User:Ageary/Chapter 13
| 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 |
Chapter 13 Specialized techniques
13.1 S-wave conversion in marine surveys
13.1a In a marine survey, the water depth is 100 m and a reflector is 3 km below the seafloor. Use Figure 13.1a to determine the optimum range of offsets for S-wave generation. Take Poisson’s ratio just below the seafloor as 0.35 and the P-wave velocity as 2.8 km/s. The velocity in sea water is 1.5 km/s.
Solution
Referring to Figure 13.1a(i), we see that the offset $ x $ for the 3-km reflector is
$ {\begin{aligned}x=2(0.1\times \tan \theta _{1}+3\times \tan \delta _{2})=0.2\tan \theta _{1}+6\tan \delta _{2},\end{aligned}} $ ()
where $ \theta _{1} $ is the angle of incidence of the P-wave at the bottom of the water layer, and $ \delta _{2} $ is the angle of refraction of the converted S-wave.
To find $ \delta _{2} $ for a given $ \theta _{1} $, we need the S-wave velocity $ \beta _{2} $. Using equation (1,8) in Table 2.2a, we find that $ \beta /\alpha =0.48 $ for $ \sigma =0.35 $, so $ \beta _{2}=0.48\alpha _{2}=0.48\times 2.8=1.3\ {\rm {km/s}} $. To get the angles of incidence $ \theta _{1} $ and of refraction $ \delta _{2} $, we have from equation (3.1a),
$ {\begin{aligned}\sin \theta _{1}/1.5=\sin \delta _{2}/1.3\;,\;\delta _{2}={\rm {sin}}^{-1}\left(0.87\sin \theta _{1}\right).\end{aligned}} $
The optimum angles of incidence $ \theta _{1} $ for S-wave conversion for $ \alpha =2.8\ {\rm {km/s}} $ are obtained by interpolating between the curves for $ \alpha =2.5 $ and 3.0 in Figure 13.1a(ii). This gives a range of about $ 38^{\circ } $ to $ 80^{\circ } $ for $ \theta _{1} $. The corresponding values of $ \delta _{2} $ are $ 32^{\circ } $ and $ 59^{\circ } $. Equation (13.1a) now gives offsets of 4 and 11 km for $ \theta _{1}=38^{\circ } $ and $ 80^{\circ } $.
13.1b Most marine S-wave surveys that wish to deal with S-waves utilize conversion at the reflector and recording with three-component geophones laid on the seafloor (OBC, ocean-bottom cables), so that only one mode conversion is involved. Assume a ray leaving an airgun source at $ 10^{\circ } $ to the vertical in water 100 m deep, an increase in P-wave velocities from 1.5 km/s at the seafloor to 3.0 km/s at a reflector 3 km below the sea floor (average velocity in the sediments of 2.25 km/s). Take $ \sigma $ in the sediments as 0.3. What will be the source-geophone offset?
Solution
The P-wave gives an incident angle at the reflector of $ \theta =\sin ^{-1}[(3.0/1.5)\sin 10^{\circ }]=20.3^{\circ } $. Using equation 1,8 in Table 2.2a, the S-wave velocity here is about 0.53 $ \alpha $ or 1.59 km/s. Hence the angle of reflection is $ \delta =\sin ^{-1}(0.53\sin 20.3^{\circ })=10.6^{\circ } $. The average direction of the P-wave ray in the sediments is $ \sin ^{-1}[(2.25/1.5)\sin 10^{\circ }]=15.1^{\circ } $. With constant $ \sigma $ in the sediments, $ \beta /\alpha $ is constant and, hence, $ \sin \delta /\sin \theta =0.53 $. Hence, the average direction of the S-wave is $ \sin ^{-1}(0.53\sin 15.1^{\circ }=7.9^{\circ }) $. The offset is thus
$ {\begin{aligned}100\tan 10^{\circ }+3000\tan 15.1^{\circ }+3000\tan 7.9^{\circ }=1240\ {\rm {m}}.\end{aligned}} $
This is a very reasonable offset.
13.2 Equally inclined orthogonal geophones
Determine the inclination angles for three orthogonal geophones that are equally inclined to the vertical.
Solution
The three orthogonal geophones define a rectangular coordinate system with the $ x $-, $ y $-, and $ z $-axes along the geophones axes. We take a straight line through the origin of this system such that it is equally inclined to each axis, and rotate the coordinate system so that the straight line is vertical. The direction cosines of the vertical line are $ (l,m,n) $ and $ l=m=n $. Because $ l^{2}+m^{2}+n^{2}=1 $ [Sheriff and Geldart, problem (15.9a)],
$ {\begin{aligned}3l^{2}=1,\quad l=1{\sqrt {3}},\cos ^{-1}(1/{\sqrt {3}})=54.74^{\circ }.\end{aligned}} $
Thus the geophones must be inclined $ 54.74^{\circ } $ to the vertical.
13.3 Guided (Channel) waves; Normal-mode propagation
13.3a In Figure 13.3a the first arrival ($ f_{o}=40\ {\rm {Hz}} $) has traveled at the velocity 2.7 km/s; find the water depth.
Background
A wave guide is a layer in which a wave can propagate with little loss of energy. In a water layer nearly total reflection occurs at both boundaries, at the surface because of the very large impedance contrast and at the bottom reflection beyond the critical angle. The phase is inverted at the surface because the reflectivity is nearly $ -1 $, but not at the seafloor (beyond the critical angle) until the angle becomes very large.
Figure 13.3b(i) shows waves bouncing back and forth in a wave guide. For certain incident angles $ \theta $ and frequencies $ f $, they interfere constructively. In Figure 13.3b(ii), $ AC $ is a wavefront traveling upward at the angle $ \theta $. The previous cycle of a parallel wavefront that passed through the position earlier followed paths such as EFGH and BDA and thus coincides with $ AC $. Clearly $ (EF+FG+GH)=BD+DA $. $ DA=h/\cos \theta $, $ BD=DA\cos 2\theta $, so $ (BD+DA)=2h/\cos \theta $. Taking into account the phase reversal at the surface the condition for constructive interference is
$ {\begin{aligned}(BD+DA)=2h\cos \theta =(2n+1)\lambda /2,\end{aligned}} $
$ {\begin{aligned}{\mbox{so}}\qquad \qquad h=(2n+1)\lambda /4\cos \theta .\end{aligned}} $ ()
Writing $ V $ for the phase velocity, the frequencies that are reinforced are
$ {\begin{aligned}f_{n}=V/\lambda =(2n+1)V/4h\cos \theta ,\quad n=0,1,2,\ldots .\end{aligned}} $ ()
In addition to the upgoing waves parallel to $ AC $ and $ A^{\prime }C^{\prime } $ in Figure 13.3b(iii), a second downgoing set $ PQ $ and $ P^{\prime }Q^{\prime } $ will combine with the set $ AC $ to build up the energy traveling in the direction $ RR^{\prime } $ along the wave guide. Energy travels from $ R $ to $ R^{\prime } $ in the time that wavefront $ AC $ moves to $ A^{\prime }C^{\prime } $, so the phase velocity of the energy traveling along $ RR^{\prime } $, $ V_{g} $, is
$ {\begin{aligned}V_{g}=V/\sin \theta .\end{aligned}} $ ()
Since both $ V_{g} $ and $ f $ are functions of $ \theta $, $ V_{g} $ is dispersive with a group velocity $ U $ given by equation (2.7a):
$ {\begin{aligned}U=V_{g}+f{\frac {{\rm {d}}V_{g}}{{\rm {d}}f}}.\end{aligned}} $ ()
The derivative is always negative for a water channel wave (Figure 13.3c), so $ U<V_{g} $.
If the wave-guide effect had been due to reflection beyond the critical angle at both boundaries, as with a low-velocity coal seam, the phase would not have been reversed at either boundary and equations (13.3a,b) would reduce to
$ {\begin{aligned}h=n\lambda /2\cos \theta ,\quad f_{n}=nV/2h\cos \theta .\end{aligned}} $ ()
Solution
Equation (13.3c) gives
$ {\begin{aligned}\sin \theta =V/V_{g}=1.5/2.7,\quad \theta =33.7^{\circ },\end{aligned}} $
and we use equations (13.3b,e) with $ n=0 $ to get $ h $:
$ {\begin{aligned}h=V/4f_{0}\cos \theta =1500/4\times 40\cos 33.7^{\circ }=11.3\ {\rm {m}}.\end{aligned}} $
13.3b What frequency is reinforced when $ \theta =40^{\circ } $?
Solution
We use equation (13.3b) with $ n=0 $, $ h=11.3\ {\rm {m}} $, $ \theta =40^{\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/":): \begin{align} f=V/4h\cos \theta =1500/4\times 11.3\cos 40^{\circ} =43.3\ {\rm Hz}. \end{align}
13.4 Vertical seismic profiling
13.4a (a) A source is offset 1000 m from a vertical well in which a geophone is suspended, and a horizontal reflector is present at a depth of 2000 m. Calculate the traveltimes for the reflection when the geophone is 800, 1200, and 1600 m deep if the velocity =3000 m/s
Solution
Using a coordinate system with origin at the source 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/":): z -axis positive downward, the image 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/":): I (see problem 4.1) in Figure 13.4a is at (0, 4000) and the geophone coordinates are (1000, Failed to parse (SVG (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 ); 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} t_{c} =[1000^{2} +(4000-z)^{2} ]^{1/2} /3000. \end{align}
For 800 m 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/":): t=1.118\ {\rm s} ; for 1200 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/":): t=0.991\ {\rm s} ; for 1600 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/":): t=0.867\ {\rm s} .
13.4b Repeat for reflectors dipping Failed to parse (SVG (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 7^{\circ} , where the reflector intersects the well at the same point as in part (a) (see Figure 13.4b).
Solution
i) Dip $ +7^{\circ } $ down toward the well
Failed to parse (SVG (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&= \hbox {vertical depth of reflector at source S}\\ &=(2000-1000\tan 7^{\circ} )=1880\ {\rm m},\\ b&= \hbox {slant depth of reflector} =a\cos 7^{\circ} \\ &=1870\ {\rm m}, \\ \hbox {Coordiantes of image}&=(-2\hbox {b} \sin 7^{\circ} ,2\hbox {b} \cos 7^{\circ} )\\ &= (-460, 3710), \\ \hbox{time}_+&=[(1000+460)^{2} +(3710-Z)^{2} ]^{1/2} /3000. \end{align}
ii) 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/":): -7^{\circ} down toward 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
Failed to parse (SVG (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=(2000-1000\tan 7^{\circ} )=2120\ {\rm m}, b=2100\ {\rm m}, \end{align}
Coordinates of image 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/":): +510 , 4170),
Failed to parse (SVG (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}=[(1000+510)^{2} +(4170-z)^{2} ]^{1/2} /3000. \end{align}
Table 13.4a shows calculated traveltimes 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/":): z=800 , 1200, and 1600 m.
| Depth(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/":): \to | 800 | 1200 | 1600 | |
| Failed to parse (SVG (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 Dip} =+7^{\circ} | time (s) | $ \to $ | 1.09 | 0.97 | 0.86 |
| Failed to parse (SVG (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 Dip}=-7^{\circ} | time (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/":): \to | 1.14 | 1.00 | 0.87 |
| Failed to parse (SVG (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(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/":): x_{d} (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/":): Z_{d}(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/":): t_{d} (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) |
|---|---|---|---|---|
| 800 | 958 | 799 | 1.114 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -0.004 |
| 1200 | 937 | 1198 | 0.985 | $ -0.006 $ |
| 1600 | 916 | 1598 | 0.857 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -0.010 |
13.4c By how much do the values in part (a) change if the well deviates 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/":): 3^{\circ} towards the source (see Figure 13.4c)?
Solution
Well inclination changes the geophone coordiantes from (1000, Failed to parse (SVG (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 ) where the in-hole 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/":): z=800 , 1200, 1600, 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_{d} , z_{d}) 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} x_{d} =(1000-z\sin 3^{\circ} ),\quad z_{d} =z\cos 3^{\circ} , \end{align}
where subscript Failed to parse (SVG (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 denotes values for the deviated well. The image point remains at (0, 4000), so 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_{d} 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} t_{d} =[x_{\rm d}^{2} +(4000-z_{\rm d} )^{2} ]^{1/2} /3000. \end{align} ()
Substituting the values of $ z $, we get the results in Table 13.4b. Failed to parse (SVG (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 is the difference in traveltime from that calculated in part (a).
13.5 Effect of velocity change on VSP traveltime
13.5a A source is offset 3.00 km west of a vertical well in which a geophone is suspended. There is a vertical north-south fault 0.80 km west of the well with velocities Failed to parse (SVG (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=4.00 and 3.00 km/s west and east of the fault, respectively. A horizontal reflector is present west of the fault at a depth of 2.00 km. Find the reflection traveltime for geophone depths of 0.60 and 1.20 km.
Solution
This problem must be solved by trial-and-error methods, either graphical or numerical. We have chosen the latter, because it is easier, quicker, and more accurate than graphical methods.
Figure 13.5a shows the geometry. The horizontal reflector at a depth of 2 km has an image point 4.00 km below the source. A typical ray is shown going from the image point to a geophone located in the well at a 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 .
The angle of incidence Failed to parse (SVG (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 _{1} and 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 _{2} where the ray passes through the fault are related by Snell’s law: 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/":): \sin\theta_2 = (3.00/4.00) \sin \theta _{1} =0.75\sin \theta _{1} . We have a further constraint in that the vertical distance between the image and the geophone 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/":): (4.00-h) . These relations fix Failed to parse (SVG (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 :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin \theta _{2} =0.75\sin \theta _{1} ;
- $ 2.20\tan \theta _{1}+0.80\tan \theta _{2}=(4.00-h) $.
We denote the left-hand side of (b) 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/":): L . For the geophone at depth 0.60 km, the bending of the ray will have a small effect because of the shallowness and proximity to the fault, so we take as our first approximation the straight 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/":): \begin{align} \theta _{1} \approx \tan^{-1} (4.00-0.60)/3.00\approx 49^{\circ}. \end{align}
To allow for some bending, we try Failed to parse (SVG (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 _{1} =51^{\circ} 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/":): \theta _{2} =35.7^{\circ} . Substituting theses values in relation (b) 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} L=2.20\tan 51^{\circ} +0.80\tan 35.7^{\circ} =3.29\ {\rm km}. \end{align}
This 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/":): (4.00-0.60) =3.40 , so we must 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/":): \theta _{1} by a small amount; 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/":): \theta _{1} =52^{\circ} 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/":): \theta _{2} =36.2^{\circ} and $ L=3.40\ {\rm {m}} $. We now find 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/":): \begin{align} {\rm time} =(2.20/4.00\cos \theta _{1} )+(0.800/3.00\cos \theta _{2} )=1.224\ {\rm s}. \end{align}
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/":): h=1.20\ {\rm 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/":): L=2.80\ {\rm km} . We know 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/":): \theta _{1} will be less than for the shallower geophone, so 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/":): \theta _{1} =45^{\circ} 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/":): \theta _{2} =32.0^{\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/":): L=2.70 . 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/":): \theta _{1} =47.0^{\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/":): \theta _{2} =33.3^{\circ} , $ L=2.88 $. The change 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/":): 2^{\circ} increased Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): L by 0.18 m; we need an increase of 0.10, so 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/":): \theta _{1} =45^{\circ} +2^{\circ} \times (0.10/0.18)=46.1^{\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/":): \theta _{2} =32.7^{\circ} , 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/":): L=2.80\ {\rm km} . The traveltime 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} {\rm time} =(2.20/4.00\cos 46.1^{\circ} )+(0.800/3.00\cos 32.7^{\circ} )=1.110\ {\rm s}. \end{align}
13.5b What is the deepest geophone location for which the reflection can be recorded?
Solution
The deepest geophone position is that for which the ray passes through the intersection of the fault and reflector. For this 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/":): \begin{align} \theta _{1} =\tan^{-1} (2.00/2.20)=42.3^{\circ},\quad \theta _{2} =30.3^{\circ},\\ L=2.20\tan 42.3^{\circ} +0.80\tan 30.3^{\circ} =2.47\ {\rm km},\\ (400-h)=2.47,\quad h=1.53\ {\rm km}. \end{align}
13.6 Mapping the vertical flank of a salt dome
13.6a Two media of velocities Failed to parse (SVG (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 $ V_{2} $ are separated by a vertical plane interface. A source 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/":): S on the surface of the high-velocity medium and a geophone 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/":): G in a well in the other medium. 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/":): t_{A} is the traveltime of a wave 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 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 along the path SAG 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/":): A is on the salt/sediment interface, discuss the locus of points located at the intersection of arcs centered 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/":): 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/":): S having radii Failed to parse (SVG (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 t_{1} and $ V_{2}\Delta t_{2} $, respectively, 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/":): \Delta t_{1} +\Delta t_{2} =t_{A} .
Solution
Any path where the interface is tangent to the locus curve satisfies the observed traveltime. To draw the locus we try a series 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/":): {\rm B}_{i} 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/":): ({\rm SB}_{i} /V_{2} +{\rm GB}_{i} /V_{1} )=t_\hbox {A} . If a number of loci can be drawn for various geophone (or source) loctions, then their common tangent must locate the interface.
While this problem illustrates the concept, it can also be applied in three dimensions to allow for situations 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/":): V_{1} \Delta t_{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} \Delta t_{2} do not lie in the same plane.
| Failed to parse (SVG (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({\rm 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/":): t({\rm 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/":): z({\rm 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/":): t({\rm s}) | $ z({\rm {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/":): t({\rm s}) |
|---|---|---|---|---|---|
| 500 | 0.44 | 1250 | 0.52 | 2000 | 0.6 |
| 750 | 0.46 | 1500 | 0.56 | 2250 | 0.67 |
| 1000 | 0.49 | 1750 | 0.60 | 2500 | 0.70 |
13.6b An outcropping salt dome has roughly vertical flanks. A source is located on the salt and a geophone is suspended in a vertical well in the sediments 1600 m from the source point. Determine the outline of the salt dome from 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/":): t-z data in Table 13.6a. Take the velocities in the salt and adjacent sediments as 5.00 and 3.00 km/s.
Solution
The usual method of resolving this problem is to prepare a vertical section through the well W and source point S, then draw a series of circles centered at S with radii equal to the distances traveled in salt for convenient time intervals such as 0.2 s, 0.4 s, etc., the circles being labeled with the time value. A second set of concentric circles using the sediment velocity is drawn on a transparency; the center of these circles is then placed over a geophone position, and intersections of the two sets of circles are marked wherever the time values of the two radii add up to give the traveltime to the geophone.
An alternative method is to dispense with the circles and use only the radii. The centers of the circles and the extremities of the radii are marked along the edges of two narrow strips of light cardboard (one for salt, one for sediments), the time values being marked as before (the marks and the centers should be on opposite sides of the two strips to permit accurate determinations of the intersections). Map pins can be used to attach the zero points of the two strips to the vertical section, one at the source point and the other at the geophone location in the well. Intersections are found as before and marked directly on the section. After curves have been plotted for each geophone position, the flank is outlined by the curve through the apices of the curves. The results are shown in Figure 13.6a.
13.7 Poission’s ratio from P- and S-wave traveltimes
Find Poisson’s ratio for the five events in Figure 13.7a.
Solution
Poisson’s 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/":): \sigma can be obtained from 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/":): V_{\hbox {P}} /V_{\hbox {S}} using equation (10.2) Table 2.2a. 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/":): z=V_{\hbox {P}} t_{\hbox {P}} =V_{\hbox {S}} t_{\hbox {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_{\hbox {P}} /V_{\hbox {S}} =t_{\hbox {S}} /t_{\hbox {P}} , we can 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_{\hbox {P}}/V_{\hbox {S}}) from the traveltimes and then get $ \sigma $ using equation (10,2) (see Table 2.2a), 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/":): \begin{align} \sigma =\frac{(V_{\hbox {P}} /V_{\hbox {S}} )^{2}-2}{2[(V_{\hbox {P}}/V_{\hbox {S}} )^{2}-1]}. \end{align}
Measurements and calculations are listed in Table 13.7a.
| Event | Failed to parse (SVG (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_{\hbox {P}} | Failed to parse (SVG (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_{\hbox {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/":): \alpha /\beta | Failed to parse (SVG (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 /\beta )^{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/":): \sigma |
|---|---|---|---|---|---|
| 1 | 0.39 | 0.96 | 2.46 | 6.05 | 0.40 |
| 2 | 0.62 | 1.50 | 2.42 | 5.86 | 0.40 |
| 3 | 0.74 | 1.65 | 2.23 | 4.97 | 0.37 |
| 4 | 0.80 | 1.83 | 2.29 | 5.24 | 0.38 |
| 5 | 0.99 | 2.16 | 2.18 | 4.75 | 0.37 |