There are two distinct strategies for quantifying NDI capability for damage tolerance analyses.  These are: a) estimating POD(a) as a function of crack size and b) demonstrating capability for a fixed crack size.  To estimate a POD(a) function, the structural details to be inspected would comprise a range of crack sizes in the expected domain of increasing POD.  A parametric equation is assumed for the POD(a) function, the parameters of the equation are estimated from the inspection results, and the statistical properties of the estimates are used to place a confidence limit on the selected detection probability.  To demonstrate capability for a fixed crack size, only cracks of the size of interest are inspected.  The proportion of the cracks that are detected is the estimate of POD (for cracks of that size) and binomial theory is used to place a lower confidence bound on the detection probability.  Because of the greater utility of the POD(a) function, the approach based on estimating the entire function is preferred by many, including the Air Force [MIL-HDBK-1823].  The fixed crack size approach is used by NASA for qualifying the inspection capability of vendors [Salkowski, 1993].  It might be noted that a binomial approach to estimating POD as a function of crack size was extensively considered in the 1970’s, but later abandoned.  Very large numbers of cracked specimens were needed to ensure an adequate sample size within reasonably small intervals of crack size.

The analysis of data for demonstrating capability at a fixed crack size using the binomial approach will be discussed, but the major thrust of the capability evaluation is focused on estimating the POD(a) function.  Similar considerations apply to the planning of both types of capability demonstrations.

Inspection results are recorded in two distinct formats and the format determines the analysis method to be used in modeling the POD(a) function.  When the results of an inspection are expressed only in terms of whether or not a crack was detected, the data are known as find/no find, hit/miss, or pass/fail data.  Such dichotomous inspection results are represented by the data pair (a_i, Z_i) where a_i is the size of the i^th crack and Z_i represents the outcome of the inspection of the i^th crack: Z_i = 1 for the crack being found (hit or pass) and Z_i = 0 for the crack not being found (miss or fail).  Examples of such data would be the results of visual, magnetic particle, or fluorescent penetrant inspections or any inspection for which the magnitude of the response to the inspection stimulant was not recorded.  POD(a) analysis for data of this nature is often called hit/miss or pass/fail analysis.  Maximum likelihood estimates of the parameters of the POD(a) model are obtained from the (a_i, Z_i) data.  Asymptotic properties of the maximum likelihood estimates are used to calculate the confidence bound on the estimate of the reliably detected crack size.

When the results of the inspection are based on the quantified magnitude of a response to the inspection stimulus and the response is recorded, the POD(a) function can be estimated from the statistical scatter in the response magnitudes as a function of crack size.  The data pair comprising size and signal response are designated as (a_i, â_i) in which â_i is the response to the inspection stimulus for the i^th crack.  If â_i is greater than a pre-set threshold, â_th, a crack is indicated.  Data of this nature are often referred to as â vs a  (a-hat vs a).  Data from automated eddy current systems are of this nature.  Data from ultrasonic and liquid penetrant inspections have also been recorded and analyzed in the â vs a format.  The parameters of the POD(a) function are estimated from the scatter in â values about the mean response to cracks of size a.  Maximum likelihood is used to estimate the parameters and to place confidence bounds on the estimate of the reliably detected crack size when desired [MIL-HDBK-1823; Berens, 1988].

The demonstration of NDI capability is a consumer or quality concern.  The primary objective of such demonstrations for a particular application is to estimate the POD(a) function and, consequently, the reliably detected crack size, say a_NDI.  For damage tolerance considerations, a_NDI is commonly accepted to be the crack sizes designated as a₉₀ or a_90/95.  The a₉₀ crack size is defined as the size for which POD(a₉₀) = 0.90 and a_90/95 is the upper (conservative) 95% confidence bound on the estimate of a₉₀.  (The estimate of the a₉₀ crack size is often referred to as the a_90/50 crack size under the wrong assumption that the estimate of a₉₀ is the median of the sampling distribution of the estimates.)

NDI reliability experiments have also been conducted to optimize the inspection protocol and to ensure process control.  System optimization with respect to POD(a) would have the objective of determining system configurations that produce acceptable a₉₀ or a_90/95 values.  The design of system optimization programs is of a different character and beyond the scope of demonstrating the capability of the system.