Mathematical Geosciences

In this paper, we present a new approach for estimating spatially-distributed reservoir properties from scattered nonlinear dynamic well measurements by promoting sparsity in an appropriate transform domain where the unknown properties are believed to have a sparse approximation. The method is inspired by recent advances in sparse signal reconstruction that is formalized under the celebrated compressed sensing paradigm. Here, we use a truncated low-frequency discrete cosine transform (DCT) is redundant to approximate the spatial parameters with a sparse set of coefficients that are identified and estimated using available observations while imposing sparsity on the solution. The intrinsic continuity in geological features lends itself to sparse representations using selected low frequency DCT basis elements. By recasting the inversion in the DCT domain, the problem is transformed into identification of significant basis elements and estimation of the values of their corresponding coefficients. To find these significant DCT coefficients, a relatively large number of DCT basis vectors (without any preferred orientation) are initially included in the approximation. Available measurements are combined with a sparsity-promoting penalty on the DCT coefficients to identify coefficients with significant contribution and eliminate the insignificant ones. Specifically, minimization of a least-squares objective function augmented by an l 1-norm of DCT coefficients is used to implement this scheme. The sparsity regularization approach using the l 1-norm minimization leads to a better-posed inverse problem that improves the non-uniqueness of the history matching solutions and promotes solutions that are, according to the prior belief, sparse in the transform domain. The approach is related to basis pursuit (BP) and least absolute selection and shrinkage operator (LASSO) methods, and it extends the application of compressed sensing to inverse modeling with nonlinear dynamic observations. While the method appears to be generally applicable for solving dynamic inverse problems involving spatially-distributed parameters with sparse representation in any linear complementary basis, in this paper its suitability is demonstrated using low frequency DCT basis and synthetic waterflooding experiments.

This paper deals with kriging-based interpolation of dosimetric data. Such data typically show some inhomogeneities that are difficult to take into account by means of the usual non-stationary models available in geostatistics. By including the knowledge of suspected potential sources (such as pipes or reservoirs) better estimates can be obtained, even when no hard data are available on these sources. The proposed method decomposes the measured dose rates into a diffuse homogeneous term and the contribution from the sources. Accordingly, two random function models are introduced, the first associated with the diffuse term, and the second with the unknown and unmeasured source contribution. Estimation of the model parameters is based on cross-validation quadratic error. As a bonus, the resulting model makes it possible to estimate the source activity. The efficiency of this approach is demonstrated on data simulated according to the physical equations of the system. The method is available to researchers through an R-package provided by the authors upon request.

A numerical framework for modeling folds in structural geology is presented. This framework is based on a novel and recently published Hamilton–Jacobi formulation by which a continuum of layer boundaries of a fold is modeled as a propagating front. All the fold classes from the classical literature (parallel folds, similar folds, and other fold types with convergent and divergent dip isogons) are modeled in two and three dimensions as continua defined on a finite difference grid. The propagating front describing the fold geometry is governed by a static Hamilton–Jacobi equation, which is discretized by upwind finite differences and a dynamic stencil construction. This forms the basis of numerical solution by finite difference solvers such as fast marching and fast sweeping methods. A new robust and accurate scheme for initialization of finite difference solvers for the static Hamilton–Jacobi equation is also derived. The framework has been integrated in simulation software, and a numerical example is presented based on seismic data collected from the Karama Block in the North Makassar Strait outside Sulawesi.

Cross-wavelet transform (XWT) is proposed as a data analysis technique for geological time-series. XWT permits the detection of cross-magnitude, phase differences (= lag time), nonstationarity, and coherency between signals from different paleoclimate records that may exhibit large stratigraphic uncertainties and noise levels. The approach presented herein utilizes a continuous XWT technique with Morlet wavelet as the mother function, allows for variable scaling factors for time and scale sampling, and the automatic extraction of the most significant periodic signals. XWT and cross-spectral analysis is applied on computer generated time-series as well as two independently sampled proxy records (CO2 content approximated from plant cuticles and paleotemperature derived from δ 18O from marine fossil carbonate) of the last 290 Ma. The influence of nonstationarities in the paleoclimate records that are introduced by stratigraphic uncertainties were a particular focus of this study. The XWT outputs of the computer-models indicate that a potential causal relationship can be distorted if different geological time-scale and/or large stratigraphic uncertainties have been used. XWT detect strong cross-amplitudes (∼200 ppm ‰) between the CO2 and δ 18O record in the 20–50 Myr waveband, however, fluctuating phase differences prevent a statistical conclusion on causal relationship at this waveband.

The practice of stochastic simulation for different environmental and earth sciences applications creates new theoretical problems that motivate the improvement of existing algorithms. In this context, we present the implementation of a new version of the direct sequential co-simulation (Co-DSS) algorithm. This new approach, titled Co-DSS with joint probability distributions, intends to solve the problem of mismatch between co-simulation results and experimental data, i.e. when the final biplot of simulated values does not respect the experimental relation known for the original data values. This situation occurs mostly in the beginning of the simulation process. To solve this issue, the new co-simulation algorithm, applied to a pair of covariates Z 1(x) and Z 2(x), proposes to resample Z 2(x) from the joint distribution F(z 1,z 2) or, more precisely, from the conditional distribution of Z 2(x 0), at a location x 0, given the previously simulated value $z_{1}^{(l)}(x_{0})$ ( $F(Z_{2}|Z_{1}=z_{1}^{(l)}(x_{0})$ ). The work developed demonstrates that Co-DSS with joint probability distributions reproduces the experimental bivariate cdf and, consequently, the conditional distributions, even when the correlation coefficient between the covariates is low.

The objective of the current study is to utilize an innovative method called “change probabilities” for describing fracture roughness. In order to detect and visualize anisotropy of rock joint surfaces, the roughness of one-dimensional profiles taken in different directions is quantified. The central quantifiers, change probabilities, are based on counting monotonic changes in discretizations of a profile. These probabilities, which usually vary with the scale, can be reinterpreted as scale-dependent Hurst exponents. For a large class of Gaussian stochastic processes, change probabilities are shown to be directly related to the classical Hurst exponent, which generalizes a relationship known for fractional Brownian motion. While related to this classical roughness measure, the proposed method is more generally applicable, therefore increasing the flexibility of modeling and investigating surface profiles. In particular, it allows a quick and efficient visualization and detection of roughness anisotropy and scale dependence of roughness.