# Wiener (least-squares) inverse filters

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 |

## Contents

## Problem 9.22a

Plot cumulative energy as a function of time for wavelets **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A=[1,\; -2,\; 3]}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B=[3,\; -2,\; 1]}**

### Background

A filter that will change a signal so as to make it as close as possible to a desired signal can be designed in a least-squares sense to minimize the sum of the squares of the “errors” (differences between a desired signal and the filtered signal). Such a filter is called a *least-squares filter* or *Wiener filter*.

Using **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_{t}}**
to denote the original signal and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_{t}}**
the desired signal, the sum of the errors squared is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} E=\mathop{\sum}\limits_{t=0}^{n} (h_{t} -f_{t} *g_{t})^{2}. \end{align} }****(**)

Since all quantities on the right-hand side of equation (9.22a) are fixed except the filter elements **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{i}}**
, we vary the coefficients **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{i}}**
to minimize **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E}**
. We differentiate **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E}**
with respect to each of the elements **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{i}}**
and equate the derivatives to zero. This gives **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n+1)}**
equations which can be solved for the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n+1)}**
elements of the filter. Differentiating **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E}**
, we get

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial E}{\partial f_{i}} =0=\mathop{\sum}\limits_{t=0}^{n} (h_{t} -f_{t} *g_{t})\frac{\partial (f_{t} *g_{t})}{\partial f_{i}} \\ =\mathop{\sum}\limits_{t=0}^{n} \left(h_{t}-\mathop{\sum}\limits_{t=0}^{n} g_{k} \ f_{t-k}\right)\frac{\partial}{\partial f_{i}} \left(\mathop{\sum}\limits_{t=0}^{n} g_{k} \ f_{t-k}\right), \end{align} }**

using equation (9.2b) to replace **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t} *g_{t}}**
with a summation. The derivative now becomes

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial}{\partial f_{i}} \left(\mathop{\sum}\limits_{t=0}^{n} g_{k} \ f_{t-k}\right)=g_{t-i}, \end{align} }**

since the only nonzero term in the differentiation is that in which **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t}}**
appears, that is, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t-k=i}**
, so **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=t-i}**
. Substituting this result, we get

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathop{\sum}\limits_{t=0}^{n} \left(h_{t} -\mathop{\sum}\limits_{k}^{} g_{k} \ f_{t-k}\right)g_{t-i} =0=\mathop{\sum}\limits_{t=0}^{n} h_{t} \ g_{t-i}-\mathop{\sum}\limits_{t=0}^{n} \left(\mathop{\sum}\limits_{k}^{} (g_{k} \ f_{t-k})g_{t-i}\right). \end{align} }**

The first term is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh}}**
(**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i}**
) from equations (9.8a) and (9.8b). Interchanging the order of summation in the right-hand term, letting **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j=t-k}**
, and summing over **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j}**
, the equation becomes

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathop{\sum}\limits_{j=0}^{n} \left(f_{j} \mathop{\sum}\limits_{t=0}^{n} g_{t-j} \ g_{t-i}\right)=\mathop{\sum}\limits_{j=0}^{n} f_{j} \phi _{gh} (i-j) \end{align} }**

(see problem 9.21a). Thus we arrive at the *normal equations*

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathop{\sum}\limits_{j=0}^{n} \phi_{gg} (i-j)f_{j} =\phi _{gh} (i),\qquad i=0, 1, 2, \ldots , n. \end{align} }****(**)

### Solution

The transforms of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B}**
are **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1-2z+3z^{2})}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (3-2z+z^{2})}**
, so the roots of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A}**
are **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3} (1\pm j\sqrt{2})}**
and those of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B}**
are **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1\pm j\sqrt{2})}**
, the magnitudes of the roots being **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/\sqrt{3}}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}}**
. Thus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A}**
is maximum-phase and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B}**
is minimum-phase.

The energy of a wavelet at any instant is proportional to its amplitude squared. We thus get for the cumulative energy of

## Problem 9.22b

Calculate three-element Wiener inverse filters assuming the desired output is (i) [1, 0, 0] and (ii) [0, 1, 0], then apply the inverse filters to wavelets

### Solution

To write the normal equations in equation (9.22b) in explicit form, we give **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i}**
the values 0, 1, and 2 in succession. The result is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi_{gg} (0)f_{0} +\phi_{gg} (-1)f_{1} +\phi_{gg} (-2)f_{2} =\phi_{gh} (0) ,\\ \phi_{gg} (1)f_{0} +\phi_{gg} (0)f_{1} +\phi_{gg} (-1)f_{2} =\phi_{gh} (1) ,\\ \phi_{gg} (2)f_{0} +\phi_{gg} (1)f_{1} +\phi_{gg} (0)f_{2} =\phi_{gh} (2) .\\ \end{align} }**

*Shaping wavelet A* [1,−2, 3] *into* [1, 0, 0]

To solve these equations, we need the values of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gg} (\tau)}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (\tau)}**
. Using equations (9.8a) and (9.8e) we get

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi _{gg} (0)=1^{2} +2^{2} +3^{2} =14; \quad \phi_{gg} (1)=\phi_{gg} (-1)=-2-6=-8;\\ \phi_{gg} (2)=\phi_{gg} (-2)=3; \quad \phi_{gh} (0)=1; \quad \phi_{gh} (1)=0; \quad \phi_{gh} (2)=0. \end{align} }**

Substituting these values in the normal equations gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} &=1,\\ -8f_{0} +14f_{1} -8f_{2} &=0,\\ 3f_{0} -8f_{1} +14f_{2} &=0. \end{align} }**

The solution of these equations can be obtained using Cramer’s rule [see Sheriff and Geldart, 1995, equation (15.3b)]. We first calculate the following determinants:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta =\left|\begin{array}{ccc} {14} {-8} {3} \\ {-8} {14} {-8} \\ {3} {-8} {14} \end{array}\right|\\ =14(14\times 14-8\times 8)-8(-8\times 3+8\times 14)+3(8\times 8-14\times 3)=1210;\\ \Delta _{0} =\left|\begin{array}{ccc} {1} {-8} {3} \\ {0} {14} {-8} \\ {0} {-8} {14} \end{array}\right|=(14\times 14-8\times 8)=132;\\ \Delta _{1} =\left|\begin{array}{ccc} {14} {1} {3} \\ {-8} {0} {-8} \\ {3} {0} {14} \end{array}\right|=(-8\times 3+8\times 14)=88;\\ \Delta _{2} =\left|\begin{array}{ccc} {14} {-8} {1} \\ {-8} {14} {0} \\ {3} {-8} {0} \end{array}\right|=(8\times 8-14\times 3)=22. \end{align} }**

Then, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{0} =\Delta_0/\Delta =132/1210=0.1091}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{1} =88/1210=0.0727}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{2} =22/1210= 0.0182}**
. Applying this filter to

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *A_{t} =\left[0.1091,\; 0.0727,\; 0.0182\left]*\right[1,\; -2,\; 3\right]\\ =0.1091 , 0.0727, 0.0182\\ -0.2182,\; -0.1454,\; -0.0364\\ 0.3273, 0.2181, 0, 0546\\ =0.1091 , -0.1455 , 0.2001, 0.1817, 0.0546. \end{align} }**

Normalizing the wavelet to make the first element equal to +1, we get the wavelet

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left[1, -1.3336, 1.8341, 1.6654, 0.5005 \right], \end{align} }**

which is far from **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [1,\; 0,\; 0,\; 0]}**
, the rms difference between the two wavelets being 1.43. Thus, it appears that we cannot shape the maximum-phase wavelet **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta(t)}**
.

*Shaping wavelet* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B[3,\; -2,\; 1]}**
*into* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [1,\; 0,\; 0]}**

Using wavelet **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gg} (\tau)}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (\tau)}**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi_{gg} (0)=14; \phi_{gg} (1)=-8, \phi _{gg} (2)=3;\phi _{gh} (0)=3;\phi _{gh} (1)=0;\phi _{gh} (2)=0. \end{align} }**

The normal equations now become

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} &=3,\\ -8f_{0} +14f_{1} -8f_{2} &=0,\\ 3f_{0} -8f_{1} +14f_{2} &=0. \end{align} }**

Proceeding as before,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1210}**
from calculation for wavelet A;

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta _{0} =\left|\begin{array}{ccc} {3} & {-8} & {3} \\ {0} & {14} & {-8} \\ {0} & {-8} & {14} \end{array}\right|=3(14\times 14-8\times 8)=396;\\ \Delta _{1} =\left|\begin{array}{ccc} {14} & {3} & {3} \\ {-8} & {0} & {-8} \\ {3} & {0} & {14} \end{array}\right| =-3(-8\times 14+8\times 3)=264;\\ \Delta _{2} =\left|\begin{array}{ccc} {14} & {-8} & {3} \\ {-8} & {14} & {0} \\ {3} & {-8} & {0} \end{array}\right|=3(8\times 8-14\times 3)=66. \end{align} }**

The solution is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{0} =396/1210=0.3273 , f_{1} =264/1210=0.2182, f_{2} =66/1210=0.0545. \end{align} }**

Applying the filter [0.3273, 0.2182, 0.0545] gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *B_{t} &=\left[0.3273,\; 0.2182,\; 0.0545\left]*\right[3,\; -2,\; 1\right]\\ &=0.9819, 0.6546, 0.1635\\ &\qquad\qquad-0.6546,\; -0.4364,\; -0.1090\\ &\qquad\qquad\qquad\qquad0.3273, 0.2182, 0.0545\\ &0.9819, 0.,\quad 0.0544, 0.1092, 0.0545. \end{align} }**

Normalizing the first value to 1 gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *g_{t} =[1 , 0, 0.056, 0.114, 0.056], \end{align} }**

which is close to the desired wavelet of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [1,\; 0,\; 0]}**
beause

*Shaping wavelet* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A[1,\; -2,\; 3]}**
*into* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [0,\; 1,\; 0]}**

We use the values of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gg}(\tau)}**
from the previous calculations: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gg} (0)=14}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gg} (1)=-8,\,\phi_{gg} (2)=3}**
; and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (0)=-2}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (1)=1}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gh} (2)=0}**
. The normal equations thus become

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} =-2,\\ -8f_{0} +14f_{1} -8f_{2} =1,\\ 3f_{0} -8f_{1} +14f_{2} =0. \end{align} }**

We next calculate the determinants:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1210}**
as before,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta _{0} =\left|\begin{array}{ccc} {-2} & {-8} & {3} \\ {1} & {14} & {-8} \\ {0} & {-8} & {14} \end{array}\right|=-176,\\ \Delta _{1} =\left|\begin{array}{ccc} {14} & {-2} & {3} \\ {-8} & {1} & {-8} \\ {3} & {0} & {14} \end{array}\right|=11\\ \Delta _{2} =\left|\begin{array}{ccc} {14} & {-8} & {-2} \\ {-8} & {14} & {1} \\ {3} & {-8} & {0} \end{array}\right|=44. \end{align} }**

Thus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{0} =-0.1455}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{1} =0.0091}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{2} =0.0364}**
and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *A_{t} =\left[-0.1455,\; 0.0091,\; 0.0364\left]*\right[1,\; -2,\; 3\right]= \end{align} }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} -0.1455 , 0.0091, 0.0364\\ 0.2910, -0.0182, -0.0728\\ -0.4365 , 0.0273 0.1092\\ -0.1455 , 0.3001, -0.4183, -0.0455 , 0.1092. \end{align} }**

Normalizing the second value to 1 gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *A_{t} =[-0.485,\; 1,\; -1.394,\; -0.152,\; 0.364], \end{align} }**

and the rms difference from the desired output is 0.764. The result is poor because the input wavelet

*Shaping wavelet* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B[3,\; -2,\; 1]}**
*into* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [0,\; 1,\; 0]}**

We have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi _{gg} (0)=14,\; \phi _{gg} (1)=-8,\; \phi _{gg} (2)=3;\phi _{gh} (0)=-2,\\ \phi _{gh} (1)=3,\; \phi _{gh} (2)=0. \end{align} }**

The normal equations are

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} =-2,\\ -8f_{0} +14f_{1} -8f_{2} =3,\\ 3f_{0} -8f_{1} +14f_{2} =0. \end{align} }**

We next calculate the following determinants:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1210}**
as before;

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta _{0} =\left|\begin{array}{ccc} {-2} & {-8} & {3} \\ {3} & {14} & {-8} \\ {0} & {-8} & {14} \end{array}\right|=0,\\ \Delta _{1} = \left|\begin{array}{ccc} {14} & {-2} & {3} \\ {-8} & {3} & {-8} \\ {3} & {0} & {14} \end{array}\right|=385,\\ \Delta _{2} =\left|\begin{array}{ccc} {14} & {-8} & {-2} \\ {-8} & {14} & {3} \\ {3} & {-8} & {0} \end{array}\right|=220. \end{align} }**

We now find that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{0} =0}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{1} =-0.3182}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{2} =0.1818}**
, and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *B_{t} =[0, 0.3182, 0.18181, 0.0182]*[3,\; -2,\; 1]\\ =0, 0.9546, 0.5454\\ 0,\; -0.6364,\; -0.3636,\\ 0, 0.3182, 0.1818\\ =0, 0.9546, -0.0910, -0.0454, 0.1818. \end{align} }**

Normalizing the second value to 1 gives

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *B_{t} =[0,\; 1,\; -0.0964,\; -0.0476,\; 0.1904], \end{align} }**

and the rms difference from the desired output is 0.109. The result is not as good as for the previous case, but it is much better than for the wavelet

The original and filtered wavelets are shown in Figure 9.22b. Reviewing the errors:

To make

To make

To make

To make

## Continue reading

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Autocorrelation | Interpreting stacking velocity |

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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