An adaptive, multiscale inverse scattering approach to photothermal depth pro lometry

Photothermal depth pro lometry is formulated as a non-linear inverse scattering problem. Starting with the one dimensional heat di usion equation, we derive a mathematical model relating arbitrary variation in the depth-dependent thermal conductivity to observed thermal wave elds at the surface of a material sample. The form of the model is particularly convenient for incorporation into a non-linear optimization framework for recovering the conductivity based on thermal wave data obtained at multiple frequencies. We develop an adaptive, multi-scale algorithm for solving this highly ill-posed inverse problem. The algorithm is designed to produce an accurate, low order representation of the thermal conductivity by automatically controlling the level of detail in the reconstruction. This control is designed to re ect both (1) the nature of the underlying physics which says scale should decrease with depth and (2) the particular structure of the conductivity pro le which may require a sparse collection of ne scale components to adequately represent signi cant features such as a layering structure. The approach is demonstrated in a variety of synthetic examples representative of non-destructive evaluation problems seen in the steel industry.