An Acoustic Reflectivity Method for Laterally Varying Layered Media

Abstract

To compute the response of a stratified earth or sea-bed model it is often convenient to approximate the model by a stack of homogeneous plane parallel layers. The equations of motion are Fourier transformed with respect to time and Fourier-Bessel transformed with respect to the horizontal coordinates. Then one computes the exact response of the layer stack using Haskell¹ matrices, or Kennett² iteration, or the global matrix method of Schmidt³ and Jensen. Such approaches are referred to in seismology as “reflectivity methods” after the work of Fuchs and Muller⁴ in the early 1970’s. The Kennett^2,5 form of the reflectivity method is the one that we shall generalize here. As a form of invariant imbedding it has the advantage that each layer of the model is needed only once; thus the effective dimensionality of the model is reduced by one. In our new method the earth or sea-bed is still described as a stack of plane parallel layers; however, within each layer the velocity, density, and quality factor, Q, are now functions of horizontal position x. These functions are all independent from layer to layer so any earth or sea-bed model can be parameterized in this way. The two foundations of the new method are factorization^7,8 and imbedding⁶. Both of these topics are explored in detail elsewhere in this volume so the mathematics in this paper will be kept at a very low level. To simplify notation we treat propagation in two spatial dimensions.