Second approximation for dip moveout

Problems in Exploration Seismology and their Solutions

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

${\begin{aligned}t=t_{0}[1+(r/2)(r+2\sin \xi )-(r^{2}/8)(r+2\sin \xi )^{2}].\end{aligned}}$

(4.5a)

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
Table of Contents (book)

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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