Deghosting and reconstructing a seismic wavefield
A technique includes receiving data, which are indicative of measurements acquired by seismic sensors of a seismic wavefield. The seismic wavefield is formed from a combination of an upgoing wavefield and a downgoing wavefield. The technique includes providing a statistical estimator, which relates the upgoing wavefield to a linear combination of the measurements and a reconstruction error. The technique includes determining the upgoing wavefield based on the statistical estimator. The determination includes substantially minimizing a variance of the reconstruction error.
1. A method comprising:
receiving data indicative of measurements acquired by seismic sensors of a seismic wavefield, the seismic wavefield being formed from a combination of an upgoing wavefield and a downgoing wavefield;
processing the data on a processor-based machine to provide a statistical estimator relating the upgoing wavefield to a linear combination of the measurements and a reconstruction error; and
based on the statistical estimator, determining the upgoing wavefield, the determining comprising substantially minimizing a variance of the reconstruction error and interpolating the upgoing wavefield for at least one location other than one of the locations of the sensors, wherein the reconstruction error comprises an error attributable to the interpolation.
2. The method of claim 1 , wherein the act of receiving comprises receiving seismic data indicative of pressure measurements and particle motion measurements.
3. The method of claim 1 , wherein the act of determining the upgoing wavefield comprises performing a least squares-based minimization of the variance.
4. The method of claim 1 , wherein the act of determining the upgoing wavefield comprises determining correlations between the measurements.
5. The method of claim 1 , wherein the act of determining the upgoing wavefield comprises determining correlations between the measurements and the upgoing wavefield.
6. The method of claim 5 , wherein the act of determining the correlations between the measurements and the wavefield comprises estimating a wavenumber spectrum of the upgoing wavefield.
7. The method of claim 6 , wherein the act of estimating the wavenumber spectrum comprises estimating the wavenumber spectrum at a higher frequency based on the wavenumber spectrum at a lower frequency.
8. The method of claim 6 , wherein the act of estimating the wavenumber spectrum comprises assuming part of the wavenumber spectrum is constant.
9. The method of claim 6 , further comprising:
determining an expected reconstruction error; and
adjusting the estimation of the wavenumber spectrum based on the determined expected reconstruction error.
10. The method of claim 1 , further comprising:
towing the seismic sensors on a spread of at least one streamer.
11. The method of claim 1 , wherein the measurements are associated with a sampling that introduces spatial aliasing.
12. The method of claim 1 , wherein the sensors are regularly-spaced in a grid pattern or irregularly-spaced.
13. The method of claim 1 , wherein the act of receiving comprises receiving data indicative of measurements acquired by sensors of seismic streamers towed at different depths.
14. The method of claim 1 , wherein the act of receiving comprises receiving data indicative of measurements acquired by particle motion and pressure sensors of seismic streamers towed at different depths.
15. A system comprising:
an interface to receive data indicative of measurements acquired by seismic sensors of a seismic wavefield, the seismic wavefield being formed from a combination of an upgoing wavefield and a downgoing wavefield; and
a processor to process the data to:
provide a statistical estimator relating the upgoing wavefield to a linear combination of the measurements and a reconstruction error; and
based on the statistical estimator, determine the upgoing wavefield, the determining comprising substantially minimizing a variance of the reconstruction error and interpolating the upgoing wavefield for at least one location other than one of the locations of the sensors, wherein the reconstruction error comprises an error attributable to the interpolation.
16. The system of claim 15 , wherein the measurements comprise pressure measurements and particle motion measurements.
17. The system of claim 15 , wherein the processor processes the data to perform a least squares-based minimization of the variance.
18. The system of claim 15 , wherein the processor processes the data to determine correlations between the measurements.
19. The system of claim 15 , wherein the processor processes the data to determine correlations between the measurements and the upgoing wavefield.
20. The system of claim 19 , wherein the processor process the data to determine correlations between the measurements and the wavefield based at least in part on an estimated wavenumber spectrum of the upgoing wavefield.
21. The system of claim 20 , wherein the estimated wavenumber spectrum is associated with a higher frequency and is based at least in part on a wavenumber spectrum associated with a lower frequency.
22. The system of claim 20 , wherein the estimated wavenumber spectrum is nonzero and constant over a first portion of the spectrum and zero otherwise.
23. The system of claim 15 , further comprising:
a spread of at least one streamer, comprising the seismic sensors.
24. The system of claim 23 , further comprising:
at least one vessel to tow the spread.
25. The system of claim 15 , further comprising:
a spread of streamers towed at different depths, the streamers comprising the seismic sensors.
26. The system of claim 25 , wherein the seismic sensors comprise particle motion sensors and pressure sensors.
27. The system of claim 15 , wherein the measurements are associated with a sampling that introduces spatial aliasing.
28. The system of claim 15 , wherein the seismic sensors are regularly-spaced in a grid pattern or irregularly-spaced.