Spatio-temporal resolution improvement via weighted time-reversal

Abstract We formulate resolution enhancement as a modified Backus–Gilbert inverse problem and determine the optimal complex weights that improve focusing of waves in space and time. The optimization corrects for receiver geometry. If we accurately know the location of a control point in the subsurface we can use the corresponding optimal weights to achieve enhanced focusing in a prescribed target zone surrounding the control point. Errors in the back propagation velocity and noisy data degrade the quality of focusing. The optimal wave field shows a blow-up behavior outside the optimization area. We show different measures of resolution to estimate the compression of the focal spot. The optimized weights amplify the high frequencies, but the algorithm also improves the focusing for monochromatic waves. At all frequencies our algorithm improves the resolution of the focal spot. We also show that for a uniformly sampled line array and a homogeneous medium, the weights used to enhance resolution have a negligible imaginary part and that they are oscillatory across the array used. To fully test the robustness of our algorithm, we also consider focusing in a heterogeneous medium with embedded scatterers and an irregular receiver line, and show that in this scenario we are also able to attain focusing improvement.