A probability of detection model for a sensor-based monitoring method against local wall thinning

Abstract

In this study, a multivariable probability of detection (POD) model is developed to quantify the detection capability of a low frequency electromagnetic (LFEM) testing method with permanently installed sensors to monitor the thinning of pipe walls. Numerical simulations are utilized to predict the signal response by modeling LFEM testing against local wall thinning with different profiles. By probabilistically calibrating the simulated signal response with a limited number of experimental samples, signal response distributions due to various flaw profiles are inferred. Subsequently, a model is developed using Monte Carlo simulation to determine the distribution of the signal response affected by sensor placement and to calculate the POD. The resultant POD contours reflect the effect of multiple flaw parameters and sensor placement on the detection capability of the LFEM testing method.

Keywords

Electromagnetic non-destructive evaluation probability of detection sensor network

1. Introduction

Carbon steel pipes are used in various industries. However, they are prone to aging effects, such as pipe wall thinning (PWT), which is mainly due to flow-accelerated corrosion and liquid droplet impingement [1]. In a recent study, a low-frequency electromagnetic (LFEM) testing method using permanently installed sensors was proposed to monitor critical pipe segments and detect PWT [2,3]. This method enables more frequent inspections compared to conventional, manual, nondestructive testing (NDT) methods (such as ultrasonic testing).

In practical applications of the LFEM testing method, the signal response is unavoidably affected by various factors, leading to confusion over the presence or absence of flaws. Since the information on PWT provided by the LFEM testing method is leveraged for pipe reliability analysis, the detection capability of the method requires quantification.

The probability of detection (POD) represents the detection capability of an NDT method based on an assumption that the probability that a flaw is detected can be expressed as a function of the dimension of the flaw [4,5]. In contrast, a classical POD model assumes that the probability depends on a single flaw parameter, commonly denoted as “defect size”. An earlier study by the authors evaluated the applicability of such a classical POD model to the LFEM testing method targeting full-circumferential PWT, whose results confirmed that using only a single flaw parameter led to an irrational POD [6]. Then, in a previous study, the authors proposed a POD model that considered sensor placement and multiple flaw parameters to quantify the capability of LFEM testing methods [7]. However, the model considered only one-dimensional sensor placement and postulated uniform and full-circumferential thinning. As it is not always reasonable to assume that the actual PWT is circumferentially uniform, a more sophisticated POD model is necessary to quantify the detection capability of the LFEM testing method when monitoring local PWT. Therefore, this study aims to develop a new POD model that considers two-dimensional sensor placement and multiple flaw parameters.

2. LFEM testing method

Numerical simulations to model LFEM testing were performed to predict the signal response of various profiles of local wall thinning. The simulated signal response was calibrated using experimental data to infer the practical signal response distribution.

2.1. Numerical simulation

Figure 1 illustrates the cross-sections of a numerical model of the LFEM testing method for inspecting local PWT in a carbon steel pipe. The pipe model was built according to the nominal values of the geometric parameters, including the inner radii, lengths, and thicknesses of the pipe samples used in the experiments. An excitation coil surrounds the pipe to induce magnetic fields by carrying alternating currents. The local PWT was positioned at the midpoint of the pipe and shaped by subtracting one cylinder from the pipe. Moreover, the deepest position of a flaw is located at its circumferential center. Three parameters were used to characterize a flaw: axial length (l = 10–70 mm), circumferential angle (θ = 60°–120°), and residual thickness (t_r = 1.2–4.7 mm) at the deepest position. The number of flaws evaluated by the numerical simulations is 168. The axial component (z component) of the magnetic flux density was obtained along a two-dimensional surface (red dotted line) 0.75 mm above the outer surface of the pipe. Furthermore, given the symmetry of the model, it was cut in half to save computational time.

Fig. 1.

Schematic illustration of simulation model of LFEM testing of local PWT.

The simulations were implemented on COMSOL Multiphysics ver. 5.2 and its associated AC/DC module (Comsol Inc., USA). The Magnetic Fields interface was used. The excitation frequency was 1 Hz, and the resultant magnetomotive force of the coil was 20 Ampere-turn. Because the magnetic fields generated by the excitation coil is weak enough [3,10], the pipe was assumed to have a constant relative permeability of 160. The electrical conductivity and relative permittivity were 5.2 × 10⁶ S/m and 1, respectively. The coil was discretized using hexahedral elements; other domains, including the air, were discretized using tetrahedral elements. The boundary condition used in the simulations is n × A = 0 on the symmetric plane cutting the pipe into half as illustrated in Fig. 1 and outmost surfaces 1 m away from the center of the pipe, where n and A denote a unit normal vector and magnetic vector potential, respectively.

It should be noted that the purpose of the numerical simulations in this study is not to quantitatively reproduce signals measured by the experiment but to roughly evaluate the general dependence of the signal due to a flaw on the dimension of the flaw as explained in 3.1. Signals measured by the experiment are used to evaluate the scattering of signals caused by various factors that cannot be modeled by numerical simulations [8,9]. This indicates that the ranges of flaw parameters in the numerical simulations need to be wide enough to cover the domain where POD is presented, because an interpolation is more preferable than an extrapolation in general. For this reason, the ranges of the flaw parameters in the numerical simulations are wider than those of grooves used in the experiments in 2.2.

2.2. Experiment

Local grooves with the same profiles as those designed in the numerical model were machined on the inner surface of carbon steel pipes (STPG 370). In total, 27 pipe samples with grooves of different axial lengths (l = 30–70 mm), circumferential angles (θ = 60°–120°), and residual thicknesses (t_r = 1.8–4.1 mm) were prepared. In addition, two flawless pipe samples were used to collect noise. Figure 2 shows a simplified diagram of the experimental system for LFEM testing. Function generator 1 (WF1973; NF Corporation, Yokohama, Japan) provided a voltage of 0.9 V_p−p at 1 Hz, which was subsequently amplified 10× using a power amplifier (HSA4104; NF Corporation, Yokohama, Japan) to an excitation coil wound around the pipe. The excitation coil had 20 turns and was fabricated using copper wire with a diameter of 1 mm. The current flowing through each turn (0.9–1.0 A) was indirectly monitored with lock-in amplifier 1 (LI5640; NF Corporation, Yokohama, Japan) by measuring the voltage across the shunt resistor of 1 Ω. A magneto-impedance (MI) sensor (MI-CB-1DM (A type); Aichi Micro Intelligent Corporation, Tokai, Japan) was employed to measure the signal. The sensitivity of the sensor is approximately 4.0 mV/μT up to 300 μT and 10 kHz, according to the information offered by the manufacturer. In one measurement, the sensor was mounted on a customized support (fabricated using a 3D printer) to measure the axial (z) component of the magnetic flux density with a constant lift-off of 0.75 mm. The sensor was activated by a 5 V DC voltage (provided by the DC power) and square wave AC characterized by a frequency of 2,000,000/3 Hz, an amplitude of 5 V_p−p, and an offset of 2.5 V, which was produced by function generator 2 (WF 1974; NF Corporation, Yokohama, Japan). The sensors were kept in position for 5 s after each movement to stabilize the signals. The z component (as measured by the sensor) was filtered by lock-in amplifier 2 (LI5640; NF Corporation, Yokohama, Japan), given the reference frequency. Finally, the output voltages of each MI sensor, together with the excitation current, were recorded by a PC through an A/D converter (TUSB-0216ADMZ; Turtle Industry Co. Ltd.) and the attached software.

The measurements were performed on a grid (−50 to +50 mm in the axial direction and −60° to +60° in the circumferential direction) centered on the grooves. Because the excitation coil was situated 100 mm away from the center of the groove, the distance of the scanning area shown in the figure from the excitation coil is from 50 to 150 mm. The reason why signals in this area, somewhat away from the exciter coil, were considered is that an earlier study by the authors [3] confirmed that signals in this area can be regarded as almost independent from the distance from the exciter if they are properly normalized. The pitches between the grid nodes were 2 mm and 10° in the axial and circumferential directions, respectively. As a result, a 13 × 51 matrix of signal responses was obtained for each pipe sample.

Fig. 2.

Diagram of experimental LFEM testing system.

2.3. Signal response analysis

The signal responses (z component of the magnetic flux density) obtained in the numerical simulations or experiments were normalized as B = (B_flawed∕I)∕(B_flawless∕I). Here, B_flawed denotes the signal amplitude acquired from a flawed pipe; B_flawless denotes the signal amplitude acquired from a flawless pipe; and I is the current flowing in the excitation coil. It has been confirmed that normalization can enhance signal features due to PWT [3]. When normalizing the experimental signal response, B_flawless was calculated as the average of all axial line measurements (columns of the metrices) obtained from the two flawless pipe samples, to reduce any effects of variations on normalization. Hereafter, B^exp and B^sim are employed to represent the normalized signal response generated by the experiments and numerical simulations, respectively. Figure 3 illustrates B^exp and B^sim, where the bright region at the center indicates the area of local wall thinning. The sensor’s sensitive area is smaller than the pixel size of the figure. By comparing the plots, general agreement between the profiles of B^exp and B^sim can be observed, although there are discrepancies in the magnitudes. In fact, in addition to the variables explicitly considered, the experimental signal response was unavoidably affected by several factors, including changes in wall thickness (varying within the manufacturing tolerance), inhomogeneity of the magnetic permeability of the pipe material, misalignment of the magnetic sensors, and seamless pipe noise [10 ]. In addition, some factors, such as the skill of an inspector and instrumental noise, cannot be quantitatively considered in numerical simulations although they often have a large influence on measured signals. Therefore, an appropriate method for quantifying such discrepancies is required.

Fig. 3.

Demonstration of normalized signal response obtained from (a) numerical simulation and (b) the experiment when flaw parameters are: l = 30 mm, θ = 60°, t_r = 2.1 mm.

3. Probability of detection analysis

3.1. Calibration model

As mentioned previously, to leverage simulation data to perform POD analysis for evaluating detection capability, it is necessary to calibrate B^sim by evaluating the discrepancy between B^sim and B^exp. Thus, the calibration model proposed in the previous study was reshaped to represent the relationship between $B_{i,j}^{\exp }$ and $B_{i,j}^{\text{sim}}$ obtained at the jth location relative to the center of the ith local wall thinning: $\begin{eqnarray}\displaystyle B_{i,j}^{\exp }=c_{1}B_{i,j}^{\text{sim}}+c_{2}+e, & & \displaystyle\end{eqnarray}$ (1) where c₁ and c₂ are calibration coefficients employed to represent the bias of the discrepancy; e denotes a random term and was assumed to follow a normal distribution, $e\sim N(0,{\sigma}_{e}^{2})$ , to represent the stochastic characteristic of the discrepancy. The standard deviation of the error (σ_e) was assumed to satisfy $\begin{eqnarray}\displaystyle {\sigma}_{e}=c_{3}B_{i,j}^{\text{sim}}+c_{4}, & & \displaystyle\end{eqnarray}$ (2) where c₃ and c₄ are coefficients used to correlate the scatter of $B_{i,j}^{\exp }$ with $B_{i,j}^{\text{sim}}$ . Consequently, $B_{i,j}^{\exp }$ follows a normal distribution, whose mean (μ_{i, j}) and standard deviation (σ_{i, j}) are functions of $B_{i,j}^{\text{sim}}$ . The coefficients (c₁, c₂, c₃, and c₄) are unknown and postulated as being independent of the flaw parameters. They were estimated using the maximum likelihood method with the particle swarm optimization algorithm, and their confidence intervals were constructed by employing the bootstrap method, which was implemented on MATALB R2020b.

The data measured from pipe samples (n = 27) in the experiments were stored as 13 × 51 matrices. As a result, 17901 observations (27 × 13 × 51) of experimental or simulated signal responses were used for the estimation. The confidence intervals of the estimates of the parameters were determined with 1000 bootstrap samples. The estimation results and corresponding 95% confidence intervals are summarized in Table 1. All the confidence intervals were narrow, indicating the high reliability of the estimation results. It should be noted that whether or not the coefficients are single-valued does not affect the validity of the method this study proposes. It is probable that there are other combinations of the coefficient that are as “good” as the ones shown in the table. However, the purpose of the estimation is not to obtain “correct” values of the coefficients but to obtain a model that fits the experimental data well.

Table 1

Estimation results of the coefficients

Coefficient	Estimated value	95% confidence interval
c ₁	0.863	[0.852, 0.875]
c ₂	0.116	[0.102, 0.148]
c ₃	−0.027	[−0.030, −0.025]
c ₄	0.072	[0.070, 0.073]

Fig. 4.

Distribution of rescaled experimental signal response with means and standard deviations calculated by the estimated values of c₁, c₂, c₃, and c₄.

The validity of the calibration dominated the effectiveness of the approximation of practical signal response (based on the calibrated simulated signal response). To validate the calibration results, the idea of standardization (z-score normalization) was leveraged. If the experimental signal response (B^exp) follows a normal distribution before considering the effect of random sensor location, and assuming the distribution coefficients (c₁, c₂, c₃, and c₄) were correctly estimated, the means (μ_{i, j}) and standard deviations (σ_{i, j}) should be able to rescale $B_{i,j}^{\exp }$ to ensure that $B_{i,j}^{\exp }$ follows a standard normal distribution. To confirm this, the 17901 observations of $B_{i,j}^{\exp }$ were rescaled using the equation of standardization (with the resultant μ_{i, j} and σ_{i, j}) and compared with the standard normal distribution in Fig. 4. The figure confirms that the distribution of the rescaled signals was very close to the standard normal distribution, which validated the estimated values of the distribution parameters.

3.2. Determination of the POD

In practical inspections, a positive indication of a flaw occurs when the signal response exceeds a predefined decision threshold. Accordingly, the capability of the LFEM testing method to detect flaws was quantified by evaluating the probability that the signal response became larger than the decision threshold (the POD). As the signal response of the LFEM testing is continuous, the distribution of the signal response is described by a probability density function, which is conditional on the variables of concern. Therefore, the POD can be calculated as follows: $\begin{eqnarray}\displaystyle \text{POD}(\boldsymbol{\boldsymbol{\it\tau}})=\int _{B_{th}}^{\infty }g(B|\boldsymbol{\boldsymbol{\it\tau}})dB, & & \displaystyle\end{eqnarray}$ (3) where B denotes the signal response (the z component of magnetic flux density), 𝝉 denotes the vector of influential variables of B, g denotes the probability density function of B (conditional on 𝝉), and B_th denotes the decision threshold.

Obviously, a probability density function that can reasonably represent the signal response distribution is essential to ensure the accuracy of the calculated POD. As shown in Fig. 1, the sensor placement for LFEM testing was defined by the spacings in the two directions s_a and s_c. The signal response varied according to sensor locations x_a and x_c, which were random and were supposed to follow the uniform distributions of x_a ∼ U (−s_a∕2, s_a∕2) and x_c ∼ U (−s_c∕2, s_c∕2). Therefore, in this study, 𝝉 contains all significant variables, including the flaw parameters and sensor spacings, 𝝉 = (l, t_r, θ, s_a, s_c). However, it is difficult to deduce the explicit probability density function of the multiple variables. Hence, a method based on Monte Carlo simulation was developed to determine the POD. The procedure of the developed method is as follows:

Define the input parameters, including flaw sizes (l, t_r, and θ) and sensor spacings (s_a and s_c);

Sample the values of the sensor location, x_{a, k} ∼ U (−s_a∕2, s_a∕2) and x_{c, k} ∼ U (−s_c∕2, s_c∕2), and the random error, $e_{k}\sim N(0,{\sigma}_{e}^{2})$ . This is based on the estimated values of c₃ and c₄, where k = 1, …, N, and N is the total number of simulations;

Obtain the simulated signal response ( $B_{k}^{\text{sim}}$ ) by numerical simulation according to l, t_r, θ, x_{a, k}, and x_{c, k};

Acquire the experimental signal response ( $B_{k}^{\exp }$ ), which is not measured experimentally. Rather, it is calculated according to Eq. (1) with $B_{k}^{\text{sim}}$ and e_k obtained from (2)–(3);

If $B_{k}^{\exp }$ is larger than the decision threshold (B_th), a unit is added to the number of detection (n_d);

Repeat (3)–(5) N times, and the POD is calculated as the ratio of n_d to N.

The decision threshold was defined based on the noise distribution to calculate the POD. The noise was obtained by normalizing one axial line measurement with the other matrices obtained by scanning the two flawless pipe samples. As a result, 1326 observations of noise were generated to analyze the empirical noise distribution. The 90% quantile of the empirical noise distribution was selected as the decision threshold (B_th = 1.15). Hence, the corresponding false positive rate was 10%.

The sensor spacings were initially assumed to be s_a = 50 mm and s_c = 90°. Subsequently, 1,000,000 Monte Carlo simulations (N = 1,000,000) were performed to determine the POD for each flaw profile, which took tens of seconds. To display the change in POD with the three flaw sizes lucidly, 2D contours were adopted (Fig. 5). The LFEM testing method achieved greater detection capability for local wall thinning with a larger circumferential angle (θ) or smaller residual thickness (larger depth) (t_r). In addition, the POD increased with axial length (l) until it reached 50 mm (=s_a) and decreased as l further increased. This is attributed to the axial span of the uplift area of the signal response of LFEM testing being proportional to l [11]. Moreover, the probability that a flaw is detected by the separated sensors significantly depends on the axial span when l < s_a. Further, when l > s_a, POD decreases with l due to the decreased magnitude of the signal response. Therefore, the POD contours were confirmed to correctly reflect the effect of flaw sizes on the detection capability of LFEM testing.

The ability of the new multivariable model to evaluate the effect of sensor placement on the detection capability of LFEM testing was also examined by individually altering s_a and s_c. Figure 6 demonstrates changes in the POD according to senor placement. First, s_c was changed from 90° to 60°, and the resultant POD are presented in Fig. 6(a). Compared with the results shown in Fig. 5, the reduction of s_c led to an overall increase in the POD. Then, s_a was changed from 50 to 30 mm, and the corresponding POD contours shown in Fig. 6(b) indicates that the turning points becomes 30 mm, and the POD increased compared with the results shown in Fig. 5, especially for flaws with smaller axial lengths.

It should be noted that the contour lines in the figures are obviously dependent on all the parameters: residual thickness, axial length and circumferential angles of PWT, and sensor spacings. This indicates that using a classical POD model, which represents POD as a single parameter named “defect size”, leads to an improper evaluation, and supports the validity of the POD model proposed in this study.

Fig. 5.

2D contours showing the change of POD with flaw sizes when s_a = 50 mm and s_c = 90°.

Fig. 6.

2D contours showing the change in POD when θ = 60°, (a) s_a = 50 mm and s_c = 60°, (b) s_a = 30 mm and s_c = 90°.

4. Conclusion

In this study, a new multivariable model of POD determination is proposed to quantify the detection capability of an LFEM testing method that uses sensors to monitor local wall thinning in carbon steel pipes. The signal response distribution due to various flaw profiles and random sensor locations can be properly inferred based on simulation data by calibrating the discrepancy between the experimental and simulated signal responses and by employing Monte Carlo simulations. The POD determined by the new multivariable model is able to accurately reflect the joint effect of 3D profiles of local wall thinning and 2D sensor placement on the detection capability of LFEM testing of local wall thinning. The application of the POD determination model is not limited to sensor-based monitoring, as it can also be used for ultrasonic testing, which is commonly utilized to gauge thickness at grid nodes. In addition, by incorporating the model to comprehensively quantify the detection capability of an NDT method, a more reasonable reliability analysis is expected for pipes suffering from wall thinning. It should be emphasized that the parameters considered by the POD model developed in this study are categorized into two groups: those about flaw size and those about sensor spacings. The main contribution of this study, namely how to quantify the uncertainty caused by sensor spacings, is independent from how to represent a flaw. Therefore, it would be probable that the procedure explained in 3.2 is basically applicable to other flaw types such as cracking, while how to represent a flaw profile, which is dependent on the type of the flaw is an important issue in making a proper POD model.

Footnotes

Acknowledgement

This work was supported by JSPS Grant-in-Aid for JSPS Fellows (JP20J11957).

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