Multiple-point Simulation Research Articles

Sub-pixel land-cover mapping with improved fraction images upon multiple-point simulation

Outputs of soft classification inherently contain uncertainty. As an input for the sub-pixel mapping (SPM) method, the uncertainty is propagated to SPM result especially the boundary region between classes. Therefore, reducing the uncertainty within the outputs of soft classification is worth exploring. This paper firstly utilizes multiple-point simulation (MPS) through training images for characterizing the spatial structural properties of a surface object/class. Consequently, MPS results are used to increase the accuracy of the fraction image of the surface object/class. The improved fraction image then inputs to the SPM method for producing the land cover map with finer spatial resolution. In order to validate the proposed method, a remotely sensed image from Landsat TM 30m over the Qianyanzhou red earth hill region in China is used. This experimental study not only compares the results from SPM with improved fraction images with MPS and results from SPM with original fraction images, but also investigates the performances of different soft classifiers. It has been demonstrated that this proposed method is an effective way to reduce the uncertainty in outputs of different soft classification, increase the recognition accuracies of boundary regions and thus increase the accuracies of SPM simulated images.

Comparison of three geostatistical methods for hydrofacies simulation: a test on alluvial sediments

The hydrodispersive properties of porous sediments are strongly influenced by the heterogeneity at fine scales, which can be modeled by geostatistical simulations. In order to improve the assessment of the properties of three different geostatistical simulation methods (Sequential indicator simulation, SISIM; Transition probability geostatistical simulation, T-PROGS; Multiple point simulation, MPS) a comparison test at different scales was performed for a well-exposed aquifer analogue. In the analysed volume (approximately 30,000 m3) four operative hydrofacies have been recognised: very fine sand and silt, sand, gravelly sand and open framework gravel. Several equiprobable realizations were computed with SISIM, MPS and T-PROGS for a test volume of approximately 400 m3 and for the entire volume, and the different outcomes were compared with visual inspection and connectivity analysis of the very or poorly permeable structures. The comparison of the different simulations shows that the geological model is best reproduced when the simulations are realised separately for each highest rank depositional element and subsequently merged. Moreover, the three methods yield different images of the volume; in particular MPS is efficient in mapping the geometries of the most represented hydrofacies, whereas SISIM and T-PROGS can account for the distribution of the less represented facies.

Three-dimensional high resolution fluvio-glacial aquifer analog – Part 2: Geostatistical modeling

► A hierarchical framework to simulate highly heterogeneous fluvio-glacial aquifers. ► Realistic geological heterogeneity of a fluvio-glacial aquifer is reproduced. ► Flow properties of the 3D analog and observed 2D sections are comparable. ► The model can be used for testing and validating hydrogeological modeling concepts. ► The simulated 3D aquifer analog is provided as additional material. The heterogeneity of sedimentary structures at the decimeter scale is crucial to the understanding of groundwater flow and transport. In a series of two papers, we provide a detailed analysis of a fluvio-glacial aquifer analog: the Herten site. The geological data along a series of 2D sections in a quarry, the corresponding GPR measurements, and their sedimentological interpretation are described in the companion paper. In this paper, we focus on the three-dimensional reconstruction of the heterogeneity. The resulting numerical model is provided as an electronic supplementary material for further studies. Furthermore, the geostatistical parameters derived from this analysis and the methodology described in the paper could be used in the future for the simulation of similar deposits where less data would be available. To build the 3D model, we propose a hierarchical simulation method which integrates various geostatistical techniques. First, we model the subdivision of the domain into regions corresponding to main sedimentological structures (e.g. a sedimentation event). Within these volumes, we use multiple-point statistics to describe the internal heterogeneity. What is unusual here is that we do not try to use a complex training image for the multiple-point algorithm accounting for all the non-stationarity and complexity, but instead use a simple conceptual model of heterogeneity (ellipsoidal shapes as a training image) and constrain the multiple point simulations within the regions by a detailed interpolation of orientation data derived from the 2D sections. This method produces realistic geological structures. The analysis of the flow and transport properties (hydraulic conductivity and tracer breakthrough curves) of the resulting model shows that it is closer to the properties estimated directly from the 2D geological observations rather than those estimated from a model of heterogeneity based on probability of transitions and not including the modeling of the large-scale structures.

Multiple-point simulation-based method for extraction of objects with spatial structure from remotely sensed imagery

Classification of Combining Spectral information and Spatial information upon Multiple-point statistics (CCSSM) is a method for information extraction that introduces multiple-point simulation (MPS) to increase the classification accuracy of remotely sensed imagery data by incorporating structural information through a training image. This paper focuses on (1) applying CCSSM using a multigrid approach to a Satellite Pour l'Observation de la Terre (SPOT) 5 image, (2) adopting consensus-based fusion to combine two different information sources, the spectral information from supervised classification and spatial structure information from the MPS and (3) analysing the change trend for the accuracy of information extraction and optimizing the proportions in the combination of the two different information sources. We demonstrate that, even if the spectral information from the SPOT 5 image used in the classification results in better classification accuracy, with the introduction of spatial structure information from MPS the accuracy of the information extraction can still be increased significantly.

MPS-based information extraction method for remotely sensed imagery: a comparison of fusion methods

Recently, multiple-point simulation (MPS) was introduced to increase the accuracy of information extraction from remotely sensed imagery by incorporating structural information through a training image. An important procedure in the MPS-based information extraction method is the fusion of two probability fields from two different classifiers, extracting spectral and spatial structure information. In previous studies the fusion process was accomplished using the theory of evidence and the theory of consensus. It has been shown that these fusion methods each have their own capabilities and characteristics for different data under different circumstances. This paper investigates primarily the advantages and disadvantages of three different types of fusion methods: evidence-based, consensus-based, and probability-based, and then compares the fusion results through an accuracy assessment. For validation purposes, we selected two remotely sensed images taken in different areas and with distinct structural characteristics of roads. Both images, from Satellite Pour l'Observation de la Terre 5 (SPOT5) with spatial resolution of 10 m, were used to investigate the performance of the three fusion methods in extracting road information with distinct structural characteristics from the images. A comparison of the different fusion methods can assist users in selecting the appropriate fusion method for the given data characteristics. Based on the results of two experiments, the relationships between these fusion methods are further investigated.

Multiple-point Simulation and Its Application and Development in RS Image Classification

Multiple-point simulation(MPS) is one of the applications of multiple-point statistics in geology.It can simulate the complex geometry-structure by getting the conditional probability from training image through search-tree while the traditional geostatistics has little ability to capture the complex geometry-structure.When MPS is applied into remote sensing classification,it can help us to improve the traditional-pixel-by-pixel-classification accuracy by combining the structural and spectral information.This article firstly introduces the development of new classification method which is called CCSSM,and then discusses the revision of this methods.Furthermore,it extends CCSSM's application to the classification of the multiple-class from in the perspects of a theoretical and feasible way.

A Methodology for the Assessment of Unconventional (Continuous) Resources with an Application to the Greater Natural Buttes Gas Field, Utah

The Greater Natural Buttes tight natural gas field is an unconventional (continuous) accumulation in the Uinta Basin, Utah, that began production in the early 1950s from the Upper Cretaceous Mesaverde Group. Three years later, production was extended to the Eocene Wasatch Formation. With the exclusion of 1100 non-productive (“dry”) wells, we estimate that the final recovery from the 2500 producing wells existing in 2007 will be about 1.7 trillion standard cubic feet (TSCF) (48.2 billion cubic meters (BCM)). The use of estimated ultimate recovery (EUR) per well is common in assessments of unconventional resources, and it is one of the main sources of information to forecast undiscovered resources. Each calculated recovery value has an associated drainage area that generally varies from well to well and that can be mathematically subdivided into elemental subareas of constant size and shape called cells. Recovery per 5-acre cells at Greater Natural Buttes shows spatial correlation; hence, statistical approaches that ignore this correlation when inferring EUR values for untested cells do not take full advantage of all the information contained in the data. More critically, resulting models do not match the style of spatial EUR fluctuations observed in nature. This study takes a new approach by applying spatial statistics to model geographical variation of cell EUR taking into account spatial correlation and the influence of fractures. We applied sequential indicator simulation to model non-productive cells, while spatial mapping of cell EUR was obtained by applying sequential Gaussian simulation to provide multiple versions of reality (realizations) having equal chances of being the correct model. For each realization, summation of EUR in cells not drained by the existing wells allowed preparation of a stochastic prediction of undiscovered resources, which range between 2.6 and 3.4 TSCF (73.6 and 96.3 BCM) with a mean of 2.9 TSCF (82.1 BCM) for Greater Natural Buttes. A second approach illustrates the application of multiple-point simulation to assess a hypothetical frontier area for which there is no production information but which is regarded as being similar to Greater Natural Buttes.

High-order Stochastic Simulation of Complex Spatially Distributed Natural Phenomena

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.

Multiple Point Metrics to Assess Categorical Variable Models

Geostatistical models should be checked to ensure consistency with conditioning data and statistical inputs. These are minimum acceptance criteria. Often the first and second-order statistics such as the histogram and variogram of simulated geological realizations are compared to the input parameters to check the reasonableness of the simulation implementation. Assessing the reproduction of statistics beyond second-order is often not considered because the “correct” higher order statistics are rarely known. With multiple point simulation (MPS) geostatistical methods, practitioners are now explicitly modeling higher-order statistics taken from a training image (TI). This article explores methods for extending minimum acceptance criteria to multiple point statistical comparisons between geostatistical realizations made with MPS algorithms and the associated TI. The intent is to assess how well the geostatistical models have reproduced the input statistics of the TI; akin to assessing the histogram and variogram reproduction in traditional semivariogram-based geostatistics. A number of metrics are presented to compare the input multiple point statistics of the TI with the statistics of the geostatistical realizations. These metrics are (1) first and second-order statistics, (2) trends, (3) the multiscale histogram, (4) the multiple point density function, and (5) the missing bins in the multiple point density function. A case study using MPS realizations is presented to demonstrate the proposed metrics; however, the metrics are not limited to specific MPS realizations. Comparisons could be made between any reference numerical analogue model and any simulated categorical variable model.

Stationarity Scores on Training Images for Multipoint Geostatistics

This research introduces a novel method to assess the validity of training images used as an input for Multipoint Geostatistics, alternatively called Multiple Point Simulation (MPS). MPS are a family of spatial statistical interpolation algorithms that are used to generate conditional simulations of property fields such as geological facies. They are able to honor absolute “hard” constraints (e.g., borehole data) as well as “soft” constraints (e.g., probability fields derived from seismic data, and rotation and scale). These algorithms require 2D or 3D training images or analogs whose textures represent a spatial arrangement of geological properties that is presumed to be similar to that of a target volume to be modeled. To use the current generation of MPS algorithms, statistically valid training image are required as input. In this context, “statistical validity” includes a requirement of stationarity, so that one can derive from the training image an average template pattern. This research focuses on a practical method to assess stationarity requirements for MPS algorithms, i.e., that statistical density or probability distribution of the quantity shown on the image does not change spatially, and that the image shows repetitive shapes whose orientation and scale are spatially constant. This method employs image-processing techniques based on measures of stationarity of the category distribution, the directional (or orientation) property field and the scale property field of those images. It was successfully tested on a set of two-dimensional images representing geological features and its predictions were compared to actual realizations of MPS algorithms. An extension of the algorithms to 3D images is also proposed. As MPS algorithms are being used increasingly in hydrocarbon reservoir modeling, the methods described should facilitate screening and selection of the input training images.

New classification method for remotely sensed imagery via multiple-point simulation: experiment and assessment

There has been substantial effort dedicated to the issue of how to incorporate spatial information to improve the classification accuracy in past decades and some excellent methods have been developed. Each method has its own advantages and disadvantages for different images and user requirements. This paper proposes a new classification method, which introduces multiple-point simulation to improve the classification of remotely sensed imagery data by incorporating structural information through a training image. This new method named CCSSM is the derivation of two classifications and based on spectral and spatial information, which then are fused. For validation purpose, a real-life example of road extraction from Landsat TM is used to substantiate the conceptual arguments. An assessment of the accuracy of the proposed method compared with results using a maximum likelihood classifier shows the overall accuracy improves from 48.9% to 82.6%, and the kappa coefficient improves from 0.12 to 0.55 and therefore, the new method has superior overall performance on the classification of remotely sensed data.

Multiple-Point Simulations Constrained by Continuous Auxiliary Data

An important issue of using the multiple-point (MP) statistical approach for reservoir modeling concerns the integration of auxiliary constraints derived, for instance, from seismic information. There exist two methods in the literature for these non-stationary MP simulations. One is based on an analytical approximation (the “τ-model”) of the conditional probabilities that involve auxiliary data. The degree of approximation with this method depends on the parameter τ, whose inference is difficult in practice. The other method is based on the inference of these conditional probabilities directly from training images. This method classifies the auxiliary data into a few classes. This classification is in general arbitrary and therefore inconvenient in practice, especially in the case of continuous auxiliary constraints. In this paper, we propose an alternative method for performing non-stationary MP simulations. This method accounts for the data support in the modeling procedure and allows, in particular, continuous auxiliary data to be integrated into MP simulations. This method avoids the major limitations of the previous methods, namely the use of an approximate analytical model and the reduction of the auxiliary data into a limited number of classes. This method can be easily implemented in the existing MP simulation codes. Numerical tests show good performance of this method both in reproducing the geometrical features of the training image and in honouring the auxiliary data.

The Necessity of a Multiple-Point Prior Model

Any interpolation, any hand contouring or digital drawing of a map or a numerical model necessarily calls for a prior model of the multiple-point statistics that link together the data to the unsampled nodes, then these unsampled nodes together. That prior model can be implicit, poorly defined as in hand contouring; it can be explicit through an algorithm as in digital mapping. The multiple-point statistics involved go well beyond single-point histogram and two-point covariance models; the challenge is to define algorithms that can control more of such statistics, particularly those that impact most the utilization of the resulting maps beyond their visual appearance. The newly introduced multiple-point simulation (mps) algorithms borrow the high order statistics from a visually and statistically explicit model, a training image. It is shown that mps can simulate realizations with high entropy character as well as traditional Gaussian-based algorithms, while offering the flexibility of considering alternative training images with various levels of low entropy (organized) structures. The impact on flow performance (spatial connectivity) of choosing a wrong training image among many sharing the same histogram and variogram is demonstrated.

Using the Snesim program for multiple-point statistical simulation

Traditionally, there are two mainstream avenues for geostatistical modeling: pixel-based two-point simulation and object-based simulation. Each is good at either data conditioning or reproducing geological shapes, but none is good at both. Multiple-point simulation combines the strengths of these two avenues. As an advanced pixel-based technique, it inherits the flexibility of pixel-based techniques by building the model one pixel at a time, hence data conditioning is easily achieved; it is also capable of reproducing curvilinear geological shapes through borrowing multiple-point statistics from a training image. The snesim code provides such a multiple-point simulation program. A training image is used to represent the prior geological knowledge, which is scanned to obtain the conditioning probability values for the central node belonging to a facies category given any multiple-point conditioning data event. These training probability values are stored in a search tree a prior to simulation. Then in a sequential simulation mode, at each uninformed node, according to its specific conditioning data event a probability value is retrieved from the search tree, and a value is simulated from it. There are many input parameters to the snesim program, the impact of which might not be immediately clear to people who are not familiar with the code. In this paper, we aim at bringing important aspects of this program and providing practical guidelines to using the program. Sensivity analyses are performed on the important input parameters. The results are analyzed and recommendations are provided on how to set these parameters appropriately.

Filter-Based Classification of Training Image Patterns for Spatial Simulation

Multiple-point simulation, as opposed to simulation one point at a time, operates at the pattern level using a priori structural information. To reduce the dimensionality of the space of patterns we propose a multi-point filtersim algorithm that classifies structural patterns using selected filter statistics. The pattern filter statistics are specific linear combinations of pattern pixel values that represent directional mean, gradient, and curvature properties. Simulation proceeds by sampling from pattern classes selected by conditioning data.

Improving Sequential Simulation with a Structured Path Guided by Information Content

Multiple-point simulation is a newly developed geostatistical method that aims at combining the strengths of two mainstream geostatistical methods: object-based and pixel-based methods. It maintains the flexibility of pixel-based algorithms in data conditioning, while enhancing its capability of reproducing realistic geological shapes, which is traditionally reserved to object-based algorithms. However, the current snesim program for multiple-point simulation has difficulty in reproducing large-scale structures, which have a significant impact on the flow response. To address this problem, we propose to simulate along a structured path based on an information content measure. This structured path accounts for not only the information from the data, but also some prior structural information provided by geological knowledge. Various case studies show a better reproduction of large-scale structures. This concept of simulating along a structured path guided by information content can be applied to any sequential simulation algorithms, including traditional variogram-based two-point geostatistical algorithms.

Multiple-point simulation integrating wells, three-dimensional seismic data, and geology

There are two significant challenges in building a reservoir model integrating all available information. One challenge is that wells and seismic data measure the reservoir at different scales of resolution. The other challenge lies in how to account for conceptual geological knowledge with resolution at multiple scales.In this paper, we present a case study of integrating well data, seismic data, and conceptual geologic models. The well and seismic data are of good quality, but conventional well-seismic data calibration indicates that the seismic data are unable to fully differentiate sand from shale. The reason for this poor well-seismic data calibration is that well log and seismic data measure the reservoir at different scales. Well logs are able to differentiate sand from shale, whereas seismic data are better at detecting larger scale depositional geometries.A new workflow is presented to deal with this problem. First, principal component analysis clustering is used to identify characteristic patterns of certain depositional facies, from which sandy and shaly channels are interpreted. Next, multiple-point geostatistical simulation is performed to build a depositional-facies model, which integrates both hard and soft data but also incorporates realistic depositional-facies geometries provided by our geological knowledge of this reservoir. Finally, different lithofacies (sand and shale) indicators and corresponding petrophysical properties are simulated honoring the limited well data.The results show that not only are the geological features better reproduced, but also is the uncertainty about the reservoir significantly reduced because of a better integration of corresponding three-dimensional seismic data.

Field scale variability of soil structure and its impact on crop growth and nitrate leaching in the analysis of fertilizing scenarios

Soil structure variability was inventoried on a field scale, and translated to basic input data for a solute transport and crop production simulation model. After validation of the model, it was used to simulate the spatially varying effect of slurry application and N-fertilizing scenarios by multiple point-simulations. Feasibility of scenarios was determined by comparing nitrate leaching concentrations with current and pursued threshold levels. Disjunctive kriging was used to estimate and map the probability that these threshold levels were exceeded. Emphasis was given to the effect of soil-specific slurry application rates and N-fertilizer levels on leaching probabilities and crop yields. Soil-specific N-treatments increased field yields, because crop response to N, both in slurry and in fertilizer, was soil dependent. The leaching response to N-treatment was also soil dependent.