Surface multiple attenuation via eigenvalue decomposition

Claim 1

A method for substantially attenuating surface multiples from a marine seismic signal, comprising the steps of: transforming said marine seismic signal from time domain into frequency domain to generate a transformed signal represented by matrix D, truncating said marine seismic signal in time to generate a truncated signal and transforming said truncated signal from time domain into frequency domain to generate a transformed truncated signal represented by truncated matrix D T ; computing eigenvalue decomposition D T =S·Λ·S -1 of said truncated matrix D T ; where Λ is the diagonal matrix whose elements are the eigenvalues of D T , S is the square matrix whose rows are the corresponding eigenvectors of D T and S -1 is the inverse of S, solving for source wavelet w by minimizing total energy in matrix product [D·S][I-w -1 Λ] -1 S -1 in an iterative loop; solving for primary matrix P by inserting said solved value for said source wavelet w into the expression P=[D·S][I-w.sup.-1 Λ].sup.-1 S.sup.-1 ; and inverse transforming said primary matrix P from frequency domain into time domain.

Claim 2

A method for substantially attenuating surface multiples from a marine seismic signal, comprising the steps of: transforming said marine seismic signal from time domain into frequency domain to generate a transformed signal represented by matrix D, truncating said marine seismic signal in time to generate a truncated signal and transforming said truncated signal from time domain into frequency domain to generate a transformed truncated signal represented by truncated matrix D T ; computing eigenvalue decomposition D T =S·Λ·S -1 of said truncated matrix D T ; where Λ is the diagonal matrix whose elements are the eigenvalues of D T , S is the square matrix whose rows are the corresponding eigenvectors of D T and S -1 is the inverse of S, solving for source wavelet w by minimizing total energy in matrix product [D·S][I-w -1 Λ] -1 S -1 in an iterative loop; solving for primary matrix P by inserting said solved value for said source wavelet w into the expression P=[D·S][I-w.sup.-1 Λ].sup.-1 S.sup.-1 ; and inverse transforming said primary matrix P from frequency domain into time domain.

Claim 5

A method for substantially attenuating surface multiples from a marine seismic signal, comprising the steps of: (a) transforming said marine seismic signal from time domain into frequency domain to generate a transformed signal represented by matrix D, truncating said marine seismic signal in time to generate a truncated signal and transforming said truncated signal from time domain into frequency domain to generate a transformed truncated signal represented by truncated matrix D T ; (b) computing eigenvalue decomposition D T =S·Λ·S -1 of said truncated matrix D T ; where Λ is the diagonal matrix whose elements are the eigenvalues of D T , S is the square matrix whose rows are the corresponding eigenvectors of D T and S -1 is the inverse of S, (c) computing matrix product D·S and saving said matrix product D·S in memory; (d) computing matrix inverse S -1 of said matrix S and saving said matrix inverse S -1 in memory; (e) making initial estimate of source wavelet w; (f) computing diagonal matrix [I-w -1 Λ] and computing matrix inverse [I-w -1 Λ] -1 of said diagonal matrix [I-w -1 Λ]; (g) retrieving said matrix product D·S and said matrix inverse S -1 from said memory; (h) computing matrix product [D·S][I-w -1 Λ] -1 S -1 in terms of said source wavelet w; (i) iterating said steps (f) through (h) to solve for said source wavelet w by minimizing total energy in said matrix product [D·S][I-w -1 Λ] -1 S -1 ; (j) solving for primary matrix P by inserting said solved value for said source wavelet w into the expression P=[D·S][I-w.sup.-1 Λ].sup.-1 S.sup.-1 ; and (k) inverse transforming said primary matrix P from frequency domain into time domain.

Claim 6

A method for substantially attenuating surface multiples from a marine seismic signal, comprising the steps of: (a) transforming said marine seismic signal from time domain into frequency domain to generate a transformed signal represented by matrix D, truncating said marine seismic signal in time to generate a truncated signal and transforming said truncated signal from time domain into frequency domain to generate a transformed truncated signal represented by truncated matrix D T ; (b) computing eigenvalue decomposition D T =S·Λ·S -1 of said truncated matrix D T ; where Λ is the diagonal matrix whose elements are the eigenvalues of D T , S is the square matrix whose rows are the corresponding eigenvectors of D T and S -1 is the inverse of S, (c) computing matrix product D·S and saving said matrix product D·S in memory; (d) computing matrix inverse S -1 of said matrix S and saving said matrix inverse S -1 in memory; (e) making initial estimate of source wavelet w; (f) computing diagonal matrix [I-w -1 Λ] and computing matrix inverse [I-w -1 Λ] -1 of said diagonal matrix [I-w -1 Λ]; (g) retrieving said matrix product D·S and said matrix inverse S -1 from said memory; (h) computing matrix product [D·S][I-w -1 Λ] -1 S -1 in terms of said source wavelet w; (i) iterating said steps (f) through (h) to solve for said source wavelet w by minimizing total energy in said matrix product [D·S][I-w -1 Λ] -1 S -1 ; (j) solving for primary matrix P by inserting said solved value for said source wavelet w into the expression P=[D·S][I-w.sup.-1 Λ].sup.-1 S.sup.-1 ; and (k) inverse transforming said primary matrix P from frequency domain into time domain.

Claim 9

A method for substantially eliminating surface multiples from a marine seismic signal, comprising the steps of: (a) truncating said marine seismic signal in time to generate a truncated signal; (b) transforming said marine seismic signal from time domain into frequency domain to generate a transformed signal and transforming said truncated signal from time domain into frequency domain to generate a transformed truncated signal; (c) modifying said transformed truncated signal to include obliquity, scale and phase factors required by the Kirchhoff integral, thereby generating a modified signal; (d) representing said transformed signal by data cube D(T,R,f) and representing said modified signal by modified data cube D M (T,R,f), where T is source location, R is receiver location, and f is frequency; (e) extracting data matrix D(T,R) from said data cube D(T,R,f) for a frequency f and extracting modified matrix D M (T,R) from modified data cube D M (T,R,f) for said frequency f; (f) computing eigenvalue decomposition D M =S·Λ·S -1 of said modified matrix D M ; where Λ is the diagonal matrix whose elements are the eigenvalues of D T , S is the square matrix whose rows are the corresponding eigenvectors of D T and S -1 is the inverse of S, (g) computing matrix product D(T,R)·S and saving said matrix product D(T,R)·S in memory; (h) computing matrix inverse S -1 of said matrix S and saving said matrix inverse S -1 in memory; (i) making initial estimate of source wavelet w; (j) computing diagonal matrix [I-w -1 Λ] and computing matrix inverse [I-w -1 Λ] -1 of said diagonal matrix [I-w -1 Λ]; (k) retrieving said matrix product D(T,R)·S and said matrix inverse S -1 from said memory; (l) computing matrix product [D(T,R)·S][I-w -1 Λ] -1 S -1 in terms of said source wavelet w; (m) iterating said steps (j) through (1) to solve for said source wavelet w by minimizing total energy in said matrix product [D(T,R)·S][I-w -1 Λ] -1 S -1 ; (n) solving for primary matrix P(T,R) by inserting said solved value for wavelet w into the expression P=[D(T,R)·S][I-w.sup.-1 Λ].sup.-1 S.sup.-1 ; (o) inserting said primary matrix P(T,R) into data cube D(T,R,f) for said frequency f; (p) repeating said steps (e) through (o) for each frequency f in said data cube D(T,R,f); and (q) inverse transforming said data cube D(T,R,f) from frequency domain into time domain.

Claim 10

A method for substantially eliminating surface multiples from a marine seismic signal, comprising the steps of: (a) truncating said marine seismic signal in time to generate a truncated signal; (b) transforming said marine seismic signal from time domain into frequency domain to generate a transformed signal and transforming said truncated signal from time domain into frequency domain to generate a transformed truncated signal; (c) modifying said transformed truncated signal to include obliquity, scale and phase factors required by the Kirchhoff integral, thereby generating a modified signal; (d) representing said transformed signal by data cube D(T,R,f) and representing said modified signal by modified data cube D M (T,R,f), where T is source location, R is receiver location, and f is frequency; (e) extracting data matrix D(T,R) from said data cube D(T,R,f) for a frequency f and extracting modified matrix D M (T,R) from modified data cube D M (T,R,f) for said frequency f; (f) computing eigenvalue decomposition D M =S·Λ·S -1 of said modified matrix D M ; where Λ is the diagonal matrix whose elements are the eigenvalues of D T , S is the square matrix whose rows are the corresponding eigenvectors of D T and S -1 is the inverse of S, (g) computing matrix product D(T,R)·S and saving said matrix product D(T,R)·S in memory; (h) computing matrix inverse S -1 of said matrix S and saving said matrix inverse S -1 in memory; (i) making initial estimate of source wavelet w; (j) computing diagonal matrix [I-w -1 Λ] and computing matrix inverse [I-w -1 Λ] -1 of said diagonal matrix [I-w -1 Λ]; (k) retrieving said matrix product D(T,R)·S and said matrix inverse S -1 from said memory; (l) computing matrix product [D(T,R)·S][I-w -1 Λ] -1 S -1 in terms of said source wavelet w; (m) iterating said steps (j) through (1) to solve for said source wavelet w by minimizing total energy in said matrix product [D(T,R)·S][I-w -1 Λ] -1 S -1 ; (n) solving for primary matrix P(T,R) by inserting said solved value for wavelet w into the expression P=[D(T,R)·S][I-w.sup.-1 Λ].sup.-1 S.sup.-1 ; (o) inserting said primary matrix P(T,R) into data cube D(T,R,f) for said frequency f; (p) repeating said steps (e) through (o) for each frequency f in said data cube D(T,R,f); and (q) inverse transforming said data cube D(T,R,f) from frequency domain into time domain.