Convolution and correlation calculations

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 9.12a

Of the four wavelets given in problem 9.8, which are minimum-phase?

Background

Minimum-phase wavelets are discussed briefly in problem 9.11 and in more detail in Sheriff and Geldart, 1995, section 9.4 and section 15.5.6); $z$-transforms are discussed in Sheriff and Geldart, 1995, section 15.5.3.

Solution

The four wavelets are: $a_{t}=[2,\;-1]$, $b_{t}=[4,1]$, $c_{t}=[6,\;-7,\;2]$, and $d_{t}=[4,9,2]$. The roots of the wavelets $a_{t}$ and $b_{t}$ are 2 and $-4$, so both are minimum-phase because the magnitudes of the roots are greater than unity. Since $c_{t}\leftrightarrow (6-7z+2z^{2})=(3-2z)(2-z)$, the roots are 3/2 and 2; $c_{t}$ is therefore also minimum-phase. Finally, $d_{t}\leftrightarrow (4+9z+2z^{2})=(4+z)(1+2z)$, the roots being $-4$ and $-1/2$. Because $|-1/2|<1$, $d_{t}$ is mixed-phase.

Problem 9.12b

Find $a_{t}*b_{t}$ and $a_{t}*c_{t}$ by calculating in the time domain.

Solution

${\begin{aligned}a_{t}*b_{t}=[2,\;-1]*[4,\;1]\;=8,\;-4\\2,\;-1\\=[8,\;-2,\;-1];\end{aligned}}$ ${\begin{aligned}a_{t}*c_{t}=[2,\;-1]*[6,\;-7,\;2]\;=12,\;-6\\-14,\;7\\4,\;-2\\=[12,\;-20,\;11,\;2].\end{aligned}}$

Problem 9.12c

Repeat part (b) except using transforms.

Solution

${\begin{aligned}A(z)=(2-z)\;,\;B(z)=(4+z)\;,\;C(z)=(6-7z+2z^{2});\\a_{t}*b_{t}\leftrightarrow (2-z)(4+z)=8-2z-z^{2}\leftrightarrow [8,\;-2,\;-1];\\a_{t}*c_{t}\leftrightarrow (2-z)(6-7z+2z^{2})=12-20z+11z^{2}-2z^{3}\\=[12,\;-20,\;11,\;-2].\end{aligned}}$

Problem 9.12d

Find $a_{t}*b_{t}*c_{t}$.

Solution

${\begin{aligned}a_{t}*b_{t}*c_{t}=(a_{t}*b_{t})*c_{t}=[8,\;-2,\;-1]*[6,\;-7,\;2]\\=48,\;-12,\;-6\\-56,\;14,\;7\\16,\;-4,\;-2\\=[48,\;-68,\;24,\;3,\;-2].\\\end{aligned}}$

Problem 9.12e

Does the largest value of a minimum-phase wavelet have to come at $t=0$?

Solution

The wavelet $(a+z)(b+z)=[ab+(a+b)z+z^{2}]$ is minimum-phase if $|a|>1$ and $|b|>1$. The ratio of the first two terms is $ab/(a+b)=a/(1+a/b)$ and it has its minimum absolute value when $a$ and $b$ have the same signs. When $a$ and $b$ have the same sign and are both slightly larger than unity, the ratio is close to 1/2 and the second term is larger than the first. As $a$ and/or $b$ increase, the ratio increases; the first and second terms are equal when $a=b=2$. If $a$ and $b$ differ significantly in magnitude, the second term can be larger than the first for large values of $a$ or $b$; e.g., if $b\approx 1$, the ratio is $\approx 0.9$ when $a=9$.

If $a$ and $b$ have opposite signs, the ratio cannot be smaller than 1 since the two terms in the denominator have opposite signs and the denominator cannot exceed the numerator.

When the wavelet has three factors, the ratio of the first to second term takes the form $abc/(ab+ac+bc)$. When $a$, $b$, and $c$ are all close to unity and of the same polarity, the magnitude of the ratio is $\approx 1/3$ and the second term is larger than the first. Generalizing to $n$ terms the ratio can be $\approx 1/n$.

Problem 9.12f

Can a minimum-phase wavelet be zero at $t=0$?

Solution

If a wavelet is zero at $t=0$, it is of the form ($0,a,b,c,\ldots$}, so

${\begin{aligned}\left[0,\;a,\;b,\;c,\;.\;.\;.\right]\;\leftrightarrow (0+az+bz^{2}+cz^{3}+\cdots \cdot )=z(a+bz+cz^{2}+\cdots ).\end{aligned}}$

Since one of the roots is $z=0<1$, the wavelet is not minimum-phase.

When we deal with an individual wavelet, we avoid the root $z=0$ by taking $t=0$ as the time when the first nonzero value occurs.

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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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