Depth and dip calculations using velocity functions

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Problem 4.15a

Repeat the calculations of problem 4.14 assuming horizontal velocity layering and dip moveout of 104 ms/km, and find the dips.


While velocity generally follows the layering, especially in structurally deformed areas, isovelocity surfaces may not parallel interfaces. Where the section has not been uplifted significantly, isovelocity surfaces are apt to be nearly horizontal in spite of structural relief.


The depths will be the same as those calculated in problem 4.14 except that is now slant depth. Vertical depths are , where the dip is given by , usually being either or . Using the values of , and from Table 4.14a, we obtain the results in Table 4.15a.

Table 4.15a. Calculated depths and dips.
Time Velocity Slant Dip Vert. depth
i) Assuming average velocities:
1.00 s 2.00 km/s 1000 m 5.97 990 m
2.00 2.50 2500 7.47 2480
2.10 2.67 2800 7.98 2770
3.10 3.10 4800 9.28 4740
ii) Assuming rms velocities:
1.00 s 2.99 km/s 1000 m 5.97 990 m
2.00 2.55 2550 7.62 2530
2.10 2.81 2950 8.40 2920
3.10 3.24 5020 9.70 4950
iii) Assuming best-fit depth function:
1.00 s 2.02 km/s 1010 m 6.03 1000 m
2.00 2.54 2540 7.59 2520
2.10 2.59 2720 7.74 2700
3.10 3.12 4840 9.34 4780
iv) Assuming best-fit traveltime function:
1.00 s 2.03 km/s 1020 m 6.00 1010 m
2.00 2.62 2620 7.83 2800
2.10 2.68 2810 8.01 2780
3.10 3.27 5270 9.79 5000

Problem 4.15b

Trace rays assuming (i) the velocity layering given in Figure 4.13a, and (ii) that the velocity is constant at the values of and listed in Figure 4.13b. Find the arrival times and reflecting points of reflections at each of the interfaces.


i) We first calculate the angle of approach [using equation (4.2d)] and then use Snell’s law to find the other angles:

Next we find -coordinates of intersections of rays and interfaces:

Figure 4.15a.  Raypath.

ii) Assuming (see Figure 4.13b) and the given depths,

Assuming and the given depths,

Assuming and times in part (i):

Assuming and times in part (i):

The use of any functional form involves approximation, so it is not surprising that values of the depths and horizontal displacements depend upon the way in which they are calculated.

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