*4.5. Quantitative Analysis Using Error Coefficient*

The accuracy of the tomographic reconstruction was quantified by using an error coefficient comparing the reference image with the obtained tomographic maps. Each of the prepared tomographic maps was replaced by a simplified model with a reduced spectrum, indicating high and low velocities as well as large and small differences. In the case of direct measurement, pixels with velocities below the 25th percentile of the maximum scale value are considered to be low-velocity indications and are assigned a value of 0, while others are assigned a value of 1. An example of spectrum reduction for direct measurement is shown in Figure 17. In the case of maps based on signal differences for an undamaged structure and a damaged structure, pixels with a difference above the 75th percentile of differences in a scale are assigned a value of 0, and the remaining pixels are assigned a value of 1.

**Figure 17.** Error coefficient values for maps with reduced spectrum based on direct travel time measurement method: (**a**) reference model with a reduced spectrum, (**b**) topographic map with a full spectrum, (**c**) tomographic map with a reduced spectrum.

Figures 18 and 19 present maps with the reduced spectrum for direct measurement and for the travel time difference method, respectively. The first column indicates the reference model for which the defects have a value of 0, while the remaining area has a value of 1. The next two columns show the experimental results for straight and curved rays in the element divided into 64 pixels. Maps from numerical tests are summarized in columns 4–6 for straight and curved rays and for the element divided into 225 pixels. Below each map, the value of the error coefficient calculated by the following equation is shown:

$$\gamma = \text{mean}\left( \left| I^j - I^j\_{ref} \right| \times 100\% \right), \tag{5}$$

where: *I<sup>j</sup>* is the value in i-th pixel of the considered ultrasound tomography map, and *I<sup>j</sup> ref* denotes the value in the i-th pixel of the reference ultrasound tomography map.

**Figure 18.** Error coefficient values for maps with reduced spectrum based on direct travel time measurement method.

**Figure 19.** Error coefficient values for maps with reduced spectrum based on the travel time difference method.

In the case of defects with diameters of 2 and 5 cm, the error coefficient clearly indicates the improvement of the image quality, in the case of using the hybrid ray-tracing method and the densification of the pixel grid. The use of the hybrid method for experimental data improved the possibility of estimating the size of the defect with a diameter of 2 cm from 15.34 to 1.84% and 7.39 to 3.84%, respectively, for the direct measurement and for the comparative measurement of the damaged and undamaged model signals. For damage with a diameter of 5 cm, the improvement was from 7.28 to 2.10% and 1.70 to 1.37%. In the case of numerical tests, the error coefficient was calculated for the defect with a diameter of 2 cm as 5.25 and 3.83%, respectively using straight and curved radii in the direct measurement. At the same time, the value of the coefficient decreased to 2.50% in the case of the pixel grid density. The error coefficient value for the same measurements, but from the comparative method, was 4.09, 2.59 and 0.66%. In the case of a defect with a diameter of 5 cm, the use of the hybrid method improved the coefficient value from 11.17 to 3.85%. The error coefficient, when dividing the element into 225 pixels, was 1.60% for the direct measurement. The error coefficient for the comparative measurement and the 5 cm diameter defect is 2.35, 1.49 and 1.42% for straight wave paths, for curved paths and for a dense pixel grid, respectively. In the case of damage with a diameter of 10 cm, the value of the error coefficient slightly decreases. This is due to the accumulation of low-speed values in the case of applying both proposed methods to improve the image quality.

#### **5. Conclusions**

In tests carried out on steel plates, the use of ultrasound tomography to locate surface damage was assessed. Laboratory and numerical tests were carried out on four plates: one intact and three with surface damage of varying intensities. The performed ultrasound tomography was based on the reconstruction of the Lamb wave propagation velocity. Moreover, analyses were performed utilizing signal differences between the reference and defective plates. The conducted research allowed for the formulation of the following conclusions:

• Surface defects in the form of a circular hole were visualized effectively on tomograms as areas with reduced wave propagation velocity using both the TOF for the current state and the difference of the TOF between the current and reference state;


Lamb waves and their processing by the technique of ultrasound tomography proved to be an effective technique for imaging defects in thin plates. The presented approach is suitable for diagnosing defects in elements of real metal structures. Assessing the occurrence of damage can be particularly useful for monitoring plate structures for which the reference state is known, and the SHM system is designed to detect emerging and developing surface discontinuities.

**Author Contributions:** Conceptualization, M.Z. and M.R.; formal analysis, M.Z. and M.R.; funding acquisition, M.Z.; investigation, M.Z.; methodology, M.Z. and M.R.; software, M.Z.; supervision, M.R.; validation, M.R.; visualization, M.Z.; writing—original draft, M.Z.; writing—review and editing, M.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research work was carried out within project No. 2017/27/N/ST8/02399, financed by the National Science Centre, Poland.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** Abaqus calculations were carried out at the Centre of Informatics Tricity Academic Supercomputer & Network.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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