Time-domain finite-difference solutions to the wave equation are a standard method for modeling seismic wave propagation, but they see limited use in data processing because they are expensive to compute. Given the resurgence of interest in reverse-time migration, anything that improves the performance and accuracy of the finite-difference method may be worthwhile. Additionally, the design problem for optimal finite-difference operators is a "simple and approachable" problem for teaching non-linear least squares. The standard solution for improving the finite-difference method is to use long spatial difference operators. However, there are computational reasons why it might be preferable to use spatially compact stencils, assuming they can be made accurate enough.