Preconditioning

Earthquake depth illustrates a null space

In the dawn of the era of computerized earthquake seismology, someone decided to add earthquake depth to their catalog. Traditionally, they had solved for only three unknowns, latitude, longitude, and time of source at the source, i.e., origin time. Now, they would add a fourth, the depth. They wrote down the $4\times 4$ system of equations and solved it. Erratic results. So then, they froze the depth at zero, solved for the old three variables; only then introducing the depth. Problem solved. (Compared to seismograph station separation, zero depth is an excellent approximation.)

I first understood the earthquake experience as an issue with nonlinear problems. True that earthquake travel time is not a linear function of distance, so the nonlinearity could lead to difficulty. But, something more is going on. When all seismometers are far from the earthquake, the waves arrive propagating nearly vertically (Earth curvature and $v(z)$ ray bending). Source depth affects such data in much the same way as time origin shift, so they are near a null space. Whenever near a null space, especially with a nonlinear problem, a good starting solution is needed.

Preconditioning