Surfa ce-wave dispersion computations: Knopoff's method

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Abstract
The optimization of Knopoff's method computer computation of surface-wave dispersion, and the comparison of this method with the Thomson-Haskell technic and its extensions are studied. Of the various versions of the Thomson-Haskell formulation for Rayleigh-wave dispersion computations, the reduced-δ-matrix extension is the most powerful, i.e., the fastest which contains the feature controlling the loss-of-precision problem occasionally encountered by the original formulation. It is shown that the results of the δ-matrix extensions are actually contained in Knopoff's work, which appeared earlier than these extensions, and that these results are obtainable directly from his formulation without recourse to δ-matrix theory. The flexibility of Knopoff's method is used to devise a new representation of the Rayleigh-wave dispersion function which is more powerful than the most powerful of the Thomson-Haskell versions, i.e., it contains the loss-of-precision control feature and is about 38 per cent faster than the reduced-δ-matrix extension; in fact, it is about 12 per cent faster than the fastest of the Thomson-Haskell versions. Explicit forms of the new representation are given for the layered-half-space analogs of continental and oceanic models of the Earth, which can be terminated by a homogeneous half-space beginning in the mantle, a liquid outer core, or a solid inner core. The representations for symmetric and antisymmetric modes of the symmetric-plate analog of the Earth are also given. The Rayleigh-wave dispersion functions for the product form of the original Thomson-Haskell formulation, the δ-matrix and reduced-δ-matrix extensions of this formulation, and the representations developed from Knopoff's method, are identical when written out fully. The various technics differ only in their matrix representations of the dispersion function, which, of course, govern the speeds of computation of the various methods. The Knopoff and Thomson-Haskell technics for computing Love-wave dispersion are compared. Although the matrix representations of the dispersion function differ for the two technics, the functions are identical, except for a possible sign reversal, when written out fully. The speeds of computation are also identical. The representations derived from Knopoff's method are given for the two possible flat-layer analogs of the Earth, which are applicable to Love-wave propagation.

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