Method for improved processing of data with time overlapping recordings of energy sources
A method for deblending seismic signals includes entering as input to a computer recorded signals comprising seismic energy from a plurality of actuations of one or more seismic energy sources. A model of deblended seismic data and a blending matrix are initialized. A blending matrix inversion is performed using the initialized model. The inversion includes using a scaled objective function. The inversion is constrained by a thresholding operator. The thresholding operator is arranged to recover coefficients of the model of the deblended seismic data that are substantially nonzero, against a Gaussian white noise background. The thresholded model is projected into data space. Performing the blending matrix inversion is repeated if a data residual exceeds a selected threshold and the inversion is terminated if the data residual is below the selected threshold. At least one of storing and displaying an output of the blending matrix inversion is performed when the blending matrix inversion is terminated.
1. A method for deblending seismic signals, comprising:
(a) entering as input to a computer recorded signals comprising time overlapping seismic energy from a plurality of actuations of one or more seismic energy sources;
(b) in the computer, initializing a model of deblended seismic data and initializing a blending matrix, the model comprising a plurality of seismic signals each having energy corresponding to only one of the plurality of actuations, the blending matrix comprising times of each of the plurality of actuations;
(c) in the computer, initializing a blended output as a product of the blending matrix and the model;
(d) in the computer, calculating a residual as a difference between the blended output and the input recorded signals;
(e) in the computer, transforming the model into a thresholding domain and applying a thresholding function only to the transformed model to generate a deblended portion, the deblended portion comprising a plurality of seismic signals each having energy corresponding to only one of the plurality of actuations;
(f) in the computer, inverse transforming the deblended portion to generate a thresholded model;
(g) in the computer, applying the blending matrix to the thresholded model to generate an updated blended output;
(h) determining whether a difference between the updated blended output and the input recorded signals falls below a selected limit, wherein the selected limit is related to a scaled objective function, a scale factor for the scaled objective function related to a number of the plurality of actuations; and
when the difference exceeds the selected limit, in the computer, adding the thresholded model to a back-projected residual comprising a product of the residual and a complex conjugate transpose of the blending matrix to generated an updated model, in the computer, replacing the model at (b) with the updated model and repeating (d) through (h); and
when the selected difference is below the limit, in the computer, at least one of storing and displaying the updated model as a final output, the final output comprising a plurality of seismic signal sets each having energy corresponding to only one of the plurality of actuations.
2. The method of claim 1 wherein the thresholding function is performed in overlapping N-dimensional windows of predetermined size and wherein N>2, wherein individual results of the thresholding function in the N-dimensional windows are linearly recombined.
3. The method of claim 1 wherein the thresholding function comprises quadratic thresholding.
4. The method of claim 1 wherein the thresholding function comprises a smooth function with continuous first and second derivatives in a region above a threshold level.
5. The method of claim 1 wherein the thresholding function comprises a function of the form:
T
τ
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x
i
)
=
{
x
i
(
1
-
τ
n
/
x
i
n
)
x
i
>
τ
0
x
i
≤
τ
for n>1, and wherein T represents the thresholding function, τ represents a threshold value, and x represents a coefficient.
6. The method of claim 1 wherein a thresholding schedule for the thresholding function comprises lowering a value of a threshold exponentially to zero.
7. The method of claim 1 wherein an N-dimensional transform for the thresholding function comprises a Fourier transform.
8. The method of claim 1 wherein an N-dimensional transform for the thresholding function comprises a curvelet transform.
9. The method of claim 1 wherein an N-dimensional transform for the thresholding function comprises a tau-p transform.
10. The method of claim 1 wherein an N-dimensional transform for the thresholding function comprises a parabolic radon transform.
11. The method of claim 1 wherein an N-dimensional transform for the thresholding function comprises a hyperbolic radon transform.
12. The method of claim 1 wherein an N-dimensional transform for the thresholding function comprises a single or double focal transform.
13. The method of claim 1 wherein an N-dimensional transform for the thresholding function is based upon at least one of the following functions: Activelet, AMlet, Armlet, Bandlet, Barlet, Bathlet, Beamlet, Binlet, Bumplet, Brushlet, Caplet, Camplet, Chirplet, Chordlet, Circlet, Coiflet, Contourlet, Cooklet, Craplet, Cubelet, CURElet, Daublet, Directionlet, Dreamlet, Edgelet, FAMlet, FLaglet, Flatlet, Fourierlet, Framelet, Fresnelet, Gaborlet, GAMlet, Gausslet, Graphlet, Grouplet, Haarlet, Haardlet, Heatlet, Hutlet, Hyperbolet, Icalet (Icalette), Interpolet, Loglet, Marrlet, MIMOlet, Monowavelet, Morelet, Morphlet, Multiselectivelet, Multiwavelet, Needlet, Noiselet, Ondelette, Ondulette, Prewavelet, Phaselet, Planelet, Platelet, Purelet, QVlet, Radonlet, RAMlet, Randlet, Ranklet, Ridgelet, Riezlet, Ripplet (original, type-I and II), Scalet, S2let, Seamlet, Seislet, Shadelet, Shapelet, Shearlet, Sinclet, Singlet, Slantlet, Smoothlet, Snakelet, SOHOlet, Sparselet, Spikelet, Splinelet, Starlet, Steerlet, Stockeslet, SURE-let (SURElet), Surfacelet, Surflet, Symmlet, S2let, Tetrolet, Treelet, Vaguelette, Wavelet-Vaguelette, Wavelet, Warblet, Warplet, Wedgelet, Xlet.
14. The method of claim 1 wherein the thresholding domain enhances a sparse representation of predictable signals.
15. A method for seismic surveying, comprising:
actuating at least one seismic energy source;
recording signals generated by at least one seismic sensor in response to the actuating, the actuating performed such that energy from different actuations of the at least one source is present in the recorded signals;
(a) entering the recorded signals as input to a computer;
(b) in the computer, initializing a model of deblended seismic data and initializing a blending matrix, the model comprising a plurality of seismic signals each having energy corresponding to only one of the plurality of actuations, the blending matrix comprising times of each of the plurality of actuations;
(c) in the computer, initializing a blended output as a product of the blending matrix and the model;
(d) in the computer, calculating a residual as a difference between the blended output and the input recorded signals;
(e) in the computer, transforming the model into a thresholding domain and applying a thresholding function only to the transformed model to generate a deblended portion, the deblended portion comprising a plurality of seismic signals each having energy corresponding to only one of the plurality of actuations;
(f) in the computer, inverse transforming the deblended portion to generate a thresholded model;
(g) in the computer, applying the blending matrix to the thresholded model to generate an updated blended output;
(h) determining whether a difference between the updated blended output and the recorded signals falls below a selected limit, wherein the selected limit is related to a scaled objective function, a scale factor for the scaled objective function related to a number of the plurality of actuations; and
when the difference exceeds the selected limit, in the computer, adding the thresholded model to a back-projected residual comprising a product of the residual and a complex conjugate transpose of the blending matrix to generated an updated model, in the computer, replacing the model at (b) with the updated model and repeating (d) through (h); and
when the selected difference is below the limit, in the computer, at least one of storing and displaying the updated model as a final output, the final output comprising for at least one of the plurality of actuations a plurality of seismic signal sets each having energy corresponding to only one of the plurality of actuations.
16. The method of claim 15 wherein the thresholding function is performed in overlapping N-dimensional windows of pre-determined size wherein N>2, wherein results of the thresholding function in the N-dimensional windows are linearly re-combined.
17. The method of claim 15 wherein the thresholding function comprises quadratic thresholding.
18. The method of claim 15 wherein the thresholding function comprises a smooth function with continuous first and second derivatives in the region above the threshold level.
19. The method of claim 15 wherein the thresholding function comprises a function of the form:
T
τ
(
x
i
)
=
{
x
i
(
1
-
τ
n
/
x
i
n
)
x
i
>
τ
0
x
i
≤
τ
for n>1, and wherein T represents a thresholding function, τ represents the threshold value, and x represents a coefficient.
20. The method of claim 15 wherein a thresholding schedule comprises lowering the value of a threshold exponentially to zero.
21. The method of claim 15 wherein an N-dimensional transform for the thresholding function comprises a Fourier transform.
22. The method of claim 15 wherein an N-dimensional transform for the thresholding function comprises a curvelet transform.
23. The method of claim 15 wherein an N-dimensional transform for the thresholding function comprises a tau-p transform.
24. The method of claim 15 wherein an N-dimensional transform for the thresholding function comprises a parabolic radon transform.
25. The method of claim 15 wherein an N-dimensional transform for the thresholding function comprises a hyperbolic radon transform.
26. The method of claim 15 wherein an N-dimensional transform for the thresholding function comprises a single or double focal transform.
27. The method of claim 15 wherein an N-dimensional transform for the thresholding function is based upon at least one of the following functions: Activelet, AMlet, Armlet, Bandlet, Barlet, Bathlet, Beamlet, Binlet, Bumplet, Brushlet, Caplet, Camplet, Chirplet, Chordlet, Circlet, Coiflet, Contourlet, Cooklet, Craplet, Cubelet, CURElet, Daublet, Directionlet, Dreamlet, Edgelet, FAMlet, FLaglet, Flatlet, Fourierlet, Framelet, Fresnelet, Gaborlet, GAMlet, Gausslet, Graphlet, Grouplet, Haarlet, Haardlet, Heatlet, Hutlet, Hyperbolet, Icalet (Icalette), Interpolet, Loglet, Marrlet, MIMOlet, Monowavelet, Morelet, Morphlet, Multiselectivelet, Multiwavelet, Needlet, Noiselet, Ondelette, Ondulette, Prewavelet, Phaselet, Planelet, Platelet, Purelet, QVlet, Radonlet, RAMlet, Randlet, Ranklet, Ridgelet, Riezlet, Ripplet (original, type-I and II), Scalet, S2let, Seamlet, Seislet, Shadelet, Shapelet, Shearlet, Sinclet, Singlet, Slantlet, Smoothlet, Snakelet, SOHOlet, Sparselet, Spikelet, Splinelet, Starlet, Steerlet, Stockeslet, SURE-let (SURElet), Surfacelet, Surflet, Symmlet, S2let, Tetrolet, Treelet, Vaguelette, Wavelet-Vaguelette, Wavelet, Warblet, Warplet, Wedgelet, Xlet.
28. The method of claim 15 wherein the thresholding domain enhances a sparse representation of predictable signals.