On the derivatives of scattering coefficients in smooth elastic models

At a planar elastic discontinuity, the Zoeppritz scattering coefficients quantify the abrupt repartitioning of seismic wave energy among the various wave modes, whether upor 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 closedform 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.