2D surface topography boundary conditions in seismic wave modelling |
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| Authors: | Bent Ruud Stig Hestholm |
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| Institution: | University of Bergen, Department of Solid Earth Physics, Allégaten 41, 5007 Bergen, Norway |
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| Abstract: | New formulations of boundary conditions at an arbitrary two-dimensional (2D) free-surface topography are derived. The top of a curved grid represents the free-surface topography while the grid's interior represents the physical medium. The velocity–stress version of the viscoelastic wave equations is assumed to be valid in this grid. However, the rectangular grid version attained by grid transformation is used to model wave propagation in this work in order to achieve the numerical discretization. We show the detailed solution of the particle velocities at the free surface resulting from discretizing the boundary conditions by second-order finite-differences (FDs). The resulting system of equations is spatially unconditionally stable. The FD order is gradually increased with depth up to eighth order inside the medium. Staggered grids are used in both space and time, and the second-order leap-frog and Crank–Nicholson methods are used for time-stepping. We simulate point sources at the surface of a homogeneous medium with a plane free surface containing a hill and a trench. Applying parameters representing exploration surveys, we present examples with a randomly realized surface topography generated by a 1D von Kármán function of order 1. Viscoelastic simulations are presented using this surface with a homogeneous medium and with a layered, randomized medium realization, all generating significant scattering. |
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