Showing papers in "Mathematical Geosciences in 2010"

Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

Abstract: The advent of multiple-point geostatistics (MPS) gave rise to the integration of complex subsurface geological structures and features into the model by the concept of training images Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction In these approaches, the training image is used to construct a pattern database Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image In this paper, an entirely different approach will be taken toward geostatistical modeling A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented In the developed methodology, patterns scanned from the training image are represented as points in a Cartesian space using multidimensional scaling The idea behind this mapping is to use distance functions as a tool for analyzing variability between all the patterns in a training image These distance functions can be tailored to the application at hand Next, by significantly reducing the dimensionality of the problem and using kernel space mapping, an improved pattern classification algorithm is obtained This paper discusses the various implementation details to accomplish these ideas Several examples are presented and a qualitative comparison is made with previous methods An improved pattern continuity and data-conditioning capability is observed in the generated realizations for both continuous and categorical variables We show how the proposed methodology is much less sensitive to the user-provided parameters, and at the same time has the potential to reduce computational time significantly

The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets

Abstract: Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.

Measurement of Areas on a Sphere Using Fibonacci and Latitude–Longitude Lattices

Abstract: The area of a spherical region can be easily measured by considering which sampling points of a lattice are located inside or outside the region. This point-counting technique is frequently used for measuring the Earth coverage of satellite constellations, employing a latitude–longitude lattice. This paper analyzes the numerical errors of such measurements, and shows that they could be greatly reduced if the Fibonacci lattice were used instead. The latter is a mathematical idealization of natural patterns with optimal packing, where the area represented by each point is almost identical. Using the Fibonacci lattice would reduce the root mean squared error by at least 40%. If, as is commonly the case, around a million lattice points are used, the maximum error would be an order of magnitude smaller.

High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

Abstract: The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian, and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants.

Reconstruction of Incomplete Data Sets or Images Using Direct Sampling

Abstract: With increasingly sophisticated acquisition methods, the amount of data available for mapping physical parameters in the geosciences is becoming enormous. If the density of measurements is sufficient, significant non-parametric spatial statistics can be derived from the data. In this context, we propose to use and adapt the Direct Sampling multiple-points simulation method (DS) for the reconstruction of partially informed images. The advantage of the proposed method is that it can accommodate any data disposition and that it can indifferently deal with continuous and categorical variables. The spatial patterns found in the data are mimicked without model inference. Therefore, very few assumptions are required to define the spatial structure of the reconstructed fields, and very limited parameterization is needed to make the proposed approach extremely simple from a user perspective. The different examples shown in this paper give appealing results for the reconstruction of complex 3D geometries from relatively small data sets.

Compressed History Matching: Exploiting Transform-Domain Sparsity for Regularization of Nonlinear Dynamic Data Integration Problems

Abstract: 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.

High-order Stochastic Simulation of Complex Spatially Distributed Natural Phenomena

Abstract: Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data. Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity in the data used, in addition to high order.

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography.

Abstract: A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.

Direct sequential co-simulation with joint probability distributions

Abstract: 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.

Improving the ensemble estimate of the Kalman gain by bootstrap sampling

Abstract: Using a small ensemble size in the ensemble Kalman filter methodology is efficient for updating numerical reservoir models but can result in poor updates following spurious correlations between observations and model variables. The most common approach for reducing the effect of spurious correlations on model updates is multiplication of the estimated covariance by a tapering function that eliminates all correlations beyond a prespecified distance. Distance-dependent tapering is not always appropriate, however. In this paper, we describe efficient methods for discriminating between the real and the spurious correlations in the Kalman gain matrix by using the bootstrap method to assess the confidence level of each element from the Kalman gain matrix. The new method is tested on a small linear problem, and on a water flooding reservoir history matching problem. For the water flooding example, a small ensemble size of 30 was used to compute the Kalman gain in both the screened EnKF and standard EnKF methods. The new method resulted in significantly smaller root mean squared errors of the estimated model parameters and greater variability in the final updated ensemble.

Towards Stochastic Time-Varying Geological Modeling

Abstract: The modeling of subsurface geometry and properties is a key element to understand Earth processes and manage natural hazards and resources. In this paper, we suggest this field should evolve beyond pure data fitting approaches by integrating geological concepts to constrain interpretations or test their consistency. This process necessarily calls for adding the time dimension to 3D modeling, both at the geological and human time scales. Also, instead of striving for one single best model, it is appropriate to generate several possible subsurface models in order to convey a quantitative sense of uncertainty. Depending on the modeling objective (e.g., quantification of natural resources, production forecast), this population of models can be ranked. Inverse theory then provides a framework to validate (or rather invalidate) models which are not compatible with certain types of observations. We review recent methods to better achieve both stochastic and time-varying geomodeling and advocate that the application of inversion should rely not only on random field models, but also on geological concepts and parameters.

The Value of Information in Spatial Decision Making

Abstract: Experiments performed over spatially correlated domains, if poorly chosen, may not be worth their cost of acquisition. In this paper, we integrate the decision-analytic notion of value of information with spatial statistical models. We formulate methods to evaluate monetary values associated with experiments performed in the spatial decision making context, including the prior value, the value of perfect information, and the value of the experiment, providing imperfect information. The prior for the spatial distinction of interest is assumed to be a categorical Markov random field whereas the likelihood distribution can take any form depending on the experiment under consideration. We demonstrate how to efficiently compute the value of an experiment for Markov random fields of moderate size, with the aid of two examples. The first is a motivating example with presence-absence data, while the second application is inspired by seismic exploration in the petroleum industry. We discuss insights from the two examples, relating the value of an experiment with its accuracy, the cost and revenue from downstream decisions, and the design of the experiment.

Ore Grade Prediction Using a Genetic Algorithm and Clustering Based Ensemble Neural Network Model

Abstract: Accurate prediction of ore grade is essential for many basic mine operations, including mine planning and design, pit optimization, and ore grade control. Preference is given to the neural network over other interpolation techniques for ore grade estimation because of its ability to learn any linear or non-linear relationship between inputs and outputs. In many cases, ensembles of neural networks have been shown, both theoretically and empirically, to outperform a single network. The performance of an ensemble model largely depends on the accuracy and diversity of member networks. In this study, techniques of a genetic algorithm (GA) and k-means clustering are used for the ensemble neural network modeling of a lead–zinc deposit. Two types of ensemble neural network modeling are investigated, a resampling-based neural ensemble and a parameter-based neural ensemble. The k-means clustering is used for selecting diversified ensemble members. The GA is used for improving accuracy by calculating ensemble weights. Results are compared with average ensemble, weighted ensemble, best individual networks, and ordinary kriging models. It is observed that the developed method works fairly well for predicting zinc grades, but shows no significant improvement in predicting lead grades. It is also observed that, while a resampling-based neural ensemble model performs better than the parameter-based neural ensemble model for predicting lead grades, the parameter-based ensemble model performs better for predicting zinc grades.

Hunting for Geochemical Associations of Elements: Factor Analysis and Self-Organising Maps

Abstract: Two approaches, factor analysis (FA) and self-organising maps (SOM), have been used for the determination of geochemical associations in the two case studies evaluated here. In both case studies, different associations of elements, derived from different anthropogenic sources (smelters, ironworks, and chemical industry), are presented, together with natural associations of elements, all representing different geological environments. FA and SOM give similar results, despite some differences. Most similarities were achieved with the group of elements affected by strong pollution caused by smelting activities. The biggest difference between the two is that SOM can combine different groups into one, especially in the case of associations of elements connected with mild pollution (ironworking, chemical industry). The biggest advantage of SOM as opposed to FA is that SOM allow us to process variables, which are not normally distributed, or even of attributive nature. The use of such variables in SOM classification to prove the origins of associations of elements is also demonstrated here.

Pebbles, Shapes, and Equilibria

Abstract: The shape of sedimentary particles may carry important information on their history. Current approaches to shape classification (e.g. the Zingg or the Sneed and Folk system) rely on shape indices derived from the measurement of the three principal axes of the approximating tri-axial ellipsoid. While these systems have undoubtedly proved to be useful tools, their application inevitably requires tedious and ambiguous measurements, also classification involves the introduction of arbitrarily chosen constants. Here we propose an alternative classification system based on the (integer) number of static equilibria. The latter are points of the surface where the pebble is at rest on a horizontal, frictionless support. As opposed to the Zingg system, our method relies on counting rather than measuring. We show that equilibria typically exist on two well-separated (micro and macro) scales. Equilibria can be readily counted by simple hand experiments, i.e. the new classification scheme is practically applicable. Based on statistical results from two different locations we demonstrate that pebbles are well mixed with respect to the new classes, i.e. the new classification is reliable and stable in that sense. We also show that the Zingg statistics can be extracted from the new statistics; however, substantial additional information is also available. From the practical point of view, E-classification is substantially faster than the Zingg method.

Orientation Distribution Within a Single Hematite Crystal

Abstract: While crystallography conventionally presumes that a single crystal carries a unique crystallographic orientation, modern experimental techniques reveal that a single crystal may exhibit an orientation distribution. However, this distribution is largely concentrated; it is extremely concentrated when compared with orientation distributions of polycrystalline specimen. A case study of a deformation experiment with a single hematite crystal is presented, where the experimental deformation induced twining, which in turn changed a largely concentrated unimodal “parent” orientation distribution into a multimodal orientation distribution with a major mode resembling the parent mode and three minor modes corresponding to the progressive twining. The free and open source software MTEX for texture analysis was used to compute and visualize orientations density functions from both integral orientation measurements, i.e. neutron diffraction pole intensity data, and individual orientation measurements, i.e. electron back scatter diffraction data. Thus it is exemplified that MTEX is capable of analysing orientation data from largely concentrated orientation distributions.

Log-Ratio Analysis Is a Limiting Case of Correspondence Analysis

Abstract: It is common practice in compositional data analysis to perform the log-ratio transformation in order to preserve sub-compositional coherence in the analysis Correspondence analysis is an alternative approach to analyzing ratio-scale data and is often contrasted with log-ratio analysis It turns out that if one introduces a power transformation into the correspondence analysis algorithm, then the limit of the power-transformed correspondence analysis, as the power parameter tends to zero, is exactly the log-ratio analysis Depending on how the power transformation is applied, we can obtain as limiting cases either Aitchison’s unweighted log-ratio analysis or the weighted form called “spectral mapping” The upshot of this is that one can come as close as one likes to the log-ratio analysis, weighted or unweighted, using correspondence analysis

Measures of Parameter Uncertainty in Geostatistical Estimation and Geostatistical Optimal Design

Abstract: Studies of site exploration, data assimilation, or geostatistical inversion measure parameter uncertainty in order to assess the optimality of a suggested scheme. This study reviews and discusses measures for parameter uncertainty in spatial estimation. Most measures originate from alphabetic criteria in optimal design and were transferred to geostatistical estimation. Further rather intuitive measures can be found in the geostatistical literature, and some new measures will be suggested in this study. It is shown how these measures relate to the optimality alphabet and to relative entropy. Issues of physical and statistical significance are addressed whenever they arise. Computational feasibility and efficient ways to evaluate the above measures are discussed in this paper, and an illustrative synthetic case study is provided. A major conclusion is that the mean estimation variance and the averaged conditional integral scale are a powerful duo for characterizing conditional parameter uncertainty, with direct correspondence to the well-understood optimality alphabet. This study is based on cokriging generalized to uncertain mean and trends because it is the most general representative of linear spatial estimation within the Bayesian framework. Generalization to kriging and quasi-linear schemes is straightforward. Options for application to non-Gaussian and non-linear problems are discussed.

On Mathematical Aspects of a Combined Inversion of Gravity and Normal Mode Variations by a Spline Method

Abstract: This paper provides a brief overview of two linear inverse problems concerned with the determination of the Earth’s interior: inverse gravimetry and normal mode tomography. Moreover, a vector spline method is proposed for a combined solution of both problems. This method uses localised basis functions, which are based on reproducing kernels, and is related to approaches which have been successfully applied to the inverse gravimetric problem and the seismic traveltime tomography separately.

ODSIM: An Object-Distance Simulation Method for Conditioning Complex Natural Structures

Abstract: Stochastic simulation of categorical objects is traditionally achieved either with object-based or pixel-based methods. Whereas object-based modeling provides realistic results but raises data conditioning problems, pixel-based modeling provides exact data conditioning but may lose some features of the simulated objects such as connectivity. We suggest a hybrid dual-scale approach to combine both shape realism and strict data conditioning. The procedure combines the distance transform to a skeleton object representing coarse-scale structures, plus a classical pixel-based random field and threshold representing fine-scale features. This object-distance simulation method (ODSIM) uses a perturbed distance to objects and is particularly appropriate for modeling structures related to faults or fractures such as karsts, late dolomitized rocks, and mineralized veins. We demonstrate this method to simulate dolomite geometry and discuss strategies to apply this method more generally to simulate binary shapes.

Poloidal and Toroidal Field Modeling in Terms of Locally Supported Vector Wavelets

Abstract: This paper deals with multiscale modeling of poloidal and toroidal fields such as geomagnetic field and currents. The wavelets are developed from scale-dependent regularizations of the Green function with respect to the Beltrami operator. They are constructed as to be locally compact, thus, allowing a locally reflected (zooming-in) reconstruction of the geomagnetic quantities. Finally, a reconstruction algorithm is indicated in form of a tree algorithm.

Dempster–Shafer Theory Applied to Uncertainty Surrounding Permeability

Abstract: Typically, if uncertainty in subsurface parameters is addressed, it is done so using probability theory. Probability theory is capable of only handling one of the two types of uncertainty (aleatory), hence epistemic uncertainty is neglected. Dempster–Shafer evidence theory (DST) is an approach that allows analysis of both epistemic and aleatory uncertainty. In this paper, DST combination rules are used to combine measured field data on permeability, along with the expert opinions of hydrogeologists (subjective information) to examine uncertainty. Dempster’s rule of combination is chosen as the combination rule of choice primarily due to the theoretical development that exists and the simplicity of the data. Since Dempster’s rule does have some criticisms, two other combination rules (Yager’s rule and the Hau–Kashyap method) were examined which attempt to correct the problems that can be encountered using Dempster’s rule. With the particular data sets used here, there was not a clear superior combination rule. Dempster’s rule appears to suffice when the conflict amongst the evidence is low.

Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data

Abstract: In the present paper, a new geostatistical parameterization technique is introduced for solving inverse problems, either in groundwater hydrology or petroleum engineering. The purpose of this is to characterize permeability at the field scale from the available dynamic data, that is, data depending on fluid displacements. Thus, a permeability model is built, which yields numerical flow answers similar to the data collected. This problem is often defined as an objective function to be minimized. We are especially focused on the possibility to locally change the permeability model, so as to further reduce the objective function. This concern is of interest when dealing with 4D-seismic data. The calibration phase consists of selecting sub-domains or pilot blocks and of varying their log-permeability averages. The permeability model is then constrained to these fictitious block-data through simple cokriging. In addition, we estimate the prior probability density function relative to the pilot block values and incorporate this prior information into the objective function. Therefore, variations in block values are governed by the optimizer while accounting for nearby point and block-data. Pilot block based optimizations provide permeability models respecting point-data at their locations, spatial variability models inferred from point-data and dynamic data in a least squares sense. A synthetic example is presented to demonstrate the applicability of the proposed matching methodology.

Geir Evensen: Data Assimilation—The Ensemble Kalman Filter, 2nd edn

Abstract: Operational forecasting systems, as being widely used in meteorology and oceanography, rely on high-resolution dynamical models coupled with very large data sets of observations. The methodology used to merge the numerical model with these observations has been termed “data assimilation.” A fair amount of research is currently being devoted to adapting data assimilation methods to the history-matching problem in petroleum reservoir engineering. Data assimilation, as defined by Geir Evensen, refers to the computation of the conditional probability distribution function of the output of a numerical model describing a dynamical process, conditioned by observations. The probabilistic approach is justified by the various sources of uncertainty about the initial and the boundary conditions, by the fact that the mathematical model does not integrate all aspects of the physical process, and, last but not least, by the observational errors occurring in the process. The Ensemble Kalman filter (EnKF), an algorithm that has proven to be very successful in many applications since it was proposed by Geir Evensen in 1994, provides a solution to the data assimilation problem. It is, moreover, fairly simple to implement. The present second edition of the book is subdivided into seventeen chapters, which progressively introduce different aspects of data assimilation with Kalman filters. As a rule, they contain applications to numerical data as well as careful discussions of the results. The book follows a three-part structure. The first part is a basic introduction (Chapters 1 to 6) which acquaints the reader with basic statistical concepts, linear and nonlinear Kalman filters, and variational methods. The second part (Chapters 7 to 15) addresses more advanced material, offering an exposition of

Optimizing Subsurface Field Data Acquisition Using Information Theory

Abstract: Oil and gas reservoirs or subsurface aquifers are complex heterogeneous natural structures. They are characterized by means of several direct or indirect field measurements involving different physical processes operating at various spatial and temporal scales. For example, drilling wells provides small plugs whose physical properties may be measured in the laboratory. At a larger scale, seismic techniques provide a characterization of the geological structures. In some cases these techniques can help characterize the spatial fluid distribution, whose knowledge can in turn be used to improve the oil recovery strategy. In practice, these measurements are always expensive. In addition, due to their indirect and incomplete character, the measurements cannot give an exhaustive description of the reservoir and several uncertainties still remain. Quantification of these uncertainties is essential when setting up a reservoir development scenario and when modelling the risks due to the cost of the associated field operations. Within this framework, devising strategies that allow one to set up optimal data acquisition schemes can have many applications in oil or gas reservoir engineering, or in the CO2 geological storages. In this paper we present a method allowing us to quantify the information that is potentially provided by any set of measurements. Using a Bayesian framework, the information content of any set of data is defined by using the Kullback–Leibler divergence between posterior and prior density distributions. In the case of a Gaussian model where the data depends linearly on the parameters, explicit formulae are given. The kriging example is treated, which allows us to find an optimal well placement. The redundancy of data can also be quantified, showing the role of the correlation structure of the prior model. We extend the approach to the permeability estimation from a well-test simulation using the apparent permeability. In this case, the global optimization result of the mean information criterion gives an optimal acquisition time frequency.

Kriging prediction intervals based on semiparametric bootstrap

Abstract: Kriging is a widely used method for prediction, which, given observations of a (spatial) process, yields the best linear unbiased predictor of the process at a new location. The construction of corresponding prediction intervals typically relies on Gaussian assumptions. Here we show that the distribution of kriging predictors for non-Gaussian processes may be far from Gaussian, even asymptotically. This emphasizes the need for other ways to construct prediction intervals. We propose a semiparametric bootstrap method with focus on the ordinary kriging predictor. No distributional assumptions about the data generating process are needed. A simulation study for Gaussian as well as lognormal processes shows that the semiparametric bootstrap method works well. For the lognormal process we see significant improvement in coverage probability compared to traditional methods relying on Gaussian assumptions.

Hoisting a red flag: An early warning system for exceeding subsidence limits

Abstract: We present a general framework that enables decision-making when a threshold in a process is about to be exceeded (an event). Measurements are combined with prior information to update the probability of such an event. This prior information is derived from the results of an ensemble of model realisations that span the uncertainty present in the model before any measurements are collected; only probability updates need to be calculated, which makes the procedure very fast once the basic ensemble of realisations has been set up. The procedure is demonstrated with an example where gas field production is restricted to a maximum amount of subsidence. Starting with 100 realisations spanning the prior uncertainty of the process, the measurements collected during monitoring bolster some of the realisations and expose others as irrelevant. In this procedure, more data will mean a sharper determination of the posterior probability. We show the use of two different types of limits, a maximum allowed value of subsidence and a maximum allowed value of subsidence rate for all measurement points at all times. These limits have been applied in real world cases. The framework is general and is able to deal with other types of limits in just the same way. It can also be used to optimise monitoring strategies by assessing the effect of the number, position and timing of the measurement points. Furthermore, in such a synthetic study, the prior realisations do not need to be updated; spanning the range of uncertainty with appropriate prior models is sufficient.

On the Existence of Mosaic and Indicator Random Fields with Spherical, Circular, and Triangular Variograms

Abstract: This paper focuses on two specific families of stationary random fields (namely, mosaic and indicator random fields) and examines the question of whether the variogram of a member of these two families could be spherical, circular, or triangular. It is shown that there are such examples in one-dimensional spaces, while there are counterexamples in higher-dimensional spaces for mosaic random fields, indicators of Boolean random sets and indicators of excursion sets of Gaussian random fields. Further results concerning the spherical plus nugget and circular plus nugget variograms are provided.

Fast Summation of Functions on the Rotation Group

Abstract: Computing with functions on the rotation group is a task carried out in various areas of application. When it comes to approximation, kernel based methods are a suitable tool to handle these functions. In this paper, we present an algorithm which allows us to evaluate linear combinations of functions on the rotation group as well as a truly fast algorithm to sum up radial functions on the rotation group. These approaches based on nonequispaced FFTs on SO(3) take $\mathcal{O}(M+N)$ arithmetic operations for M and N arbitrarily distributed source and target nodes, respectively. In this paper, we investigate a selection of radial functions and give explicit theoretical error bounds, as well as numerical examples of approximation errors. Moreover, we provide an application of our method, namely the kernel density estimation from electron back scattering diffraction (EBSD) data, a problem relevant in texture analysis.

Predicting Threshold Exceedance by Local Block Means in Soil Pollution Surveys

Abstract: Soil contamination by heavy metals and organic pollutants around industrial premises is a problem in many countries around the world. Delineating zones where pollutants exceed tolerable levels is a necessity for successfully mitigating related health risks. Predictions of pollutants are usually required for blocks because remediation or regulatory decisions are imposed for entire parcels. Parcel areas typically exceed the observation support, but are smaller than the survey domain. Mapping soil pollution therefore involves a local change of support. The goal of this work is to find a simple, robust, and precise method for predicting block means (linear predictions) and threshold exceedance by block means (nonlinear predictions) from data observed at points that show a spatial trend. By simulations, we compared the performance of universal block kriging (UK), Gaussian conditional simulations (CS), constrained (CK), and covariance-matching constrained kriging (CMCK), for linear and nonlinear local change of support prediction problems. We considered Gaussian and positively skewed spatial processes with a nonstationary mean function and various scenarios for the autocorrelated error. The linear predictions were assessed by bias and mean square prediction error and the nonlinear predictions by bias and Peirce skill scores. For Gaussian data and blocks with locally dense sampling, all four methods performed well, both for linear and nonlinear predictions. When sampling was sparse CK and CMCK gave less precise linear predictions, but outperformed UK for nonlinear predictions, irrespective of the data distribution. CK and CMCK were only outperformed by CS in the Gaussian case when threshold exceedance was predicted by the conditional quantiles. However, CS was strongly biased for the skewed data whereas CK and CMCK still provided unbiased and quite precise nonlinear predictions. CMCK did not show any advantages over CK. CK is as simple to compute as UK. We recommend therefore this method to predict block means and nonlinear transforms thereof because it offers a good compromise between robustness, simplicity, and precision.