Wim A. Mulder

Time-domain modeling of electromagnetic diffusion with a frequency-domain code

We modeled time-domain EM measurements of induction currents for marine and land applications with a frequency-domain code. An analysis of the computational complexity of a number of numerical methods shows that frequency-domain modeling followed by a Fourier transform is an attractive choice if a sufficiently powerful solver is available. A recently developed, robust multigrid solver meets this requirement. An interpolation criterion determined the automatic selection of frequencies. The skin depth controlled the construction of the computational grid at each frequency. Tests of the method against exact solutions for some simple problems and a realistic marine example demonstrate that a limited number of frequencies suffice to provide time-domain solutions after piecewise-cubic Hermite interpolation and a fast Fourier transform.

Exploiting the airwave for time-lapse reservoir monitoring with CSEM on land

In the application of controlled source electromagnetics for reservoir monitoring on land, repeatability errors in the source will mask the time-lapse signal due to hydrocarbon production when recording surface data close to the source. We demonstrate that at larger distances, the airwave will still provide sufficient illumination of the target. The primary airwave diffuses downward into the earth and then is scattered back to the surface. The time-lapse difference of its recorded signal reveals the outline on the surface of the resistivity changes in a hydrocarbon reservoir under production. However, repeatability errors in the primary airwave can destroy the signal-to-noise ratio of the time-lapse data. We present a simple and effective method to remove the primary airwave from the data, which we call partial airwave removal. For a homogeneous half space and a delta-function type of source, the surface expression of the airwave does not depend on frequency. For this reason, the primary airwave can be subtracted from the data using recordings at two frequencies, one low enough with a skin depth of the order of the reservoir depth that is sensitive to the reservoir, the other high enough to only sense the near surface. The method does not affect secondary airwave components created by signals that have propagated through the earth and returned to the surface. We show that the method provides a direct indicator of production-related time-lapse changes in the reservoir. We illustrate this for several models, including a general 3D heterogeneous model and one with strong surface topography, for situations where survey repeatability errors are large.

Effects of the airwave in time-domain marine controlled-source electromagnetics

In marine time-domain controlled-source electromagnetics (CSEM), there are two different acquisition methods: with horizontal sources for fast and simple data acquisition or with vertical sources for minimizing the effects of the airwave. Illustrations of the electric field as a function of space and time for various source antenna orientations, based on analytical formulation of the electric field in two half-spaces, provide insights into the properties of the airwave and the nature of diffuse electric fields. Observing the development of the electric field over time and space reveals that diffusive fields exhibit directionality. Therefore, techniques that have thus far mostly been applied to wavefields can be adapted for CSEM. Examples range from the well-known up-down decomposition to beam steering. Vertical sources have the advantage of not creating an airwave. On the other hand, it is quite difficult to achieve perfect verticality of the source antenna. Results, using a numerically modeled data set to analyze the impact of the airwave on a signal from a subsurface reservoir in the case of a slightly dipping vertical source, indicate that already for a dip of 0:05 � , the airwave contributes 20% to the complete electric field in our configuration of reservoir depth, water thickness, and conductivity values.

Salt Reconstruction in Full-Waveform Inversion With a Parametric Level-Set Method

Seismic full-waveform inversion tries to estimate subsurface medium parameters from seismic data. Areas with subsurface salt bodies are of particular interest because they often have hydrocarbon reservoirs on their sides or underneath. Accurate reconstruction of their geometry is a challenge for current techniques. This paper presents a parametric level-set method for the reconstruction of salt-bodies in seismic full-waveform inversion. We split the subsurface model in two parts: a background velocity model and a salt body with known velocity but undetermined shape. The salt geometry is represented by a level-set function that evolves during the inversion. We choose radial basis functions to represent the level-set function, leading to an optimization problem with a modest number of parameters. A common problem with level-set methods is to fine-tune the width of the level-set boundary for optimal sensitivity. We propose a robust algorithm that dynamically adapts the width of the level-set boundary to ensure faster convergence. Tests on a suite of idealized salt geometries show that the proposed method is stable against a modest amount of noise. We also extend the method to joint inversion of both the background velocity model and the salt geometry.

Preconditioning for linearized inversion of attenuation and velocity perturbations

Summary We consider the linearized constant density viscoacoustic wave equation, which involves simultaneous inversion both for velocity and attenuation contrasts or perturbations. T he medium parameters can be characterized by a complex-valued velocity that includes wave speed as well as attenuation. The least-squares error measures the squared norm of the difference between modeled and observed data. Its gradient with respect to the medium parameters represents a migration image. We can use a gradient-based minimization algorithm to invert for the model parameters. Convergence rates will improve by using a suitable preconditioner, which usually is some approximation of the Hessian. For the linearized, constant density viscoacoustic wave equation we derive an exact Hessian that differs from the more conventional Hessian by including the complex-valued part. For the inverse problem, we consider a single point scatterer and investigate the one-dimensional vertical line as preconditioner. We observe that this preconditioner improves the convergence and accuracy. However, the question remains if the suggested preconditioner is feasible, since computing and inverting it is computationally costly.

Getting Around the Ambiguity In Attenuation Imaging

Migration of seismic data in a constant-density visco-acoustic model provides reflectors that consist of velocity and of attenuation perturbations relative to a given background model. Earlier work has shown that simultaneous imaging of these perturbations leads to an ambiguity: a velocity perturbation may produce nearly the same data as an attenuation perturbation. There are various approaches that can circumvent the ambiguity. An acquisition geometry with sources and receivers all around the scattering object removes the ambiguity, but is useless in geophysical exploration. Including causality in the scattering model when using the Born approximation will help if data with a sufficient broad frequency band are available. Here we consider the question if nonlinear inversion is able to get around the ambiguity. A numerical experiment with a visco-acoustic frequency-domain finite-difference code suggests a positive answer if causality is included, although the number of iterations required to reach an acceptable solution is very large. Without causality, nonlinear inversion appears to provide better results than linearized inversion with the Born approximation, but it seems that the ambiguity still hampers convergence to the true solution.