*4.2. Calculation Example 2*

The purpose of the second calculation example is to show that the desired resolution of the image of changes in ultrasound wave velocity around localized damage can be obtained by using a reduced number of transmitting/receiving points. To this end, so-called fictitious transmitting/receiving points should be introduced for which propagation times will be interpolated on the basis of data from the original set of points (e.g., Reference [33,47]). In a real measurement situation, this may often result in a significant reduction of transmitting/receiving points, costs and time consumption of tests. The simulation of such a measurement strategy for the data of the first calculation example is shown below. The same geometry of the beam model and the method of its damage as in the first example were adopted, but the number of points for which the original information about wave propagation times is available was reduced in such a way that the gaps between them amount to Δ<sup>p</sup> = 10 cm. There are fictitious points every 6.25 mm between them at the same intervals as the points in the first example. For each group of paths, depending on the location of their transmitting/receiving points relative to each other (the opposite points, the points lying diagonally at an angle of 45◦ and 135◦ in relation to the axis *x*) times for paths connecting fictitious points were interpolated. The interpolation in the function of transmitting points position was carried out with the use of a cubic Hermite spline assuming continuity to the first derivative (*pchip* function in the MATLAB program environment was used). The results are shown in Figure 14, where interpolation nodes were marked with circles. They were also compared with the propagation times calculated with Dijkstra's algorithm in the first example. Figure 14 does not show the variability of times for the paths between the points lying diagonally to each other at an angle of 135◦ relative to the axis *x* because it is the same as for those lying at an angle of 45◦ taking into account the symmetry of the problem. In order to make the graphs easier to read, the interpolated times and those of the first example are presented in the form of continuous curves. Relative global interpolation errors are summarized in Table 4, i.e., *e*<sup>g</sup> = √( *<sup>I</sup> <sup>i</sup>*=<sup>1</sup> ( *t*path,*<sup>i</sup>* − *t*path int,*<sup>i</sup>* ) 2 / *<sup>I</sup> <sup>i</sup>*=<sup>1</sup> *t* 2 path,*i* ) was calculated where *t*path int,*<sup>i</sup>* (s) is the interpolated propagation time of the wave on the *i*-th path.

**Table 4.** Relative global interpolation errors of propagation times from the first example using a cubic Hermite spline in different phases of defect growth.


**Figure 14.** Times of wave propagation *t*path,*<sup>i</sup>* and *t*path int,*<sup>i</sup>* on the fastest paths between transmitting/receiving points in different phases of defect evolution: min(*E*D/*E*0) = (**a**) 0.9, (**b**) 0.8, (**c**) 0.6, and (**d**) 0.2. The results are presented as a function of the position of the transmitting points: a black line for the fastest paths determined by Dijkstra's algorithm (for Δ<sup>n</sup> = 3.125 mm); a grey line in case of interpolation with nodes marked with circles. The diagrams on the left refer to the paths connecting the opposite points and those on the right to the points lying diagonally at an angle of 45◦ to each other in relation to the axis *x*.

In the same way as in the first example, the propagation times for the fastest ultrasound wave paths connecting opposite transmitting/receiving points were modified to bring them as close as possible to the propagation times of straight rays, but with the use of times *t*path int,*i*, i.e.,:

$$t\_{\text{ray},i} \approx t\_{\text{ray}\,\text{approx},i} = t\_{\text{path\, int},i} + \beta \left( t\_{\text{path\, int},i} - \min \left( t\_{\text{path\, int},i} \right)\_{i=1,2,...,l} \right) \tag{28}$$

Then, using formulas (23)–(25), the values of βopt were obtained for data from Figure 14 as in Table 5. Illustratively, the variability of the wave propagation time *t*ray approx,*<sup>i</sup>* modified according to relation (28) between opposite transmitting/receiving points at β = βopt and min(*E*D/*E*0) = 0.2 is shown in Figure 15. At the same time, it was compared with *t*ray,*<sup>i</sup>* and *t*path,*i*, where *t*ray approx,*<sup>i</sup>* has also a much more similar course to that of *t*ray,*<sup>i</sup>* than *t*path,*i*.

**Table 5.** βopt calculated on the basis of data from Figure 14 in case of the opposite transmitting/receiving points in different phases of defect growth.


**Figure 15.** Propagation times *t*path,*i*, *t*ray,*<sup>i</sup>* and *t*ray approx,*<sup>i</sup>* between the opposite transmitting/receiving points for a defect with min(*E*D/*E*0) = 0.2. *t*ray approx,*<sup>i</sup>* were calculated according to relation (28) with β = βopt.

In order to assess the correctness of the proposed propagation time interpolation method, tomographic reconstructions of the wave velocity maps in the beam model longitudinal section from Figure 8 are presented in Figure 16. The reconstructions were determined by the randomized Kaczmarz method in accordance with the information presented in point 2 where *c*L0 was adopted as *c*L ref. The results are shown only for the central area of the beam section separated by a red dashed line through which all types of rays passed due to their inclination. For comparison purposes, reconstructions with the use of times *t*ray approx,*<sup>i</sup>* according to relation (28) and *t*path int,*<sup>i</sup>* were presented. For the individual reconstructions, their relative errors were also calculated: global—mean square *e*g, local—maximal *e*l max. The relative maximal difference in average speed over rays *d*cL ray max was also determined due to the use of *t*ray,*<sup>i</sup>* = *t*ray approx,*<sup>i</sup>* or *t*ray,*<sup>i</sup>* = *t*path int,*i*, i.e., in this particular case:

$$d\_{\text{cL\\_ray max}} = \max\left(1 - \frac{t\_{\text{path int},i}}{t\_{\text{ray approx},i}}\right)\_{i=1,2,\ldots,l} \tag{29}$$

A summary of *eg*, *e*l max and *d*cL ray max is presented in Table 6, depending on the calculation strategy adopted, where the exact value of *d*cL ray max is also shown for a comparison as in Table 3.

**Table 6.** Global—mean square (*e*g) and local—maximal (*e*l max) relative tomographic reconstruction error using the propagation times for the opposite points in Equation (3) *t*ray,*<sup>i</sup>* = *t*ray approx,*<sup>i</sup>* or *t*ray,*<sup>i</sup>* = *t*path int,*i*. Maximal relative difference in average wave velocities over the rays *d*cL ray max due to the use of *t*ray,*<sup>i</sup>* = *t*ray approx,*<sup>i</sup>* or *t*ray,*<sup>i</sup>* = *t*path int,*i*.


**Figure 16.** Tomographic reconstructions of the wave velocity distribution from Figure 13a according to Equation (3): (**a**) calculated with the propagation times *t*ray,*<sup>i</sup>* = *t*ray approx,*<sup>i</sup>* for the paths connecting the opposite points and *t*ray,*<sup>i</sup>* = *t*path int,*<sup>i</sup>* for the paths connecting the points diagonally, (**b**) calculated only with the propagation times *t*ray,*<sup>i</sup>* = *t*path int,*<sup>i</sup>* (red dotted lines—explanation in the text).

Analyzing the presented results one can draw general conclusions similarly to the reconstructions from the first example. In addition, it can be noted that reconstruction errors, compared to those in Table 3 for damaged zones with Young's modulus drop of more than 10%, were decreased by up to 2 times at most. Again, it can also be stated that, in the case of a defect in a concrete member with a higher degree of elastic degradation, the introduction into the vector **b** in Equation (3) of wave propagation times after appropriate scaling (Equation (28) with the optimal value of β) allows for effective, even several fold reduction of calculation errors and more correct evaluation of the defect shape. The obtained results also confirm that the use of so-called fictitious points with interpolated propagation times allows to increase the resolution of tomographic reconstructions of elastically degraded concrete areas without the need to use "too dense" system of real transmitting/receiving points. In this case, the value of *d*cL ray max also allows rough estimation of differences in reconstructed wave velocity maps, which can occur due to the adoption of the fastest propagation paths as straight in calculations. However, the use of fictitious points causes that it deviates from the exact values much more than in the first example. Nevertheless, it is important at this point that it starts to increase noticeably when Young's modulus in the defect drops above 20%, and, for example, such an estimation can be rationally increased by 2–3 times for safety reasons.
