# Vertical depth calculations using velocity functions

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 4 79 - 140 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 4.14

Assuming flat bedding, calculate depths corresponding to ${\displaystyle t_{0}=1.0}$, 2.0, 2.1, and 3.1 s using the velocity functions determined in problem 4.13b,c. What depth errors are created?

### Solution

The velocity functions determined are

1. average velocity ${\displaystyle {\bar {V}}}$ versus depth (in problem 4.13b),
2. rms-velocity ${\displaystyle V_{\rm {rms}}}$ versus depth (in problem 4.13b),
3. the best-fit ${\displaystyle {\bar {V}}}$ versus depth function (in problem 4.13c),
4. the best-fit ${\displaystyle V_{\rm {rms}}}$ versus traveltime function (in problem 4.13c).

Using these, we obtain the depths in Table 4.14a.

No depth errors are present in (i) because ${\displaystyle t_{0}}$ and ${\displaystyle {\bar {V}}}$ were derived from the the given data. The errors in calculated depths in (ii), (iii), and (iv) are tabulated in Table 4.14b. Using ${\displaystyle V_{\rm {rms}}}$ gives ${\displaystyle z}$-values 2–5% too large. The best-fit depth function in (iii) gives the best results overall while the best-fit traveltime function in (iv) has errors of the same order of magnitude as those in (ii).

Table 4.14b. Errors in depth calculations.
1.00 km 2.50 km 2.80 km 4.80 km
ii) ${\displaystyle V_{\rm {rms}}}$ 0.0% 2.5% 5.4% 4.6%
iii) best-fit ${\displaystyle {\bar {V}}}$ 1.0% 1.6% –2.9% 0.8%
iv) best-fit ${\displaystyle V_{\rm {rms}}}$ 2.0% 4.8% 0.4% 5.6%