Linearization of the matched field processing approach to inverse tomography

Abstract

To determine the speed of propagation of a signal in a medium, signals with known harmonic components are transmitted from one or more sources within the region. One or more detectors within the region measure wave amplitude over time at so as to detect the signals transmitted at the sources and so as to identify the source from which each detected signal was transmitted. Beta bar coefficients are calculated so as to maximize the correlation of the calculated matched field processing power and the measured matched field processing power for each source-detector combination. Beta coefficients for all cells through which a path from a source to a detector passes are calculated by representing all positions in the region by cell number i and position coordinate z, the region being partitioned into M discrete, indexed cells C i ; expressing the wave speed as a known function w of the beta coefficients; and calculating the beta coefficients based on the beta bar coefficients, on the path lengths between each source and each detector combination, and on the lengths of those paths passing through each cell. The wave speed is then calculated by using the function w and the beta coefficients.

What is claimed is: 
     
       1. A method for determining the wave speed within a region of a medium, comprising the steps of: (a) transmitting signals from one or more sources disposed within the region, each signal having a known harmonic component:   (b) measuring wave amplitude over time at one or more detectors disposed within the region so as to detect the signals transmitted at the sources and so as to identify the source from which each detected signal was transmitted;   (c) calculating beta bar coefficients so as to maximize the correlation of the calculated matched field processing power and the measured matched field processing power for each source-detector combination;   (d) calculating beta coefficients for all cells through which a path from a source to a detector passes, comprising the steps, (i) representing all positions in the region by cell number i and position coordinate z, the region being partitioned into M discrete, indexed cells C i  ;   (ii) expressing the wave speed as a known function w of the beta coefficients;   (iii) calculating the beta coefficients based on the beta bar coefficients, on the path lengths between each source and each detector combination, and on the lengths of those paths passing through each cell; and     (e) calculating the wave speed by using the function w of part (d)(ii) and the beta coefficients calculated in part (d)(iii).   
     
     
       2. A wave speed determination method according to claim 1 wherein the beta bar calculation step further comprises the steps of calculating perturbation normal mode wave number bars and calculating the beta bar coefficients therefrom. 
     
     
       3. A wave speed determination method according to claim 1 wherein the wave speed characterization step further comprises the steps of characterizing the wave speed in cell C i  as the inverse of the square root of the sum of a function h of depth and the sum over all values of j of the product of the j-th beta coefficient for cell C i  and the function g j  of the depth, and   determining the functions h and g j .   
     
     
       4. A wave speed determination method according to claim 1 wherein the signals are acoustic and the medium is ocean water. 
     
     
       5. A wave speed determination method according to claim 4 wherein the angular frequencies of the signals harmonic components are between 20π radians/sec and 60π radians/sec. 
     
     
       6. A wave speed determination method according to claim 4 wherein the signals are explosives deployed to generate signals near the surface of the water. 
     
     
       7. A wave speed determination method according to claim 6 wherein the explosives are air-deployed. 
     
     
       8. A wave speed determination method according to claim 1 wherein the locations of the sources and detectors are known to within about one wavelength.