Review of imaging condition

In passive seismic imaging, if the start time is known, the time-reversal imaging (TRI) condition states that a true seismic source will correspond to the focus of all back-propagated events at start time. If we treat each receiver wavefield individually, we can formulate this imaging condition as

$$I(\mathbf{x},t) = \sum\limits_{i=0}^{N-1} R_i(\mathbf{x},t) \; .$$ (1)

However, the time information is usually unavailable, e.g. for microseismic events that happen during hydraulic fracturing. To deal with this situation, most existing algorithms need to perform scanning and picking over a group of potential start times (Maxwell, 2014; Fink, 2006). An alternative cross-correlation time-reversal imaging (CTRI) condition states that a true seismic source can be indicated by the coincidence of all the wavefields in both space and time. This condition can be expressed as (Sun et al., 2015; Nakata and Beroza, 2016)

$$I(\mathbf{x},t) = \prod\limits_{i=0}^{N-1} R_i(\mathbf{x},t) \; .$$ (2)

The key difference between equations 1 and 2 is two-fold. One difference lies in image resolution. Summing over all the receiver wavefields will result in an image that contains non-zeros across the wave propatation paths, while multiplication will lead to non-zeros corresponding only to the focuses. The other difference lies in computational cost. The summation indicated in equation 1 is implicit, and in practice that entire data volume can be back-propagated at once. However, the multiplication in equation 2 has to be carried out explicitly, leading to an $N$ times increase in computational cost. In order to leverage the efficiency of TRI, but to retain the high-resolution of CTRI, a hybrid cross-correlation time-reversal imaging (HCTRI) condition was proposed by Sun et al. (2015):

$$I(\mathbf{x},t) = \prod\limits_{j=0}^{N/n-1} \sum\limits_{k=0}^{n-1} R_{j \times n+k}(\mathbf{x},t) \; ,$$ (3)

where $n$ is the local summation window length. In practice, we find that, with a dense array coverage, three to four groups of separately propagated receiver wavefield suffice to attenuate the artifacts of TRI and improve the imaging resolution. The HCTRI is also connected with the auto-correlation time-reversal imaging (ATRI) (Artman et al., 2010): instead of correlating the entire wavefield with itself as done in ATRI, HCTRI effectively correlate different parts of the wavefield to reduce low-frequency artifacts.