ISSN 1061-8309, Russian Journal of Nondestructive Testing, 2016, Vol. 52, No. 9, pp. 520–527. © Pleiades Publishing, Ltd., 2016. Original Russian Text © M.S. Sudakova, A.Yu. Kalashnikov, E.B. Terent’eva, 2016, published in Defektoskopiya, 2016, No. 9, pp. 50–59.
ELECTROMAGNETIC METHODS
Studying the Possibilities of Georadar Tomography in Searching for Air Cavities in Engineering Constructions M. S. Sudakova, A. Yu. Kalashnikov, and E. B. Terent’eva Moscow State University, Moscow, 119991 Russia e-mail:
[email protected] Received July 16, 2016
Abstract⎯Possibilities of a tomographic version of the georadar method of searching for air cavities in engineering constructions and estimating cavity sizes are considered. Cavity sizes vary from 10 to 70 cm. Results of mathematical modeling and in-situ measurements for a pillar with a known structure are provided in order to verify the results of calculations. It is shown that the problem of searching for and outlining cavities in engineering constructions can be successfully solved with georadar tomography. Despite high labor expenditures that are involved in the method of tomographic raying, the results are of a higher quality and reliability than the results of traditional georadiolocation with a combined source–receiver. Keywords: nondestructive testing, electromagnetic methods, solutions of direct and inverse problems, mathematical modeling, georadiolocation DOI: 10.1134/S1061830916090059
INTRODUCTION Nondestructive methods are the main and, in many cases, the only possible tools for testing the condition of engineering constructions such as pillars, building supports, and piers. Carrying capacity of engineering constructions largely depends on various flaws in their internal structure: cracks; foreign inclusions; and cavities of different sizes and origins, e.g., intended cavities such as air ducts and cavities due to integrity violations that indicate an emergency state of a construction. Georadiolocation is one of the methods of nondestructive evaluation of the state of buildings and constructions. The high resolving ability of the method makes it possible to determine the structure of a building and abnormal zones due to humidification and fractured condition. The most common modification of the method is the one with combined source and receiver. It is also most economical and least laborious. The interpretation of acquired data is in many cases ambiguous, however, the information on the internal structure of a medium can be trusted in most cases. Using georadiolocation tomography allows one to get rid of the above shortcomings of the method in which source and receiver are combined. It seems logical to resort to more complicated research methods in order to gather more information and transit from a qualitative to quantitative level. The complications involve using transmitted signal and increasing the number of observation points in order to obtain quantitative electromagnetic characteristics at each point of the tested space and, as a result, the propagation speeds and amplitudes of electromagnetic waves. The latter can be used not only to reveal “abnormal zones” but also to determine the humidity of construction layers and the volume of cavities. The goal of this work is to analyze the possibilities that are offered by georadiolocation tomography in the search for cavities of different sizes. By way of example, the analysis is carried out for an isometric engineering construction. To this end, mathematical modeling was performed and in-situ measurements were taken for a pillar of a known structure with an embedded cavity. Georadar tomography is a counterpart of seismic tomography as applied to the solution of the problem of reconstructing the speeds of electromagnetic waves that pass through a selected volume of a medium from their arrival times, with the positions of sources and receivers being known. For over 20 years, seismic tomography has been the choice when solving complex geological problems. The method is described in detail in [1–4] and elsewhere. Aki, Lee, and Richards [5, 6] are considered to be pioneers in the field of tomography. Their first work was devoted to determining the speed of seismic waves in the California region using an earthquake as a source. Seismic tomography has been modernized since then by improve520
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ments in the methods of solving direct and inverse problems, the growth of the volume of processed data, and upward spiral of hardware. Existing tomographic algorithms produce good imaging results in most geophysical models. However, some of the problems are still believed to be too difficult to implement and analyze further, with the example being the problem of separating and outlining domains in which speed significantly differs from that in the embedding medium towards higher or lower values [7]. The difficulty consists in both estimating the size of and outlining an abnormality. The speed anomaly leads to the difference of speeds in embedding and embedded rocks being distributed over the entire medium and to the interface between domains with different speeds becoming blurred [8]. Chao-Ying Bai and Greenhalgh [9] managed to reconstruct images of high-speed anomalies (speed jumps of more than 20%) by using “irregular” algorithms that were suggested in [10–12]. Researchers are often aided by mathematical or physical modeling. For example, Chen Guo-Jin et al. [13] analyzed the reason for poor quality of produced seismic-tomography results in the case of a high speed contrast. Using physical- and numerical-modeling examples, they showed that “with a speed contrast of more than 30% the tomography result is significantly distorted, within the interval 30 to 15% the quality is acceptable, and for contrasts less than 15% the quality is very good”. They ascribed this to irregular ray coverage and insufficient grid discretization, although they used approximately 100 emission points and the same number of reception points for the anomaly size of about 50% of the area of the model. The same mathematical algorithms are used for tomographic inversion of georadar data as the ones that are employed in seismic prospecting, therefore, all the above conclusions also hold true in the former case. The speed of propagation of electromagnetic waves in air is 30 cm/ns, whereas in concrete it is only 10–12 cm/ns. That is to say, the observed speed contrast for an air cavity in concrete is much greater than the value of 30% that is considered in seismic prospecting. Despite this, there are solitary examples of applying radar tomography to the search of cavities in particular types of constructions among foreign publications. The researchers managed to determine the sizes of cavities accurately enough, given the fact that they were assuming the speed within cavities to be 20 cm/ns, which is 30% lower than its true value. It seems to be interesting and topical to consider possibilities of georadiolocation in its tomographic version for searching for cavities of different sizes and locations. This is possible with the use of physical and mathematical modeling and in-situ measurements for objects with known structures. In terms of the data used, tomography is divided into ray, diffraction, and full waveform ones. In ray tomography, wavelengths are mush shorter that the size of inhomogeneities that are studied, and in diffraction tomography, on the contrary, long-wave approximation is used. Ray tomography has recently become most common when studying three-dimensional speed inhomogeneities. Ray tomography presents an inverse problem in which initial data are represented by functionals along spatial lines (rays). In the ray approximation, it is assumed that energy is propagating in ray tubes of some thickness that depends on the wavelength. Given a large enough number of rays, the ray tubes cover the entire domain that is being studied [4]. Eikonal equation that relates the wave arrival time at a point with the correspondent speed distribution is among the fundamental equations of ray tomography. A consequence of the eikonal equation is the integral that expresses the travel time of a wave along a certain ray P from the i-th source to the j-th receiver
Tij =
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Then, the integral is calculated along the entire ray P0 that was obtained for the background model. This approach facilitates tracing the rays by choosing a layered background model for calculating the path P0
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When subtracting, a linearized relationship is obtained for the travel-time delay δT
δ T = T − T0 ≈
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0
where δV(r) is the speed anomaly of the model. As a result, an approximate linear relationship is established between δV and a deviation between the times T and T0 under the assumption that the ray path remains the same. The problem is now reduced to reconstructing δV, given the known time deviation. RESEARCH METHODS A 1.5 × 1.5 m engineering construction with an embedded square cavity was chosen as a model. The speed of propagation of electromagnetic waves in the solid part is 12 cm/ns. On the average, such a speed corresponds to concrete, brick, or stone constructions. A standard georadar antenna with a maximum frequency of 1000 to 5000 MHz is usually used for testing engineering constructions. This is explained by the requirement for the highest resolving ability that is possible in the georadar frequency band and rather small depths (on order of 1–2 m). If a 2 GHz antenna is used in the research, the wavelength in the realistic medium will be approximately 10 cm. On this basis, the dimensions of the cavity varied from 10 (approximately one wavelength, i.e., at the limit of the method’s resolving ability) to 70 cm (half the linear size of the whole model, on order of several wavelengths). Three possible positions of the cavity were tested during modeling: at the center, in a corner, or close to one of the walls of the model. The modeled variations of the size and position of the cavity describe almost every possible simple case for a single cavity inside a concrete construction. Figure 1 schematically presents the modeled media and shows cavities of all sizes from 10 to 70 cm, with the cavity dimensions being numbered one through seven in increasing order. The calculations were carried out in the twodimensional approximation. Emission and reception points were situated on mutually opposite sides of the square model. The pitch between sources and receivers was 10 cm, with each side accommodating 14 emission and reception points. For each version of the model, 392 times of arrival at the receiver were calculated. Figure 2 shows a ray scheme with straight rays (paths of rays in a medium with a constant speed) and with the positions of emitters and receivers. The sources are marked with S and the receivers with R. The calculations were carried out with the GeoTomCG software by LLC GeoTom (the code author Dr Daryl Tweeton). The general principles and the solution algorithm are described in [15]. A regular mesh with rectangular cells is used in this program. Times of advent were calculated for a given model with a cavity at the first stage, and the inverse problem was solved at the second stage using the calculated times. As an initial model for the solution, two approximations were used with a constant speed, viz., with V = 12 cm/ns (the speed in the construction without cavities) and Vrms, where Vrms is the root-mean-square speed over all times of advent under the assumption that all the rays are straight. With the cavity present, the root-mean-square speed grew to 12.4 for a 10-cm cavity and to 13.4 for a 70-cm cavity. Although this change is within 12% of the speed for the model, a speed growth can indirectly indicate the presence of a cavity, given no noise pollution of the data. The number of iterations was 40 for every inverse problem, and the problem was solved on curvilinear rays. The cell size was 10 × 10 cm. The iterative process was fine-tuned so that the root-mean-square deviation and the sum of all the deviations decrease as the iteration number grows. When analyzing the calculation errors, we established that when the cavity size grew from 10 to 70 cm, the deviation went up two orders of magnitude from 0.0025 ns for a 10-cm cavity to 0.5 ns for a 70-cm cavity (both in a corner). Variations in the cavity position also affect the calculation error, viz., the minimum root-mean-square deviation corresponds to the cavity in a corner and the maximum one to the cavity on a side of the construction. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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RESEARCH RESULTS AND DISCUSSION Some of the results of mathematical modeling are presented in Fig. 3. The calculated distribution of speeds is shown for different sizes (10, 20, 40, and 70 cm) and spatial positions of cavities. The calculation for model 3 with a 10-cm cavity is omitted, as the final result was no different from the initial model. The color palette includes all the speeds that were obtained in a range of 5 to 30 cm/ns, with the black frame indicating the position of the cavity in the model. Figure 4 shows ray trajectories that were obtained for different positions of the 40-cm cavity. Theoretical ray trajectories from the solution of the direct problem (the second column from the left) are different from the ones that were calculated for the inverse problem (the two columns on the right); however, refraction on the anomaly boundaries can be clearly seen. There are no principal distinctions in the ray paths for different initial speed approximations. It follows from our calculations that cavities with the size of 20 cm and more can be reliably outlined regardless of their positions (see Fig. 3). The presence of cavities is indicated, first of all, by a speed RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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increase of more than 30% for the 20-cm cavity and more than 100% for the 70-cm cavity and, secondly, by bending of the rays on the boundary of the high-speed domain that coincides with the boundary of the theoretical cavity (Fig. 4). The 10-cm cavity is not rendered correctly on speed cross sections (see the leftmost column in Fig. 3); this is possibly related to the dimensions of the observation grid. Dependences of the root-mean-square error of determining the speed inside the cavity and in the solid part on the cavity size are provided in Fig. 5 for three different cavity positions. The smaller the cavity, the greater the difference between calculated and prescribed speeds in the solid part (the root-mean-square deviation of 20 cm corresponds to 7% of all cavity positions) and in the cavity (approximately 50%). When the cavity size is increased to the maximum size that has been modeled, the speed difference grows to 44% for the solid part and falls to 25% for the cavity. If the root-mean-square–speed model is used as an initial approximation, the errors in determining the speed reduce but not much, viz. the difference is no more than 4%. The minimum errors in determining the speed were obtained for the cavity in a corner, and the maximum ones for the cavity on one of the square sides. In order to verify the calculations on real models, georadar tomographic raying was performed for a hollow column. The horizontal cross section of the column was a square with a side of 1.5 mm, with a 70 × 70 cm square cavity inside. The column was granite-faced, with several layers of concrete, possibly reinforced with metal constructions, underneath. The thickness of the granite-concrete layer was approximately 40 cm. A Zond 12e two-channel georadar by Radar Systems Ltd was used in the experiments. The radar is equipped with two antennas with a carrier frequency of 2 GHz in the air. The antennas were positioned on two opposite sides of the column according to a scheme that replicates the one that is presented in Fig. 2, with one antenna being used as a source and the other as a receiver. Georadar raying of the column by the traditional technique with combined source–receiver was also carried out. However, the results of such an observation did not reveal the presence of the cavity. We took Vrms = 15 cm/ns as an initial model when calculating the inverse tomographic transform. The calculation was performed for curved rays, with the root-mean-square discrepancy being approximately 0.08 ns and the sum of all discrepancies being 1.3. Figure 6 shows a tomographic model with a constant speed of 12 cm/ns and a 70-cm cavity at the center; the result of tomographic inversion of the times that were measured with the column (C); and ray paths for these speed distributions (B and D, respectively). The known cavity is framed in black. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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It can be seen from Fig. 6 (C) that the speeds vary within an interval of 10 to 30 cm/ns, with a highspeed anomaly clearly seen at the center of the speed cross section. The anomaly can be interpreted as an air cavity (a domain with the speeds 25–30 cm/ns that correspond to the speed of propagation of electromagnetic waves in air is shown in white). The appearance of such an anomaly indicates that the problem of revealing an air cavity can be reliably solved. When outlining the air cavity, one can also use signs that were defined in the course of modeling, viz., refraction of rays is observed on the cavity boundary and the speed inside is more than 30% higher than that in the solid part. However, the average speed in the solid part was measured to be 17 cm/ns; this is somewhat higher than normal values for concrete. CONCLUSIONS Mathematical modeling and in-situ measurements have shown the following: 1. cavities with sizes from two wavelengths to 20 cm can be reliably outlined regardless of their positions by using such signs as refraction of rays on the cavity boundary and a speed increase by more than 30%; 2. cavities with sizes less than 10 cm for the source–receiver distances of 10 cm cannot be revealed with georadar tomography; 3. determining the speed distributions of electromagnetic waves in the solid part of an engineering construction concurrently with the speed inside the cavity seems to be difficult due to large errors; 4. increasing the number of known parameters of the object that is being studied may lead to improvements in model’s fine-tuning; this, in turn, will lead to the increased stability and reliability of the problem solution and to the improved accuracy of revealing anomalies and determining their dimensions; 5. the problem of searching for and outlining cavities inside engineering constructions can be successfully solved by means of georadar tomography. Despite high labor costs of tomographic raying, the results are of a higher quality and reliability than in georadiolocation probing. REFERENCES 1. Billette, L.B. and Podvin, L., Practical aspects and applications of 2D stereotomography, Geophys., 2002, vol. 68, no. 3, pp. 1008–1021. 2. Zhou H., Multiscale traveltime tomography, Geophys., 2003, vol. 68, no. 5, pp. 1639–1649. 3. Yanovskaya, T.B. and Porokhova, L.N., Obratnye zadachi geofiziki (Inverse Geophysics Problems), St. Petersburg: St. Petersburg State Univ., 2004. 4. Yanovskaya, T.B., Problemy seismicheskoi tomografii. Problemy geotomografii (Problems of Seismic Tomography. Problems of Geotomography), Moscow: Nauka, 1997. 5. Aki, K. and Lee, W.H., Determination of the three-dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes. A homogeneous initial model, J. Geophys. Res., vol. 81, pp. 4381– 4399. 6. Aki, K. and Richards, P.G., Quantitative Seismology: Theory and Methods, WH Freeman Company, 1980. 7. Flecha, I., Marti, D., Carbobell, R., Escuder-Viruete, J., and Perez-Estaun, A., Imaging low-velocity anomalies with the aid of seismic tomography, Tectonophys., 2004, vol. 388, pp. 225–238. 8. Mathewson, J.C., Evans, D., Leone, C., Leathard, M., Dangerfield, J., and Tonning, S.A., Improved imaging and resolution of overburden heterogeneity by combining amplitude inversion and tomography, SEG, 2012. 9. Chao-Ying, B. and Stewart G., 3-D non-linear travel-time tomography: imaging high contrast, Pure Appl. Geophys., 2005, vol. 162, pp. 2029–2049. 10. Fischer, R. and Lees, J.M., Shortest path ray tracing with sparse graphs, Geophys.,, vol. 58, pp. 987–996. 11. Gruber, T. and Greenhalgh, S.A., Short note: precision analysis of first-break times in grid models, Geophys., vol. 63, pp. 1062–1065. 12. Moser, T.J., Shortest path calculation of seismic rays, Geophys., vol. 56, pp. 59–67. 13. Chen G.J., Cao H., Wu T.S., Zou F., and Yao Z.Z., Effects of velocity contrast on the quality of crosswell traveltime tomography and an improved method, Chin. J. Geophys., 2006. vol. 49, no. 3, pp. 810–818. 14. Nolet, G., A breviary of Seismic Tomography, Imaging the Interior of the Earth and Sun, Cambridge: Cambridge Univ. Press, 2008. 15. Tweeton, D.R., Jackson, M.J., and Roessler, K.S., BOMCRATR – A Curved Ray Tomographic Computer Program for Geophysical Applications,. USBM RI, 1992, v. 9411, 39 p.
Translated by V. Potapchouck RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
SPELL: OK
Vol. 52
No. 9
2016
