FORMULATING FOCUSING AS A BACKUS-GILBERT PROBLEM
Standard time-reversal experiments typically achieve a focal spot in the dimensions of the dominant wavelength. In exploration seismology, more precisely in microseismic imaging, a focal spot of such size does not provide much information about the source location. To remedy this disadvantage we can assign complex weights to each of the receivers as shown in Figure 1 so that focusing is optimized inside the target area, without regard to what happens outside. Ignoring the rest of the medium allows us to have sharp focusing inside the target area but triggers a blow-up behavior when we look at other points in the medium. Figure 2, which shows a cross-section at the source level that extends beyond the original target area, shows this blow-up behavior.