Derivative and integral operators

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 9.31a

The operator $f_{t}=[-1,\;+1]$ is called the “derivative operator”; explain why.

Solution

In terms of finite differences (see problem 9.29), the derivative of a function is approximately equal to the change in values between two adjacent points divided by the distance between the two points. Convolving $f_{t}$ with $g_{t}$ gives values of the finite-difference derivative of $g_{t}$ for different values of $t$. To show this, we find the $(n+1)^{th}$ term of $h_{t}=f_{t}*g_{t}$. From equation (9.2b), we have

${\begin{aligned}h_{n}=\mathop {\sum } \limits _{k=0}^{}f_{k}g_{n-k}=[-g_{0},\;(-g_{1}+g_{0}),\;.\;.\;.\;(-g_{n}+g_{n-1})].\end{aligned}}$

Figure 9.30a.  Seismic section. (i) Unmigrated; (ii) migrated.

Figure 9.30b.  Interpreted migrated seismic section.

Each term (except the first) equals $-\Delta g$ at $=n\Delta$. Taking $\Delta =1$, we see that $h_{n}$ is minus the finite difference derivative at $t=n\Delta$.

Problem 9.31b

What is the integral operator?

Solution

The integral, $\int _{g_{0}}^{g_{n}}\ g(t)\,\mathrm {d} t$, for continuous functions becomes a summation for digital functions, the equivalent of the above integral being $\Delta \mathop {\sum } \limits {_{g_{0}}^{g_{n}}}g_{t}$. If we take the operator $f_{t}=[1,1,1,\ldots ..,1]$ where both $f_{t}$ and $g_{t}$ have $(n+1)$ terms, we obtain the following for the terms of $f_{t}*g_{t}$ (omitting $\Delta$):

${\begin{aligned}&[(g_{0}),(g_{0}+g_{1}),...,(g_{0}+g_{1}+\cdots +g_{r}),(g_{0}+g_{1}+\cdots +g_{n}),\\&\quad (g_{1}+g_{2}+\cdots +g_{n}),(g_{2}+g_{3}+\cdots +g_{n}),...,(g_{n-1}+g_{n}),(g_{n}).\end{aligned}}$

The terms are equivalent to the integrals between 0 and $r$ for $0\leq r\leq n$. After the value $r=n$ is reached, the upper limit remains $n$ while the lower limit is successively 1, 2, . . . , $n$. Thus we conclude that the operator [1, 1, 1, . . . , 1] is an integral operator. By taking $f_{t}$ with fewer terms than $g_{t}$, we can get the equivalent of an integral over selected parts of $g_{t}$.

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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

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