Finite difference modeling of 2D GPR data considering attenuation

Abstract

We use a finite-difference scheme to simulate 2D ground penetrating radar data by solving the damped wave equation. We show that the stability condition for the damped scalar wave equation does not depend on the damping factor. As an example application of our modeling tool we investigate a typical situation in granite prospecting, where a conductive clay overburden masks the real position of the fractures in the granite. The results of the modeling have a good agreement with the actual data. Also, the algorithm shows well the increase in attenuation with the increase of the frequency. The algorithm shows to be a good basic modeling tool that can be used for further applications and for comparisons with other modeling methods like pseudospectral or finite elements. Introduction 2D forward numerical modeling of GPR data is an important tool to validate the interpretation of actual radargrams. Goodman (1994) and Cai and McMechan (1995) presented forward modeling routines using raytracing techniques. The drawback of these techniques is that they do not simulate diffraction events, as outlined by Zeng et al. (1995), who presented modeling by Fourier methods and compared them with the ray methods. Casper and Kung (1996) applied the pseudospectral forward modeling algorithm on GPR based on an explicit solution of the 2-D lossy electromagnetic wave equation and Chen and Huang (1996) used finite-differences following the same approach. All these methods have in common the assumption of a dielectric behavior of the earth, which is valid for most geologic applications of GPR, figuring in the frequency range between 10 and 1000 MHz, and imaging materials with conductivities lower than 100 mS/m (Davis and Annan, 1989). Under such conditions the velocity of propagation remains constant and the attenuation may be considered seperately from the velocity, what gives the electromagnetic radar waves the same behavior as the seismic acoustic waves. In the present work we use this electromagnetic-acoustic analogy and employ classical finite-differences to solve the damped scalar wave equation following Chen and Huang (1996) and Alford et. al. (1974). We show that the stability condition for the damped scalar wave equation does not depend on the damping factor, therefore reducing to the same condition presented by Alford et. al. (1974) for the wave equation without attenuation. Also, we study the effect of attenuation on radargrams by computing 2D synthetic zero-offset sections. As an example application of our modeling tool we investigate a typical situation in granite prospecting, where a conductive clay overburden masks the real position of the fractures in the granite. The results of the modeling have a good agreement with the actual data and show the usefulness of our approach. The algorithm shows to be a good basic modeling tool that can be used for further applications in migration or for comparisons with other modeling methods like the pseudospectral or finite elements. The Electromagnetic Wave Equation Our derivation of the 2D electromagnetic damped waveequation follows Casper and Kung (1996). Maxwell’s equations are t ∂ ∇× = − − ∂ s B E M (1) t ∂ ∇× = + + ∂ c D H J s J (2)