*5.1. Numerical Simulation*

To evaluate the localization method described, the reconstruction of the location of a Gaussian pulse source of 50 μs is simulated from the reception of four sensors located on the lateral surface of different coordinates. Figure 6 shows the position of the sensors and the source in the space for the simulated model [6]. In this simulation, the sources are always into the volume covered by the coordinates of the sensors. Despite it being possible to locate the source with three sensors, the use of at least four sensors is incorporated to both improve the reliability and quality of the results.


**Table 1.** Positions of the sensors and the source in the simulations.

**Figure 6.** Volume proposed to evaluate the localization algorithm. In this case, four sensors (black points) have been situated on the sides of the cube. Inside, three events will be simulated in different positions. The positions of the sensors and sources are shown in Table 1. This figure shows the point source (blue point).

To evaluate the algorithm, the volume of the cube has been modified between 27.0 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>m</sup><sup>3</sup> and 512.0 <sup>×</sup> 10−<sup>3</sup> <sup>m</sup>3. Positions of the sensors are shown in Table 1, where *<sup>H</sup>* represents the length of the cube, whose values were 200, 300, 400, 500, and 600 mm [6].

Table 2 shows the deviation of the position of the simulated source with respect to the real position of the source. The results can be expressed as a function of the distance between the reconstructed position and the real position of the source. Figure 7 shows the results, where the abscissa axis shows the volume in m3, while the ordinates axis shows the distance in mm between the prediction of the algorithm and the real position. In addition, a fitting line to the results is shown.


**Table 2.** Real and estimated positions of three different sources depending on the volume case.

**Figure 7.** The positions of sources 1, 2, and 3 are shown in red, blue, and black, respectively. The dotted line is a fitting line to the results.

These reconstructed positions do not exceed 5 mm of the real position for all the studied cases. In addition, it is important to calculate the algorithm for a future application in real time. For this reason, Figure 8 shows the calculation times for 4, 6, 8, 10, and 12 sensors for a source position, depending on changes in volume.

**Figure 8.** Calculation time with an Intel i5 processor for different sensors and volume.

Once satisfactory results of the localization algorithm have been obtained by simulating different known source points in known sensor positions, the localization method has been evaluated experimentally in the next section.

### *5.2. Experimental Localization with Thermoacoustic Signals*

To validate the localization method, 12 reception positions (sensors) have been configured for a single source position. The signal emitted is the pulse obtained by the simulation of the thermoacoustic model, given by Equation (15). Table 3 shows the positions of the measuring points, which have all been referenced to the lower corner of the tank.


**Table 3.** Position of the source and sensor reception points.

Two inputs channels were used. Channel 1 recorded the emitted signal passed to the amplification system while channel 2 recorded the received signal of the emitter Reson TC4014. Both signals were used afterward for correlation. The receiver is fixed to the MOCO programmed axis that moves the hydrophone to each of the measurement positions from position 1 to position 12 in the direction shown in Figure 9. By doing this, we have the same information as when we using a 12-element sensor array.

**Figure 9.** This figure shows the configuration of the receiver positions. The source is represented by a sphere (white) that is inside the volume generated by the sensors (cylinders of grey color with black tip) whose route is shown with the black lines and the conical red marks. In this figure, a smaller volume is represented inside the tank for its correct visualization.

The analysis of the measurements has taken into account the calibration of the measurement of the speed of sound 1492 m/s. This correction allows for a better detection precision in the TOA. In addition, the speed of the localization algorithm and the accuracy of the results will be evaluated, according to the number of reception points. Figure 10 shows an example of the signal emitted and the signal received as well as the correlation between them in order to obtain the TOA as the time of maximum amplitude of the correlation [9].

**Figure 10.** The figure shows the process of detecting the signal from the correlation between the emitted and received signal. (**a**) The signal generated in the simulation, shown in black, is emitted by the transmitter. The captured signal for point 1 is shown in red; (**b**) The arrival time can be extracted from the maximum value of the correlation of signals.

Table 4 shows the estimated location of the source (mean and standard deviation) for different groups of sensors labelled, according to Figure 9 and Table 3. From Table 4, we observe that the results obtained with this configuration are satisfactory and below 1 mm. Furthermore, similar accurate results are also possible with fewer sensors such as four sensors surrounding the source and covering different directions isotropically. In cases that sensors are only in a specific region, larger offsets (several mm) may appear in specific coordinates.


**Table 4.** Reconstructed position for different groups of sensors.
