The seismic traveltime inversion algorithm presented here requires few limiting assumptions and is computationally efficient.
The traveltime for a raypath in a layered earth (uniform or linearly varying sound speeds in each layer) can be written in closed form if the coordinates of the intersections between the raypath and the layer interfaces are known. The traveltime on an actual raypath is minimal with respect to neighboring paths. The raypath can thus be determined by equating the derivatives of the traveltimes with respect to the intersection coordinates to zero. This approach can be applied to the inverse problem, which seeks those layer parameter values which “best fit” the observed traveltimes from a CMP reflection gather. The result is a system of equations in which the intersection coordinates and layer parameters are independent variables, with the constraint that the traveltimes be minimal with respect to the coordinates. When this systemis linearized with a step-size constraint, an efficient inversion algorithm results. The equations are solved without numerical differentiation or iterative solution of the forward problem at each step in the inversion. The number of operations required in each iteration is proportional to the number of moveouts and the third power of the number of interfaces.
The algorithm has been applied to the estimation of sound speed in thin (e.g., 3−15 m) sea-floor sediment layers from CMP reflection data obtained with a 3 kHz system bandwidth. Errors were reduced by fitting traveltime differences determined by crosscorrelation of events at different moveouts. In favorable conditions, the scatter in estimated sound speeds was 2 percent for a 15 m layer.