Second approximation for dip moveout
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| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 4 |
| Pages | 79 - 140 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | SEG Online Store |
Problem 4.5
The expressions for dip in terms of dip moveout, equation (4.2b), involves the approximation of dropping higher-order terms in the expansion of equation (4.2a). What is the effect on equation (4.2b) if an additional term is carried in this expansion? What is the percentage change in dip?
Solution
In problem 4.2b we obtained the dip equation (4.2b) by taking the first approximation of equation (4.2a), that is, using
$ {\begin{aligned}t&=(1/V)(x^{2}+4h^{2}+4hx\sin \xi )^{1/2}\\&=t_{0}[1+r(r+2\sin \xi )]^{1/2},\end{aligned}} $
where $ r=x/2h $. Expanding and taking the second approximation gives
$ {\begin{aligned}t=t_{0}[1+(r/2)(r+2\sin \xi )-(r^{2}/8)(r+2\sin \xi )^{2}].\end{aligned}} $ ()
If we take two offsets, $ \Delta x $ and $ -\Delta x $, and let $ r=|\Delta x/2h| $, then
$ {\begin{aligned}\Delta t_{d}&=t_{1}-t_{2}=t_{0}(2r\sin \xi -r^{3}\sin \xi )\\&=t_{0}[\Delta x/h-(\Delta x/h)^{3}/8]\sin \xi \\&=(t_{0}\Delta x/h)\sin \xi [1-(\Delta x/h)^{2}/8]\\&=(2/V)(\Delta x\sin \xi )[1-(\Delta x/h)^{2}/8],\end{aligned}} $
$ {\begin{aligned}{\hbox{so}}\qquad \qquad \sin \xi =(V/2)\left({\frac {\Delta t_{d}}{\Delta x}}\right)[1+(2\Delta x/h)^{2}/32].\end{aligned}} $
Comparing this result with equation (4.2b), we see that the second approximation increases the calculated dip by the fraction $ (2\Delta x/h)^{2}/32 $, that is, by the approximate percentage
$ {\begin{aligned}3(2\Delta x/h)^{2}=3({\hbox{spread length/depth}})^{2}.\end{aligned}} $
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| Reflector dip in terms of traveltimes squared | Calculation of reflector depths and dips |
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| Partitioning at an interface | Seismic velocity |
Also in this chapter
- Accuracy of normal-moveout calculations
- Dip, cross-dip, and angle of approach
- Relationship for a dipping bed
- Reflector dip in terms of traveltimes squared
- Second approximation for dip moveout
- Calculation of reflector depths and dips
- Plotting raypaths for primary and multiple reflections
- Effect of migration on plotted reflector locations
- Resolution of cross-dip
- Cross-dip
- Variation of reflection point with offset
- Functional fits for velocity-depth data
- Relation between average and rms velocities
- Vertical depth calculations using velocity functions
- Depth and dip calculations using velocity functions
- Weathering corrections and dip/depth calculations
- Using a velocity function linear with depth
- Head waves (refractions) and effect of hidden layer
- Interpretation of sonobuoy data
- Diving waves
- Linear increase in velocity above a refractor
- Time-distance curves for various situations
- Locating the bottom of a borehole
- Two-layer refraction problem
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