I prove three theorems on earthquake location in the laterally homogeneous, radially concentric layered earth showing that (1) epicenters of earthquakes are locatable without using numerical travel-time information, (2) focal-depth residuals change sign at some epicentral distance that is a function of the focal depth but not of the distance residuals, and (3) the distance Δ between two random points on the surface of a sphere is distributed as sinΔ. Theorem (1) implies that the epicenter can be located without simultaneously solving for hypocentral depth or origin time. Theorem (2) suggests that the focal depth should never be determined by joint regression on all four hypocentral parameters. Theorem (3) suggests a likely reason why late readings tend to overpower accurate readings when a standard least-squares technique is used. Together these theorems suggest that the problem of earthquake location is ill posed.
Joint least-square regression will lead to the true hypocenter only if well-posedness is restored by restricting the class of admissible solutions using a priori knowledge, such as a travel-time table. But if the travel-time table is derived from the hypocentral solutions there is a feedback between location errors and errors in the travel-time tables, and local minima cannot be eliminated. Gutenberg instructively attempted to correct for bias due to azimuthal clustering, by assuming that residuals were nonnegative vectors.