Theoretical relationship between back-projection imaging and classical linear inverse solutions

Abstract

We clarified the theoretical relationship between the back-projection (BP) imaging and classical linear inverse solutions via the hybrid back-projection (HBP) imaging. In the HBP imaging, which is mathematically similar to the time-reversal source imaging, cross correlations of observed waveforms with the corresponding Green's functions are calculated. The key condition for BP to work well is that the Green's function is sufficiently close to the delta function. Then, the BP image represents the slip motion on a fault, and approximately equals to the least-squares solution (LSS). In HBP, instead of the Green's function in BP, the stacked autocorrelation function of the Green's function must be close to the delta function to obtain a fine image. Because the autocorrelation function is usually closer to the delta function than the original function, we can expect that HBP works better than BP, if we can reasonably estimate the Green's function. With additional condition that the stacked cross-correlation function of the Green's functions for different source locations is small enough, the HBP image is also approximately equal to the LSS. If these assumptions are not satisfied, however, the HBP image corresponds to a damped LSS with an extremely large damping parameter, which is clearly inferior to usual inverse solutions. On the other hand, the advantages of the BP method are much less computation and no necessity of Green's functions.