Theoretical Explanation for the Empirical Probability of Detection ( POD ) Curve : A Neural Network - Motivated

|For non-destructive testing of aerospace structures, it is extremely important to know how the probability of detecting a fault depends on its size. Recently, an empirical formula has been found which described this dependence. In this paper, we provide the theoretical justiication for this formula by using methods motivated by the neural network approach. I. Formulation of the Problem For non-destructive testing of aerospace structures (see, e.g., 3]{{7], 9], 11]), it is extremely important to know how the probability p(a) of detecting a fault of linear size a depends on this size a. This dependence is called a probability of detection (POD) curve. Recently, an empirical formula has been found which described this dependence 2], 7], 8]: p(a) = A a 1 + A a : (1) Since important decisions are based on this formula, it is desirable to nd out how reliable it is, i.e., whether it is a crude empirical approximation or a precise formula which has deep theoretical justiications. II. What We Are Planning to Do In this paper, we show that this formula (1) can indeed be theoretically justiied. Our justiication for this formula will use methods motivated by the neural network approach (see, e.g., 10]). III. We Must Choose a Family of Functions, Not a Single Function A. POD can be, in principle, experimentally determined For practical applications, we need the function p(a) which would determine the probability that if a sample with a fault size a is presented to a certain NDE technique, then this fault will be detected. In order to determine this function empirically, we must have a statistics of samples which were presented to this techniques and for which, later on, the fault was discovered; from this statistics, we can determine the desired probability. B. POD depends on the pre-selection procedure This probability, however, depends on how we select the samples presented to the NDE techniques. For example , most structures are inspected visually before using a more complicated NDE technology. Some aerospace structures are easier to inspect visually, so we can detect more faults visually, and only harder-than-usual faults are presented to the NDE technique; as a result of this pre-selection, for such structures, the success probability p(a) is lower than in other cases. Other structures are more diicult to inspect visually; for these structures, all the faults (including easy-to-detect ones) are presented to the NDE techniques, and the success …