A Cramér–Rao resolution limit for structural-parameter estimation in electrical resistivity tomography: interface dip, data noise, and model error

Summary Dip is often read from a regularized electrical resistivity tomography (ERT) image without a parameter-specific uncertainty. Image-resolution measures describe how an inversion blurs the resistivity field, but they do not give a noise-dependent limit for the dip of a geological interface. We derive that limit from the 2.5-D forward operator using Fisher information and a Cramér–Rao bound (CRB). The background resistivity is profiled out in log-data space. For a specified structural family, the dip standard error satisfies σ _θ ≥ σ / ∂ g₀/∂ θ _PC₀. We also derive a marginal bound for unknown geometry and a first-order bias caused by structural misspecification. Tests with a 30-electrode dipole–dipole array show that the maximum-likelihood estimate attains the CRB when the fitted family is correct. At 3 % noise, the single-parameter floor is 0.14–0.50○ and the geometry-marginalised floor is 0.33–0.72○. Holding a coupled geometry parameter one cell away from its true value instead produces a 3–5○ dip bias. This is 6–25 times the formal floor. An ablation attributes most of the bias to thickness, width, or depth rather than resistivity contrast. The result is conditional on the chosen model family: the CRB measures noise-limited precision within that family, whereas the bias measures the cost of using the wrong geometry. For the cases tested here, improving an independent geometric constraint is therefore more useful than further reducing data noise.