Seismic waves in a stratified half space—III. Piecewise smooth models

Abstract

Summary. A recursive procedure is introduced for the calculation of the properties of an earth model consisting of regions with smoothly varying properties separated by discontinuities in seismic parameters or parameter gradients. Within each region the solution is constructed from a Langer uniform approximation in terms of Airy functions supplemented by a series of terms representing multifold wave interaction with the parameter gradients. This procedure allows an efficient treatment of turning point problems and in general it is an adequate approximation to retain only the leading order term in the interaction series. At an interface between varying media the solution may be expressed in terms of generalized interface coefficients or alternatively recast into a form which separates the effects of gradients and the interface itself. In the latter case the plane wave reflection and transmission coefficients are to be used. The resulting calculation scheme for the reflection matrices is an extension of the recursive scheme for uniform layers. The simple phase delay transmission effect of a uniform region is replaced by the reflection and transmission from a gradient zone sandwiched between two uniform media. This recursive scheme gives good results for crustal and upper mantle models, and only about a dozen subdivisions of the stratification are required down to 950 km. Checks on the accuracy of the computation from unitarity relations between reflection coefficients show that for periods less than 20 s, in the upper mantle, the error associated with neglect of the interaction series is less than 0.1 per cent except for a few cases. When a turning point is just at a structural boundary the error can increase to 2 per cent. For a low‐velocity zone where close turning points occur for slownesses within the range of the inversion, rather larger errors may occur but these do not affect waves which turn well above or well below the zone. The leading order approximation allows no conversion of wave types except at a discontinuity in elastic properties and the error introduced by neglecting conversions at steep gradient zones of limited vertical extent (as at upper mantle discontinuities) may reach a few per cent. Show more