Michael Oristaglio

Finite-difference time-domain simulation of ground penetrating radar on dispersive, inhomogeneous, and conductive soils

A three-dimensional (3D) time-domain numerical scheme for simulation of ground penetrating radar (GPR) on dispersive and inhomogeneous soils with conductive loss is described. The finite-difference time-domain (FDTD) method is used to discretize the partial differential equations for time stepping of the electromagnetic fields. The soil dispersion is modeled by multiterm Lorentz and/or Debye models and incorporated into the FDTD scheme by using the piecewise-linear recursive convolution (PLRC) technique. The dispersive soil parameters are obtained by fitting the model to reported experimental data. The perfectly matched layer (PML) is extended to match dispersive media and used as an absorbing boundary condition to simulate an open space. Examples are given to verify the numerical solution and demonstrate its applications. The 3D PML-PLRC-FDTD formulation facilitates the parallelization of the code. A version of the code is written for a 32-processor system, and an almost linear speedup is observed.

Diffusion of electromagnetic fields into a two-dimensional earth; a finite-difference approach

We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space...

Reverse-time wave-field extrapolation, imaging, and inversion

The scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background, which are expressed as a scattering potential. In general, there is no closed‐form expression for the kernel of this integral representation, although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array. When the receiver array completely surrounds the scatterer, the kernel reduces to the imaginary part of the Green’s function for the background medium. This integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential. Two such relations are given: (1) for the migrated image, obtained by deconvolving the extrapolated field with the incident field; and (2) for the reconstructed image, obtained by applying a one‐way wave operator to the extrapolated field and then deconvolving by the incident field. The migrated image highlig...

Application of perfectly matched layers to the transient modeling of subsurface EM problems

Berengers perfectly matched layers (PML) have been found to be very efficient as a material absorbing boundary condition (ABC) for finite‐difference time‐domain (FDTD) modeling of lossless media. In this paper, we apply the PML technique to truncate the simulation region of conductive media. Examples are given to show some possible applications of the PML technique to subsurface problems with lossy media. To apply the PML ABC for lossy media, we first modify the original 3-D Maxwells equations to achieve PML at the boundaries of the simulation region. The modified equations are then solved by using a staggered grid with a central‐differencing scheme. A 3-D FDTD code has been written on the basis of our PML formulation to simulate the electromagnetic field responses of a dipole source in both lossless and lossy media. The code is first tested against analytical solutions for homogeneous media of different losses and then applied to some subsurface problems, such as a geological fault and a buried gas tan...

Spatial Resolution of Migration Algorithms

This paper presents a systematic approach to the description of spatial resolution of seismic experiments and migration (or inversion) algorithms.

A modeling study of borehole radar for oil-field applications

This paper examines the suitability of borehole radar for near‐wellbore imaging. The maximum imaging range is primarily determined by the conductivity of the formation in which the borehole lies and the reflectivity of the targets. Under similar medium contrast, formation interfaces result in much stronger reflections than fractures. Complex horizontal borehole geometries are modeled with a 3‐D finite‐difference time‐domain (FDTD) code. Borehole effects, which are often almost insurmountable for acoustic methods, are very small for radar. As a result, the reflections in general are visually identifiable on the synthetic radar waveforms even before any signal processing. Therefore, borehole radar is a promising approach to map structures in the immediate vicinity of the borehole for a penetration depth of at least a few meters in relatively less‐conductive reservoirs (e.g., <0.03 S/m). As such, it complements borehole acoustic methods and has potential for geosteering applications.

Inversion of diffusive transient electromagnetic data by a conjugate‐gradient method

Inversion of three-dimensional transient electromagnetic (TEM) data to obtain electrical conductivity and permeability can be done by a time-domain algorithm that extends to diffusive electromagnetic (EM) fields the imaging methods originally developed for seismic wavefields (Claerbout, 1971; Tarantola, 1984). The algorithm uses a conjugate-gradient search for the minimum of an error functional involving EM measurements governed by Maxwells equations without displacement currents. The connection with wavefield imaging comes from showing that the gradient of the error functional can be computed by propagating the errors back into the model in reverse time and correlating the field generated by the backpropagation with the incident field at each point. These two steps (backpropagation and cross correlation) are the same ones used in seismic migration. The backpropagated TEM fields satisfy the adjoint Maxwells equations, which are stable in reverse time. With magnetic field measurements the gradient of the error functional with respect to conductivity is the cross correlation of the backpropagated electric field with the incident electric field, whereas the gradient with respect to permeability is the cross correlation of the backpropagated magnetic field with the time derivative of the incident magnetic field. Tests on two-dimensional models simulating crosswell TEM surveys produce good images of a conductive block scatterer, with both exact and noisy data, and of a dipping conductive layer. Convergence, however, is slow.

3-D electromagnetic modeling and nonlinear inversion

We describe a new algorithm for 3-D electromagnetic inversion that uses global integral and local differential equations for both the forward and inverse problems. The coupled integral and differential equations are discretized by the finite element method and solved on a parallel computer using domain decomposition. The structure of the algorithm allows efficient solution of large 3-D inverse problems. Tests on both synthetic and field data show that the algorithm converges reliably and efficiently and gives high-resolution conductivity images.

GPR imaging using the generalized Radon transform

A new algorithm for imaging ground-penetrating radar (GPR) data follows from the theory of the generalized Radon transform (GRT), which was developed and has been used extensively for seismic imaging. The algorithm enables separate reconstructions (to first-order accuracy) of subsurface permittivity and conductivity images. A pseudovector inverse adapts the original scalar Radon transform theory for vector electromagnetic data. Synthetic examples show that the algorithm can image plastic and metallic pipes buried in a half-space. In the examples, the half-space is 100 ohm-m with a relative permittivity of 10. The pipes are 2 cm thick and 15 cm in outer diameter and are buried at a depth of 2 m. The plastic pipes are assumed to be pure insulators, whereas the metallic pipes are assumed to be pure conductors with a conductivity value of 10 6 S/m. The transmitter waveform has a peak frequency of 200 MHz with a 500-MHz bandwidth. The results show that quantitative images are obtained for plastic pipes, whose scattering is weak (so the Born approximation is accurate). Good images are also obtained for metallic pipes. As with most imaging algorithms, a sharp image requires good estimates of the properties of the background medium.

A guide to current uses of vertical seismic profiles

Vertical seismic profiles (VSPs) are small‐scale seismic surveys in which geophones are lowered into a well to record waves traveling both down into the earth (direct waves from the surface source and downgoing multiples) and back toward the surface (primary reflections and upgoing multiples). VSPs thus contain information about the reflection and transmission properties of the earth with a coverage that depends upon the geometry of the VSP experiment and the structure near the well. This article describes the uses of VSPs in seismic exploration that have been published in the last three years and is designed to complement the more detailed surveys by Hardage (1983) and Balch and Lee (1984). When the earth is horizontally layered, the well is vertical, and the source is close to the wellhead, upgoing and downgoing waves recorded by the VSP travel vertically, and the VSP can be used to calibrate surface seismic sections by providing the time‐to‐depth curve and allowing a detailed analysis of reflection and...