Frequency-domain acoustic and elastic modeling and waveform inversion in the logarithmic grid set

One of the factors influencing the accuracy of the seismic modeling is the boundary condition. Several boundary conditions have been developed and have their own advantages and disadvantages. One possible method to perfectly remove edge reflections is to extend the dimension of a given model so that the edge reflections cannot be recorded within the recording duration. To make this idea feasible without increasing computational costs, we propose acoustic and elastic modeling algorithms performed in the logarithmic grid set, where grid size increases logarithmically from the middle of model surface. This method has an advantage to reduce the number of grids by the property of logarithmic scale. For acoustic and elastic wave modeling in the logarithmic grid set, the wave equations are first converted from the uniform scale to the logarithm scale. Then we apply the conventional node-based finite-difference method for the acoustic case and the cell-based finite-difference method for the elastic case. Numerical examples show that the new modeling algorithms yield solutions comparable to those of the conventional modeling algorithm, although they can suffer from numerical dispersion when the source is located in the coarse grids (far from the origin). Inversion results for the simple layered model and the modified version of the Marmousi-2 model show that the logarithmic inversion algorithms provide results comparable to those obtained by the conventional inversion achieving computational efficiency when the recording duration is not too long and the influence of numerical dispersion is almost negligible in the inversion. We expect that computational efficiency achieved by the logarithmic grid set would be greater in 3D than in 2D.