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.

我们将分辨率增强公式化为改进的 Backus-Gilbert 逆问题，并确定改善波在空间和时间上的聚焦的最佳复杂权重。优化校正接收器几何形状。如果我们准确地知道控制点在地下的位置，我们就可以使用相应的最佳权重在控制点周围的指定目标区域内实现增强聚焦。反向传播速度的误差和噪声数据会降低聚焦质量。最佳波场在优化区域外表现出爆破行为。我们展示了不同的分辨率度量来估计焦斑的压缩。优化的权重放大了高频，但该算法还改进了单色波的聚焦。在所有频率下，我们的算法都提高了焦点的分辨率。我们还表明，对于均匀采样的线阵列和均匀介质，用于提高分辨率的权重具有可以忽略不计的虚部，并且它们在所使用的阵列中是振荡的。为了充分测试我们算法的稳健性，我们还考虑在具有嵌入式散射体和不规则接收器线的异构介质中聚焦，并表明在这种情况下我们也能够实现聚焦改进。