Derivative and integral operators
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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 | 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}} $
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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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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