A Stable Subgridding 2D-FDTD Method for Ground Penetrating Radar Modeling

Implementation of ADI-FDTD subgrids in ground penetrating radar FDTD models

Numerical modelling of Ground Penetrating Radar (GPR) using the Finite-Difference Time-Domain (FDTD) method can be computationally intensive when large volumes must be discretised at fine spatial resolutions. This requirement exists because of the conditionally explicit nature of the FDTD algorithm, and the need to simulate fine geometrical details as commonly found in GPR antennas or in regions of high permittivity dielectric materials. We have developed and implemented a Huygens subgridding (HSG) approach in the open source software gprMax. The HSG can be applied to multiple localised regions of the FDTD grid where it is required, whilst the rest of the grid can be discretised at a more appropriate spatial resolution. An example of applying the HSG to a GPR model that includes a detailed antenna model, demonstrates the accuracy and efficiency of the approach. The relative error compared with a uniform fine grid is generally less than 1.3%. The computational time is reduced by a factor of 16. This level of computational saving is especially important for optimisation and inversion algorithms where many GPR forward models are computed in a single iteration.

Ground penetrating radar (GPR) forward modeling is one of the core geophysical research topics and also the primary task of simulating ground penetrating radar system. It is a process of simulating the propagation laws and characteristics of electromagnetic waves in simulated space when the distribution of internal parameters in the exploration region is known. And the finite-difference-time-domain (FDTD) method has the characteristics of simulating the space-time transient evolution of electromagnetic wave, whose numerical method is simple and easy to program, so it has become one of the most extensively utilized methods in GPR forward modeling. It is generally known that the conventional FDTD approach requires finer uniform Yee cell all the time to produce satisfactory accuracies from numerical simulations of the GPR. However, the smaller temporal incremental has to be adopted due to the lower spatial incremental, which would dramatically weaken the advantage of the FDTD method. To solve this issue, the subgridding-technique-based hybrid local-one-dimensional FDTD (LOD-FDTD) is applied in this work to modeling the classical GPR scenarios. In this method, the unconditional-stable LOD-FDTD is employed in the fine-grid domain, while the traditional FDTD is used in the coarse-grid domain, which could avoid the oversampling problem in the local domain if the uniform fine-grid scheme is adopted. Meanwhile due to the unconditional stability of the LOD-FDTD, the larger time step, derived from the coarse grid which satisfies the Courant-Friedrichs-Lewy (CFL) stability condition, could be utilized in the whole domain so that the long-time interpolation process could be circumvented. Additionally, the proposed approach could be arbitrarily adjusted by means of different ratio of both coarse- and fine-grid, and hence it holds much higher generality. As compared with the auxiliary differential equation (ADE) technique, the Z-transform method is integrated into FDTD methods for modeling multi-pole Debye-based dispersive media in this method, resulting in more direct numerical implementations and fewer computing steps. Finally, three different classical GPR problems are carried out to validate accuracies and efficiencies of the proposed method.

Ground penetrating radar (GPR) forward is one of the geophysical research directions.Through the forward of geological model,the database of radar model can be enriched and the characteristics of typical geological radar echo images can be understood,which in turn can guide the data interpretation of GPR measured profile,thereby improving the GPR data interpretation level.In this article,the interpolating wavelet scale function by using iterative interpolation method is presented,and the derivative of scale function is used in spatial differentiation of discrete Maxwell equations. The forward modeling formula of GPR based on the interpolation wavelet scale method is derived by using fourth-order Runge-Kutta method (RK4) for calculating the higher time derivative.Compared with the conventional finite difference time domain (FDTD) algorithm based on the central difference method,the interpolation wavelet scale algorithm improves the accuracy of GPR wave equation in both space and time discretization.Firstly,the FDTD algorithm and the interpolation wavelet scale method are applied to the forward modeling of a layered model with analytic solution. Single channel radar data and analytical solution fitting indicate that the interpolation wavelet scale method has higher accuracy than FDTD,with the same mesh generation used.Therefore,auxiliary differential equation perfectly matching layer (ADE-PML) boundary condition is used on an interpolation wavelet scale,and the comparisons between reflection errors obtained using CPML (FDTD),RK4ADE-PML (FDTD),and RK4ADE-PML (interpolating wavelet scales) in a homogeneous medium model show that the absorption effect of RK4ADE-PML (interpolating wavelet scales) is better than the other two absorbing boundaries.Finally,interpolation wavelet scale method,with both UPML,FDTD and RK4ADE-PML loaded,is used for two-dimensional GPR forward modeling,showing good absorption effect for evanescent wave.From all the experimental results,the following conclusions are obtained.1) Using the derivative of the interpolating wavelet scale function instead of central difference schemes for the spatial derivative discretization of Maxwell equations and time derivative calculated using the fourth-order Runge Kutta algorithm,the interpolating wavelet scale algorithm has higher accuracy than regular FDTD algorithm due to the improvement in the spatial and time accuracy of GPR wave equation.2) The best absorption layer parameters of interpolating wavelet scale RK4ADE-PML are selected, when the maximum value of the reflection error is the minimum.The maximum reflection error can reach-93 dB,which increases 20 dB compared with that of UMPL boundary in FDTD algorithm.And the higher simulation accuracy of interpolating wavelet scale algorithm than FDTD algorithm is confirmed after calculating single channel radar data.3) Comparing wave field snapshots of GPR forward modeling,radar pictures from wide-angle method and section method indicates that interpolating wavelet scale RK4ADE-PML reduces reflection error of absorption boundary,improves both spatial and time accuracy,is more effective than UPML boundary in eliminating false reflection of large angle incidence, and has better absorption effect for evanescent wave and low-frequency wave.

Forward modeling of ground penetrating radar (GPR) is an important part to the inversion/modeling of the observed data. The aim of this study is to establish specific numerical schemes for forward modeling of GPR data by finite difference frequency domain (FDFD) method which were originally developed for seismic or finite difference time domain (FDTD) method. A total number of six modified and improved FDFD techniques have been used to discretize the two-dimensional (2D) transverse electric (TE)-mode scalar wave equation in order to find the suitable method for this. These techniques include five-point classical to nine-point mixed unstaggered-grid configurations. The numerical schemes for three unsplit perfectly matched layer (PML) for nine-point mixed unstaggered-grid configurations are also presented. The applicability of these techniques is tested by using the underground models of relative permittivity and conductivity for the two cases of homogeneous and 2-cross models. GPR shot gather data for these two models are also produced for this study. The relative reflection errors of the numerical schemes are also estimated for the homogeneous model to comprehend the appropriate method for the modeling. The algorithm for complex-frequency shifted PML (CFSPML) gives the least error in case of the forward modeling of the GPR data.

Analysis of thin-wire ground penetrating radar systems for buried target detection, using a hybrid MoM-FDTD technique

Traditional subgridding finite difference time-domain (FDTD) method can greatly save computation time and memory consumption. However, the limits of the Courant-Friedrichs-Lewy (CFL) stability condition still be met. In this paper, an explicit unconditionally stable technique is introduced into the subgridding FDTD approach and applied in the ground penetrating radar (GPR) modeling. Thus, a relatively large and unified time step can be used in coarse and fine grids, which breaks through the method's CFL condition limit.

Ground penetrating radar (GPR) is a powerful tool for characterisation of the subsurface in a range of applications. Finite-difference time-domain (FDTD) forward modelling is often used to gain a more quantitative understanding of the interaction between GPR systems and the region of interest. This can be undertaken in 2D, where simulations fail to model multiple polarisations and require a number of simplifying assumptions for both electromagnetic fields and for modelling the environment. Alternatively full 3D modelling may be used, but this can be very computationally expensive. Here we present an idea, in contrast to the more formal 2.5D FDTD GPR modelling approach, which is based on using a thin slice of a full 3D model in cases that we want full 3D fields but we have a 2D modelling environment. The key issue with this approach is minimising, if possible, the error introduced by the very close proximity of the perfectly matched layer (PML) absorbing boundaries of the model in the invariant direction of the modelled geometry. We use gprMax, an open source FDTD GPR modelling package, to check the viability of this idea. An assessment of domain size required is made considering the error produced and the computational demands. Optimising the PMLs will be key in making this approach viable for future GPR modelling that provides a full 3D field solution in cases where a 2D geometry is a reasonable assumption.

A036 NUMERICAL MODELLING OF GROUND PENETRATING RADAR ANTENNAS Introduction 1 Numerical modelling of ground penetrating radar (GPR) can significantly enhance our understanding of the GPR’s detection mechanism especially in complex environments. In addition models could aid GPR data interpretation. There are a number of numerical methods that can be used to develop a forward GPR model. One of the most versatile and powerful technique is the Finite-Difference Time-Domain (FDTD) method pioneered by Kane Yee (1966). FDTD has been used successfully to model GPR responses from simple targets assuming either a two dimensional or a three dimensional geometries (Holliger et. al.

We modeled and implemented a joint airborne system integrating ground penetrating radar (GPR) and magnetometer (MAG) models specifically for landmine detection applications. We conducted both simulations and experimental analyses of the joint airborne GPR and MAG models, with a focus on detecting the metallic components of different types of landmines, including antitank (AT) M15 metallic, antipersonnel (AP) M16 metallic, and AT M19 plastic (minimum-metal) landmines. The GPR model employed the finite-difference time-domain (FDTD) method and was evaluated using a singular value decomposition (SVD) and Kirchhoff migration (KM) with matched filtering (MF). These advanced techniques enabled the automatic identification and precise focusing of the reflected hyperbolic signals emitted by the landmines while considering cross-range resolution. Additionally, the MAGs were utilized based on the magnetic dipole model with a de-trend and a spatial median filtering method to estimate the magnetic anomaly of the landmines while considering various data spatial intervals. The joint airborne GPR and MAG system was implemented by combining and integrating the GPR and MAG models for experimental validation. Through this comprehensive approach, which included experiments, simulations, and data processing, the design parameters of the final system were obtained. These design parameters can be used in the development and application of landmine detection systems based on airborne GPR and MAG technology.

The complex envelope (CE) finite-difference time-domain (FDTD) method has been proposed for solving electromagnetic fields of limited frequency-bands. However, it has been formulated often in terms of wave equations. In addition, its numerical properties have not been thoroughly studied. In this paper, a full-wave CE-FDTD method is presented and its numerical stability and dispersion properties are analyzed. It is found that if the carrier frequency is sufficiently high or numerical cell sizes are adequately large, the time step is no longer bounded by the stability condition; otherwise the time step is constrained by a CFL like stability condition. However, through the numerical dispersion analysis, the errors caused by large time step are not acceptable. Thus the CE-FDTD does not present much gain in computation efficiency over the conventional FDTD.

Ground penetrating radar (GPR) is an efficient and no damage method which can be used in look-ahead detection of various obstacles like large boulders, concrete pile foundations and metal pipes or plates in the construction of slurry balance shield machine tunnels. Based on finite difference time domain method (FDTD), GPR responses were investigated using synthetic forward modellings. Numerical simulations show that obstacles can be identified in images of GPR.

The simple source model used in the conventional finite difference time domain (FDTD) algorithm gives rise to large errors. Conventional second-order FDTD has large errors (order h **2/ 12), h = grid spacing), and the errors due to the source model further increase this error. Nonstandard (NS) FDTD, based on a superposition of second-order finite differences, has been demonstrated to give much higher accuracy than conventional FDTD for the sourceless wave equation and Maxwell’s equations (h**6 / 24192). Since the Green’s function for the wave equation in free space is known, we can compute the field due to a point source. This analytical solution is inserted into the NS finite difference (FD) model and the parameters of the source model are adjusted so that the FDTD solution matches the analytical one. To derive the scattered field source model, we use the NS-FD model of the total field and of the incident field to deduce the correct source model. We find that sources that generate a scattered field must be modeled differently from ones radiate into free space. We demonstrate the high accuracy of our source models by comparing with analytical solutions. This approach yields a significant improvement inaccuracy, especially for the scattered field, where we verified the results against Mie theory. The computation time and memory requirements are about the same as for conventional FDTD. We apply these developments to solve propagation problems in subwavelength structures.

gprMax is open source software that simulates electromagnetic wave propagation, using the Finite-Difference Time-Domain (FDTD) method, for the numerical modelling of Ground Penetrating Radar (GPR). gprMax was originally developed in 1996 when numerical modelling using the FDTD method and, in general, the numerical modelling of GPR were in their infancy. Current computing resources offer the opportunity to build detailed and complex FDTD models of GPR to an extent that was not previously possible. To enable these types of simulations to be more easily realised, and also to facilitate the addition of more advanced features, gprMax has been redeveloped and significantly modernised. The original C-based code has been completely rewritten using a combination of Python and Cython programming languages. Standard and robust file formats have been chosen for geometry and field output files. New advanced modelling features have been added including: an unsplit implementation of higher order Perfectly Matched Layers (PMLs) using a recursive integration approach; diagonally anisotropic materials; dispersive media using multi-pole Debye, Drude or Lorenz expressions; soil modelling using a semi-empirical formulation for dielectric properties and fractals for geometric characteristics; rough surface generation; and the ability to embed complex transducers and targets. Program summaryProgram title: gprMaxCatalogue identifier: AFBG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFBG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v3No. of lines in distributed program, including test data, etc.: 627180No. of bytes in distributed program, including test data, etc.: 26762280Distribution format: tar.gzProgramming language: Python.Computer: Any computer with a Python interpreter and a C compiler.Operating system: Microsoft Windows, Mac OS X, and Linux.RAM: Problem dependentClassification: 10.External routines: Cython[1], h5py[2], matplotlib[3], NumPy[4], mpi4py[5]Nature of problem: Classical electrodynamicsSolution method: Finite-Difference Time-Domain (FDTD)Running time: Problem dependent

Laguerre polynomials to circumvent the Courant-Friedrich-Lewy (CFL) stability condition in the conventional Finite-Difference Time-Domain (FDTD) method are dealt with in this paper. The programming codes based on Laguerre-FDTD (LFDTD) method, FDTD method and Perfectly Matched Layers (PMLs) Absorbing Boundary Conditions (ABCs) are applied to analyze microstrip fed-rectangular patch antennas. The patch antenna of interest is fed by uniform and non uniform microstrip lines, respectively. In this paper, sinus tapered microstrip line configurations are adopted in order to achieve broadband impedance matching on a frequency band ranging form 6 GHz to 19 GHz. The total and incident waves, the return loss S 11 (f) and the input impedance Z in (f) of the microstrip line fed-rectangular patch antenna, obtained from the LFDTD formulation, are compared to those of the conventional FDTD method.