US6038197A
Expired / Lapsed
US
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Efficient inversion of near singular geophysical signals
Assignee: Westerngeco LLC
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Filing: 1998-07-14
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Granted: 2000-03-14
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Expires: 2018-07-14
Abstract
Abstract A method for inverting a seismic signal, wherein an input seismic signal is represented as an input vector i(t) of length m+n-1, an output seismic signal is represented as an output vector o(t) of length m, and a finite impulse response filter is represented as a solution vector u(t) of length n, satisfying a convolutional equation i(t)Xu(t)=o(t). An m×n Toeplitz matrix T(t,τ) corresponding to i(t) is calculated, satisfying a matrix equation T(t,τ)·u(t)=o(t). Both sides of the matrix equation are transformed to the frequency domain, to generate a transformed equation. Both sides of the transformed equation are spectrally pruned, to generate a pruned equation. The pruned equation is then solved.
Inventors
- 👤 Gary A. Sitton
- 👤 Zeljko Jericevic
Bibliographic Details
Application Number
—
Jurisdiction
US
Data Source
google_patents
Last Synchronized
2026-11-09 12:42
CPC Classifications
G; G01; G01V; G01V1/00; G01V1/28; G01V1/36; G01V1/364; G01V1/368
Claims Analysis
Total Claims: 16
Claim 1
Independent Claim
103 words
A method for inverting a seismic signal, comprising the steps of: representing an input seismic signal as an input vector i(t) of length m+n-1, an output seismic signal as an output vector o(t) of length m, and a finite impulse response filter as a solution vector u(t) of length n, satisfying a convolutional equation i(t)Xu(t)=o(t); calculating an m×n Toeplitz matrix T(t,τ) corresponding to i(t), satisfying a matrix equation T(t,τ)·u(t)=o(t); transforming both sides of the matrix equation to the frequency domain, to generate a transformed equation; spectrally pruning both sides of the transformed equation, to generate a pruned equation; and solving the pruned equation.
Claim 2
Independent Claim
103 words
A method for inverting a seismic signal, comprising the steps of: representing an input seismic signal as an input vector i(t) of length m+n-1, an output seismic signal as an output vector o(t) of length m, and a finite impulse response filter as a solution vector u(t) of length n, satisfying a convolutional equation i(t)Xu(t)=o(t); calculating an m×n Toeplitz matrix T(t,τ) corresponding to i(t), satisfying a matrix equation T(t,τ)·u(t)=o(t); transforming both sides of the matrix equation to the frequency domain, to generate a transformed equation; spectrally pruning both sides of the transformed equation, to generate a pruned equation; and solving the pruned equation.
Claim 3
Dependent Claim
35 words
The method of claim 1, wherein the step of transforming is done by applying a column-wise discrete Fourier transform to the Toeplitz matrix T(t,τ) and to the output vector o(t), generating the transformed equation {T}·u={o}.
Claim 4
Dependent Claim
35 words
The method of claim 1, wherein the step of transforming is done by applying a column-wise discrete Hartley transform to the Toeplitz matrix T(t,τ) and to the output vector o(t), generating the transformed equation [T]·u=[o].
Claim 5
Dependent Claim
30 words
The method of claim 2, wherein the step of spectrally pruning is applied to rows of the both sides of the transformed equations, generating the pruned equation {T.sup.- }·u={o.sup.- }.
Claim 6
Dependent Claim
29 words
The method of claim 3, wherein the step of spectrally pruning is applied to rows of the both sides of the transformed equations, generating the pruned equation [T.sup.-]·u=[o- ].
Claim 7
Dependent Claim
49 words
The method of claim 1, wherein the step of solving the pruned equation further comprises the steps of: converting the pruned equation to a normal form equation; calculating the QR decomposition of the normal form equation, to generate a decomposed equation; and solving the decomposed equation by back substitution.
Claim 8
Dependent Claim
36 words
The method of claim 1, wherein the step of solving the pruned equation further comprises the steps of: converting the pruned equation to a normal form equation; solving the normal form equation by singular value decomposition.
Claim 9
Dependent Claim
46 words
The method of claim 1, wherein the step of solving the pruned equation further comprises the steps of: applying a second transform to the pruned equation, to generate a doubly-transformed equation; spectrally pruning the doubly-transformed equation, to generate a doubly-pruned equation; and solving the doubly-pruned equation.
Claim 10
Dependent Claim
50 words
The method of claim 8, wherein the step of applying a second transform is done by applying a row-wise discrete Fourier transform to the pruned, transformed Toeplitz matrix {T - } and a column-wise discrete Hartley transform to the filter vector u(t), generating the doubly-transformed equation {{T.sup.- }.sup.H }.sup.H ·{u}={o.sup.-}.
Claim 11
Dependent Claim
51 words
The method of claim 8, wherein the step of applying a second transform is done by applying a row-wise discrete Fourier transform to the pruned, transformed Toeplitz matrix [T - ] and a column-wise discrete Hartley transform to the filter vector u(t), generating the doubly-transformed equation {[T.sup.- ].sup.H }.sup.H ·{u}=[o.sup.- ].
Claim 12
Dependent Claim
37 words
The method of claim 9, wherein the step of spectrally pruning the doubly-transformed matrix equation is applied to columns of the both sides of the transformed equations, generating the doubly-pruned equation {{T.sup.- }.sup.H- }.sup.H ·{u.sup.- }={o.sup.- }.
Claim 13
Dependent Claim
37 words
The method of claim 10, wherein the step of spectrally pruning the doubly-transformed matrix equation is applied to columns of the both sides of the transformed equations, generating the doubly-pruned equation {[T.sup.- ].sup.H- }.sup.H ·{u.sup.- }=[o.sup.- ].
Claim 14
Dependent Claim
90 words
The method of claim 8, wherein the step of solving the doubly-pruned equation further comprises the steps of: converting the doubly-pruned equation to a normal form equation; calculating the QR decomposition of the normal form equation, to generate a decomposed equation; solving the decomposed equation by back substitution, to generate a pruned solution vector {u - }; filling in the pruned coefficients in the pruned solution vector {u - } with zeros, to generate a filled-in vector {u}; and applying an inverse discrete Fourier transform to the filled-in vector {u}.
Claim 15
Dependent Claim
90 words
The method of claim 8, wherein the step of solving the doubly-pruned equation further comprises the steps of: converting the doubly-pruned equation to a normal form equation; calculating the QR decomposition of the normal form equation, to generate a decomposed equation; solving the decomposed equation by back substitution, to generate a pruned solution vector [u - ]; filling in the pruned coefficients in the pruned solution vector [u - ] with zeros, to generate a filled-in vector [u]; and applying an inverse discrete Hartley transform to the filled-in vector [u].
Claim 16
Dependent Claim
108 words
The method of claim 8, wherein the step of solving the doubly-pruned equation further comprises the steps of: converting the doubly-pruned equation to a normal form equation; calculating the QR decomposition of the normal form equation, to generate a decomposed equation; solving the decomposed equation by back substitution, to generate a pruned solution vector [u - ]; filling in the pruned coefficients in the pruned solution vector [u - ] with zeros, to generate a filled-in vector [u]; converting the filled-in vector [u] from the Hartley to the Fourier domain, to generate a converted vector {u}; and applying an inverse discrete Fourier transform to the converted vector {u}.
References Cited (Backward Citations)
US3629800A
Cited
US5587965A
Cited
USPTO Annuity & Maintenance Windows
| Annuity Stage | Window Opens | Payment Deadline | Surcharge Period Until | Estimated Fee | Status |
|---|---|---|---|---|---|
| 3.5 Years (Year 4) | 2003-03-14 | 2003-09-13 | 2004-03-14 | $2000.00 | Window Closed / Lapsed |
| 7.5 Years (Year 8) | 2007-03-14 | 2007-09-13 | 2008-03-14 | $3760.00 | Window Closed / Lapsed |
| 11.5 Years (Year 12) | 2011-03-14 | 2011-09-13 | 2012-03-14 | $7700.00 | Window Closed / Lapsed |