Identifiability Limits in Ultrasonic Microstructure Characterisation: A Canonical and Stochastic Framework
Ultrasound for microstructure characterisation is increasingly studied and is often assessed through inversion performance. However, the framework is fundamentally constrained by the information content available in the measured response. Hence, this work examines identifiability directly by analysing the geometry of the forward operator in both a canonical pulse-echo model and a stochastic surrogate microstructure. For the canonical model, a closed-form sensitivity analysis reveals information limits arising from parameter coupling, dimensional restriction, and interface-driven saturation. For the surrogate microstructures represented by Gaussian random fields, the forward map from correlation length $D$ and texture-coherence parameter $T$ to the attenuation and velocity observables remains structurally full rank. However, the sensitivity geometry is strongly anisotropic, with uneven parameter influence across the observable space. When intrinsic microstructural variability is incorporated, practical identifiability is further reduced. A variance-weighted Fisher framework shows that recoverability is governed by the balance between sensitivity magnitude and stochastic variability, rather than by structural rank alone. Inversion results confirm this behaviour: single observables produce elongated and weakly constrained objective landscapes, whereas combined observables improve conditioning through complementary sensitivities. These results show that, within the feature-level framework considered here, identifiability limits are governed primarily by forward-map structure and intrinsic variability, with direct implications for observable selection and measurement design.