Variational Principles For Surface Wave Propagation On A Laterally Heterogeneous Earth‐Iii. Potential Representation

We present a frequency‐domain analysis of travelling and standing surface waves on a smooth, laterally heterogeneous Earth model, using a potential representation that is valid everywhere, including in the neighbourhood of surface wave caustics. the Love and Rayleigh wave displacement fields are written in the form u_L=k‐¹_LW (‐řX▿₁)χ_L and u_R=Uřχ_R+k‐¹_RV▿₁χ_R; the quantities U, V and W are the local radial eigenfunctions, k_L and k_R are the local Love and Rayleigh wavenumbers, and χ_L and χ_R are surface wave potentials that vary rapidly on the surface of the unit sphere. the natural normalization condition for the local radial eigenfunctions is cCI₁= 1, where c is the local phase velocity, C is the local group velocity and I₁ is the local radial kinetic energy integral; with this normalization, the Love and Rayleigh wave potentials satisfy the spherical Helmholtz equations ▿²1χL + k²LχL = 0 and ▿²1χR + k²RχR = 0. Standing wave eigenfunctions χL and χR can be determined either by solving a truncated matrix eigenvalue problem or by using the EBK semi‐classical method; the results incorporate multiplet coupling along a single dispersion branch but ignore cross‐branch coupling. the theory allows for slowly varying topography of the Earth's surface and the core‐mantle boundary, and incorporates the effect of self‐gravitation.