Abstract

We describe an algorithm for the computation of source-to-detector ranges from the traveltime of multiply reflected sound waves. Our technique is based on ray tracing the corresponding reflection multiple in an unfolded velocity-depth function. It requires information about water depth in the vicinity of each reflection point, and the velocity-depth function characteristic for the surveyed area. The point-wise velocity information is transformed into a continuous function by assuming a linear gradient between sampling points. For each layer (i), horizontal offsets and traveltimes are computed fromml:  
Xi(P)=(AiP)1{[1(PVi1)2]1/2[1(PVi)2]1/2}
and  
Ti(P)=(Ai)1{Cosh1[1/(PVi1)]Cosh1[1/(PVi)]}
where P=sinθi/Vi is the ray parameter (θi is the incidence angle), Ai is the velocity gradient in the ith layer, and Vi is the sound velocity at the ith point in the velocity-depth function. The solution is found by iterating for that ray parameter (P*) which minimizes |i1NTi(P)T0|. Here, T0 is the observed traveltime and N is the number of layers in the unfolded velocity-depth function. The horizontal distance (range) between the source and detector is given by i1NXi(P*). The method has been tested with data gathered off Hawaii during the anisotropy Show experiment. Depths at reflection points were estimated from a surface fit to the bathymetry data, and ranges computed from one and two bottom reflections agree within 0.1 percent.

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