The interaction of a seismic source device with the free surface of the earth makes the radiation characteristic different from that of the equivalent body point source . Usual seismic sources are built to produce enough energy to penetrate a few kilometers into the earth. There is no guarantee that when the source is activated, the surface material behavior will stay within a region that can be appropriately described using infinitesimal stress strain relationships. In this section I use finite-differences to model radiation characteristics and wave propagation when the equations of motion are no longer linear. I extend a nonlinear scalar equation that is normally used in one dimension into a three component, three-dimensional setting.
Many geophysical problems involve the repeated application of identical operators to some data.
A few example can be found in recursive schemes, recursive wave extrapolations, or multi-grid methods. For numerical implementation an approximation to an operator is usually used, which deviates from the true desired behavior of the operator. The answer lies in the existence of a Central Limit Theorem for that particular operator. How to Beat the Central Limit Theorem . I illuminate this idea further and contrast it with other optimal design criteria for approximate operators.