An adaptive, multiscale inverse scattering approach to photothermal depth profilometry

Photothermal depth profilometry is formulated as a nonlinear inverse scatte ring problem. Starting with the one-dimensional heat diffusion equation, we derive a mathematical model relating arbitrary variation in the depth-depe ndent thermal conductivity to observed thermal wavefields at the surface of a material sample. The form of the model is particularly convenient for in corporation into a nonlinear optimization framework for recovering the cond uctivity based on thermal wave data obtained at multiple frequencies. We de velop an adaptive, multiscale algorithm for serving this highly ill-posed i nverse problem. The algorithm is designed to produce an accurate, low-order representation of the thermal conductivity by automatically controlling th e lever of detail in the reconstruction. This control is designed to reflec t both (I) the nature of the underlying physics, which says that scale shou ld decrease with depth, and (2) the particular structure of the conductivit y profile, which may require a sparse collection of fine-scale components t o adequately represent significant features such as a layering structure. T he approach is demonstrated in a variety of synthetic examples representati ve of nondestructive evaluation problems seen in the steel industry.