining delay times.

Note that equations (11.9a,b) apply at any point on the profile where the dip is very small, not merely at souce points.

To find the geophone delay times we have from equation (11.8c)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta_{AQ} &=\delta_{A} +\delta_{Q},\\ \delta_{AR} &=\delta_{A} +\delta_{R},\\ \Delta \delta &= \delta_{AQ} - \delta_{AR} =\delta_{q} - \delta_{R} = differential \ delay \ time. \end{align}

For zero dip, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{B} =\delta_{Q} , so we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{1}{2} \left(\delta_{BR} +\Delta \delta\right)=\frac{1}{2} \left(\delta_{Q} +\delta_{R} +\delta_{Q} -\delta_{R} \right)\delta_{Q} -\delta_{R} =\delta_{Q}, \end{align}

(11.9c)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{hence}\quad \quad \frac{1}{2} \left(\delta_{BR} -\Delta \delta\right)=\delta_{R}. \end{align}

(11.9d)

Table 11.9a. Time-distance data.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} (s)	$ t_{B} $ (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{B} (s)
2.6	1.02	0.25	5.4	1.62	1.28
2.8	1.05	0.34	5.6	1.66	1.31
3.0	1.10	0.43	5.8	1.72	1.36
3.2	1.24	0.52	6.0	1.75	1.42
3.4	1.18	0.61	6.2	1.80	1.47
3.6	1.20	0.70	6.4	1.85	1.53
3.8	1.26	0.78	6.6	1.91	1.56
4.0	1.32	0.87	6.8	1.97	1.59
4.2	1.35	0.96	7.0	2.00	1.63
4.4	1.39	1.05	7.2	2.02	1.67
4.6	1.45	1.10	7.4	2.05	1.70
4.8	1.50	1.14	7.6	2.10	1.73
5.0	1.56	1.20	7.8	2.13	1.78
5.2	1.59	1.22	8.0	2.16	1.81

To use these equations we must find the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q , preferably by expressing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BQ in terms of delay times. From Figure 11.9a and equation (11.9b) we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} BQ = 2h_{N} \tan \theta _{C} =2\left(V_{1} \delta_{B} /\cos \theta _{c} \right)\tan \theta _{c} =2V_{2} \ \delta_{B} \ \tan ^{2} \theta _{c}. \end{align}

(11.9e)

Interpretation involves the following steps:

1. The traveltimes are corrected for weathering and elevation (problem 8.18)
2. Total delay times are calculated and plotted at the geophone positions
3. The distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PP' in Figure 11.9a is calculated for each geophone using equation (11.8a), and the total delay times shifted the distances Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PP' toward Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A
4. The curves in (b) and (c) should be parallel; if not, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} is adjusted until the curves are sufficiently close to being parallel
5. The total delay times in (b) are separated into source and geophone delay times and then plotted above points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N , and $ P $. Delays times can be converted into depths using equation (11.9b)

Solution

The data are plotted in Figure 11.9b. Measurements give an average value of 4.60 km/s for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} and intercept times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{iA} =0.45 s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{iB} =0.55 s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{B} =0.55/2=0.28 s. The critical angle is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} =\sin ^{-1} \left(2.50/4.60\right)=32.9^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \cos \theta _{c} =0.840 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan \theta _{c} =0.647 . Using equation (11.9e), we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} BQ=2V_{2} \delta_{B} \tan ^{2} \theta _{c} =2\times 4.60\times 0.28\times 0.647^{2} =1.1\ {\rm km}. \end{align}

Figure 11.9b.  Plot of data.

Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q is located at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=3.1 km. Also, we need Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{AQ} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta_{AQ} =t_{AQ} -x_{AQ} /V_{2} =1.12-3.1/4.60=0.45\ {\rm s}. \end{align}

Figure 11.9b shows that we observe refraction data from both sources only for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x\ge 4.6 km. We show the calculations in Table 11.9b. Column 1 is the offset measured from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A , columns 2 and 6 are traveltime for sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B , columns 3 and 7 are the source-to-geophone distances divided by $ V_{2} $, columns 4 and 8 are the total delay times [the differences between columns 2 and 3, 6 and 7, respectively—see equation (11.8b)], column 5 is the differential delay time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \delta between geophones at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R=\delta_{AQ} -\delta_{AR} =\left(0.45-\delta_{AR} \right) , column 9 is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{R} =\left(\delta_{BR} -\Delta \delta\right)/2 [see equation (11.9c)], column 10 is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PP' , column 11 is column 1 minus column 10 = location of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P' in Figure 11.9a. Depth values can be obtained by mutliplying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{PR} in column 9 by $ V_{1}/\cos \theta _{c}=2.98 $ [see equation (11.9a)].

We used a new value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{AQ} =1.123-3.10/4.60=0.449s . Comparing the new and old values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{PR} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=5.0 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=5.6 , we see that rounding errors are not responsible for the anomalies. The anomaly at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=5.6 km is 0.01 s whereas the original data are also given to the nearest 0.01 s, so this anomaly could be the result of rounding off of the original time values; however, the anomaly at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=5.0 is too large to be due to this.

Table 11.9b. Delay-time calculations.

1	2	3	4	5	6	7	7	8	10	11
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x/V_2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{AR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta\delta	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{BR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'/V_2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{BR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_R	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PP'	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x - PP'
4.6	1.45	1.00	0.45	0.00	1.10	0.57	0.53	0.27	0.52	4.08
4.8	1.50	1.04	0.46	–0.01	1.14	0.61	0.53	0.27	0.52	4.28
5.0	1.56	1.09	0.47	–0.02	1.20	0.65	0.55	0.29	0.56	4.44
5.2	1.59	1.13	0.46	–0.01	1.22	0.70	0.52	0.27	0.52	4.68
5.4	1.62	1.17	0.45	0.00	1.28	0.74	0.54	0.27	0.52	4.88
5.6	1.66	1.22	0.44	0.01	1.31	0.78	0.53	0.26	0.50	5.10
5.8	1.72	1.26	0.46	–0.01	1.36	0.83	0.53	0.27	0.52	5.28
6.0	1.73	1.30	0.43	0.02	1.42	0.87	0.55	0.27	0.52	5.48
6.2	1.80	1.35	0.45	0.00	1.47	0.91	0.56	0.28	0.54	5.66
6.4	1.85	1.39	0.46	–0.01	1.53	0.96	0.57	0.29	0.56	5.84
6.6	1.91	1.43	0.48	–0.03	1.56	1.00	0.56	0.30	0.58	6.02
6.8	1.97	1.48	0.49	–0.04	1.59	1.04	0.55	0.30	0.58	6.22
7.0	2.00	1.52	0.48	–0.03	1.63	1.09	0.54	0.29	0.56	6.44
7.2	2.02	1.57	0.45	0.00	1.67	1.13	0.54	0.27	0.52	6.68
7.4	2.05	1.61	0.44	0.01	1.70	1.17	0.53	0.26	0.50	6.90
7.6	2.10	1.65	0.45	0.00	1.73	1.22	0.51	0.26	0.50	7.10
7.8	2.13	1.70	0.43	0.02	1.78	1.26	0.52	0.25	0.48	7.32
8.0	2.16	1.74	0.42	0.03	1.81	1.30	0.51	0.24	0.46	7.54

Note. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'=x-2 .

Table 11.9c. Part of Table 11.9b with increased precision.

1	2	3	4	5	6	7	8	9
$ x $	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x/V_2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{AR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta\delta	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{BR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'/V_2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{BR}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{PR}
4.8	1.50	1.043	0.457	–0.008	1.14	0.609	0.531	0.270
5.0	1.56	1.087	0.473	–0.024	1.20	0.652	0.548	0.286
5.2	1.59	1.130	0.460	–0.011	1.22	0.696	0.524	0.268
5.4	1.62	1.174	0.446	0.003	1.28	0.739	0.541	0.269
5.6	1.66	1.217	0.443	0.006	1.31	0.783	0.527	0.261
5.8	1.72	1.261	0.459	–0.010	1.36	0.826	0.534	0.272

11.10 Parallelism of half-intercept and delay-time curves

Prove that a half-intercept curve is parallel to the curve of the total delay time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta (see Figure 11.10a).

Solution

Referring to Figure 11.10a, we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{i} /2=h\left(\cos \theta _{c} /V_{1} \right) \end{align}

Figure 11.10a.  Delay-time and half-intercept curves.

[see equations (11.9b)]. Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{i} /2 is a linear function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h with slope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\cos \theta _{c} /V_{1} \right) . The total delay time is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta=\delta_{s} +\delta_{g}, \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{s} being constant. If we substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N=h\tan \theta _{c} (see Figure 11.9a) in equation (11.8a), we obtain the result

$ {\begin{aligned}\delta _{g}=h\left(\cos \theta _{c}/V_{1}\right),\\{\hbox{so}}\quad \quad \delta =\delta _{s}+h\left(\cos \theta _{c}/V_{1}\right).\end{aligned}} $

Thus the total delay-time curve is parallel to the half-intercept time curve and lies above it the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{s} .

11.11 Wyrobek’s refraction interpretation method

Sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G in Figure 11.11a are 5 km a part. The data in Table 11.11a are for three profiles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CE , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DF , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): EG with sources at $ C $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E , no data being recorded for offsets less than 3 km. For profiles from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G the intercepts were 1.52 and 1.60 s, respectively. Use Wyrobek’s method (Wyrobek, 1956) to interpret the data.

Background

Wyrobek’s method is based on a series of unreversed profiles such as those shown in Figure 11.11a. The steps in the interpretation are as follows:

Figure 11.11a.  Unreversed refraction profiles.

1. The traveltimes are measured, corrected, and plotted, and apparent velocities and intercepts are measured. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} cannot be measured, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} is calculated from an assumed value.
2. The total delay times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta are calculated [see equation (11.8b)] for each geophone location for each profile. The curves for the different profiles are displaced up or down to obtaina composite curve covering the entire range.
3. The half-intercept times are plotted at the source locations and a curve drawn through them. This curve is compared with the composite curve in (d); if the curves are not sufficiently parallel, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} is adjusted to achieve parallelism. The composite delay-time curve is also used to interpolate or extrapolate the half-intercept curve to cover the complete range. Delay times are now converted into depths using equation (11.9a), i.e., by multiplying half-intercept times by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} /\left(\cos \theta _{c} \right) .

Table 11.11a. Time-offset data for three refraction profiles.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{CE} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{DF} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{EG} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{CE} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{DF} (s)	$ t_{EG} $ (s)
3.00	1.18	1.20	1.19	6.60	1.90	2.12	2.49
3.20	1.22	1.29	1.28	6.80	1.94	2.16	2.54
3.40	1.24	1.38	1.35	7.00	1.97	2.20	2.57
3.60	1.28	1.45	1.43	7.20	2.01	2.25	2.60
3.80	1.35	1.54	1.50	7.40	2.06	2.30	2.65
4.00	1.38	1.60	1.58	7.60	2.10	2.33	2.68
4.20	1.41	1.70	1.68	7.80	2.14	2.37	2.71
4.40	1.47	1.74	1.76	8.00	2.17	2.41	2.74
4.60	1.51	1.77	1.82	8.20	2.20	2.45	2.77
4.80	1.53	1.80	1.89	8.40	2.24	2.47	2.82
5.0	1.58	1.82	2.00	8.60	2.30	2.52	2.85
5.20	1.63	1.85	2.06	8.80	2.32	2.55	2.89
5.40	1.65	1.91	2.15	9.00	2.35	2.61	2.93
5.60	1.69	1.95	2.21	9.20	2.38	2.64	2.97
5.80	1.74	1.97	2.29	9.40	2.44	2.68	3.00
6.00	1.78	1.99	2.38	9.60	2.47	2.73	3.04
6.20	1.82	2.03	2.43	9.80	2.50	2.78	3.07
6.40	1.87	2.08	2.46	10.00	2.54	2.82	3.10

Solution

The traveltimes in Table 11.11a are plotted in the upper part of Figure 11.11b. The values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} have different accuracies since different numbers of points are used for each value, so we obtain weighted averages using as weights the horizontal extent of the data for each value. Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{1} &= \left(2.50\times 1+2.52\times 3\right)/4=2.52\ {\rm km/s}, \\ V_{2} &= \left(5.13\times 7+5.08\times 6+5.59\times 4\right)/17=5.22\ {\rm km/s}, \\ \hbox{so}\quad \quad \theta _{c} =\sin ^{-1} \left(2.52/5.22\right)=28.9^{\circ}. \end{align}

Figure 11.11b.  Time-distance plot (top half ) and plot of delay-times and half-intercept times (bottom).

Table 11.11b. Delay times for profiles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CE , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DF , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): EG .

1	2	3	4	5	6	7	8	9
		5.22			6.25		7.7	5.6
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{CE} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{DF} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{EG} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{CE} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{DF} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{EG} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{CE} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{DF} (s)
3.0	0.61			0.70
3.2	0.61			0.71
3.4	0.59			0.70
3.6	0.59			0.70
3.8	0.62			0.74
4.0	0.61			0.74
4.2	0.61	0.90		0.74	1.03
4.4	0.63	0.90		0.77	1.04
4.6	0.63	0.89		0.77	1.03
4.8	0.61	0.88		0.76	1.03
5.0	0.62	0.86		0.78	1.02			0.93
5.2	0.63	0.85		0.80	1.02			0.92
5.4	0.62	0.88		0.79	1.05			0.95
5.6	0.62	0.88		0.79	1.05			0.95
5.8	0.63	0.86		0.81	1.04			0.93
6.0	0.63	0.84		0.82	1.03			0.92
6.2	0.63	0.84	1.24	0.83	1.04	1.44		0.92
6.4	0.64	0.85	1.23	0.85	1.06	1.44		0.94
6.6	0.64	0.86	1.23	0.84	1.06	1.43	1.04	0.94
6.8	0.64	0.86	1.24	0.85	1.07	1.45	1.06	0.95
7.0	0.63	0.86	1.23	0.85	1.08	1.45	1.06	0.95
7.2	0.63	0.87	1.22	0.86	1.10	1.45	1.07	0.96
7.4	0.64	0.88	1.23	0.88	1.12	1.47	1.10	0.98
7.6	0.64	0.87	1.22	0.88	1.11	1.46	1.11	0.97
7.8	0.65	0.88	1.22	0.89	1.12	1.46	1.13	0.98
8.0	0.64	0.88	1.21	0.89	1.13	1.46	1.13	0.98
8.2	0.63	0.88	1.20	0.89	1.14	1.46	1.14	0.99
8.4	0.63	0.86	1.21	0.90	1.13	1.48	1.15	0.97
8.6	0.65	0.87	1.20	0.92	1.14	1.47	1.18	0.98
8.8	0.63	0.86	1.20	0.91	1.14	1.48	1.18	0.98
9.0	0.63	0.89	1.21	0.91	1.17	1.49	1.18	1.00
9.2	0.62	0.88	1.21	0.91	1.17	1.50	1.19	1.00
9.4	0.64	0.88	1.20	0.94	1.18	1.50	1.22	1.00
9.6	0.63	0.89	1.20	0.93	1.19	1.50	1.22	1.02
9.8	0.62	0.90	1.19	0.93	1.21	1.50	1.23	1.03
10.0	0.62	0.90	1.18	0.94	1.22	1.50	1.24	1.03

The intercept times from the data in Table 11.11a are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{C} =0.60 s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{D} =0.82 s, $ t_{E}=1.31 $ s, and we are also given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{F} =1.52 s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{G} =1.60 s. Obviously the refractor is dipping down from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C towards Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} above is in fact Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{d} . However, initially we shall ignore dip and use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} =5.22 km/s.

The calculated delay times are listed in Table 11.11b; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x is the offset distance from the sources for profiles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CE , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DF , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): EG , while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{CE} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{DF} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{EG} are total delay times. These were obtained in the same way as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{AR} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{BR} in Table 11.9b using the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} =5.22 km/s to get columns 2, 3, and 4 in Table 11.11b.

The delay times can also be obtained by drawing straight lines through sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D , and $ E $ with slopes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/V_{2} (the lines Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): HJ , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): KL , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN in Figure 11.11b) and then measuring the time differences between these lines and the observed times.

The delay times in columns 2, 3, and 4 are plotted in the lower part of Figure 11.11b using small circles (o). The half-intercept times for sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G are also plotted (solid line at top of the lower figure) but using a different scale from that used for delay times.

The next step is to shift the delay-time values to form a continuous composite curve; we achieve this by moving the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CE curve up and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): EG curve down. Since this is merely a preliminary step we do not move individual values but displace the average straight lines through the points, giving the composite curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ .

The delay-time curve is not parallel to the half-intercept line and, to achieve parallelism, we must change Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} to increase the delay times at large values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x relative to those at small values. For profile Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CE we need to change Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): J moves downward about 0.2 s more than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H ; this gives the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H'J' with slope equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/V_{2} = 1/6.25 km/s, the other two curves becoming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K'L' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M'N' . We recalculate the delay times using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} =6.25 km/s; the new values are given in columns 5, 6, and 7 of Table 11.11b and plotted as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x's in Figure 11.11b. The new curves do roughly parallel the half-intercept curve, and we obtain a new composite delay-time curve by moving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{DF} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{CE} upward by 0.2 s and 0.3 s, respectively, to join the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{EG} values to form a continuous curve. The values agree exactly except for the first and last overlapping values, which differ by 2 ms; we used the average values at these two points.

Comparison of the composite delay-time curve with the half-intercept time curve shows reasonably good agreement at the two ends but significant divergence in the central part. We might assume that the intercept time at source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E is in error but the value 1.31 s would have to decrease to about 1.15 s (for a half-intercept time of about 0.58 s) to agree with the delay-time curve. Although the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): EG -curve is short, it is regular so that it is difficult to fit a line having an intercept of 1.15 s. A more likely source of error is variations of velocity; these could be of two kinds: (i) the actual value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} could be 6.25 at the two ends but higher than 6.25 in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 7<x_{c} <10 km and lower than 6.25 in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10<x_{c} <15 km, (ii) velocity changes due to dip (the intercepts show an overall dip down from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} is the apparent velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{d} . While velocity variations due to changes in dip are the more likely explanation, we can proceed with the interpretation without deciding which velocity effect is the cause.

To reduce the gap between the two curves, we change Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} so that the difference between the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{CE} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): X_{C} =10.0 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{C} =6.6 km increases by 0.1 s. Letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V be the required velocity and using equation (11.8b), we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left(2.54-10.0/V\right)-\left(1.90-6.6/V\right)=\left(0.94-0.84\right)+0.10, \\ V=3.4/\left(0.64-0.20\right)=7.7\ {\rm km/s}. \end{align}

We also need a new velocity that will increase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{DF} about 0.1 s more at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{C} =5.0 than at $ x_{C}=10.0 $. Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \left(1.82-5.0/V\right)-\left(2.82-10.0/V\right)=\left(1.02-1.22\right)+0.10, \\ V=5.0/0.90=5.6\ {\rm km/s}. \end{align}

These two velocities were used to calculate revised delay times in columns 8 and 9 of Table 11.11b, and the revised values are plotted in Figure 11.11b (using small squares).

The final interpreted curve is represented by inverted triangles (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \nabla ) from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{c} =3.0 to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{c} =15.0 and by crosses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(\times \right) from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{c} =16.2 to 20.0. The values can be changed to depths by multiplying the half-intercept times by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} /\left(\cos \theta _{c} \right) [see equation (11.9b)].

We now get approximate dip Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi by finding depths at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G using equation (11.9b); then we use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{d} to calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{u} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} which give a more accurate depth factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} /\cos \theta _{c} . Thus, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{1} =2.52\ {\rm km/s},\quad V_{2} =5.22\ {\rm km/s},\quad \theta _{c} =28.9^{\circ},\quad \delta_{c} =0.60/2,\quad \delta_{G} = 1.60/2. \end{align}

Using these values, the depths become

$ {\begin{aligned}h_{C}&=0.30\times 2.52/\cos 28.9^{\circ }=0.86\ {\rm {km}},\\h_{G}&=0.80\times 2.52/\cos 28.9^{\circ }=2.30\ {\rm {km}},\\\xi &=\tan ^{-1}\left[\left(2.30-0.86\right)/20\right]=4.1^{\circ }.\end{aligned}} $

Using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{d} =6.25 km/s, we solve equation (4.24d) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{u} , giving

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \xi =\left(1/2\right)[\sin ^{-1} \left(2.52/6.25\right)-\sin ^{-1} (2.52/V_{u})], \\ \hbox{so}\quad \quad 4.1^{\circ} &=\left(1/2\right)[23.8^{\circ} - \sin ^{-1} (2.52/V_{u})] \\ \sin ^{-1} \left(2.52/V_{u} \right)&=23.8^{\circ} -8.2^{\circ} =15.6^{\circ},\\ \left(2.52/V_{u} \right)&=\sin 15.6^{\circ} =0.269, V_{u} =9.37\ {\rm km/s}, \\ V_{2} \approx \frac{1}{2} \left(V_{d} +V_{u} \right) &=\frac{1}{2} \left(6.25+9.37\right)=7.81\ {\rm km/s}, \\ \theta _{c} &\approx \sin ^{-1} \left(2.52/7.81\right)\approx 18.8^{\circ},\; \cos \theta _{c} \approx 0.947,\\ \hbox{depth factor} &\approx V_{1} /\cos \theta _{c} \approx 2.52/0.947\approx 2.66. \end{align}

Thus, the refractor is nearly flat over the region where we used Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} =6.25 km/s, so local dip is mainly in the places where we carried out the second revision using velocities of 7.7 and 5.6 km/s.

We shall not refine our interpretation further because of the limited acccuracy of the data.

11.12 Properties of a coincident-time curve

11.12a A coincident-time curve connects points where waves traveling by different paths arrive at the same time. In Figure 11.12a, the curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): AC is where the head wave and direct wave arrive simultaneously. On a vertical section through the source with constant-velocity above a refractor, head-wave wavefronts are parallel straight lines. In Figure 11.12b, show that the virtual wavefront Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 is at a slant depth $ SD=2h=2z\cos \theta _{c} $.

Background

Figure 11.12a shows first-arrival wavefronts at intervals of 0.1 s generated by the source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S for a three-layer situation where the velocities are in the ratio 2:3:4. The critical angle at the first interface is reached at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A , so head waves are generated to the right of this point, the wavefronts in the upper layer being straight lines that join with the direct wavefronts having the same traveltimes. The locus of the junction point where the first-arrival wavefronts abruptly change direction is a coincident time curve. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ABC is a coincident-time curve. In general a coincident-time curve (for example, DEFG) is the locus of the junction points where two wavefronts having the same traveltimes but have traveled different paths.

A curve that is equidistant from a fixed point and a fixed straight line is a parabola.

Solution

In Figure 11.12a, the wave generated at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 arrives at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} =SA/V_{1} , the angle of incidence being the critical angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} . Head waves traveling upwards at the critical angle are generated to the right of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A . We assume that a fictitious source generates plane wavefronts traveling parallel to the head-wave wavefronts with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE being their position at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 . This wavefront arrives at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CD=SA . Hence,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} SD &=SC+CD=SC+SA\\ &=SA\left(1+\cos 2\theta _{c} \right)\\ &=2SA{\rm cos}^{2} \theta _{c} =2z\cos \theta _{c}. \end{align}

Figure 11.12a.  First-arrival wavefronts at 0.1-s intervals.

Figure 11.12b.  Deriving properties of a coincident-time curve.

11.12b Show that after Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE reaches Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A , wavefronts such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BF coincde with the head-wave wavefronts.

Solution

If the wavefront Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CA arrives at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): FB at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} +\Delta t , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): AF=V_{1} \Delta t . During the time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t , the headwave travels from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B at velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} , that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): AB =V_{2} \Delta t . Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): AF/AB=V_{1} /V_{2} =\sin \theta _{c} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BF parallels the refracted wavefronts.

11.12c Show that the coincident-time curve is a parabola.

Solution

At any point on the coincident-time curve, the traveltime of the direct wave equals that of a wavefront coming from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE . Since both wavefronts travel with the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} , the point on the curve is equidistant from $ S $ and from the straight line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE , hence the curve is a parabola.

11.12d Show that, taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DS as the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y -axes, the equation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): AH is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 4hy=x^{2} +4h^{2} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DS=2h .

Solution

We take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H as $ H\left(x,\;y\right) $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S\left(0,2h\right) . We know from part (c) that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): SH= distance from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H to the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): DE . The squares of these distances are also equal, so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} SH^{2} = HE^{2},\ \hbox{that is}, x^{2} +(y-2h)^{2} =y^{2}, \\ \hbox{and}\quad \quad x^{2} +4h^{2} =4hy. \end{align}

11.12e Show that the coincident-time curve is tangent to the refractor at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A .

Solution

We must show that the coincident-time curve passes through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A with the same slope as the refractor. Obviously the curve starts at $ A $ because the head wave starts at the instant the direct wave reaches Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A . We use the equation of the curve in part (d) to get the slope and then substitute the coordinates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A . Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x^{2} +4h^{2} =4hy,\qquad dy/dx=x/2h. \end{align}

The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -coordinate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x_{A} &= AS\sin 2\theta _{c} =\left(z/\cos \theta _{c} \right)\sin 2\theta _{c} =2z\sin \theta _{c} \\ &=\left(2h/\cos \theta _{c} \right)\sin \theta _{c} =2h\tan \theta _{c}, \end{align}

where we used the result in (a) in the last step. Substitution in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): dy/dx gives the slope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan \theta _{c} which is the same as the refractor slope. Therefore, the coincident-time curve is tangent to the refractor at $ A $.

11.13 Interpretation by the plus-minus method

Interpret the data in Table 11.13a using the plus-minus method.

Background

Fermat’s principle (problem 4.13) states that the raypath between two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B is such that the traveltime is either a minimum (e.g., direct waves, reflections and head waves) or a maximum. Therefore, the raypath between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B is unique so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AB} =t_{BA} =t_{r} = reciprocal time . As a result, when we have reversed profiles, we can locate the refractor by drawing wavefronts from the two sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B ; when the sum of the traveltimes for two intersecting wavefronts equals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} , the point of intersection must lie on the refractor (see problem 11.14c). This is the basic concept of the plus-minus method (Hagedoorn, 1959).

Construction of wavefronts is discussed in problem 11.14c.

Based on the recorded data, we draw and label wavefronts at intervals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta as in Figure 11.13a. If the dip is zero, they are at the angles $ \pm \theta _{c} $ to the refractor and the intersections give diamond-shaped parallelograms. The horizontal diagonal of a parallelogram is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} \Delta and the vertical diagonal is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} \Delta /\cos \theta _{c} . Lines of constant sum of the traveltimes minus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} (plus values) gives the refractor configuration and differences (minus values) give a check on the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} . The refractor lies at plus value = 0.

Table 11.13a. Time-distance data for plus-minus interpretation.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{B} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} (s)	$ t_{B} $(s)
0.0	0.00	2.30	6.0	1.30	1.32
0.4	0.15	2.23	6.4	1.33	1.28
0.8	0.28	2.15	6.8	1.40	1.24
1.2	0.44	2.09	7.2	1.51	1.18
1.6	0.52	2.04	7.6	1.57	1.10
2.0	0.63	1.98	8.0	1.60	1.04
2.4	0.70	1.92	8.4	1.72	0.96
2.8	0.76	1.85	8.8	1.78	0.90
3.2	0.84	1.80	9.2	1.80	0.83
3.6	0.91	1.72	9.6	1.91	0.76
4.0	0.95	1.64	10.0	1.93	0.66
4.4	1.04	1.60	10.4	2.04	0.52
4.8	1.12	1.55	10.8	2.07	0.39
5.2	1.16	1.47	11.2	2.17	0.25
5.6	1.25	1.40	11.6	2.20	0.12
—	—	—	12.0	2.30	0.00

Figure 11.13a.  Illustrating the plus-minus method.

Figure 11.13b.  Solution by the plus-minus method.

Solution

The traveltime curves are shown in Figure 11.13b. From the figure we obtained the following values: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} = 2.90 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} =6.25 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} =2.28 s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} =\sin ^{-1} \left(2.90/6.25\right)=27.6^{\circ} .

We next draw straight-line wavefronts at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm 27.6^{\circ} spaced at intervals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta =0.20 s. Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} =2.30 s and the refraction from source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B starts around 0.8 s, we draw wavefronts for source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B for $ t_{B}=0.80 $, 1.00, 1.20, 1.40, and 1.60 s. For source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A we draw wavefronts for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} =1.48 , 1.28, 1.08, 0.88, and 0.68 s. We interpolate to find the starting points of these wavefronts.

The horizontal and vertical diagonals of the parallelograms have lengths of 1.24 and 0.66 km, so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{2} \Delta =1.24,\quad V_{2} =1.24/0.20=6.20\ {\rm km/s}, \\ V_{1} \Delta /\cos \theta _{c} =0.66,\quad V_{1} =0.66\cos 27.6^{\circ} /0.20=2.92\ {\rm km/s}. \end{align}

These values agree with the values in Figure 11.13b within the limits of error.

Table 11.14a. Refraction time-distance data.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_A (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_B (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t^*_A (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_A (s)	$ t_{B} $(s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t^*_B (s)
0.00	0.000	3.310		6.40	2.330	2.003
0.40	0.182	3.182		6.80	2.422	1.862
0.80	0.320	3.140		7.20	2.504	1.743
1.20	0.504	3.063		7.60	2.602	1.622
1.60	0.680	2.917		8.00	2.658	1.610
2.00	0.862	2.839		8.40	2.720	1.482	1.561
2.40	0.997	2.714		8.80	2.744	1.329	1.440
2.80	1.170	2.681	1.682	9.20	2.760	1.140	1.288
3.20	1.342	2.570	1.760	9.60	2.855	1.018	1.202
3.60	1.495	2.505	1.858	10.00	2.920	0.863	1.177
4.00	1.677	2.442	1.881	10.40	2.980	0.660	1.082
4.40	1.821	2.380	1.962	10.80	3.065	0.503
4.80	1.942	2.318	2.053	11.20	3.168	0.340
5.20	2.103	2.220		11.60	3.230	0.198
5.60	2.150	2.125		12.00	3.310	0.000
6.00	2.208	2.030

The refractor is indicated in Figure 11.13b by the dashed line. The variation in the spacing of the vertical minus lines is very slight so that we can assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} is constant.

11.14 Comparison of refraction interpretation methods

The data in Table 11.14a show refraction traveltimes for geophones spaced 400 m a part between sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B which are separated by 12 km. The columns in the table headed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A}^{*} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{B}^{*} give second arrivals.

11.14a Interpret the data using the basic refraction equations (4.24a) to (4.24f).

Solution

The data are plotted in Figure 11.14a and best-fit lines suggest that this is a two-layer problem. Measurements give the following values:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{1} = 2.88\ \hbox{km/s (average value)}, \\ V_{d} =4.65\ {\rm km/s},\quad V_{u} =5.71\ {\rm km/s},\quad t_{iu} =1.21\ {\rm s},\quad t_{id} =0.73\ {\rm s}. \end{align}

Equation (4.24d) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V_{d} =4.65=2.38/\sin \left(\theta _{c} +\xi \right),\quad V_{u} =5.71=2.38/\sin \left(\theta _{c} -\xi \right), \\ \left(\theta _{c} +\xi \right)=30.8^{\circ},\quad \left(\theta _{c} -\xi \right)=24.6^{\circ},\quad \theta _{c} =27.7^{\circ},\quad \xi =3.1^{\circ}. \end{align}

Figure 11.14a.  Plot of the time-distance data.

From equation (4.24f), we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} \approx 2(1/V_{d} +1/V_{u} )^{-1} \approx 5.13 km/s. From equation (4.24b) we get for the slant depths,

$ {\begin{aligned}h_{d}=V_{1}t_{id}/2\cos \theta _{c}=0.98\ {\rm {km}},\quad h_{u}=V_{1}t_{iu}/2\cos \theta _{c}=1.63\ {\rm {km}}.\end{aligned}} $

Checking the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi , we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \theta _{c} =\sin ^{-1} \left(2.38/5.13\right)=27.6^{\circ},\quad \xi =\tan ^{-1} \left[\left(1.63-0.98\right)/12.0\right]=3.1^{\circ}. \end{align}

11.14b Interpret the data using Tarrant’s method.

Background

Tarrant’s method (Tarrant, 1956) uses delay times (problem 11.8) to locate the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q [see Figure 11.14b(i)] where the refracted energy that arrives at geophone Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R leaves the refractor. The refractor is defined by finding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q for a series of geophone positions. Tarrant’s method is based on the properties of the ellipse.

The delay time for the path Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): QR in Figure 11.14b(i) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{g} = \rho/V_{1} -\left(\rho\cos \theta \right)/V_{2} . Solving for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho , we get

$ {\begin{aligned}\rho =V_{1}\delta _{g}/\left(1-\sin \theta _{c}\cos \phi \right).\end{aligned}} $

(11.14a)

This is the polar equation of an ellipse. An ellipse is traced out by a point moving so that the ratio of the distance from a straight line (directrix) to that from a fixed point (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R in Figure 11.14b(ii) is a constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon (eccentricity)).

The standard polar equation of an ellipse is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \rho=\varepsilon h/\left(1-\varepsilon \cos \phi \right). \end{align}

(11.14b)

Figure 11.14b.  Illustrating Tarrant’s method. (i) Relation between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q ; (ii) locus of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q is ellipse, focus at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R ; (iii) geometry of ellipse through Q.

In Figure 11.14b(ii) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q moves so that the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): QR/QM=\varepsilon = \rho/\left(\rho\cos \phi +h\right)<1 . The major axis 2a of the ellipse is

$ {\begin{aligned}2a=\rho _{\phi =0}+\rho _{\phi =\pi }\\=\varepsilon h/\left(1-\varepsilon \right)+\varepsilon h/\left(1+\varepsilon \right)\\=2\varepsilon h/\left(1-\varepsilon ^{2}\right).\end{aligned}} $

(11.14c)

To get the minor axis, we set the first derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2b=2\rho\sin \phi equal to zero. Using equation (11.14b), we find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 2b=2\varepsilon h/(1-\varepsilon ^{2} )^{1/2}, \end{align}

(11.14d)

The distance from the focal point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R to the center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): O is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OR &=\left(\rho\phi =0-a\right) \\ &=\varepsilon h/\left(1-\varepsilon \right)-\varepsilon h/\left(1-\varepsilon ^{2}\right)\varepsilon a. \end{align}

If we substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varepsilon =\sin \theta _{c} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h=V_{2} \delta_{g} in equation (11.14b), we get equation (11.14a). Also these values give the following results.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a=V_{1} \delta_{g} /{\rm cos}^{2} \theta _{c} \\ &=V_{2} \delta_{g} \tan \theta _{c} /\cos \theta _{c} \end{align}

(11.14e)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &=OQ=V_{2} \delta_{g} \sin \theta _{c} /\cos \theta _{c} =V_{2} \delta_{g} \tan \theta _{c}, \end{align}

(11.14f)

$ {\begin{aligned}OR&=\varepsilon a=V_{2}\delta _{g}\tan ^{2}\theta _{c},\end{aligned}} $

(11.14g)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OQR&=\tan ^{-1} \left(OR/b\right)=\tan ^{-1} \left(V_{2} \delta_{g} \tan ^{2} \theta _{c} /V_{2} \delta_{g} \tan \theta _{c} \right)=\theta _{c} \;, \end{align}

(11.14h)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OC&=OR\; \tan =V_{2} \delta_{g} \tan ^{3} \theta _{c}. \end{align}

(11.14i)

To approximate the ellipse in the vicinity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q with a circle, we need to find the center of curvature of the ellipse. The general equation of an ellipse is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} (x/a)^{2} +(y/b)^{1} =1. \end{align}

(11.14j)

The equation for the radius of curvature of a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y\left(x\right) is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} r=[1+\left(y)^{2} \right]/y''. \end{align}

Differentiating, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} y'=-(b/a)^{2} \left(x/y\right);\quad y''=-(b/a)^{2} \left[1/y-\left(x/y^{2} \right)y'\right]. \end{align}

$ {\begin{aligned}{\hbox{At Q,}}\quad \quad x=0,y=-b,\ {\rm {so}}\ y'=0,y''=\left(b/a^{2}\right),r=a^{2}/b=V_{2}\delta _{g}\tan \theta _{c}/\cos ^{2}\theta _{c}.\end{aligned}} $

The center of curvature is a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r above Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q , so the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y -coordinate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C [Figure 11.14b(iii)] is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(r-b\right)=\left(V_{2} \delta_{g} \tan \theta _{c} /\cos^{2} \theta _{c} -V_{2} \delta_{g} \tan \theta _{c} \right)=V_{2} \delta_{g} \tan ^{3} \theta _{c} . A circle with center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C and radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r will approximate the ellipse in the vicinity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q .

Solution

We need the total delay time at source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A , $ \delta _{SA} $, and the delay times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{g} at the geophones where the head wave is observed.

We have from part (a): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} =2.38 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} =5.13 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} =27.7^{\circ} ; from equation(11.9b), we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{SA} =t_{iA} /2=0.60 s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{SB} =0.36 s. For a geophone Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R at a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A , equation (11.8b) gives for source $ A $,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta_{g} =t_{R} -x/V_{2} -\delta_{SA} =t_{R} -\left(x/5.13+0.60\right), \end{align}

and for source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta_{g} =t_{R} -x/V_{2} -\delta_{SB} =t_{R} -\left(x/5.13+0.36\right) \end{align}

(note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x is measured from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{SA} and from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{SB} ).

We can obtain values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{g} either by using the above equations or graphically by drawing straight lines with slope $ 1/V_{2} $ starting at the half-intercept values (there by subtracting it); the vertical distances between these lines and the traveltime curves give Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{g} . The values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{g} in Tables 11.14b,c were calculated.

Table 11.14b. Calculations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r for source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_A (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T (s)	$ \delta _{g} $ (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OQ (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OR (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r (km)
2.80	1.15	0.53	1.43	0.75	0.39	1.82
3.20	1.22	0.54	1.45	0.76	0.40	1.85
3.60	1.30	0.56	1.51	0.79	0.41	1.92
4.00	1.38	0.50	1.34	0.70	0.37	1.71
4.40	1.46	0.50	1.34	0.70	0.37	1.71
4.80	1.54	0.51	1.37	0.72	0.38	1.75
5.20	1.62	0.48	1.29	0.68	0.36	1.65
5.60	1.69	0.46	1.24	0.65	0.34	1.58
6.00	1.77	0.44	1.18	0.62	0.33	1.51
6.40	1.85	0.48	1.29	0.68	0.36	1.65
6.80	1.93	0.49	1.32	0.69	0.36	1.68
7.20	2.01	0.49	1.32	0.69	0.36	1.68
7.60	2.08	0.52	1.40	0.73	0.39	1.79
8.00	2.16	0.50	1.34	0.70	0.37	1.71
8.40	2.24	0.48	1.29	0.68	0.36	1.65
8.80	2.32	0.42	1.13	0.59	0.31	1.44
9.20	2.40	0.36	0.97	0.51	0.27	1.24
9.60	2.48	0.38	1.02	0.54	0.28	1.30
10.00	2.55	0.37	1.00	0.52	0.27	1.27
10.40	2.63	0.35	0.94	0.49	0.26	1.20
10.80	2.71	0.36	0.97	0.51	0.27	1.24
11.20	2.79	0.38	1.02	0.54	0.28	1.30
11.60	2.87	0.36	0.97	0.51	0.27	1.24
12.00	2.94	0.37	1.00	0.52	0.27	1.27

The last step is to find the center of curvature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C in Figure 11.14b(iii) and to draw an arc with radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r=CQ . We first calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OQ and then find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C by calculating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC or $ OR $ and drawing a line normal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RQ . The first method was used to get Tables 11.14b,c (although Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OR is given in the tables, it was not used). We repeat equations (11.14f,g,h) and get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OQ&=V_{2} \delta_{g} \tan \theta _{c} =2.69\delta_{g}, \end{align}

(11.14l)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OR&=V_{2} \delta_{g} \tan ^{2} \theta _{c} =1.41\delta_{g}, \end{align}

(11.14m)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} OC&=V_{2} \delta_{g} \tan ^{3} \theta _{c} =0.74\delta_{g}. \end{align}

(11.14n)

The calculations are shown in Tables 11.14b,c. The quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T=\left(x_{A} /5.12+0.60\right) in Table 11.14b and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(x_{B} /5.12+0.36\right) in Table 11.14c. Because complete reversed profiles were obtained, there is considerable duplication of the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r . Rather than plot all of the arcs, we used the average values of $ OC $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r (calculated in Table 11.14d). The results are shown in Figure 11.14c.

11.14c Interpret the data in Table 11.14a using the wavefront method illustrated in Figure 11.14d

Background

In Figure 11.14d(i) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MCD and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PCE are two wavefronts generated at sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B , respectively, and meeting at C. Clearly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AC} +t_{BC} =t_{AB} =t_{r} . If wavefronts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MC and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): CP continue upward to the surface at velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} and are recorded, we can project them backwards using the method shown in Figure 11.14d(ii). Point $ C $ where they meet locates a point on the refractor. This is the basis of the wavefront method.

Table 11.14c. Calculations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r for source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{A} (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta_{g} (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OQ (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OR (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC (km)	$ r $(s)
1.60	0.67	0.41	1.10	0.58	0.30	1.40
2.00	0.75	0.43	1.16	0.61	0.32	1.48
2.40	0.83	0.37	1.00	0.52	0.27	1.27
2.80	0.91	0.38	1.02	0.54	0.28	1.30
3.20	0.98	0.46	1.24	0.65	0.34	1.58
3.60	1.06	0.50	1.34	0.70	0.37	1.71
4.00	1.14	0.47	1.26	0.66	0.35	1.61
4.40	1.22	0.40	1.08	0.56	0.30	1.38
4.80	1.30	0.44	1.18	0.62	0.33	1.51
5.20	1.38	0.48	1.29	0.68	0.36	1.65
5.60	1.45	0.55	1.48	0.78	0.41	1.89
6.00	1.53	0.50	1.34	0.70	0.37	1.71
6.40	1.61	0.51	1.37	0.72	0.38	1.75
6.80	1.69	0.53	1.43	0.75	0.39	1.82
7.20	1.77	0.55	1.48	0.78	0.41	1.89
7.60	1.84	0.54	1.45	0.76	0.40	1.85
8.00	1.92	0.52	1.40	0.73	0.39	1.79
8.40	2.00	0.50	1.34	0.70	0.37	1.71
8.80	2.08	0.49	1.32	0.69	0.36	1.68
9.20	2.16	0.52	1.40	0.73	0.39	1.79
9.60	2.24	0.47	1.26	0.66	0.35	1.67
10.00	2.31	0.53	1.45	0.75	0.39	1.82
10.40	2.39	0.53	1.43	0.75	0.39	1.82
10.80	2.47	0.59	1.59	0.83	0.44	2.03
11.20	2.55	0.59	1.59	0.83	0.44	2.03
11.60	2.63	0.55	1.48	0.78	0.41	1.89
12.00	2.70	0.61	1.64	0.86	0.45	2.09

Table 11.14d. Calculating average values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r.

	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Av	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Av
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_{A}	$ OC $	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): OC	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r
0.00	0.00	0.00	0.45	2.09	0.45	2.09
0.40			0.41	1.89	0.41	1.89
0.80			0.44	2.03	0.44	2.03
1.20			0.44	2.03	0.44	2.03
1.60			0.39	1.82	0.39	1.82
2.00			0.39	1.82	0.39	1.82
2.40			0.35	1.61	0.35	1.61
2.80	0.39	1.82	0.39	1.79	0.39	1.80
3.20	0.40	1.85	0.36	1.68	0.38	1.76
3.60	0.41	2.92	0.37	1.71	0.39	1.82
4.00	0.37	1.71	0.39	1.79	0.38	1.75
4.40	0.37	1.71	0.40	1.85	0.38	1.78
4.80	0.38	1.75	0.41	1.89	0.39	1.82
5.20	0.36	1.65	0.39	1.62	0.38	1.64
5.60	0.34	1.58	0.38	1.75	0.36	1.66
6.00	0.33	1.51	0.37	1.71	0.35	1.61
6.40	0.36	1.65	0.41	1.89	0.38	1.77
6.80	0.35	1.68	0.36	1.65	0.36	1.66
7.20	0.36	1.68	0.33	1.51	0.34	1.60
7.60	0.39	1.79	0.30	1.38	0.34	1.59
8.00	0.37	1.71	0.35	1.61	0.36	1.66
8.40	0.36	1.65	0.37	1.71	0.36	1.68
8.80	0.39	1.44	0.34	1.58	0.36	1.51
9.20	0.27	1.24	0.28	1.30	0.28	1.27
9.60	0.28	1.30	0.27	1.27	0.28	1.28
10.00	0.27	1.27	0.32	1.48	0.30	1.38
10.40	0.26	1.20	0.30	1.40	0.28	1.30
10.80	0.27	1.24			0.27	1.24
11.20	0.28	1.30			0.28	1.30
11.60	0.27	1.24			0.27	1.24
12.00	0.27	1.27			0.27	1.27

Figure 11.14c.  Solution by Tarrant’s method; the small circles at the top are centers of the arcs. The wavefront solution is shown by small squares, the formula solution by the dashed line.

Figure 11.14d.  Illustrating the wave-front method. (i) Two wavefronts where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AC} +t_{BC} =t_{r} ; (ii) reconstructing wavefronts.

Solution

The earliest refracted wavefront from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A that we can reconstruct is at 1.60 s (using the best-fit line to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=1.60 ) and the last is about 3.00 s; the corresponding limits for source $ B $ are 1.10 and 3.00 s. We take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta =0.20 s and draw wavefronts such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(t_{A} +t_{B} \right)=t_{r} =3.31 \ {\hbox {s}} . We reconstruct the four wavefront pairs Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (t_{A}, t_{B}) = (1.60, 1.71) , (1.80, 1.51), (2.00, 1.31), (2.20, 1.11) using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} =2.38 km/s [from part (a)].

We swing arcs from points on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis as in Figure 11.14d(ii). Because we are interested only in the portions near the points of intersection, we determine the refractor depth. Using the intercept times at the sources in part (a) to obtain slant depths, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{A} \approx 1.6 km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{B} \approx 1.0 km; multiplying by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \cos \theta _{c} \approx 0.9 to get vertical depths; the maximum vertical depth is about 1.40 km.

The radius of the arcs is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R=V_{1} \Delta t=2.38\Delta t . The maximum value of $ \Delta t $ is about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (1.40/2.38) \approx 0.6 \ {\hbox {s}} but, to be on the safe side, we calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R for values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t ranging from 0.45 to 0.75 s for the pair (1.60, 1.71), then adjust the range as necessary to achieve wavefront intersections for the other three pairs.

Table 11.14e. Calculation of wavefront radii for sources Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B .

Wavefront: (A-1.600)				Wavefront: (B-1.710)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_A (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_A(s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t(s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R (km)	$ x_{A} $ (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_A (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R (km)
4.80	1.942	0.453	1.08	4.00	2.442	0.732	1.74
5.20	2.103	0.503	1.20	4.40	2.380	0.670	1.59
5.60	2.150	0.550	1.31	4.80	2.318	0.608	1.45
6.00	2.208	0.608	1.45	5.20	2.220	0.510	1.21
6.40	2.330	0.730	1.74	5.60	2.125	0.415	0.99
Wavefront: (A-1.800)				Wavefront: (B-1.510)
6.00	2.208	0.408	0.97	4.80	2.318	0.808	1.92
6.40	2.330	0.530	1.26	5.20	2.220	0.710	1.69
6.80	2.422	0.622	1.48	5.60	2.125	0.615	1.46
7.20	2.504	0.704	1.68	6.00	2.030	0.520	1.24
7.60	2.602	0.802	1.91	6.40	2.003	0.493	1.17
Wavefront: (A-2.000)				Wavefront: (B-1.310)
6.80	2.422	0.422	1.00	6.00	2.030	0.720	1.71
7.20	2.504	0.504	1.20	6.40	2.003	0.693	1.65
7.60	2.602	0.602	1.43	6.80	1.802	0.552	1.31
8.00	2.658	0.658	1.57	7.20	1.743	0.433	1.03
8.40	2.720	0.720	1.71	7.60	1.622	0.312	0.74
Wavefront: (A-2.200)				Wavefront: (B-1.110)
8.40	2.720	0.520	1.24	6.40	2.003	0.893	2.13
8.80	2.744	0.544	1.29	6.80	1.862	0.752	1.79
9.20	2.760	0.560	1.33	7.20	1.743	0.633	1.51
9.60	2.855	0.655	1.56	7.60	1.622	0.512	1.22
10.00	2.920	0.720	1.71

The calculations are shown in Table 11.14e. Columns 1 and 2 in each sub-table come from table 11.14a. Column 3 is the difference between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{A} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{B} and the wavefront traveltime shown above each subtable; the radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R=V_{1} \Delta t=2.38\Delta t .

Figure 11.14e shows the result for the wavefront pair (1.60, 1.71). We leave to the reader the construction of the other three pairs of wavefronts and determination of the points of intersection. Our results are shown by the four small squares in Figure 11.14c and, more clearly, in Figure 11.14g.

File:Ch11 fig11-14e.png

Figure 11.14e.  Determining the point of intersection of wavefronts (1.60,1.71).

11.14d Interpret the data in Table 11.14a using Hales’s method

Background

Hales’s (1958) method is a graphical method based on reversed profiles. It enables us to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q in Figure 11.14f(i) in terms of data recorded at geophones Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S . The circumscribed circle through points $ R $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q is shown in Figure 11.14f(iv), all of the angles inside the circle being expressible in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta_{c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi .

Point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C is on the perpendicular bisector of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RS ; writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho=CQ , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \rho\cos \theta _{c} = QN = RQ - RN,\quad {\rm but}\quad QN=QG, \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\hbox {so}}\qquad \qquad \rho\cos \theta _{c} = QG = SQ + SG. \end{align}

Adding the two expressions, we get

$ {\begin{aligned}2\rho \cos \theta _{c}=\left(RQ+SQ\right)+\left(SG-RN\right).\end{aligned}} $

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RN=CN\tan \xi =CG\tan \xi =SG ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \rho=\left(RQ+SQ\right)/2\cos \theta _{c}. \end{align}

(11.14o)

Also we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{AS} +t_{BR} =t_{r} +\left(RQ+SQ\right)/V_{1} =t_{r} +t', \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\hbox {so}}\qquad \qquad t_{AS} =\left(t_{r} -t_{BR} \right)+t{'}, \end{align}

(11.14p)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} = reciprocal time (see problem 11.13). Thus we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t'=\left(RQ+SQ\right)/V_{1},\quad \rho=V_{1} t'/2\cos \theta _{c}, \end{align}

(11.14q)

Figure 11.14f.  Hales’s graphical method. (i) Relation between two receivers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S having a common emergent point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q ; (ii) geometrical properties of points on the traveltime curves corresponding to sources $ A $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B ; (iii) construction for locating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q ; (iv) properties of circumscribed circle through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q ; (v) offset due to errors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x' in part (ii).

using equation (11.14g). Using the law of sines and equation (11.14i) we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x'=RS=RH+HS=\frac{RQ\sin \theta _{c} }{\sin \left(\pi /2+\xi \right)} +\frac{SQ\sin \theta _{c} }{\sin \left(\pi /2-\xi \right)}\\ =\frac{\left(RQ+SQ\right)\sin \theta _{c} }{\cos \xi } =V_{1} t'\sin \theta _{c} /\cos \xi, \end{align}

(11.14r)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\hbox {so}}\qquad \qquad x'/t'=\tan \alpha =V_{1} \sin \theta_{c} /\cos \xi =V_{1} \sin \theta _{c},\quad {\rm when}\ \xi =0. \end{align}

(11.14s)

The traveltime curves are shown in Figure 11.14f(ii). We draw the vertical through point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S on the $ x $-axis and locate the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): KS=\left(t_{r} -t_{AS} \right) . Assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi =0 , we draw a line at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K at the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha to meet the other traveltime curve. From equations (11.14h,j,k) we see that the line from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K at the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha must intersect the traveltime curve for source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{BR} , which locates the point $ R $ and gives the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{BR}, x' , and, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t' . At point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis we draw a line at angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta_{c} to the horizontal [see Figure 11.14f(iii)] and locate point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C vertically above the midpoint of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RS , i.e., at the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'/2 from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R ; With center $ C $ and radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho=V_{1} t'/2\cos \theta _{c} . [see equation (11.14i)], we draw an arc. Repeating this process for a series of points S, we obtain several arcs to define the refracting surface.

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi \ne 0, x', t' , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho change, but it can be shown (see Sheriff and Geldart, 1995, 444) that the change in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho is negligible and the only effect of dip is to displace the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q updip the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta x'/2 [Figure 11.14e(v)], which is usually negligible for moderate dips.

Figure 11.14g.  Illustrating Hales’s solution. (i) Determining values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{BR}, x' , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t' ; (ii) locating $ C $ and drawing arcs of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho .

Solution

From part (a) we have: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} =2.38 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} = 5.12 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} =27.7^{\circ} , the reciprocal time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} =3.310 s.

Figure 11.14g shows details of the solution. We select points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S at intervals of 0.80 km, then find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(t_{r} -t_{AS} \right) and the points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K [shown as small triangles (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta ) in Figure 11.14g(i)]. Points $ R $ in Figure 11.14f(i) are found by laying off at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): K the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha =\tan ^{-1} \left(V_{1} \sin \theta _{c} \right)= \tan ^{-1} \left(2.38\times \sin 27.7^{\circ} \right)=\tan ^{-1} 1.11 [see equation (11.14s)]; to take into account the scale factors, we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha =\tan ^{-1} (2.22/2.00), then draw a vertical line equal to 2.00 s, then a lineto the left equivalent to 2.22 km [see lines going up at about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10^{\circ} -15^{\circ} from the vertical from the triangles in Figure 11.14g(i)]. This gives the locations of points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R and values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{BR}, x' , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t' ; we can now calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho from equation (11.14g). The calculations are shownin Table 11.14f.

Finally we locate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C as in Figure 11.14g(ii), with $ C $ as center and radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho we draw arcs which pass through point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q in Figure 11.14f(v). The refractor is fairly well defined by the arcs (except for two arcs marked with?).

11.14f On the basis of your results, compare the methods in terms of (1) time involved; (2) effect of refractor curvature; (3) effect of random errors; (4) suitability for routine production; and (5) for special effort where high accuracy is essential.

Solution

1) Time involved: The formula method is by far the quickest, the wavefront method is next, and Tarrant’s and Hales’s methods are the most time consuming, being more-or-less the same in this respect.

2) Effect of refractor curvature: The formula method does not take curvature into account (except on a broad scale over two or more profiles). The remaining methods all work well for curved refractors. Tarrant’s and Hales’s methods give good results over the commonly observed angle of curvature, while the wavefront method is useable over a smaller range of curvatures.

3) Effect of refractor random errors: These errors affect the measured slopes and intercepts and therefore affect the formula solution; however, the effects are usually minimized by the use of best-fit lines which utilize most of the available data. Tarrant’s and Hales’s methods use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} and the wavefront method uses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} , so that errors affect all three methods. In addition Tarrant’s and Hales’s methods use two traveltimes in each calculation so that random errors cause further errors. The wavefront method is less susceptible to this type of error because several time values enter into each wavefront determination.

4): (i) Suitability for routine production: The formula method, being the quickest, is satisfactory where refractor relief is minimal. Tarrant’s and Hales’s methods are slightly less suitable, and the wavefront method is least suitable.

4): (ii) Suitability for high accuracy: The formula method is unsuitable, the wavefront method the best, Tarrant’s and Hales’s methods being almost as good.

Table 11.14f. Calculating position of center Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C and radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho of arcs in Hales’s method.

S	S	S	R	R
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{AS}	$ (t_{r}-t_{RS}) $	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_R	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{SA}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t'	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \rho	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'/2
11.20	3.17	0.14	10.10	1.14	1.00	1.10	1.34	0.55
10.40	2.98	0.33	9.40	1.25	0.92	1.00	1.24	0.50
9.60	2.86	0.45	8.38	1.56	1.11	1.22	1.49	0.61
8.80	2.74	0.57	7.65	1.62	1.05	1.15	1.41	0.57
8.00	2.66	0.65	6.57	1.95	1.30	1.43	1.75	0.71
7.20	2.50	0.81	5.78	2.08	1.27	1.42	1.71	0.71
6.40	2.33	0.98	4.95	2.28	1.30	1.45	1.75	0.72
5.60	2.15	1.16	4.22	2.40	1.24	1.38	1.67	0.69
4.80	2.05	1.26	3.37	2.54	1.28	1.43	1.72	0.71
4.00	1.88	1.43	2.57	2.70	1.27	1.43	1.71	0.71
3.20	1.76	1.55	1.70	2.90	1.35	1.50	1.81	0.75

11.15 Feasibility of mapping a horizon using head waves

Construct the expected time-distance curve for the Java Sea velocity-depth relation shown in Figure 11.15a. Is it feasible to map the top of the relatively flat 4.25 km/s limestone at a depth of about 0.9 km by using head waves? What problems are likely to be encountered?

Figure 11.15a.  Java Sea velocity-depth relation.

Table 11.15a. Java Sea layering.

Depth range (km)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_t (km/s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t_T (s)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_i = \sum \Delta t_i	$ V_{t}^{2}\Delta t_{i} $	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sum V_t^2\Delta_{t_i}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{\rm rms} (km/s)
	0.00–0.03	1.53	0.039	0.039	0.091	0.091
	0.03–0.16	1.9	0.137	0.176	0.495	0.586
	0.16–0.28	1.97	0.122	0.298	0.472	1.059
	0.28–0.50	2.25	0.196	0.494	0.992	2.051
	0.50–0.70	2.15	0.186	0.68	0.86	2.911
	0.70–0.90	2.67	0.15	0.83	1.069	3.98	2.19
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm L}_{s}\to	0.90–0.97	4.25	0.033	0.863	0.596	4.576	2.30
	0.97–1.10	5.27	0.049	0.912	1.361	5.937

Solution

The time-depth data in Figure 11.15a are listed in the first two columns of Table 11.15a. We calculated the data in columns 3 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{i} two-way traveltime through the layer) to 6 to determine reflection arrival times (column 4) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{\rm rms} [using equation (4.13a)]. We take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} \approx 1.90 km/s (see Figure 11.15a) to plot the direct wave.

We must take into account other events that might interfere, primarily the reflection and head wave from the 5.27 km/s layer. To plot the refraction curves, we need their slopes, one point on each curve, and the critical distances—where a head wave is tangent to the reflection (see Figure 4.18a). We also calculate the intercept times as a check.

The slope of the limestone refractor (assumed to be flat) is 1/4.25 s/km; taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z = 0.90 km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} = 2.19, \theta _{c} =\sin ^{-1} (2.19/4.25)=31^{\circ} . From Figure 4.18a the critical distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'=2z\tan \theta _{c} =2\times 0.90\times \tan 31^{\circ} =1.1 km; at this point $ t'=2z/V_{1}\cos \theta _{c}=2\times 0.90/2.19\times \cos 31^{\circ }=0.96 $ s. The intercept time given by equation (4.18a) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{i} = 2z\cos \theta _{c} /V_{1} =0.70 s. Thus the head-wave curve passes through the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(t,\; x\right)= \left(0.96,1.1\right) with slope 1/4.25, is tangent to its reflection at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=1.1 km, and the intercept time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{i} =0.70 s. The reflection arrives at 0.863 s at zero offset and also passes through (0.96,1.1). These curves are shown in Figure 11.15b.

Carrying out similar calculations for the 5.27 km/s layer and using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} \approx 2.30 km/s (estimated from Figure 11.15a), we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\rm slope} = 1/5.27,\quad \theta _{c} =26^{\circ},\quad x'=0.95\ {\rm km},\quad t'=0.94\ {\rm s},\quad t_{i} =0.86\ {\rm s}. \end{align}

The reflection arrives at zero offset at 0.912 s and also passes through (0.95, 0.94). These curves are also plotted in Figure 11.15b.

The 4.25 km/s head wave is always a second arrival. It also follows very closely the reflection from the 5.27 km/s layer. It will almost certainly not be observed as a distinctly separate arrival because later cycles of earlier events will mask it.

11.16 Refraction blind spot

In early refraction exploration for salt domes, a “blind spot” (the region B–A in Figure 11.16a(ii)) was found when the salt dome lay directly on the line between the source and the geophone, that is, arrivals were often too weak to detect. This was called absorption of the wave” by the salt dome. What is the true explanation of the absorption?

Figure 11.15b.  Time-distance curves.

Solution

The vertical section in Figure 11.16a(i) shows rays being bent upon entering and leaving the salt dome. In the plan view of Figure 11.16a(ii), the bending of the rays in the horizontal plane is evident. This spreading of the wave, both vertically and horizontally, but especially the latter, lowers the energy density. Thus, the cause of the low amplitude is the raypath bending.

11.17 Interpreting marine refraction data.

How many distinct separate head waves are indicated in Figure 11.17a, and what are their apparent velocities? Calculate the depths and velocities of the respective refractors, assuming (i) no dip, (ii) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5^{\circ} dip to the right, (iii) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5^{\circ} dip to the left.

Figure 11.16a.  Wave travel in a salt dome (from Barton, 1929). (i) Vertical section; (ii) plan view.

Solution

We take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S at $ x=0 $ as the source and the geophone at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=12 km as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R . Four wavetrains are observed (Figure 11.17b). The slowest (which one has to view at grazing incidence to pick) has an apparent velocity of about 1.40 km/s; the arrival time at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R is off scale. This is the direct water wave, but its velocity is low for water, perhaps because of interference with a dispersive channel wave, loss of early cycles, or because the distance (calculated from water-borne direct waves seen at a floating sonobuoy) is not exact. Because of the uncertainty in the traveltime, we shall use the value 1.50 km/s for the water velocity.

The next event (L) has an apparent velocity of about 1.70 km/s and an arrival time of 6.84 s; the intercept time is difficult to determine because the event seems to be curved. It is probably a head wave at or near the sea floor.

The next fastest refraction (M) has an apparent velocity of about 2.19 km/s between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x = 1 and 4 km and an apparent velocity of 2.67 km/s for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x > 4 km; this change in velocity may be due to a change in dip or it may indicate the presence of two separate events. We shall consider only the velocity 2.67 km/s, the travel time being 5.38 s, and the intercept time 0.95 s.

Refraction N has a linear alignment and an apparent velocity of 5.45 km/s, traveltime of 3.97 s, and an intercept of 1.77 s; it is probably from a basaltic basement surface.

The measured data are:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathbf{L:}&\quad V_{\rm L} =1.70\ {\rm km/s},\quad t_{R{\rm L}} = 6.84\ {\rm s},\quad t_{iL} =?; \\ \mathbf{M:}&\quad V_{\rm M} =2.67\ {\rm km/s},\quad t_{R{\rm M}} = 5.38\ {\rm s},\quad t_{iM} =0.95\ {\rm s}; \\ \mathbf{N:}&\quad V_{\rm N} =5.45\ {\rm km/s},\quad t_{R{\rm N}} = 3.97\ {\rm s},\quad t_{iN} =1.77\ {\rm s}. \\ \end{align}

If L is a head wave, the intercept time should be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (6.84-12.0/1.70) = 0.22 s. Thus we conclude that L is not a head wave from a planar refractor; it may be a dispersive water wave or part of a reflection, but we have insufficient data to identify it. Disregarding L, we are left with only the water layer and the two refractors M and N.

Figure 11.17a.  Marine refraction profile (from Ingham, 1975, 130).

Figure 11.17b.  Interpreted marine refraction profile (from Ingham, 1975,130).

(i) Assuming no dip:

Refraction M

We have the following data: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{\rm I} =1.50 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{\rm M} =2.67 km/s, $ t_{i{\rm {M}}}=0.95 $ s.

Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sin \theta _{c} /1.50 &= 1/2.67,\quad \theta _{c} =34.2^{\circ};\\ h_{\rm M} &= V_{1} t_{i{\rm M}} /2\cos \theta _{c} = 1.50\times 0.95/2\times \cos 34.2^{\circ} =0.82\ {\rm km}. \end{align}

Refraction N:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sin \theta _{1} /1.50 &=\sin \theta _{c} /2.67= 1/5.45;\quad \theta _{1} =16.0^{\circ},\quad \theta _{c} =29.3^{\circ};\\ 1.77 &=2\times 0.82\cos 16^{\circ} /1.50+2h_{2} \cos 29.3^{\circ} /2.67;\\ h_{2} &= \left(1.77-1.10\right)/0.65=0.67/0.65=1.03\ {\rm km}; \\ z_{\rm N} &=0.83+1.03=1.86\ {\rm km}. \end{align}

(ii) Assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5^{\circ} dip to the right

In this case, the profile is in the downdip direction, the arrival times, velocities, and intercept times are unchanged except that the velocities are now apparent velocities and the intercept times give slant depths normal to the beds. Note that the water depth increases to the right because horizon M dips to the right.

Refractor M

We have: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{1} =1.5 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{d{\rm M}} =2.67 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi =5^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{i{\rm M}} =0.95 s. Using equation (4.24d), we write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &\sin \left(\theta _{c} +5^{\circ} \right)=V_{1} /V_{d{\rm M}} = 1.50/2.67;\\ &\left(\theta _{c} +5^{\circ} \right)=34.2^{\circ},\quad \theta _{c} =29.2^{\circ};\\ &V_{\rm M} =V_{1} /\sin \theta _{c} =1.50/\sin 29.2^{\circ} =3.07\ {\rm km/s}; \\ &h_{\rm M} =V_{1} t_{i{\rm M}} /2\cos \theta _{c} =1.50\times 0.95/2 \cos 29.2^{\circ} =0.82\ {\rm km} \end{align}

(this is the slant water depth at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S ).

Refractor $ N $

We could use Adachi’s method (problem 11.5), but the given data are not in a form suitable for this method. Instead we shall strip off (problem 11.6) the surface layer after which we have two parallel horizons, and therefore can use equation (4.18a) to solve for horizon N.

The slant depth of horizon M at the source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S is 0.82 km. At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R the slant depth is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{\rm M} =0.82+12.0\times \sin 5^{\circ} =1.87\ {\rm km}.

Thus to locate horizon M we swing arcs at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R with radii 0.82 and 1.87 km, respectively, then draw horizon M tangent to the arcs (see Figure 11.17c).

Figure 11.17c.  Construction for stripping off the water layer.

To solve for N, we need the total time that the refraction from N spends in the water layer. For this we need the angle of approach to the surface of head wave N. Equation (4.2d) states that the angle of approach Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \sin \alpha =V_{1} \left(\Delta t/\Delta x\right)=V_{1} /V_{d{\rm N}} = 1.50/5.45,\quad \alpha =16.0^{\circ}. \end{align}

Figure 4.2c shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha is relative to the vertical, so if the ray emerges from horizon M at the angle of refraction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta +5^{\circ} =16.0^{\circ} , whereas the angle at which the ray left source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta -5^{\circ} =\theta^{\circ} . Therefore we draw a ray from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S down to M at the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 6^{\circ} to the vertical and another down from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R to M at the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 16^{\circ} to the vertical (see rays Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): SS' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RR' in Figure 11.17c). The total length (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm SS}' + {\rm RR}' ) is (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0.80+1.80 ) km; dividing by 1.50 gives 1.73 s to be subtracted from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{N} , leaving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{N'} =3.97-1.73=2.24 s for the traveltime relative to M. Also the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S'R'=x'=11.5 km. The intercept times for both M and N are based on the normals to the beds; since the beds are parallel, we can subtract Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{iM} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{in} [see part (i)] to get $ t_{N'} $, the intercept time of N for the virtual source Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S' . This gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{iN'} =\left(1.77-0.95\right)=0.82 s.

We must correct the apparent velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{d{\rm {N}}} =5.45 km/s. We write this as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/5.45=\Delta t/\Delta x=2.20/12.0 s/km. The event N in Figure 11.17b is linear from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=12 to beyond Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=6 km, so we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/5.45=0.110/6 s/km. The numerator is the difference between $ t_{N} $ at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=12 and 6 km, and Figure 11.17c shows that the correction is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): RT=0.50/1.50=0.33 s. Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t'=1.10-0.33=0.77 s. The correction to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta x is negligible, so we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{N} \approx 6.00/0.77=7.79 km/s.

We can now get the depth of N below M. We have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \theta _{c} =\sin ^{-1} \left(V_{\rm M} -V_{\rm N} \right)=3.07/7.79=23.3^{\circ}. \end{align}

Because horizons M and N are parallel, we find the intercept time of N relative to M by subtracting the intercept times in part (i), so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{iN^{'}} = 1.77-0.95=0.82\ {\rm s}. \end{align}

Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h_{N^{'}} = 3.07\times 0.82/2 \cos 23.3^{\circ} =1.37\ {\rm km}. \end{align}

Then, since the depth of M at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S is 0.82 km,

$ {\begin{aligned}z_{N}=0.82+1.37=2.19\ {\hbox{km}}\ {\hbox{(slant depth at S')}}.\end{aligned}} $

(iii) Assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5^{\circ} dip to the left

Because of the shallow depth of M we check to see where it outcrops. The horizon M passes through the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(x,\; z\right)=\left(0,0.82\cos 5^{\circ} \right)=(0, 0.817) , (see part (ii)) with slope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan 5^{\circ} , Thus it will outcrop at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=0.817/ \tan 5^\circ =9.34 km. But Figure 11.17c shows that event M exists at offset of 12 km, hence the assumption of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 5^{\circ} dip to the left is not consistent with the given data.