2D approach for modelling self-potential anomalies. Application to synthetic and real data

The aim of this work is to present a 2-D Matlab code based on the finite element method for providing numerical modelling of both groundwater flow and self-potential signals. The distribution of the self-potential is obtained by starting with the solution of the groundwater flow, then computing the source current density, and finally calculating the electrical potential. The reliability of the algorithm is tested with synthetic case studies in order to simulate both the electric field resulting from the existence of a leak in the dam and SP signals associated with a pumping test in an unconfined aquifer. In addition, the algorithm was applied to field data for the localization of piping sinkholes. The results show that the outputs of the algorithm yielded satisfactory solutions, which are in good agreement with those of previous studies and field investigations. In details, the synthetic data and SP anomalies calculated by using the code are very close in terms of sign and magnitude, while real data tests clearly indicated that the computed SP signals were found to be consistent with the measured values

Estimation of the hydraulic parameters of unsaturated samples by electrical resistivity tomography

In situ and laboratory experiments have shown that electrical resistivity tomography (ERT) is an effective tool to image transient phenomena in soils. However, its application in quantifying soil hydraulic parameters has been limited. In this study, experiments of water inflow in unsaturated soil samples were conducted in an oedometer equipped to perform three-dimensional electrical measurements. Reconstructions of the electrical conductivity at different times confirmed the usefulness of ERT for monitoring the evolution of water content. The tomographic reconstructions were subsequently used in conjunction with a finite-element simulation to infer the water retention curve and the unsaturated hydraulic conductivity. The parameters estimated with ERT agree satisfactorily with those determined using established techniques, hence the proposed approach shows good potential for relatively fast characterisations. Similar experiments could be carried out on site to study the hydraulic behaviour of the entire soil deposi

Imaging of a fluid injection process using geophysical data - A didactic example

In many subsurface industrial applications, fluids are injected into or withdrawn from a geologic formation. It is of practical interest to quantify precisely where, when, and by how much the injected fluid alters the state of the subsurface. Routine geophysical monitoring of such processes attempts to image the way that geophysical properties, such as seismic velocities or electrical conductivity, change through time and space and to then make qualitative inferences as to where the injected fluid has migrated. The more rigorous formulation of the time-lapse geophysical inverse problem forecasts how the subsurface evolves during the course of a fluid-injection application. Using time-lapse geophysical signals as the data to be matched, the model unknowns to be estimated are the multiphysics forward-modeling parameters controlling the fluid-injection process. Properly reproducing the geophysical signature of the flow process, subsequent simulations can predict the fluid migration and alteration in the subsurface. The dynamic nature of fluid-injection processes renders imaging problems more complex than conventional geophysical imaging for static targets. This work intents to clarify the related hydrogeophysical parameter estimation concepts

EIT Reconstruction Algorithms: Pitfalls, Challenges and Recent Developments

We review developments, issues and challenges in Electrical Impedance Tomography (EIT), for the 4th Workshop on Biomedical Applications of EIT, Manchester 2003. We focus on the necessity for three dimensional data collection and reconstruction, efficient solution of the forward problem and present and future reconstruction algorithms. We also suggest common pitfalls or ``inverse crimes'' to avoid.Comment: A review paper for the 4th Workshop on Biomedical Applications of EIT, Manchester, UK, 200

Iterative Updating of Model Error for Bayesian Inversion

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations. We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.Comment: 39 pages, 9 figure

Nonlinear Inversion from Partial EIT Data: Computational Experiments

Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small $k$ frequencies.Comment: 24 pages, 12 figure