## 3 Results and Discussions  
After finding optimum positions for the detectors, orientation angle of \(0^{\circ}\) for the first detector and \(7^{\circ}\) for the second detector, counts under photo peaks of \({}^{241}\)Am and \({}^{133}\)Ba radioisotopes were recorded in both detectors for different scale layer thicknesses and volume fractions. Ternary contour plots of the recorded counts in both detectors for different volume fractions when the scale thickness is 0 and 3 cm, are shown in Figures 7-10. Comparing Figures 7 and 8 that correspond to the recorded counts in the first detector for \({}^{241}\)Am and \({}^{133}\)Ba radioisotopes, respectively, it can be said that dynamic range of registered counts relative to changes of gas volume fraction, or, better to say, sensitivity, for \({}^{133}\)Ba is more than \({}^{241}\)Am. A same response is also observed for the second detector. Comparing Figure 7a,b, it could be observed that, when scale layer is 0, sensitivity of detector relative to changes of gas, oil, and water components is much more than when the scale thickness is 3. In other words, by increasing thickness of scale layer, somehow information about the flow of inside the pipe starts fading. This manner can be also seen for both detectors and radioisotopes. Comparing Figures 7a and 9a, it can be observed that sensitivity of the second detector relative to changes of volume fractions is a little bit more than the first detector. Regression diagrams of actual data and predicted data using presented MLP-LM models are shown in Figures 11 and 12. In Table 1, data number, scale layer thickness, actual outputs, and measured outputs for test data set are tabulated. Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) of presented metering system were calculated using Equations (5) and (6). \[\text{MAE}=\frac{1}{\text{N}}\sum_{i=1}^{\text{Z}}\lvert X_{\text{i}}(\text{ Actual})-X_{\text{i}}(\text{Measured})\rvert \tag{5}\]  
\[\text{RMSE}=\left[\frac{\sum_{i=1}^{\text{N}}((X_{\text{i}}(\text{Actual})-X _{\text{i}}(\text{Measured}))^{2}}{\text{N}}\right]^{0.5} \tag{6}\]  
Figure 7: Recorded counts under photo peaks of \({}^{241}\)Am in the first detector versus gas, oil, and water volume fraction: (a) scale thickness is 0, (b) scale thickness is 3 cm. where N, Xi (Actual), and Xi (Measured) are the data number, real values, and estimated values, respectively. Performance criteria of the developed models were tabulated in Table 2 using Equations (5) and (6). It can be found from the obtained errors that the presented gauging system is reliable. This novel meter could be used in different industries for metering volume fraction of each phase independent of scale layer thickness. Radioisotope sources, detectors type, detectors position, extracted features from output signals, used data analysis algorithms and ANN architectures were selected appropriately in order to achieve the optimum performance for the proposed system. Figure 8: Recorded counts under photo peaks of \({}^{133}\)Ba in the first detector versus gas, oil, and water volume fraction: (**a**) scale thickness is 0, (**b**) scale thickness is 3 cm. Figure 9: Recorded counts under photo peaks of \({}^{241}\)Am in the second detector versus gas, oil, and water volume fraction: (**a**) scale thickness is 0, (**b**) scale thickness is 3 cm. Figure 11: Regression diagrams of first model results (gas volume fraction) for (**a**) train data (**b**) test data. Figure 12: Regression diagrams of second model results (oil volume fraction) for (**a**) train data (**b**) test data. Figure 10: Recorded counts under photo peaks of \({}^{133}\)Ba in the second detector versus gas, oil, and water volume fraction: (**a**) scale thickness is 0, (**b**) scale thickness is 3 cm.