Computation of partial derivatives of Rayleigh-wave phase velocity using second-order subdeterminants

Rayleigh-wave propagation in a layered, elastic earth model is frequency-dependent (dispersive) and also function of the S-wave velocity, the P-wave velocity, the density and the thickness of the layers. Inversion of observed surface wave dispersion curves is used in many fields, from seismology to earthquake and environmental engineering. When normal-mode dispersion curves are clearly identified from recorded seismograms, they can be used as input for a so-called surface wave 'modal' inversion, mainly to assess the 1-D profile of S-wave velocity. When using 'local' inversion schemes for surface wave modal inversion, calculation of partial derivatives of dispersion curves with respect to layer parameters is an essential and time-consuming step to update and improve the earth model estimate. Accurate and high-speed computation of partial derivatives is recommended to achieve practical inversion algorithms. Analytical methods exist to calculate the partial derivatives of phase-velocity dispersion curves. In the case of Rayleigh waves, they have been rarely compared in terms of accuracy and computational speed. In order to perform such comparison, we hereby derive a new implementation to calculate analytically the partial derivatives of Rayleigh-mode dispersion curves with respect to the layer parameters of a 1-D layered elastic half-space. This method is based on the Implicit Function Theorem and on the Dunkin restatement of the Haskell recursion for the calculation of the Rayleigh-wave dispersion function. The Implicit Function Theorem permits calculation of the partial derivatives of modal phase velocities by partial differentiation of the dispersion function. Using a recursive scheme, the partial derivatives of the dispersion function are derived by a layer stacking procedure, which involves the determination of the analytical partial derivatives of layer matrix subdeterminants of order two. The resulting algorithm is compared with methods based on the more widely used variational theory in terms of accuracy and computational speed.