# مدل‌سازی عددی داده‌های رادار نفوذی زمین (GPR) با استفاده از روش اجزاء محدود

نوع مقاله : مقاله تحقیقی‌ (پژوهشی‌)

نویسندگان

1 گروه فیزیک زمین، مؤسسة ژئوفیزیک، دانشگاه تهران، تهران، ایران

2 دانشکده مهندسی نفت، دانشگاه صنعتی امیر کبیر، تهران، ایران

3 موسسه ژئوفیزیک، دانشگاه تهران، تهران، ایران

چکیده

در این پژوهش از روش­ اجزاء محدود (FEM) به‌منظور مدل­سازی امواج الکترومغناطیس استفاده شده است. با توجه به قابلیت­های این روش، ابتدا معادلات ماکسول در حیطه مکان گسسته‌سازی می­شوند، سپس شرایط مرزی به‌منظور جذب امواج در کرانه­های مدل اعمال می­شود که از روش مرز جاذب مرتبه­ اول کلایتون و انگکوئیست استفاده شده است. در روش FEM عبارت مرز یک جمله­ جداگانه می­باشد، به همین دلیل اعمال شرایط مرزی در این روش بسیار آسان‌تر از روش تفاضل محدود (FDM) است. پس از گسسته‌سازی مکانی با استفاده از روش FEM، گسسته‌سازی زمانی معادلات با استفاده از روش تفاضل محدود مرکزی صورت می­گیرد. گسسته­سازی زمانی معادلات، حجیم­ترین و زمان­برترین بخش محاسبات در مدل­سازی هستند که نحوه گسسته‎سازی مکانی نقش بسزایی در این فرآیند ایفا می­کند. با توجه به تُنُک بودن و متقارن بودن ماتریس­های تشکیل شده در روش FEM، درصورتی‌که از الگوریتم­های بهینه به‌منظور محاسبات و ذخیره­سازی ماتریس­ها در این روش استفاده شود، زمان و هزینه محاسباتی به‌طور قابل ملاحظه­ای کاهش خواهد یافت که در این تحقیق چند تکنیک به‌منظور کاهش حجم و زمان محاسبات در نرم‌افزار متلب ارائه شده است. فرمول­ها و روابط ارائه شده در این تحقیق به شکل ماتریسی هستند که به‌راحتی در نرم‌افزار متلب قابل کد نویسی می­باشند. به‌منظور بررسی روش FEM در مدل­سازی داده­های GPR، الگوریتم­های توسعه داده شده بر روی مدل­های زمین‌شناسی فرضی آزمایش شده است که نتایج حاصل از مدل­سازی دارای دقت قابل قبولی هستند.

کلیدواژه‌ها

عنوان مقاله [English]

### Numerical modeling of round-penetrating radar (GPR) usingfinite-element method

نویسندگان [English]

• َAmin Rahimi Dalkhani 2
• Navid Amini 3
1 Institute of Geophysics, University of Tehran, Tehran, Iran
2 Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran
3 Institute of Geophysics, University of Tehran, Tehran, Iran
چکیده [English]

Ground-penetrating radar (GPR) is a popular geophysical method for high-resolution imaging of the shallow subsurface structures. Numerical modeling of radar waves plays a significant role in interpretation, processing, and imaging of GPR data. A number of different approaches have been presented for the numerical modeling of GPR data. The most common approach for GPR modeling is the finite-difference method (FDM) because the FDM approach is conceptually simple and easy to program. The difficulties in applying boundary conditions at non-linear boundaries and the lack of sufficient accuracy in complex geometries are the most important drawbacks of FDM.
This paper presents a finite-element method, for simulation of ground-penetrating radar (GPR) in two dimensions in the time-domain. FEM is a powerful and versatile numerical technique for handling problems involving complex geometries and inhomogeneous media. The technique is based on a weak formulation of Maxwell’s equations. In the FEM method, the wavefield is discretized on the elements using Lagrange interpolation, and integration over an element is accomplished based upon the Gaussian-quad integration rule. The major difference between the various numerical methods is in the spatial discretization. In the elemental-based methods, the complex geometry of the problem is divided into several smaller and simpler elements, then the integrals are calculated for each element. These methods have no with any regular or irregular geometry. The responses of the model in the finite-element methods are approximated in nodal points, so nodal polynomials of Lagrange are used for interpolation of the model response. Besides, the systematic generality of the method makes it possible to construct general-purpose computer programs for solving a wide range of problems. In this paper, at first, Maxwell’s equations are discretized, then the boundary condition is applied to minimize artificial reflections from the edges of the computation domain. Although the governing equations and mechanisms are completely different between radar and seismic waves, most of GPR data processing approaches are derived from seismic data processing. Due to similarities in these two techniques, accordingly, we implement the first-order Clayton and Engquist absorbing boundary conditions (firstorder CE-ABC) introduced in the numerical finite-difference modeling of seismic wave propagation. This boundary condition is simple to apply. The presented formulations are in matrix notation that simplifies the implementation of the relations in computer programs, especially in MATLAB application. After spatial discretization with FEM, time discretization is done by Finite-Central Difference (FCD). The time discretization is the most massive and time-consuming step in modeling, which spatial discretization has an important role in this process. The stiffness, mass and damping matrices are sparse and symmetrical in FEM; so if we use the optimized numerical algorithms and storages strategies, computational costs and processing-time can be reduced significantly. To investigate the efficiency of FEM, the computer program has been written in MATLAB and has been tested on two models. The results show that the radar wave simulation via FEM is an accurate and effective approach in complex models.

کلیدواژه‌ها [English]

• finite-element method (FEM)
• Numerical modeling
• absorbing boundary conditions (ABC)

#### مراجع

Bergmann, T., Robertson, J. O. A., and Holliger, K., 1998, Finite-difference modelling of electromagnetic wave propagation in dispersive and attenuating media: Geophysics, 63(3), 856–867.
Cassidy, N. J., 2007b, A review of practical numerical modelling methods for the advanced interpretation of ground- penetrating radar in near-surface environ­ments: Near Surface Geophysics, 5(1), 5–22.
Clayton, R., and Engquist, B., 1977, Absorbing boundary conditions for acoustic and elastic wave equations: Bulletin of the Seismological Society of America, 67(6), 1529-1540.
Di, Q., and Wang, M., 2003, Migration of ground-penetrating radar data with a finite-element method that considers attenuation and dispersion: Geophysics, 69(2), 472−477.
Ellefsen, K. J., 1999, Effects of layered sediments on the guided wave in crosswell radar data: Geophysics, 64(6), 1698–1707.
Fang, W. Z., Li, Y. G., and Li, X., 1993, Principle of Transient Geomagnetic Sounding Method: University of Northwest Industry Press.
Feng, D., Chen, C., and Wang, H., 2012, Finite element method GPR forward simulation based on mixed boundary condition: Chinese Journal of Geophysics, 55(11), 3774−3785 (in Chinese).
Feng, D., Guo, R., and Wang, H., 2015, An element-free Galerkin method for ground penetrating radar numerical simulation: Journal of Central South University, 22(1), 261−269.
Fichtner, A., 2011, Full Seismic Waveform Modelling and Inversion: Springer.
Irving, J., and Knight, R., 2006, Numerical modeling of ground-penetrating radar in 2-D using MATLAB: Computers and Geosciences, 32(9): 1247−1258.
Jian-Ming, J., 2014, The Finite Element Method in Electromagnetics: Wiley-IEEE Press.
Joll, H. M., 2009, Ground Penetrating Radar Theory and Application: Elsevier.
Sadiku, M. N. O., 2001, Numerical Techniques in Electromagnetics: CRC Press, LLC.
Roberts, R. L., and Daniels, J. J., 1997, Modelling near-field GPR in three dime­n­sions using the FDTD method: Geophysics, 62(4), 1114–1126.
Yee, K. S., and Chen, J. S., 1997, The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations: IEEE Transactions on Antennas and Propagation, 45(3), 354-363.