On the derivatives of scattering coefficients in smooth elastic models

Abstract

At a planar elastic discontinuity, the Zoeppritz scattering coefficients quantify the abrupt repartitioning of seismic wave energy among the various wave modes, whether up- or downgoing, or compressional or shear modes. In smooth (differentiable) elastic models, the depth derivatives of the Zoeppritz scattering coefficients quantify a continuous repartitioning of energy per unit depth. The derivatives of the Zoeppritz scattering coefficients have simple closed-form expressions that are exact in smooth elastic models. Although the form of the depth derivatives resembles the small-contrast approximations of Zoeppritz equations, their derivation does not require a small-contrast assumption. The scattering derivatives play a fundamental role in smooth models that corresponds with the role of the Zoeppritz scattering coefficients in blocked models. The matrix of scattering derivatives is identical to a row permutation with sign changes of the coupling matrix of the wave vector differential equation for smooth elastic models.