Benign misfit landscapes in one-dimensional magnetotelluric inversion: a 1992 result revisited, tested and extended

In 1992 the author's diploma thesis posed one-dimensional magnetotelluric (MT) inversion as least-squares minimization over layered conductivity models, derived an adjoint gradient, and argued that for exact data the misfit has a unique stationary point, the true model. We summarize that work, identify gaps in its uniqueness proof, and report a numerical verification: after a units convention and one complex conjugation are corrected, the adjoint gradient agrees with finite differences to about 6×10⁻⁸, and for a five-layer model 300 random starts with exact data all reach the true model. We then introduce the critical noise of a model, the smallest data perturbation that makes it a stationary (KKT) point, show that at interior points it equals the norm of the tangential component of the residual, and prove a data-space theorem: if the exact-data misfit has no false stationary point and the noise is below sup over ρ of min(δ_far(ρ), 1/κ_ρ − ρ), the noisy misfit has exactly one stationary point, where κ_ρ is the normal curvature of the data manifold. For the five-layer model, sampled estimates of this bound give 0.059% relative noise (0.63% ignoring parameter bounds), conditional on exact-data uniqueness, which is checked only by sampling. The binding constraint is physical: worst-case noise pins a poorly resolved resistive layer at its bound. No second local minimum was found; worst-case noise instead displaces the single minimum along a flat valley. A parameter-space version of the theorem is shown to be correct but vacuous for this ill-conditioned problem. Non-peer-reviewed preprint.