Hyperbola Fitting Research Articles

A novel approach for estimating root orientation by surface fitting of the hyperbola signature

In this study, we introduce a novel approach to determine root orientation from Ground Penetrating Radar (GPR) data, employing a dual-fitting strategy involving hyperbola fitting and polynomial regression surface fitting. The initial hyperbola fitting serves to capture essential parameters, while the subsequent polynomial regression surface fitting seeks to unveil nuanced relationships between root orientation angles and hyperbola characteristics. This dual-fitting framework represents a significant advancement in root analysis methodologies, offering a more refined and comprehensive understanding of root orientation derived from GPR data.

High‐frequency wide‐angle reflection and refraction method for structural engineering ground‐penetrating radar surveys

AbstractElectromagnetic wave velocity in ground‐penetrating radar (GPR) constant offset data can be estimated via the diffraction hyperbola fitting method. This method is applicable when radargrams contain diffraction events (hyperbolic patterns) caused by scatters in the host smaller or equal to the dominant wavelength. An alternative method for velocity estimation, if no intrusive information is available for a direct correlation, requires the collection of multi‐offset data. The method is quite common for broad geophysical applications; however, it seems not to be fully utilized for engineering applications, such as slabs/walls where thickness estimation and depth of the embedded features are critical requirements for structural assessments. This method would also overcome the limitations in velocity calibration in environments with no hyperbolic signal signatures. The aim of this study is to explore multi‐offset high‐frequency GPR applications, specifically the wide‐angle reflection and refraction method, for structural engineering, to understand whether it is feasible, possible limitations, and advantages. Numerical models reproducing reinforced concrete elements and a cavity wall were analysed to understand the wave behaviour and predict the response prior to testing on real cases. The main purpose is to explore how reinforcing bars can affect the velocity spectra derived from semblance analysis and what the response would be in a case of multi‐layered structure with increasing velocity with depth (cavity wall). The comparison with real cases showed that, despite some intrinsic limitations, high‐frequency multi‐offset approach could be part of standard workflow for all those surveys where no other velocity estimation method is feasible.

Object Detection and Localization in GPR B- Scan images using Hyperbola Fitting Algorithm

Ground-penetrating radar [GPR] is a non-invasive method, that generates an electromagnetic wave to map a subsurface object. It finds applications in various fields such as civil / structural engineering, environmental, Geotechnical and Military applications. This research is conducted at the Civil Engineering laboratory, IISc, Bangalore to collect the experimental data, using the Mala Pro-ex ground-coupled antenna GPR system from Mala Geoscience. GPR technology emits electromagnetic pulses into the ground and captures the reflected echoes from the target,enabling a detailed visualization of the subsurface targets.This paper presents the Hyperbola fitting algorithm for accurate detection and localization of the underground objects such as metallic and nonmetallic. It involves several steps, including preprocessing, identifying hyperbolas, fitting hyperbolas, determining target locations and depths, and validating the results. Following the fitting of the hyperbola, the algorithm proceeds to estimate the target's position and depth using the hyperbola's parameters. These estimated values for the target location and depth are subsequently validated through a model generated using the gprMax numerical modeling tool and is also tested on experimental data. There are numerous iterations of hyperbola fitting algorithms, and their implementation varies according to the attributes of the GPR data and the subject of interest. This method provides significant advantages for engineers before excavation of the ground. The algorithm has been developed and implemented using MATLAB. It has been rigorously tested using both experimental data and simulated data generated through gprMax Numerical modeling tool. The algorithm achieves a minimum error of 0.02 in the lateral position (x) and 0.01 in the depth (Y) direction. These results demonstrate the algorithm's accuracy and reliability in identifying and locating buried objects.

Stochastic hyperbola fitting, probabilistic inversion, reverse-time migration and clustering: A novel interpretation toolbox for in-situ planetary radar

Ground-penetrating radar (GPR) is becoming a mainstream tool in planetary exploration, and one of the few in-situ planetary geophysical methods. There are currently three missions (Perseverance, Tianwen-1, Chang’E-4) with GPR-equipped rovers, and two future missions (Chang’E-7, ExoMars) that will include GPR in their scientific payload. The large number of GPR data, combined with the novel setup of the measurements, creates the need for new data processing and interpretation techniques to address the unique challenges of in-situ planetary radar. The current paper proposes an interpretation pipeline that starts with a novel stochastic hyperbola fitting that estimates the probability kernel density of the bulk permittivity at different depths. Subsequently, the bulk permittivity distribution is transformed via a novel probabilistic inversion to a 1-dimensional (1D) permittivity profile. The inverted 1D permittivity profile is then used as an input to a bespoke reverse-time migration (RTM) using the finite-difference time-domain (FDTD) method. RTM using FDTD does not assume a clinical homogeneous half-space; instead, it accounts for the expected layered structure of the investigated medium. Lastly, the migrated radargram is clustered in order to identify subsurface targets and distinguish them from the background medium. Each of the processing steps has never been reported in planetary radar; and together act as a complete processing toolbox tuned for planetary science. The suggested interpretation pipeline is validated numerically in a 1D case study with a complex layered structure and multiple subsurface targets. The proposed processing scheme is then applied to the GPR data from the Chang’E-4 mission at the Von Kármán crater, revealing a previously unseen layered structure and a complex distribution of rocks/boulders.

Finetuning ground penetrating radar velocity analysis from hyperbola fitting using migration

AbstractThe widely used tool, ground penetrating radar (GPR), has proven to be an excellent research method for glaciological studies. The total ice thickness and englacial structures can be studied, and the method can give information on the temperature regime within the ice. Good velocity profiles are needed to convert measured two‐way travel time to depth information. In addition, a good velocity analysis can be used to improve data processing. A number of methods can be used to find the GPR wave velocity, which are briefly described. This study used hyperbola fitting to estimate the velocity in a glaciological study. In addition, Kirchhoff's migration was used to fine‐tune this velocity estimate. Within the Longyearbreen glacier, situated next to Longyearbyen on Spitsbergen, Svalbard archipelago, objects in the ice act as scattering points that create hyperbolas. Using the hyperbola‐fitting method, a GPR wave velocity equal to 0.170 ± 0.005 m/ns was estimated. By doing Kirchhoff's migration and studying the collapse of hyperbolas with variations of the migration velocity, a more accurate velocity estimate can be achieved; 0.172 ± 0.002 m/ns. A more accurate velocity estimate like this opens possibilities for better characterization of the ice body and ice thickness variations, as well as the variation of these as a function of time. In March 2022, the maximum ice thickness in the investigated area was calculated to be 88 ± 1 m.

Hyperbola Detection with RetinaNet and Comparison of Hyperbola Fitting Methods in GPR Data from an Archaeological Site

Hyperbolic diffractions in Ground Penetrating Radar (GPR) data are caused by a variety of subsurface objects such as pipes, stones, or archaeological artifacts. Supplementary to their location, the propagation velocity of electromagnetic waves in the subsurface can be derived. In recent years, it was shown that deep learning tools can automatically detect hyperbola in radargrams using data measured over urban infrastructure, which are relatively clear. In contrast, in this study, we used an archaeological dataset with diverse underground structures. In the first step we used the deep learning network RetinaNet to detect hyperbola automatically and achieved an average precision of 0.58. In the next step, 10 different approaches for hyperbola fitting and thus velocity determination were applied. The derived information was validated with manually determined velocities and apex points. It was shown that hyperbola extraction by using a threshold and a column connection clustering (C3) algorithm followed by simple hyperbola fitting is the best method, which had a mean velocity error of 0.021 m/ns compared to manual determination. The average 1D velocity-depth distribution derived in 10 ns intervals was in shape comparable to the manually determined one, but had a systematic shift of about 0.01 m/ns towards higher velocities.

On the Limitations of Hyperbola Fitting for Estimating the Radius of Cylindrical Targets in Nondestructive Testing and Utility Detection

Hyperbola fitting is a mainstream interpretation technique used in ground penetrating radar (GPR) due to its simplicity and relatively low computational requirements. Conventional hyperbola fitting is based on the assumption that the investigated medium is a homogeneous half-space, and that the target is an ideal reflector with zero radius. However, the zero-radius assumption can be easily removed by formulating the problem in a more generalized way that considers targets with arbitrary size. Such approaches were recently investigated in the literature, suggesting that hyperbola fitting can be used not only for estimating the velocity of the medium, but also for estimating the radius of subsurface cylinders, a very challenging problem with no conclusive solution to this day. In this paper, through a series of synthetic and laboratory experiments, we demonstrate that for practical GPR survey, hyperbola fitting is not suitable for simultaneously estimating both the velocity of the medium and the size of the target, due to its inherent non-uniqueness, making the results unreliable and sensitive to noise.

A New B-Scan Interpretation Model in Complex Environments

In the scenario where the echo hyperbolas of different targets intersect or the opening structure of the hyperbola is destroyed, the current B-scan image interpretation model has the problems of missing detection and unabling to make full use of the hyperbolic information. The hyperbola needs to be identified manually or by utilizing neural network methods. The recognition accuracy is difficult to guarantee. In order to solve the above problem, a new Ground-Penetrating Radar (GPR) B-scan interpretation model based on hyperbolic trend is proposed in this paper. The model consists of four parts, including preprocessing method, hyperbolic trend clustering algorithm (HTCA), restricted algebraic distance fitting and cluster center algorithm (RADFCC) and Hierarchical Filtered Back Projection (HFBP). Firstly, the GPR B-scan image is converted into a binarized image based on the ratio of the main lobe and side lobe of the echo. Secondly, the points in the binarized image are classified into hyperbolic column-segment clusters based on the hyperbolic trend, and further denoised according to the main lobe width of the ricker wave. Thirdly, by analyzing the characteristics of the GPR in the ground coupling mode, the restricted hyperbola fitting algorithm is used to fit the classified hyperbola column-segment clusters to speed up the convergence of the fit and avoid the result from falling into a local optimum. Fourthly, the target’s location is preliminarily estimated by utilizing the parameters of the hyperbola and the cluster center is established through the location and radius of the target to obtain the accurate estimation of the target and the dielectric constant of the underground. Finally, an imaging algorithm is applied to invert the subsurface image based on the dielectric constant of the subsurface medium. The effectiveness of the proposed model has been demonstrated on both simulated and real datasets.

A Novel Approach for Permittivity Estimation of Lunar Regolith Using the Lunar Penetrating Radar Onboard Chang’E-4 Rover

Accurate relative permittivity is essential to the further analysis of lunar regolith. The traditional hyperbola fitting method for the relative permittivity estimation using the lunar penetrating radar generally ignored the effect of the position and geometry of antennas. This paper proposed a new approach considering the antenna mounting height and spacing in more detail. The proposed method is verified by numerical simulations of the regolith models. Hence the relative permittivity of the lunar regolith is calculated using the latest high-frequency radar image obtained by the Yutu-2 rover within the first 24 lunar days. The simulation results show that the relative permittivity is underestimated when derived by the traditional method, especially at the shallow depth. The proposed method has improved the accuracy of the estimated lunar regolith relative permittivity at a depth of 0–3 m, 3–6 m, and 6–10 m by 35%, 14%, and 9%, respectively. The thickness of the lunar regolith at the Chang’E 4 landing site is reappraised to be 11.1 m, which improved by ~8% compared with previous studies.

Analysis of Imaging Internal Defects in Living Trees on Irregular Contours of Tree Trunks Using Ground-Penetrating Radar

The outer contours of living trees are often considered as a standard circle during non-destructive testing (NDT) of internal defects using ground-penetrating radar (GPR). However, the detection of classical cross-sections (circular) lacks consideration of irregular contours, making it difficult to accurately locate the radar image of the target. In this paper, we propose a method based on the image affine transformation and the Riemann mapping principle to analyze the effect of irregular detection routes on the geometric characteristics of target reflection hyperbola. First, for the similar output phenomenon in the “hyperbola fitting”, geometric analysis and numerical simulation were performed. Then, the conversion of irregular trunk radar images and physical domain radar images was implemented using the method of image affine transformation and the Riemann mapping principle. Finally, the influence of irregular detection routes on the geometry of the target reflection curve was investigated in detail through numerical simulations and actual experiments. The numerical simulation and measurement results demonstrated that the method in this study could better reflect the imaging characteristics of the target reflection hyperbola under the irregular detection pattern. This method provides assistance to further study the defects of irregular living trees and prevents the misjudgment of targets as a result of hyperbolic distortion, resulting in a greater prospect of application.

Near Surface Velocity Estimation Using GPR Data: Investigations by Numerical Simulation, and Experimental Approach with AVO Response

The velocity of near-surface materials is one of the most important for Ground-Penetrating Radar (GPR). In the study, we evaluate the options for determining the GPR velocity to measure the accuracy of velocity approximations from the acquired GPR data at an experimental site in Hangzhou, China. A vertical profile of interval velocities can be estimated from each common mid-point (CMP) gather using velocity spectrum analysis. Firstly, GPR data are acquired and analyzed using the popular method of hyperbola fitting which generated surprisingly high subsurface signal velocity estimates while, for the same profile, the Amplitude variation with offset (AVO) analysis of the GPR data (using the same hyperbola fitting method) generate a more reasonable subsurface signal velocity estimate. Several necessary processing steps are applied both for CMP and AVO analysis. Furthermore, experimental analysis is conducted on the same test site to get velocities of samples based on dielectric constant measurement during the drilling process. Synthetic velocities generated by AVO analysis are validated by the experimental velocities which confirmed the suitability of velocity interpretations.

Low Complexity Lane Detection Methods for Light Photometry System

The aim of the paper was to analyze effective solutions for accurate lane detection on the roads. We focused on effective detection of airport runways and taxiways in order to drive a light-measurement trailer correctly. Three techniques for video-based line extracting were used for specific detection of environment conditions: (i) line detection using edge detection, Scharr mask and Hough transform, (ii) finding the optimal path using the hyperbola fitting line detection algorithm based on edge detection and (iii) detection of horizontal markings using image segmentation in the HSV color space. The developed solutions were tuned and tested with the use of embedded devices such as Raspberry Pi 4B or NVIDIA Jetson Nano.

Research on Hyperbola Fitting Algorithm for Turbulence Level Measurement Test Data

Hyperbola fitting of test data is an extremely important process in turbulence level measurement test in wind tunnels. The solution of the overdetermined equations (SOE) method is often used to solve hyperbola fitting parameters to obtain turbulence level. However, due to unsteady flow characteristics, the SOE method often results in overfitting phenomena, which makes it impossible to solve turbulence level accurately. This paper proposes using the constrained least-squares (CLS) method to convert the problem of hyperbola fitting of test data into the inequality constrained optimization problem and then using the Lagrange programming neural network (LPNN) method to solve turbulence level iteratively. The stability of the LPNN method is analysed, and three sets of typical turbulence level measurement test data are processed using the LPNN method. The results verify the feasibility of applying the LPNN method to iteratively solve the turbulence level of wind tunnels.

An Effective Sharp Curve Lane Detector

Sharp bend path location is one of the difficulties of visual condition discernment innovation for self-governing driving. Right now, new hyperbola fitting based technique for bend path recognition is proposed. The strategy for the most part incorporates three sections: extraction, bunching, and hyperbola fitting of path highlight focuses. We analysed our technique with the Bezier bend fitting based, the least squares bend fitting based, the spline fitting based techniques, and a current hyperbola fitting based strategy. Examinations show that our strategy performs superior to these technique.

Evaluating soil moisture estimation from ground‐penetrating radar hyperbola fitting with respect to a systematic time‐domain reflectometry data collection in a boreal podzolic agricultural field

AbstractThe upper 30 cm of the soil profile, which hosts the majority of the root biomass, can be considered as the shallow agricultural root zone of most temperate crops. The electromagnetic wave velocity in the soil obtained from reflection hyperbolas in ground‐penetrating radar (GPR) data can be used to estimate soil moisture (SM). Finding shallow hyperbolas in a radargram and minimizing the subjective error associated with the hyperbola fitting are the main challenges in this approach. Nevertheless, we were motivated by the recent improvements of hyperbola fitting algorithms, which can reduce the subjective error and processing time. To overcome the difficulty of finding very shallow hyperbolas, we applied the hyperbola fitting method to reflections ranging from 27‐ to 50‐cm depth using a 500‐MHz centre‐frequency GPR and compared the estimated moisture with vertically installed, 30‐cm‐long time‐domain reflectometry (TDR) sensors. We also compared TDR and GPR sample areas in a 2‐D plane using different GPR survey types and different hyperbola depths. SM measured with TDR and GPR were not significantly different according to Mann–Whitney's test. Our analyses showed that a root mean square error of 0.03 m3 m−3 was found between the two methods. In conclusion, the proposed method might be suitable to estimate SM with an acceptable accuracy within the root zone if the soil profile is fairly uniform within the application depth range.

Health Monitoring of Tree Trunks Using Ground Penetrating Radar

Ground penetrating radar (GPR) is traditionally applied to smooth surfaces in which the assumption of half-space is an adequate approximation that does not deviate much from reality. Nonetheless, using GPR for internal structure characterization of tree trunks requires measurements on an irregularly shaped closed curve. A typical hyperbola fitting has no physical meaning in this new context since the reflection patterns are strongly associated with the shape of the tree trunk. Instead of a clinical hyperbola, the reflections give rise to complex-shaped patterns that are difficult to be analyzed even in the absence of clutter. In this paper, a novel processing scheme is described which can interpret complex reflection patterns assuming a circular target subject to any arbitrary shaped surface. The proposed methodology can be applied using commercial hand-held antennas in real time, avoiding computationally costly tomographic approaches that require the usage of custom-made bespoke antenna arrays. The validity of the current approach is illustrated both with numerical and real experiments.

Using Ground Penetrating Radar Methods to Investigate Reinforced Concrete Structures

This paper provides an overview of the existing literature on the subject of ground penetrating radar (GPR) methods for the investigation of reinforced concrete structures. An overview of the use of concrete and reinforced concrete in civil engineering infrastructures is given. A review of the main destructive and non-destructive testing methods in the field is presented, and an increase in the use of GPR to reinforced concrete structures is highlighted. It was also observed that research in some application areas has been predominantly or exclusively carried out at a laboratory scale, and that similarly, other more application-oriented research has been developed only on real-life structures. The effectiveness of GPR in these areas is demonstrated. Furthermore, a case study is presented on a new methodological and data processing approach for the assessment of reinforced concrete structures using a high-frequency dual-polarised antenna system. Results have proven the advantages of using the proposed methodology and GPR system in order to improve the detectability of rebars, including secondary bottom lines of reinforcement. The horizontal polarisation was proven to be more stable compared to the vertical. Finally, it has been demonstrated that a more accurate location of the rebars in a high-density grid mesh arrangement can be obtained by means of data migration processing with a scan spacing of 5 cm and wave velocity information through the use of the hyperbola fitting method from at least 30% of the targets.

Automatic hyperbola detection and fitting in GPR B-scan image

Detecting buried objects from ground penetrating radar (GPR) profiles often requires manual interaction and plenty of time. This paper presents an automatic scheme for buried objects detection and localization. First, a trained deep learning framework — Faster R-CNN with data augmentation strategy is applied to identify hyperbolic signatures from a gray GPR B-scan image, which is capable of not only recognizing whether a B-scan profile contains traces of buried object, but also detecting candidate hyperbola region. Then, the detected rectangle region is extracted and transformed to a binary image, a novel double cluster seeking estimate (DCSE) algorithm is proposed to separate object point cluster from each other and enable the identification of hyperbolic signatures. Subsequently, a column-based transverse filter points (CTFP) method is utilized to extract hyperbola fitting points automatically from the validated point cluster. Downward opening hyperbola fitting is carried out and their respective peaks are obtained finally. The proposed scheme is able to extract information from GPR B-scan images automatically and efficiently; it is validated significant performance in the analysis of synthetic and on-site GPR data sets. • Faster R-CNN with data augmentation strategy is applied to pick hyperbola region from the gray GPR B-scan image. • A novel double cluster seeking estimate (DCSE) algorithm is proposed to separate object point cluster from each other. • A column-based transverse filter points (CTFP) method is utilized to extract hyperbola fitting points automatically. • The experiments show significant performance in the analysis of synthetic and on-site GPR data sets.

Diffraction imaging of ground-penetrating radar data

Diffractions caused by, e.g., faults, fractures, and small-scale heterogeneity localized near the surface are often used in ground-penetrating radar (GPR) reflection studies to constrain the subsurface velocity distribution using simple hyperbola fitting. Interference with reflected energy makes the identification of diffractions difficult. We have tailored and applied a diffraction imaging method to improve imaging for surface reflection GPR data. Based on a plane-wave destruction algorithm, the method can separate reflections from diffractions. Thereby, a better identification of diffractions facilitates an improved determination of GPR wave velocities and an optimized migration result. We determined the potential of this approach using synthetic and field data, and, for the field study, we also compare the estimated velocity structure with crosshole GPR results. For the field data example, we find that the velocity structure estimated using the diffraction-based process correlates well with results from crosshole GPR velocity estimation. Such improved velocity estimation may have important implications for using surface reflection GPR to map, e.g., porosity for fully saturated media or soil moisture changes in partially saturated media because these physical properties depend on the dielectric permittivity and thereby also the GPR wave velocity.

Semi-automatic determination of layer depth, permittivity and moisture content for unbound granular pavements using multi-offset 3-D GPR

ABSTRACT A semi-automated multi-offset ground penetrating radar (GPR) analysis method called Ray-path Modelling Coherence (RM–C) was developed and field tested. It was used to quasi-continuously predict the depth, permittivity and volumetric moisture content of unbound granular (UBG) layers along a pavement site during two investigations using ground-coupled measurements from a 3-D GPR array. Predictions from the later site visit were compared to physical measurements of layer depth and moisture content at a number of locations along the site, which in general agreed well. RM-C permittivity predictions were compared to results from time domain reflectometery (TDR) sensors and diffraction hyperbola fitting of buried reflectors. The new approach reported higher permittivity values compared to these other methods, although showed similar trends in permittivity change with respect to pavement depth and ⁠variation over time. Reproducibility of the RM–C analysis was confirmed by comparing layer depth and moisture content predictions for repeat runs along the site, although occasional discrepancies were observed.