Optimal finite-difference operators for arbitrarily sampled data

ABSTRACT We have developed a general method to obtain the equiripple and the least-squares finite-difference (FD) operator weights to compute arbitrary-order derivatives from arbitrary sample locations. The method is based on the complex-valued Remez exchange algorithm applied to three cost functions: the total error, the relative error, and the group-velocity error. We evaluate the method on three acoustic FD modeling examples. In the first example, we assess the accuracy obtained with the optimal coefficients when propagating acoustic waves through a medium. In the second example, we propagate a wave through an irregular grid. In the final example, we position a source and receiver at arbitrary locations in-between the modeling grid points. In the examples using regular grids, the equiripple solution to the relative cost function performs best. It obtains marginally (4%–10%) better results compared to the second-best option, the least-squares solution for the relative cost function. The least-squares solution for the relative error produced the only stable and accurate results also in the example of modeling on an irregular grid.