This ‘weak enough’ condition, which is closely related to the ‘short enough’ time condition, is not clearly defined and should be checked in each particular situation. This approximation is only valid for weak enough lateral heterogeneities and for short enough time-series. When dealing with laterally heterogeneous models, a standard method is the first-order perturbation of the normal mode basis obtained in a spherically symmetric reference earth model. The normal mode solution of the wave equation is well known and widely used in global seismology when dealing with spherically symmetric models in the 20 s and longer period range. In the latter case, we show that non-linearity can be significant even at long periods and when using existing smooth global tomographic models.įréchet derivatives, Global seismology, normal modes 1 Introduction Validation tests with respect to the spectral element method solution both in the Frechet derivative case and as a synthetic seismogram tool shows a good agreement. This algorithm is similar to the one obtained when solving the adjoint problem. Another interesting point is, when used for Frechet kernel, the computing cost is (almost) independent of the number of parameters used for the inversion. For one scattering point and for a given number of time steps, the numerical cost of such a scheme grows as (number of receivers + the number of sources) × (corner frequency) 2 to be compared to (number of receivers × the number of sources) × (corner frequency) 4 when the classical normal mode coupling algorithm is used. Doing so, the perturbed displacement is obtained, for each scattering point, as the convolution of a forward wavefield from the source to the scattering point with a backward wavefield from the scattering integration point to the receiver. For each scattering point, ‘source to scattering point’ and ‘scattering point to receivers’ expressions are separated before applying a Fourier transform to return to the time domain. We start from the first-order Born solution in the frequency domain and we use a numerical scheme for the volume integration, which means that we have to compute the effect of a finite number of scattering points and sum them with the appropriate integration weight. We present an alternative to the classical mode coupling method scheme often used in global seismology to compute synthetic seismograms in laterally heterogeneous earth model and Frechet derivatives for tomographic inverse problem with the normal modes first-order Born approximation.
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