Statistics for Spatial Data.

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

In many earth sciences applications, the geological objects or structures to be reproduced are curvilinear, e.g., sand channels in a clastic reservoir. Their modeling requires multiple-point statistics involving jointly three or more points at a time, much beyond the traditional two-point variogram statistics. Actual data from the field being modeled, particularly if it is subsurface, are rarely enough to allow inference of such multiple-point statistics. The approach proposed in this paper consists of borrowing the required multiple-point statistics from training images depicting the expected patterns of geological heterogeneities. Several training images can be used, reflecting different scales of variability and styles of heterogeneities. The multiple-point statistics inferred from these training image(s) are exported to the geostatistical numerical model where they are anchored to the actual data, both hard and soft, in a sequential simulation mode. The algorithm and code developed are tested for the simulation of a fluvial hydrocarbon reservoir with meandering channels. The methodology proposed appears to be simple (multiple-point statistics are scanned directly from training images), general (any type of random geometry can be considered), and fast enough to handle large 3D simulation grids.

Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics

Applications of Supramolecular Anion Recognition

https://opus.lib.uts.edu.au/bitstream/10453/114834/4/OCC-90253_AM.pdf

A review of clustering techniques and developments

Preface, vii Acknowledgments, xi Part I Concepts I.1 Hiking in the Sierra Nevada, 3 I.2 Spatial estimation based on random function theory, 7 I.3 Universal kriging with training images, 29 I.4 Stochastic simulations based on random function theory, 49 I.5 Stochastic simulation without random function theory, 59 I.6 Returning to the Sierra Nevada, 75 Part II Methods II.1 Introduction, 87 II.2 The algorithmic building blocks, 91 II.3 Multiple-point geostatistics algorithms, 155 II.4 Markov random fields, 173 II.5 Nonstationary modeling with training images, 183 II.6 Multivariate modeling with training images, 199 II.7 Training image construction, 221 II.8 Validation and quality control, 239 II.9 Inverse modeling with training images, 259 II.10 Parallelization, 295 Part III Applications III.1 Reservoir forecasting the West Coast of Africa (WCA) reservoir, 303 III.2 Geological resources modeling in mining, 329 Coauthored by Cristian P'erez, Julian M. Ortiz, & Alexandre Boucher III.3 Climate modeling application the case of the Murray Darling Basin, 345 Index, 361

Multiple-point Geostatistics: Stochastic Modeling with Training Images

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

Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

Whereas outcrop models can provide important information on reservoir architecture and heterogeneity, it is not entirely clear how such information can be used exhaustively in geostatistical reservoir modeling. Traditional, variogram-based geostatistics is inadequate in that regard because the variogram is too limiting in capturing geologic heterogeneity from outcrops. A new field, termed multiple-point geostatistics, does not rely on variogram models. Instead, it allows capturing structure from so-called training images. Multiple-point geostatistics borrows multiple-point patterns from the training image, then anchors them to subsurface well-log, seismic, and production data. However, multiple-point geostatistics does not escape from the same principles as traditional variogram-based geostatistics: it is still a stochastic method and, hence, relies on the commonly forgotten principles of stationarity and ergodicity. These principles dictate that the training image used in multiple-point geostatistics cannot be chosen arbitrarily and that not all outcrops might be suitable training image models. In this paper, we outline the guiding principles of using analog models in multiple-point geostatistics and show that simple, so-called modular training images can be used to build complex reservoir models using geostatistics algorithms.

/pdf/multiple-point-geostatistics-a-quantitative-vehicle-for-20eatvss62.pdf

Multiple-point Geostatistics: A Quantitative Vehicle for Integrating Geologic Analogs into Multiple Reservoir Models

An entirely new approach to stochastic simulation is proposed through the direct simulation of patterns. Unlike pixel-based (single grid cells) or object-based stochastic simulation, pattern-based simulation simulates by pasting patterns directly onto the simulation grid. A pattern is a multi-pixel configuration identifying a meaningful entity (a puzzle piece) of the underlying spatial continuity. The methodology relies on the use of a training image from which the pattern set (database) is extracted. The use of training images is not new. The concept of a training image is extensively used in simulating Markov random fields or for sequentially simulating structures using multiple-point statistics. Both these approaches rely on extracting statistics from the training image, then reproducing these statistics in multiple stochastic realizations, at the same time conditioning to any available data. The proposed approach does not rely, explicitly, on either a statistical or probabilistic methodology. Instead, a sequential simulation method is proposed that borrows heavily from the pattern recognition literature and simulates by pasting at each visited location along a random path a pattern that is compatible with the available local data and any previously simulated patterns. This paper discusses the various implementation details to accomplish this idea. Several 2D illustrative as well as realistic and complex 3D examples are presented to showcase the versatility of the proposed algorithm.

Conditional Simulation with Patterns

Assessing uncertainty of a spatial phenomenon requires the analysis of a large number of parameters which must be processed by a transfer function. To capture the possibly of a wide range of uncertainty in the transfer function response, a large set of geostatistical model realizations needs to be processed. Stochastic spatial simulation can rapidly provide multiple, equally probable realizations. However, since the transfer function is often computationally demanding, only a small number of models can be evaluated in practice, and are usually selected through a ranking procedure. Traditional ranking techniques for selection of probabilistic ranges of response (P10, P50 and P90) are highly dependent on the static property used. In this paper, we propose to parameterize the spatial uncertainty represented by a large set of geostatistical realizations through a distance function measuring “dissimilarity” between any two geostatistical realizations. The distance function allows a mapping of the space of uncertainty. The distance can be tailored to the particular problem. The multi-dimensional space of uncertainty can be modeled using kernel techniques, such as kernel principal component analysis (KPCA) or kernel clustering. These tools allow for the selection of a subset of representative realizations containing similar properties to the larger set. Without losing accuracy, decisions and strategies can then be performed applying a transfer function on the subset without the need to exhaustively evaluate each realization. This method is applied to a synthetic oil reservoir, where spatial uncertainty of channel facies is modeled through multiple realizations generated using a multi-point geostatistical algorithm and several training images.

Jef Caers

Papers

Multiple-point Geostatistics: Stochastic Modeling with Training Images

Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

Multiple-point Geostatistics: A Quantitative Vehicle for Integrating Geologic Analogs into Multiple Reservoir Models

Conditional Simulation with Patterns

Representing Spatial Uncertainty Using Distances and Kernels