Interpreting marine refraction data

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 11.17

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 Failed to parse (SVG (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 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, $ V_{\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/":): 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 Failed to parse (SVG (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 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

$ {\begin{aligned}\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{aligned}} $

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 Failed to parse (SVG (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'} , 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 Failed to parse (SVG (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} 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.

Continue reading

Previous section Next section
Refraction blind spot Spatial sampling restrictions
Previous chapter Next chapter
Geologic interpretation of reflection data 3D methods

Table of Contents (book)

Also in this chapter

External links

find literature about
Interpreting marine refraction data
Retrieved from "http://wiki.seg.org/w/index.php?title=Interpreting_marine_refraction_data&oldid=133624"