# Simulation of Real Defect Geometry and Its Detection Using Passive Magnetic Inspection (PMI) Method
## 4 Discussion  
Assuming that the magnetic flux densities of different locations on the rebar are independent of one another, the probability graph method was used for fitting the magnetic flux values to a probability distribution. This method involves equating the empirical distribution of magnetic data (\(P_{i}\)) with the chosen cumulative distribution function (CDF). Next, the CDF function is written in linear form, which can be expressed as Equation (10) for Gamma CDF (GAMMADIS). The linearity is then used as a basis for determining whether the data can be modeled by a particular distribution. Additionally, the goodness-of-fit of the model is given by the coefficient of determination \(R^{2}\). \[P_{i}=GAMMADIS(magnetic\ data)\to magnetic\ data=GAMMADIS_{inverse}(P_{i}) \tag{10}\]  
The magnetic flux density data were plotted against various probability distributions (normal, log-normal, Weibull, and gamma distributions); a gamma distribution was chosen based on the minimum least-squared error (Figure 20). This distribution is based on a function of two parameters: \(\alpha\) and \(\beta\) (Equation (11)); these were calculated by the mean and standard deviations (SD), which are 87.8 \(\upmu\)T and 25.6 \(\upmu\)T, respectively. As observed in Figure 21, the gamma function correlates well with the histogram frequency of data, and this approximation may be useful for estimation in practical cases. \[f(x)=\frac{1}{\beta^{\alpha}G(\alpha)}\ x^{\alpha-1}e^{-\frac{x}{\beta}} \tag{11}\]  
According to Figure 18, a Z-component magnetic flux density of less than 76 \(\upmu\)T (without considering the edge effect and background magnetic field) corresponds to the location of Hole 2. Importing this value into the obtained CDF shows that 0.76% of the data is related to the defective locations. In other words, 0.76% of the rebar surface (at the scanned section) can be considered imperfect. This result can  
Figure 20: Probability plot for investigating the correlation of data with a gamma distribution. be verified by the Monte Carlo simulation method (based on inverse values of the obtained gamma distribution function). Figure 22 presents the probability of defects considering the mean, SD, and limit state, showing that the probability of defectiveness fluctuates until the first 300 trails are completed, and then stabilizes at the value of \(\sim\)0.75%. For our statistical investigations, we considered the magnetic data as independent variables. Those independent variables were described by the chosen probability distribution with its particular distribution parameters, knowing that distribution allowed us to estimate an interval over which the  
Figure 21: Histogram frequency of data in conjunction with gamma distribution probability density. Figure 22: Defectiveness probability for the inspected rebar based on the Monte Carlo simulation method. unknown future values may lie (with a stipulated level of confidence). Using the CDF of the gamma distribution, about 98% of all of the data are from 76 uT to 100 uT (Equation (12)). Hence, regarding the recorded magnetic data of the rebar, it can be predicted with 98% confidence that if the rebar was longer (by how much is irrelevant), the next values indicating a flawless rebar would be somewhere between 76-100 uT. Values outside this range should be reviewed as suspected defect locations. \[GAMMADIS(100\ \mu\mathrm{T})-GAMMADIS(76\ \mu\mathrm{T})=0.99-0.01=0.98 \tag{12}\]  
Figure 23 shows the values of different components of magnetic flux densities, 16 mm from the center of the rebar. For better observation of the irregularities related to Hole 2, all of the graphs are presented from 200 mm to 370 mm at the appropriate scale. The anomaly in the magnetic values at the location of Hole 2 can be detected in all of the components of the magnetic flux densities. Figure 23 indicates that the values of the X and Y-components of magnetic flux density still include some noise that is attributable to the mesh specifications.