Since the Earth is elastic, it is worth the computational burden to process multicomponent data for elastic phenomena with fully coupled time-domain wave-equation propagators. At every time sample in the back-propagated model domain, the complete wave field is decomposed exactly into compressional and shear wave components by simple spatial derivatives. Then, physically significant images are extracted from extrapolated hyper-cubes by applying appropriate imaging conditions. To locate subsurface sources (or diffractors) with the time-reverse modeling algorithm, the imaging condition required is the correlation of P and S energy since only at the source location are the two events collocated. The impulse response of the algorithm is anti-symmetric in physical space and can be enhanced through post-processing with a spatial derivative or integral.
The time-reverse modeling (TRM) algorithm was developed for locating sources within amodel domain (Fink, 1999; Gajewski and Tessmer, 2005). The method is suited for locating earthquakes, microseismic events, or tremor sources. The difference between TRMand reverse-time migration (RTM) (Levin, 1984) is the lack of a known source wave field for TRM. Otherwise, data are treated in the same manner: reversed in time and used as source functions at the acquisition locations.
The difference between a specular reflection and a stimulated heterogeneity, or diffractor, is that data contain only a direct arrival ray path: The ”from-path”. This contrasts to reflection seismic whose time delays are the sum of the ”to-path” and the ”from-path.” Without some knowledge of the to-path, imaging algorithms based on delays between a reference event and a scattering event, including RTM and interferometry, are incapable of finding sources within a domain. In contrast, the TRM algorithm exploits the ability to collapse travel-time surfaces (hyperbolic cones) using wave-equation propagators.
If data are collected as a function of time on a datum, the data space has dimensions (x, t). Using wave-equation extrapolators and a subsurface velocity model, we can create a depth axis to arrive at a hypercube with the original data dimensions and an extra spatial model dimension associated with propagation. Figure 1 shows the simple kinematic surface of an energy source within a mixed model-data domain. The extrapolation direction is defined as z, but of course could be any model domain vector. The (x, t) data recorded on the acquisition datum is a hyperbola for a homogeneous medium. The (x, z) position of the source is at the intersection of the two symmetric cones. Acausal propagation of the data, or causal propagation of time-reversed data, will collapse moveout in the data to the location of the source and then unfocus that point with subsequent extrapolation steps. Of course, the travel-time surface in Figure 1 is only a single order-zero solution to the full elastic wave equation: far field, single mode, constant velocity, etc.
Since the time and depth dimensions of the hypercube are redundant, an imaging condition which collapses the time axis provides an image defined by purely spatial coordinates. Defining an appropriate imaging condition for TRM is challenging without knowledge of absolute time.