ALBERT

Description: ABSTRACT The Sierra Grande region in northern Patagonia is considered the largest iron ore reserve in Argentina; however, the extension of the non‐outcropping deposits as well as the depth of the basins that contain them remains unknown. Utilizing 3D litho‐constrained inversion of gravity and magnetic data, we delimited an area with good prospects for iron ore deposits. In this region, high‐resolution magnetic and self‐potential profiles were acquired over the most important anomalies. Correlating both methodologies, it was possible to specify the possible existence of iron oxides (martite–hematite) in the form of 2D inclined sheets.

Description: ABSTRACT We summarize, for marine electromagnetic inverse problems, a newly developed inverse solution appraisal and non-linear uncertainty estimation method based on parameter reduction techniques and efficient posterior model space sampling. This method uses model compression methods to decorrelate parameters in an inverse solution and represent all feasible posterior models as linear combinations of a small number of model-derived basis vectors and corresponding coefficients. This allows us to reduce the posterior sampling problem by orders of magnitude. We further contract this reduced-dimensional posterior space by confining all acceptable models to a set of bounds mapped from our original parameter space. As a final step to increase efficiency, we implement a geometric sampling scheme that we use to approximate our restricted posterior by generating feasible models on adaptive, optimally-sparse grids. The sampled equi-feasible models are accepted according to a data misfit threshold and constitute an optimally-sparse representation of the restricted posterior model space. Although very efficient, our method imposes a bias in the posterior space by truncating the basis expansion during the model reduction step. To investigate this, we compare two types of fast and scalable bases, the discrete cosine transform and singular value decomposition. We demonstrate that while the choice of base does influence the type of models sampled and the model rejection rates, the posterior statistics are generally compatible between the methods providing confidence in the uncertainty estimations. For the marine electromagnetic problem, we show that a representative ensemble of equivalent inverse solutions can be generated for realistically-sized inverse problems and that solution appraisal and uncertainty inference follow directly from ensemble statistics.

Description: ABSTRACT A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse-grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse-grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis-Hastings. While both the sparse-grid and Bayesian samplers produce compatible results, in both examples, the sparse-grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse-grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.

Description: ABSTRACT Assuming Vertical Transverse Isotropy (VTI) symmetry the elastic anisotropy as a function of confining pressure of four carbonates and one evaporite from the Williston sedimentary basin in Saskatchewan, Canada is investigated using the ultrasonic pulse transmission method. Ultrasonic P- and S- wave velocities are obtained from cylindrical plugs cut from a main sample along horizontal, vertical and 45° orientations with respect to the sample's presumed vertical axis of symmetry. The elastic constants were then calculated from the measured velocities and densities. Anisotropy was quantified by estimating Thomsen parameters (Thomsen 1986) from elastic constants. The results show that the samples are at the best weakly anisotropic. The presence of microcracks and pores as well as the heterogeneity of the samples play an important role in defining the P- and S- wave velocities. The weak anisotropy found in these samples suggests that ‘intrinsic’ properties of these rocks negligibly contribute to the anisotropy observed at the seismic scale.

Description: ABSTRACT A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse-grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse-grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis-Hastings. While both the sparse-grid and Bayesian samplers produce compatible results, in both examples, the sparse-grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse-grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.

Description: ABSTRACT Fractures in fluid-saturated poroelastic media can be modeled as extremely thin, highly permeable, and compliant layers or by means of suitable boundary conditions that approximate the behavior of such thin layers. Since fracture apertures can be very small, the numerical simulations would require the use of extremely fine computational meshes and the use of boundary conditions would be required. In this work, we study the validity of using boundary conditions to describe the seismic response of fractures. For this purpose, we compare the corresponding scattering coefficients to those obtained from a thin-layer representation. The boundary conditions are defined in terms of fracture apertures that, in the most general case, impose discontinuity of displacements, fluid pressures, and stresses across a fracture. Furthermore, discontinuities of either fluid pressures, stresses, or both can be removed, or displacement jumps proportional to the stresses and/or pressures can be expressed via shear and normal dry compliances in order to simplify. In the examples, we vary the permeability, thickness, and porosity of the fracture and the type of fluid saturating the background medium and fractures. We observe good agreement of the scattering coefficients in the seismic range obtained with the two different approaches.

Description: In this paper we present a methodology to perform geophysical inversion of large scale linear systems via a covariance-free orthogonal transformation: the Discrete Cosine Transform (DCT). The methodology consists in compressing the matrix of the linear system as a digital image and using the interesting properties of orthogonal transformations to define an approximation of the Moore-Penrose pseudo-inverse. This methodology is also highly scalable since the model reduction achieved by these techniques increases with the number of parameters of the linear system involved due to the high correlation needed for these parameters to accomplish very detailed forward predictions, and allows for a very fast computation of the inverse problem solution. We show the application of this methodology to a simple synthetic 2D gravimetric problem for different dimensionalities and different levels of white Gaussian noise, and to a synthetic linear system whose system matrix has been generated via geostatistical simulation to produce a random field with a given spatial correlation. The numerical results show that the DCT pseudoinverse outperforms the classical least-squares techniques, mainly in presence of noise, since the solutions that are obtained are more stable and fit the observed data with a lowest RMS error. Besides, we show that the model reduction is a very effective way of parameter regularization when the conditioning of the reduced DCT matrix is taken into account. We finally show its application to the inversion of a real gravity profile in the Atacama Desert (north Chile) obtaining very successful results in this nonlinear inverse problem. The methodology presented here has a general character and can be applied to solve any linear and nonlinear inverse problems (through linearization) arising in technology and particularly in geophysics, independently of the geophysical model discretization and dimensionality. Nevertheless, the results shown in this paper are better in the case of ill-conditioned inverse problems for which the matrix compression is more efficient. In that sense, a natural extension of this methodology would be its application to the set of normal equations. This article is protected by copyright. All rights reserved

Description: ABSTRACT Classical least-squares techniques (Moore-Penrose pseudoinverse) are covariance-based and are therefore unsuitable for the solution of very large-scale linear systems in geophysical inversion due to the need of diagonalization. In this paper we present a methodology to perform the geophysical inversion of large-scale linear systems via the Discrete Wavelet Transform. The methodology consists of compressing the linear system matrix using the interesting properties of covariance-free orthogonal transformations, to design an approximation of the Moore-Penrose pseudoinverse. We show the application of the Discrete Wavelet Transform pseudoinverse to well-conditioned and ill-conditioned linear systems. We show the application to a general-purpose linear problem where the system matrix has been generated using geostatistical simulation techniques, and also to a synthetic 2D gravimetric problem with two different geological set-ups, in the noise-free and noisy cases. In both cases the Discrete Wavelet Transform pseudoinverse can be applied to the original linear system and also to the linear systems of normal equations and minimum norm. The results are compared to those obtained via the Moore-Penrose and the Discrete Cosine Transform pseudoinverses. The Discrete Wavelet Transform and the Discrete Cosine Transform pseudoinverses provide similar results and outperform the Moore-Penrose pseudoinverse, mainly in the presence of noise. In the case of well-conditioned linear systems this methodology is more efficient when applied to the least squares and minimum norm systems due to their higher condition number that allows for a more efficient compression of the system matrix. Also in the case of ill-conditioned systems with very high underdetermined character the application of the Discrete Cosine Transform to the minimum norm solution provides very good results. Both solutions might differ on their regularity, depending on the wavelet family that is adopted. These methods have a general character and can be applied to solve any linear inverse problem arising in technology and particularly in geophysics, and also to nonlinear inversion by linearization of the forward operator. This article is protected by copyright. All rights reserved

Description: ABSTRACT In this paper, we present the uncertainty analysis of the 2D electrical tomography inverse problem using model reduction and performing the sampling via an explorative member of the Particle Swarm Optimization (PSO) family, called the Regressive-Regressive PSO (RR-PSO). The procedure begins with a local inversion to find a good resistivity model located in the nonlinear equivalence region of the set of plausible solutions. The dimension of this geophysical model is then reduced using spectral decomposition, and the uncertainty space is explored via PSO. Using this approach, we show that it is possible to sample the uncertainty space of the electrical tomography inverse problem. We illustrate this methodology with the application to a synthetic and a real dataset coming from a karstic geological set-up. By computing the uncertainty of the inverse solution, it is possible to perform the segmentation of the resistivity images issued from inversion. This segmentation is based on the set of equivalent models that have been sampled, and makes it possible to answer geophysical questions in a probabilistic way, performing risk analysis. This article is protected by copyright. All rights reserved