Finite-difference migration

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	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 9.29

Derive the finite-difference migration equation (9.29d).

Background

The wave equation (2.5a) becomes, for two dimensions,

${\begin{aligned}{\frac {1}{V^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}=({\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}).\end{aligned}}$

(9.29a)

We can simplify equation (9.29a) by replacing $t$ with $t^{*}=t-z/V$ (see problem 9.28). To replace derivatives in the $(x,\;z,\;t)$ coordinate system with derivatives in the $(x,\;z,\;t^{*})$ system, we follow the procedure used in problem 2.6:

${\begin{aligned}{\frac {\partial \psi }{\partial t}}={\frac {\partial \psi ^{*}}{\partial t^{*}}}{\frac {\partial t^{*}}{\partial t}}={\frac {\partial \psi ^{*}}{\partial t^{*}}};\quad {\frac {\partial ^{2}\psi }{\partial t^{2}}}={\frac {\partial ^{2}\psi ^{*}}{\partial t^{*2}}};\\{\frac {\partial \psi }{\partial x}}={\frac {\partial \psi ^{*}}{\partial x}};\quad {\frac {\partial ^{2}\psi }{\partial x^{2}}}={\frac {\partial ^{2}\psi ^{*}}{\partial x^{2}}};\\{\frac {\partial \psi }{\partial z}}={\frac {\partial \psi ^{*}}{\partial z}}+{\frac {\partial \psi ^{*}}{\partial t^{*}}}{\frac {\partial t^{*}}{\partial z}}={\frac {\partial \psi ^{*}}{\partial z}}-{\frac {1}{V}}{\frac {\partial \psi ^{*}}{\partial t^{*}}};\\{\frac {\partial ^{2}\psi }{\partial z^{2}}}={\frac {\partial ^{2}\psi ^{*}}{\partial z^{2}}}+{\frac {\partial ^{2}\psi ^{*}}{\partial z\partial t^{*}}}{\frac {\partial t^{*}}{\partial z}}-{\frac {1}{V}}{\frac {\partial ^{2}\psi ^{*}}{\partial z\partial t^{*}}}-{\frac {1}{V}}{\frac {\partial ^{2}\psi ^{*}}{\partial t^{*2}}}{\frac {\partial t^{*}}{\partial z}}\\={\frac {\partial ^{2}\psi ^{*}}{\partial z^{2}}}-{\frac {2}{V}}{\frac {\partial ^{2}\psi ^{*}}{\partial z\partial t^{*}}}+{\frac {1}{V^{2}}}{\frac {\partial ^{2}\psi ^{*}}{\partial t^{*2}}}.\end{aligned}}$

Substituting these expressions into equation (9.29a), we obtain the result

${\begin{aligned}{\frac {\partial ^{2}\psi ^{*}}{\partial x^{2}}}+{\frac {\partial ^{2}\psi ^{*}}{\partial z^{2}}}-{\frac {2}{V}}{\frac {\partial ^{2}\psi ^{*}}{\partial z\partial t^{*}}}=0.\end{aligned}}$

(9.29b)

However, ${\frac {\partial ^{2}\psi ^{*}}{\partial z^{2}}}$ varies slowly because the coordinate system rides along with the wavefront and so we omit it. This gives the simplified wave equation,

${\begin{aligned}{\frac {\partial ^{2}\psi ^{*}}{\partial x^{2}}}-{\frac {2}{V}}{\frac {\partial ^{2}\psi ^{*}}{\partial z\partial t^{*}}}\approx 0.\end{aligned}}$

(9.29c)

Using the method of finite differences discussed below, this equation can be changed to

${\begin{aligned}\psi (x,\;z,\;t^{*})&={\frac {2\Delta z\Delta t^{*}(\Delta x)^{2}}{2(\Delta x)^{2}-V\Delta z\Delta t^{*}}}\left[{\frac {\psi ^{*}(x,z-\Delta z,t^{*})}{\Delta z\Delta t^{*}}}+{\frac {\psi ^{*}(x,z,t^{*}-\Delta t^{*})}{\Delta z\Delta t^{*}}}\right.\\&\quad -{\frac {V\psi ^{*}(x-\Delta x,z,t^{*})}{(\Delta x)^{2}}}-{\frac {\psi ^{*}(x,z-\Delta z,t^{*}-\Delta t^{*})}{\Delta z\Delta t^{*}}}\\&\left.+{\frac {V\psi ^{*}(x-2\Delta x,z,t^{*})}{2(\Delta x)^{2}}}\right].\end{aligned}}$

(9.29d)

Approximate solutions of differential equations can be found using the method of finite differences. If we denote the value of $f(x)$ at $x_{1}$ by the symbol $f_{1}$, an approximate value of the derivative is

${\begin{aligned}{\hbox{d}}f/{\hbox{d}}x\approx {\frac {f_{2}-f_{1}}{x_{2}-x_{1}}}={\frac {\Delta f}{\Delta x}},\end{aligned}}$

(9.29e)

the error decreasing as $\Delta x\to 0$. The second derivative at a given point can be found by finding the difference between two first derivatives close to the given point and dividing the difference by the distance between the two points. Derivatives with respect to more than one variable can be found using the same principle.

Solution

We use points spaced at intervals $\Delta x$, $\Delta z$, $\Delta t^{*}$ to evaluate the two derivatives in equation (9.29c):

${\begin{aligned}{\frac {\partial \psi ^{*}}{\partial x}}&=[\psi ^{*}\;(x,\;z,\;t^{*})-\psi ^{*}(x-\Delta x,\;z,\;t^{*})]/\Delta x;\\{\frac {\partial ^{2}\psi ^{*}}{\partial x^{2}}}&=\{[\psi ^{*}(x,\;z,\;t^{*})-\psi ^{*}(x-\Delta x,\;z,\;t^{*})]-[\psi ^{*}(x-\Delta x,\;z,\;t^{*})\\&\quad -\psi ^{*}(x-2\Delta x,\;z,\;t^{*})]\}/(\Delta x)^{2}\\&=[\psi ^{*}(x,\;z,\;t^{*})-2\psi ^{*}(x-\Delta x,\;z,\;t^{*})+\psi ^{*}(x-2\Delta x,\;z,\;t^{*})]/(\Delta x)^{2};\\{\frac {\partial \psi ^{*}}{\partial z}}&=[\psi ^{*}(x,\;z,\;t^{*})-\psi ^{*}(z,\;z-\Delta z,\;t^{*})]/\Delta z;\\{\frac {\partial ^{2}\psi ^{*}}{\partial z\partial t^{*}}}&=\{\psi ^{*}(x,\;z,\;t^{*})-[\psi ^{*}(x,\;z-\Delta z,\;t^{*})-\psi ^{*}(x,\;z,\;t^{*}-\Delta t^{*})\\&\quad -\psi ^{*}(x,\;z-\Delta z,\;t^{*}-\Delta t^{*})]\}/(\Delta z\Delta t^{*}).\\\end{aligned}}$

Writing equation (9.29c) in the form

${\begin{aligned}{\frac {\partial ^{2}\psi ^{*}}{\partial x^{2}}}={\frac {2\partial ^{2}\psi ^{*}}{V\partial z\partial t^{*}}},\end{aligned}}$

we substitute the above values of the two derivatives and obtain

${\begin{aligned}&[\psi ^{*}(x,\;z,\;t^{*})-2\psi ^{*}(x-\Delta x,\;z,\;t^{*})+\psi ^{*}(x-2\Delta x,\;z,\;t^{*})]/(\Delta x)^{2}\\&\quad =(2/V)\{[\psi ^{*}(x,\;z,\;t^{*})-\psi ^{*}(x,\;z-\Delta z,\;t^{*})]-[\psi ^{*}(x,\;z,\;t^{*}-\Delta t^{*})\\&\qquad -\psi ^{*}(x,\;z-\Delta z,\;t^{*}-\Delta t^{*})]\}/(\Delta z\Delta t^{*}).\end{aligned}}$

Rearranging, we have

${\begin{aligned}&\psi ^{*}(x,\;z,\;t^{*})\left({\frac {1}{(\Delta x)^{2}}}-{\frac {2}{V\Delta z\Delta t^{*}}}\right)\\&\quad =[2\psi ^{*}\;(x-\Delta x,\;z,\;t^{*})-\psi ^{*}(x-2\Delta x,\;z,\;t^{*})]/(\Delta x)^{2}\\&\qquad +{\frac {2}{V}}[-\psi ^{*}(x,\;z-\Delta z,\;t^{*})-\psi ^{*}(x,\;z,\;t^{*}-\Delta t^{*})\\&\qquad +\psi ^{*}(x,\;z-\Delta z,\;t^{*}-\Delta t^{*})]/(\Delta z\Delta t^{*}).\end{aligned}}$

Solving for $\psi ^{*}(x,\;z,\;t^{*})$ gives

${\begin{aligned}\psi ^{*}(x,\;z,\;t^{*})&=\left({\frac {2(\Delta x)^{2}V\Delta z\Delta t^{*}}{V\Delta z\Delta t^{*}-2(\Delta x)^{2}}}\right)\\&\quad \times \{[2\psi ^{*}\;(x-\Delta x,\;z,\;t^{*})-\psi ^{*}(x-2\Delta x,\;z,\;t^{*})]\\&\quad +{\frac {2}{V}}[-\psi ^{*}(x,\;z-\Delta z,\;t^{*})-\psi ^{*}(x,\;z,\;t^{*}-\Delta t)\\&\quad +\psi ^{*}(x,\;z-\Delta z,\;t^{*}-\Delta t^{*})]/(\Delta z\Delta t^{*})\}.\end{aligned}}$

Multiplying the first bracket by $-1/V$ and the second one by $-V/2$, we get

${\begin{aligned}\psi ^{*}(x,\;z,\;t^{*})&=\left[{\frac {2(\Delta x)^{2}\Delta z\Delta t^{*}}{(\Delta x)^{2}-V\Delta z\Delta t^{*}}}\right]\left[{\frac {\psi ^{*}(x,z-\Delta z,t^{*})}{\Delta z\Delta t^{*}}}+{\frac {\psi ^{*}(x,z,t^{*}-\Delta t^{*})}{\Delta z\Delta t^{*}}}\quad -{\frac {\psi ^{*}(x,z-\Delta z,t^{*}-\Delta t^{*})}{\Delta z\Delta t^{*}}}-{\frac {V\psi ^{*}(x-\Delta x,z,t^{*})}{(\Delta x)^{2}}}\quad +{\frac {V\psi ^{*}(x-2\Delta x,z,t^{*})}{2(\Delta x)^{2}}}\right],\end{aligned}}$

which is equation (9.29d).

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Table of Contents (book)

Also in this chapter

• Fourier series
• Space-domain convolution and vibroseis acquisition
• Fourier transforms of the unit impulse and boxcar
• Extension of the sampling theorem
• Alias filters
• The convolutional model
• Water reverberation filter
• Calculating crosscorrelation and autocorrelation
• Digital calculations
• Semblance
• Convolution and correlation calculations
• Properties of minimum-phase wavelets
• Phase of composite wavelets
• Tuning and waveshape
• Making a wavelet minimum-phase
• Zero-phase filtering of a minimum-phase wavelet
• Deconvolution methods
• Calculation of inverse filters
• Inverse filter to remove ghosting and recursive filtering
• Ghosting as a notch filter
• Autocorrelation
• Wiener (least-squares) inverse filters
• Interpreting stacking velocity
• Effect of local high-velocity body
• Apparent-velocity filtering
• Complex-trace analysis
• Kirchhoff migration
• Using an upward-traveling coordinate system
• Finite-difference migration
• Effect of migration on fault interpretation
• Derivative and integral operators
• Effects of normal-moveout (NMO) removal
• Weighted least-squares

External links

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