Interpolating seismic data

Abstract

A technique includes modeling interpolated seismic measurements as a random process characterized by seismic measurements acquired at a set of sensor locations and an interpolation error. The technique includes determining the interpolated seismic measurements based at least in part on a minimization of the interpolation error.

What is claimed is: 
     
         1 . A method comprising:
 modeling interpolated seismic measurements as a random process, wherein the seismic measurements comprise pressure measurements or particle motion measurements that are acquired at sensor locations, and an interpolation error;   processing the seismic measurements in a processor-based machine to determine the interpolated seismic measurements based at least in part on a minimization of the interpolation error; and:   further modeling the interpolated seismic measurements based on the seismic measurements and the interpolation error.   
     
     
         2 . The method of  claim 1 , wherein the sensor locations comprise non-co-located sensor locations. 
     
     
         3 . The method of  claim 1 , wherein processing the seismic measurements comprises determining the interpolated seismic measurements at locations associated with a uniform sampling. 
     
     
         4 . The method of  claim 1 , wherein the sensor locations comprise locations associated with a non-uniform sampling. 
     
     
         5 . The method of  claim 1 , wherein the act of minimizing the interpolation error comprises minimizing a variation of the interpolation error. 
     
     
         6 . The method of  claim 1 , wherein processing the seismic measurements comprises calculating autocorrelation of the seismic measurements at the sensor locations. 
     
     
         7 . The method of  claim 6 , wherein the act of calculating the autocorrelation comprises determining a power spectral density of the seismic measurements acquired at the sensor locations. 
     
     
         8 . The method of  claim 7 , wherein the autocorrelation of seismic measurements is computed as the inverse Fourier transform of an a priori model of the power spectral density of the seismic measurements. 
     
     
         9 . The method of  claim 7 , wherein the power spectral density of the seismic measurements is modeled as a constant for band-limited measurements. 
     
     
         10 . The method of  claim 7 , wherein the act of determining the power spectral density comprises determining the power spectral density at a first frequency based on a power spectral density determined at a different second frequency. 
     
     
         11 . The method of  claim 1  further comprising:
 dividing the measurements among frequency bands; and 
 performing the interpolation for each frequency band independently. 
 
     
     
         12 . The method of  claim 11 , wherein each frequency band has a different associated spatial bandwidth. 
     
     
         13 . The method of  claim 1 , wherein the measurements are acquired in a two-dimensional (2D) spatial plane. 
     
     
         14 . The method of  claim 1 , wherein the measurements are acquired in a three-dimensional (3D) spatial plane. 
     
     
         15 . The method of  claim 12 , wherein the different spatial bandwidths are determined by the speed of propagation of the signals. 
     
     
         16 . The method of  claim 1 , wherein the interpolated seismic measurements comprise three particle motion components and a pressure component. 
     
     
         17 . An article comprising instructions stored on a computer accessible storage medium that when executed by a processor-based system cause the processor-based system to:
 model interpolated seismic measurements as a random process characterized by seismic data measurements acquired at sensor locations and an interpolation error; and   determine the interpolated seismic measurements based at least in part on a minimization of the interpolation error.   
     
     
         18 . The article of  claim 17 , the storage medium storing instructions to cause the processor-based system to minimize a variation of the interpolation error to minimize the interpolation error. 
     
     
         19 . The article of  claim 17 , the storage medium storing instructions to cause the processor to further model the interpolated seismic measurements as statistics of the seismic measurements and the interpolation error. 
     
     
         20 . The article of  claim 17 , wherein the seismic measurements acquired at the sensor locations comprise pressure data and particle motion data acquired at sensor locations along at least one streamer. 
     
     
         21 . The article of  claim 17 , the storage medium storing comprising instructions to cause the processor-based system to perform autocorrelation of the seismic measurements acquired at the sensor locations. 
     
     
         22 . The article of  claim 17 , wherein the sensor locations are associated with a non-uniform sampling, and the instructions when executed cause the processor-based system to determine interpolated measurements associated with a uniform sampling. 
     
     
         23 . A system comprising:
 an interface to receive a set of seismic measurements acquired at non-uniformly spaced sensor locations; and   a processor to model interpolated seismic measurements at uniformly spaced locations as a random process characterized by the seismic data measurements acquired at the non-uniformly spaced sensor locations, an interpolation error, and statistics of the seismic measurements and interpolation error; and determine the interpolated seismic measurements at the uniformly spaced locations based at least in part on a minimization of the interpolation error.   
     
     
         24 . The system of  claim 23 , wherein the processor is adapted to further model the interpolated seismic measurements as statistics of the seismic measurements and the interpolation error. 
     
     
         25 . The system of  claim 23 , wherein the processor is adapted to minimize a variation of the interpolation error. 
     
     
         26 . The system of  claim 23 , wherein the seismic measurements acquired at the non-uniformly spaced sensor locations comprise pressure data and particle motion data acquired at sensor locations of at least one streamer. 
     
     
         27 . The system of  claim 23 , wherein the processor is adapted to perform autocorrelation of the set of seismic data measurements. 
     
     
         28 . The system of  claim 27 , wherein the processor is adapted to perform an inverse Fourier transform on a power spectral density of the seismic measurements acquired at the non-uniformly spaced sensor locations to compute autocorrelation of the seismic data measurements. 
     
     
         29 . The system of  claim 28 , further comprising:
 a survey vessel to tow at least one sensor cable to acquire said seismic measurements acquired at the non-uniformly spaced sensor locations,   wherein the processor is located on said at least one sensor cable.   
     
     
         30 . The system of  claim 29 , wherein said at least one sensor cable comprises an ocean bottom cable or a streamer. 
     
     
         31 . The system of  claim 23 , further comprising:
 a survey vessel to tow at least one streamer to acquire said seismic measurements acquired at the non-uniformly spaced sensor locations,   wherein the processor is not located on said at least one streamer.