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Synthetic seismograms by normal-mode summation: a new derivation and numerical examples



We present a new derivation of the explicit expression of displacement in response to a point dislocation source in terms of the summation of the Earth's normal modes. Starting from the well-known formal expression of normal-mode summation, we obtain the explicit expressions of displacements for the three conventional components in seismology (vertical, radial and transverse). Aided by the generalized spherical harmonics and coordinate transformation, our derivation is more comprehensive and concise and easier to handle. Numerical examples of normal-mode synthetics in different but realistic Earth models for a variety of source–receiver geometries are presented to illustrate the influence of self-gravitation, attenuation and radial anisotropy on waveforms. We also examine how waveforms of Love and Rayleigh waves are affected by the density and P-wave velocity structures in the upper mantle. Results show that the shear-wave structure obtained from surface wave tomography may be biased if the sensitivities of surface waves to the variation of density and compressional wave speed are not properly taken into account in the inversion.