Source depth and epicenter determination remains a simple but key method for discrimination between earthquakes and nuclear explosions. At near-regional distances, a variety of secondary body wave arrivals can be observed that may be useful for constraining source depth and epicenter. In this paper, we study the source location problem using regional time and arrival azimuth data from a sparse seismic network. A theoretical analysis of the source location inverse problem provides estimates of the stability and uncertainty of the solution for a particular suite of data as a function of epicenter and depth, as well as evaluating the importance of each datum to the solution. An empirical analysis with actual data complements the theoretical analysis. Our approach is applied to the three-station Natural Resources Defense Council-Soviet Academy of Sciences network in Kazakhstan. The theoretical analysis indicates that reasonable constraint on depth and epicenter is possible whenever three distinct phases (P, Pn, Pg, PzP, and PmP, or equivalent S phases) are observed at two or three stations. Our theoretical analysis for two-station locations shows that the epicenter constraints are extremely poor for sources along the line passing through the two observing stations. The use of arrival azimuth in two-station locations results in improvement in epicenter constraint for events occurring in the vicinity of the line connecting the two stations, but there is no significant improvement elsewhere. An empirical analysis of data from two of the September 1987 chemical explosions and data from an explosion in an active mining area supports the theoretical analysis, showing that two-station locations (both epicenter and depth) are quite reliable. Furthermore, the theoretical analysis also indicates that, even with data from a single observing station, satisfactory results are obtained if three or more phases plus arrival azimuth are available. ¿1991 American Geophysical Union