To advance our understanding of the earthquake rupture process, we must develop methods that will map observations into the space-time variation of seismic moment release on the fault plane. While the temporal history of overall moment release is relatively easy to obtain from P-waves, there has been no objective quantitative method available to directly determine the spatial variation in moment release. The theoretical connection between spatially varying moment release and the resultant seismogram is a linear integral relationship (specifically, a Radon transform). This paper outlines a linear inverse method that provides a high-resolution tomographic image of the space-time varying moment density function. Application of the method to P-waves from the large 1976 Mindanao earthquake demonstrates the ability of this method to reliably extract coherent pulses of moment release in space and time.