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There are often many hidden structural defects in heritage buildings. As a convenient and effective nondestructive detecting method, ground-penetrating radar (GPR) has a technical advantage in detecting and protecting heritage buildings depending on the advanced image interpretation. The analytic relationship between buried depth and radius of point object and long and short axis of hyperbolic equation was established according to derivations of formulas. The image characteristics of hyperbolic curves with different depth and radius were studied by finite-difference time-domain method (FDTD). And then, inversion models of buried depth and radius of point object were established. The buried depth and radius can be accurately deduced by long and short axis of hyperbolic image. This result was applied in the detection of pedestal defects of the heritage building, and the depth and distribution range of hidden fracture can be accurately interpreted. It provides an effective and fast method to detect hidden defects in civil engineering.

There have been many point defects in the hidden areas of civil engineering and environmental investigations, such as buried utility mapping, concrete and pavement inspection, and tunnel [

The detection principle of GPR is as follows. When emitted electromagnetic waves (frequency range is 10–2200 MHz) transmit in the underground, the electromagnetic waves in inhomogeneous relative permittivity of interface will produce the phenomenon of reflection and refraction. And then, the reflected waves, also called echo, can reflect the characteristics of underground medium [

The most common inversion methods of GPR data are linear inversion (the steepest descent method, the conjugate gradient method, the Gauss-Newton method, the gradient regularization method, etc.) and nonlinear algorithm (simulated annealing algorithm, genetic algorithm, ant colony algorithm, particle swarm optimization, fish swarm algorithm, etc.) [

Herein, a one-to-one relationship between object parameters and inversion parameters was established. We employed a forward modeling method to accurately analyze the relationship between depth and radius of object and long and short axis of hyperbolic imaging. Combined with theoretical derivation and field verification, the imaging features and regularities with different depths and radius were studied. And their multiple regression models with long and short axis of hyperbolic curves were established to achieve the purpose of interpreting the geometry of object. Furthermore, a nondestructive testing of heritage building was carried out to achieve fast interpretation of hidden fracture distribution range and the depth.

The antenna of GPR emits electromagnetic waves in the underground. When electromagnetic wave arrives at object, the reflection of signals will occur, and echo signals will be received by receiving antenna [_{0} represents the distance from _{0} to the object. _{0} point can be detected. In the time-domain recording, the reflection feature can only be recorded at the _{0}, the detection of the object belongs to the vertical detection.

Basic principle of point object imaging of GPR.

According to the geometric relationship, we can get the equilibrium
_{0} is the two-way travel time of electromagnetic wave from _{0} to O.

Formula (

Formula (

According to comparative coefficient method, we can get

Therefore, the characteristics of point object (O) with different depths and radius can be expressed by the long and short axis of the hyperbolic equation. At the same time, the imaging asymptotic lines, eccentricity, and other parameters of the GPR can also be obtained by the ratio of the long axis to short axis.

In this study, GPR imaging characteristics in different radius and depth of the object were simulated by finite-difference time-domain (FDTD) method, and we employed GprMax software. GprMax [

The influence of radius of point object on the GPR imaging is studied by FDTD method. The model design of point object is as follows. The geological background is loess, and the point object is air. The geological model scheme is shown in Figure

Geoelectric model with different radius (

The horizontal distance of this model is 2.0 m. The depth of this model is 0.7 m. The cell size is 0.0025 m by 0.0025 m. The time depth is 20 ns. The relative permittivity of geological background is 5. The electrical conductivity of geological background is 0.000001 S/m. The relative permittivity of point object is 1. The electrical conductivity of point object is 0.00 S/m. The depth of point object is 0.4 m. The radius of the point body is 0.005 m, 0.025 m, 0.050 m, and 0.100 m. Dominant frequency is set as 900 MHz. The excitation source is Ricker wavelet. In the numerical simulation, there are 180 step calculations, and each step calculation contains 3391 times. The imaging features of point object with different radius under the same depth are shown in Figure

(a–d) GPR imaging of different radius (

In Figure

(a–h) GPR imaging of different depth (

The influence of depth of point object on the GPR imaging is studied by FDTD method. The depth of object can generally be converted by the time depth and the relative permittivity of the geological background. As the depth of the object changed, the characteristics of radar imaging are bound to be changed. In this section, we established the geological model of objects with different depth, which is as shown in Figure

Geoelectric model of different depth (

The basic parameters of this model are as follows. The horizontal distance of this model is 2.0 m. The depth is 0.7 m. The cell size is 0.0025 m by 0.0025. The time depth is 20 ns. The relative permittivity of the geological background is 5, and the conductivity is 0.000001 S/m. The radius of the object is 0.05 m. The relative permittivity of the object is 1. The electrical conductivity of the object is 0 S/m. The depth of the object is set as 5 cm, 10 cm, 20 cm, 30 cm, 40 cm, 50 cm, 60 cm, and 65 cm. The main frequency of wavelet is 900 MHz, and the excitation source is Ricker wavelet. There are a total of 180 calculation steps, and each one is 3392 times. The imaging features of this model are shown in Figure

In Figure

The imaging characteristics of point objects are mainly hyperbolic curves. The coordinates of key points of hyperbolic curves can be extracted, and the fitting formulas of hyperbolic curves can be established for regression analysis. In this study, the coordinates of key points were extracted by GetData software (Figure

(a) Data conversion of time depth and distance in different object radius. (b) Data conversion of depth and distance in different object radius.

In Figure

The data of key points of hyperbolic curves are fitted by conjugate gradient method, and the hyperbolic formulas are obtained. The values of long and short axis are obtained by comparing hyperbolic formulas with (

Imaging feature of the object in different radius.

Long axis (_{r} |
Short axis (_{r} |
Hyperbolic formula | ||
---|---|---|---|---|

0.005 | 0.3 | 0.347 | 0.434 | |

0.025 | 0.3 | 0.333 | 0.388 | |

0.05 | 0.3 | 0.330 | 0.379 | |

0.100 | 0.3 | 0.342 | 0.408 |

The method for extracting data is the same with that in Section

Data conversion of time depth and depth in different object depth.

The conjugate gradient method is also used to realize hyperbolic formula fitting. And the values of long and short axis hyperbolic formula are obtained by comparing hyperbolic formulas with (

Imaging feature of the object with different depth.

Long axis ( |
Short axis ( |
Hyperbolic formula | ||
---|---|---|---|---|

0.05 | 0.05 | 0.089 | 0.104 | |

0.05 | 0.10 | 0.133 | 0.159 | |

0.05 | 0.20 | 0.231 | 0.228 | |

0.05 | 0.30 | 0.330 | 0.379 | |

0.05 | 0.40 | 0.430 | 0.487 | |

0.05 | 0.50 | 0.531 | 0.598 | |

0.05 | 0.60 | 0.632 | 0.713 |

In the forward imaging for FDTD method, there is an error between apparent depth and real depth. The main reason is that there are truncation errors between forward and backward differences of FDTD method. The value of truncation error is equal to the square of time-domain lattice (Δ

Comparison of the apparent depth and real depth with different depth.

Contents | Data | ||||||
---|---|---|---|---|---|---|---|

Real depth _{real}/m |
0.05 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 |

Apparent depth _{apparent}/m |
0.08 | 0.13 | 0.23 | 0.33 | 0.43 | 0.53 | 0.63 |

Time depth _{time}/ns |
1.23 | 1.93 | 3.39 | 4.85 | 6.37 | 7.84 | 9.36 |

It can be seen that there are errors between real depths and apparent depths. Interestingly, the errors with different buried depths are the same, and are equal to 0.03 m. So the errors can be modified and accurately real depth can be obtained.

or

Figure

(a) Correction factor of real depths and apparent depths. (b) Correction factor of real depths and time depths.

The values of the long and short axis of hyperbolic imaging can be calculated by (

Parameters of depth and radius of object, the long axis (a) and short axis (b).

_{c} |
_{c} |
_{c}_{c} |
(_{c}_{c} | ||
---|---|---|---|---|---|

1.23 | 0.05 | 0.08 | 0.10 | 1.24 | 0.76 |

1.93 | 0.10 | 0.12 | 0.15 | 1.25 | 1.21 |

3.39 | 0.20 | 0.21 | 0.26 | 1.22 | 2.07 |

4.85 | 0.30 | 0.31 | 0.37 | 1.22 | 2.96 |

6.37 | 0.40 | 0.40 | 0.48 | 1.21 | 3.85 |

7.84 | 0.50 | 0.49 | 0.60 | 1.21 | 4.74 |

9.36 | 0.60 | 0.59 | 0.71 | 1.20 | 5.64 |

_{c}_{c}

According to (_{c}_{c}

Correction factor of real depths and coefficient of the short and long axis.

The empirical (

According to

Comparison of the real radius and inversion radius in different depth.

Contents | Data | ||||||
---|---|---|---|---|---|---|---|

Real depth _{real}/m |
0.05 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 |

Real radius _{real} |
0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |

Inversion radius _{inversion}/ns |
0.089 | 0.065 | 0.054 | 0.051 | 0.049 | 0.049 | 0.047 |

Error |
78% | 30% | 8% | 2% | 2% | 2% | 6% |

In Table

The heritage building built in 1384 during the early Ming Dynasty is a symbol of the city of Xi’an and one of the grandest of its kind in China. It covers an area of 1377 square meters and has 36 meters high and consisted of a foundation, a pedestal, and a tower. The structure of the tower is mainly made up of brick and wood and was built on the square base compacted by soil. At present, the pedestal shows some apparent fractures. It may be due to artificial disturbance and natural erosion, such as earthquake and subway.

GPR method is used to detect the internal deformations and distribution area in the pedestal. A distinct advantage of GPR is the ability to provide high-resolution continuous profiling for nondestructive site investigations. In addition, GPR has become economically feasible and convenient method in the protection of the heritage. Combining with the actual situation, antenna of 400 MHz frequency is an optimal choice because of the proper precision and detection depth. The detailed parameters of this antenna are listed in Table

The antenna parameters of GPR.

Frequency | 400 MHz |

Emissivity | 100 KHz |

Ranges (ns) | 40 |

Scanning speed (scan/s) | 50 |

Gain | 5 |

(a–b) Location of the heritage buildings. (c) Leakage in external wall of the pedestal. (d) Monitoring survey area and the line layout.

After data processing and image interpretation, the results of GPR images show that abnormal regions of each survey line are distributed in the same location. So we use the data fragments of the abnormal regions to analyze the reasons of the complicated imaging. The length of each fragment is 1 m, and the slice figure consisted of the GPR images of each fragment in Figure

Geological section of anomaly region by GPR.

In Figure

Feature points are extracted from the hyperbolic curves which have the strongest reflection intensity of each GPR images in Figure

Inversion parameters of depth and radius of the defects.

RD1 line | 0.70 | 0.56 | 0.52 | 0.13 | 6.26 |

RD2 line | 0.74 | 0.58 | 0.54 | 0.13 | 6.55 |

RD3 line | 0.55 | 0.49 | 0.56 | 0.08 | 6.14 |

RD4 line | 0.50 | 0.46 | 0.54 | 0.08 | 5.56 |

RD5 line | 0.74 | 0.61 | 0.50 | 0.17 | 5.81 |

In Table

Inversion result of radius and depth of fracture.

In Figure

In summary, we established the analytic relationship between depth and radius of object and long and short axis of hyperbolic equation, according to derivations of formulas.

The image characteristics of hyperbolic curves with different depth and radius were studied by FDTD method. When

Inversion model of geometric parameters and long and short axis was established. The relationship among real depth and long and short axis (_{c}_{c}

The inversion model of object radius was obtained by nonlinear algorithm.

The practical detection in heritage building was implemented according to the above research results, and its feasibility was verified. The detecting of pedestal defects of the heritage building is obviously effective by the inversion model. It is demonstrated that this inversion model is feasible, since the observed and calculated radius and depth of pedestal defects agree fairly well with each other.

The authors declare that they have no conflicts of interest.

The authors gratefully acknowledge the financial supports by China Postdoctoral Science Foundation (Grant no. 2017M613175), Shaanxi Province Science Foundation for youths (Grant no. 2018JQ5203) and Shaanxi Province Major Science Foundation (Grant no. 2018JZ5010). And the work was supported by the National Science Foundation of China (Grant no. 11572246, 51779207) and the Open Foundation of State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area (Grant no. 2016ZZKT-8).

Formula S1: the equation of the object depth and two-way travel time. Figure S1: GETDATA obtains the key point coordinates by taking points continuously at the maximum amplitude of the reflection curve in GPR images. Table S1: data of the GPR imaging with different object radius. Table S2: transfer data of the GPR imaging with different object radius. Table S3: data of the GPR imaging with different object depth. Table S4: transfer data of the GPR imaging with different object depth.