Depth Resolution in Dam Seepage Inversion: Why L1 and Total Variation See Different Boundaries

We study the problem of localizing a subsurface source (a leak) in a 2D dam seepage model from surface pressure measurements. The problem is ill-posed: the depth of the source is not uniquely determined by the data. We test two regularization strategies: L1 (Lasso) and Total Variation (TV). L1 fails: it places the source at a shallow artifact, where the signal is stronger, rather than at the true depth. This is the "shallow mirror" effect. TV, in contrast, recovers a block-like structure but suffers from "surface attachment": it stretches the source vertically to minimize boundary length. In 1D, TV correctly identifies the block while L1 identifies only its upper edge. In 2D, with depth sensors, L1 finds the upper boundary (depth 21) and TV finds the lower boundary (depth 28). We conclude that neither regularizer alone recovers the full geometry. The divergence between their solutions is itself informative. This is a negative result for single-prior inversion and a positive result for multi-prior diagnostics.