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This analysis of eddy current testing of defects in conductive materials is divided into two parts. Part I (this paper) investigates the application of reflection-mode diffraction tomography (DT) imaging techniques. Part I1 (to appear) discusses the application of deterministic and stochastic quantitative inversion techniques to similar configurations. The defects are modeled as cylindrical inhomogeneities concealed within a homogeneous, nonmagnetic, metallic half-space. Images of the defect cross section reveal features of interest, such as location, size, shape, and contrast of conductivity. Data are the time-harmonic anomalous fields sampled on a line above and parallel with the air-metal interface of the damaged structure when a known source is also placed above this interface.Images are made via the superimposition of holograms in the spectral domain ( K space), each one obtained for a given frequency, and by using a Fourier transform to go back into real space. The paper itself is divided into two parts. First, exact synthetic data are calculated for various canonical configurations using a method of moments, and they are compared with those calculated by a finite-element code and with experimental data. Analysis of these data in particular evidences interest and some limitations of the Born approximation. Second, two imaging algorithms are proposed and investigated from numerical simulations; the first one directly originates from ultrasonic DT where most of the attenuation is neglected, and the other approximately accounts for the skin effect. In addition, an exact method is given, but not applied essentially for it replaces the Fourier transform by a numerically involved Laplace transform.

In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided inequalities for the underlying normally distributed random vector. As a main step for an algorithmic solution of such problems, we derive a derivative formula for (normal) probabilities of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a similar well-known statement for multivariate normal distribution functions. As an application, we consider a problem from water reservoir management. One of the outcomes of the problem solution is that the (still frequently encountered) use of simple individual probabilistic can completely fail. In contrast, the (more difficult) use of joint probabilistic constraints which heavily depends on the derivative formula mentioned before yields very reasonable and robust solutions over the whole time horizon considered.

The paper deals with joint probabilistic constraints defined by a Gaussian coefficient matrix. It is shown how to explicitly reduce the computation of values and gradients of the underlying probability function to that of Gaussian distribution functions. This allows to employ existing efficient algorithms for calculating this latter class of function in order to solve probabilistically constrained optimization problems of the indicated type. Results are illustrated by an example from energy production.

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