CIRCUITSSYSTEMSSIGNAL PROCESS VOL. 19, NO. 4, 2000, PP. 339-363 AN ADAPTIVE, MULTISCALE INVERSE SCATTERING APPROACH TO PHOTOTHERMAL DEPTH PROFILOMETRY* Eric L. Miller l, Ibrahim Yavuz 1, Lena Nicolaides 2, and Andreas Mandelis 2 Abstract. Photothermal depth profilometry is formulated as a nonlinear inverse scattering problem. Starting with the one-dimensional heat diffusion equation, we derive a mathe- matical model relating arbitrary variation in the depth-dependent thermal conductivity to observed thermal wavefields at the surface of a material sample. The form of the model is particularly convenient for incorporation into a nonlinear optimization framework for recovering the conductivity based on thermal wave data obtained at multiple frequencies. We develop an adaptive, multiscale 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 reflect both (1) the nature of the underlying physics, which says that scale should decrease with depth, and (2) the particular structure of the conductivity profile, which may require a sparse collection of fine-scale components to adequately represent significant features such as a layering structure. The approach is demonstrated in a variety of synthetic examples representative of nondestructive evaluation problems seen in the steel industry. Key words: Adaptive signal representation, multiscale methods, B-splines, nonlinear in- verse scattering, photothermal depth profiling. * Received November 4, 1999; revised April 15, 2000. The work of authors E. L. Miller and I. Yavuz was supported by a CAREER Award from the National Science Foundation MIP-9623721, an ODDR&E MURI under Air Force Office of Scientific Research contract F49620-96-1-0028, and the Army Research Office Demining MURI under grant DAAG55-97-1-0013. The work of authors L. Nicolaides and A. Mandelis was supported by a research contract from Material and Manufacturing Ontario (MMO). 1 Department of Electrical and Computer Engineering, Northeastern University, 235 Forsyth Building, Boston, Massachusetts 02115, USA. E-mail: elmiller@ece.neu.edu 2 Photothermal and Optoelectronic Diagnostics Laboratories, Department of Mechanical En- gineering, University of Toronto, 5 King's College Road, Toronto M5S 3G8, Canada. E-mail: mandelis @ mie.utoronto.ca