Multiplet coupling and the calculation of synthetic long‐period seismograms

Summary This paper describes a method of incorporating the effects of weak quasi‐degenerate multiplet coupling into synthetic long‐period seismogram calculations on a rotating, aspherical, anelastic earth. In the isolated multiplet approximation, the contribution of every multiplet nSt or nTl to a seismogram is of the form so(t) = Re ω −2 [R*o exp (iHoo t). ·So]exp (iω0t), where Hoo is the (2l+ 1)x (2l+ 1) matrix whose eigenvalues are the split singlet eigenfrequency perturbations, and So and Ro are 2l+ 1‐dimensional column vectors that depend respectively on the location and moment tensor of the source and the location and polarization of the receiver. The contribution of each hybrid multiplet, when coupling is considered, is of the same form, namely So(t) =Re ωo −2 [Ro*. exp(iHoot). So] exp (iHot), where the quantities Hoo, So and Ro can be expressed as perturbation expansions about Hoo, So and Ro. Both off‐path scattering and mode conversion of the equivalent propagating waves are accounted for in this perturbative hybrid multiplet summation method, and it is applicable to rough as well as smooth models of the Earth's lateral heterogeneity. Furthermore, it removes the secularity from the Born approximation coupling theory of Woodhouse & Tanimoto, and renders it uniformly valid for all times of seismological interest.