# Derivative and integral operators

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 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t}}**
with **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_{t}}**
gives values of the finite-difference derivative of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_{t}}**
for different values of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}**
. To show this, we find the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n+1)^{th}}**
term of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_{t} =f_{t} *g_{t}}**
. From equation (9.2b), we have

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} h_{n} =\mathop{\sum}\limits_{k=0}^{} f_{k} g_{n-k} =[-g_{0} ,\; (-g_{1} +g_{0}),\; .\; .\; .\; (-g_{n} +g_{n-1})]. \end{align} }**

Each term (except the first) equals **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\Delta g}**
at **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =n\Delta}**
. Taking **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1}**
, we see that **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_{n}}**
is minus the finite difference derivative at **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t=n\Delta}**
.

## Problem 9.31b

What is the integral operator?

### Solution

The integral, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta \mathop{\sum}\limits{_{g_{0}}^{g_{n}}} g_{t}}**
. If we take the operator **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t} = [1, 1, 1, \ldots.., 1]}**
where both **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t}}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_{t}}**
have **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n+1)}**
terms, we obtain the following for the terms of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t} *g_{t}}**
(omitting **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta}**
):

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &[(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{align} }**

The terms are equivalent to the integrals between 0 and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r}**
for **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0\le r\le n}**
. After the value **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=n}**
is reached, the upper limit remains **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n}**
while the lower limit is successively 1, 2, . . . , **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n}**
. Thus we conclude that the operator [1, 1, 1, . . . , 1] is an integral operator. By taking **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t}}**
with fewer terms than

## Continue reading

Previous section | Next section |
---|---|

Effect of migration on fault interpretation | Effects of normal-moveout (NMO) removal |

Previous chapter | Next chapter |

Reflection field methods | Geologic interpretation of reflection data |

## 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