An analytical gradient-based inversion method for thickness evaluation of multi-layer pipe structure via swept-frequency eddy-current testing

Multi-layered structures composed of functional coatings/claddings, and corrosion products are commonly found in pipelines operating in complex service environments, such as those in the oil and nuclear power industries. The non-destructive testing of these structures is essential for predictive maintenance, as it enables estimation of the thickness and electromagnetic properties of each layer, quantities that directly reflect the health of both the pipe and its protective coatings and/or claddings. However, conventional inspection techniques may struggle to distinguish the individual layer thicknesses, particularly when very thin films are involved. To address this issue, a swept-frequency eddy-current reconstruction strategy, based on a semi-analytical forward model, is proposed in this study. First, an existing semi-analytical forward model for a single-layer pipe is extended to an arbitrary number of layers, enabling precise computation of the electromagnetic response without increasing computational complexity. Second, an inversion procedure is established, which takes advantage of the proposed semi-analytic formulation of the gradient of the cost function with respect to physical parameters at multiple frequencies. Finally, experimental validation confirms the effectiveness of the proposed method in reconstructing the thickness and/or electrical conductivity of individual layers within multi-layer pipe structures. © 2026 Elsevier Ltd.