The variational projection method (VPM): A modern technique to simulate the seismic response of shallow alluvial valleys

Abstract The variational projection method (VPM) to compute the seismic response of three-dimensional (3D) shallow al1uvial valleys is presented. The solution is based on a weak-form variational Galerkin formulation of the problem for a smooth irregular layer overlaying an elastic half-space. The Galerkin method is applied by means of a simple set of trial functions for depth dependence. The partial differential equations of dynamic elasticity in space and time are “projected” into the horizontal free surface plane and are then solved using a pseudospectral scheme: finite differences in time and fast Fourier transform (FFT) to compute spatial horizontal derivatives. By using as reference the 1D solution, the radiation conditions are approximately fulfilled. Numerical examples of the performance and accuracy of the method in 2- and 3D are given. Possible improvements of the method are discussed.