2D Training Images Research Articles

A Multi-Point Geostatistical Modeling Method Based on 2D Training Image Partition Simulation

In this paper, a multi-point geostatistical (MPS) method based on variational function partition simulation is proposed to solve the key problem of MPS 3D modeling using 2D training images. The new method uses the FILTERSIM algorithm framework, and the variational function is used to construct simulation partitions and training image sequences, and only a small number of training images close to the unknown nodes are used in the partition simulation to participate in the MPS simulation. To enhance the reliability, a new covariance filter is also designed to capture the diverse features of the training patterns and allow the filter to downsize the training patterns from any direction; in addition, an information entropy method is used to reconstruct the whole 3D space by selecting the global optimal solution from several locally similar training patterns. The stability and applicability of the new method in complex geological modeling are demonstrated by analyzing the parameter sensitivity and algorithm performance. A geological model of a uranium deposit is simulated to test the pumping of five reserved drill holes, and the results show that the accuracy of the simulation results of the new method is improved by 11.36% compared with the traditional MPS method.

Constructing of 3D Fluvial Reservoir Model Based on 2D Training Images

Training images are important input parameters for multipoint geostatistical modeling, and training images that can portray 3D spatial correlations are required to construct 3D models. The 3D training images are usually obtained by unconditional simulation using algorithms such as object-based algorithms, and in some cases, it is difficult to obtain the 3D training images directly, so a series of modeling methods based on 2D training images for constructing 3D models has been formed. In this paper, a new modeling method is proposed by synthesizing the advantages of the previous methods. Taking the fluvial reservoir modeling of the P oilfield in the Bohai area as an example, a comparative study based on 2D and 3D training images was carried out. By comparing the variance function, horizontal and vertical connectivity in x-, y-, and z-directions, and style similarity, the study shows that based on several mutually perpendicular 2D training images, the modeling method proposed in this paper can achieve an effect similar to that based on 3D training images directly. In the case that it is difficult to obtain 3D training images, the modeling method proposed in this paper has suitable application prospects.

Reconstruction of 3D Reservoir Lithological Model Using 2D Facies Profiles in SU 36-11 Area of Ordos Basin, China

In the middle and late stages of gas field development, the establishment of a fine reservoir lithological model is an important basis for drilling well pattern adjustment and potential exploitation. The SU 36-11 area of the Ordos basin in China is developing braided channel sediment with rich gas resources. However, the success rate of drilling wells is low due to the complex reservoir heterogeneity and the lack of a fine reservoir lithological model. In this paper, the complex internal structure of the reservoir sand body is revealed using the architectural element analysis method. Three sand body models, that is, isolated channel, superimposed channel, and cut superimposed channel, can be recognized. The effective sand body is mainly the channel bar deposit with a thickness of 2–5 m, a width of 200–500 m, a length of 400–700 m, a width ratio of 50–120, and a length-to-width ratio of 1.5–2. The 2D maps of the lithofacies (architectural elements) were then digitized to create 2D training images (TI) for the construction of the 3D model. The 2D data template was selected to scan the TI to obtain the 2D multi-point probability. The 3D multi-point probability was then generated using the probability fusion theory. The Monte Carlo sampling was used to predict the lithological type between wells. Finally, the 3D reservoir lithological model was built directly using the 2D lithological profiles. From the model, the geometry of the braided channel, channel bar, and flood plain was well revealed, and the spatial distribution of effective reservoir sand bodies was accurately predicted. The cross-validation test shows that the error of the channel bar is 6.5% on average, which improves the accuracy of the prediction of lithology in the sub-surface and can be used to guide the subsequent development of residual gas.

Reconstructing a Three-Dimensional Geological Model from Two-Dimensional Depositional Sections in a Tide-Dominated Estuarine Reservoir: A Case Study of Oil Sands Reservoir in Mackay River, Canada

A tide-dominated estuarine reservoir is an important oil reservoir. However, due to the force of bidirectional water flow, its internal structure is complex, and the heterogeneity is serious. Accurately establishing the tide-dominated estuarine reservoir model is a great challenge. This paper takes the Mackay River oil sands reservoir in Canada as the research object to establish the elaborate geological model of a tide-dominated estuarine reservoir. Through the meticulous depiction of core data, 14 kinds of lithofacies and nine kinds of architectural elements are identified, and the lithological and electrical response in sedimentary architectural elements is established. On this basis, the plane and vertical distribution of architectural elements, as well as the spatial superimposition patterns, are depicted and characterized through well seismic combination and plane and section interaction, and the representative plane and section architecture maps are obtained as 2D training images (TIs) for multi-point statistical modeling. The 2D TI is scanned by 2D data template to obtain the multi-point statistical probability of the 2D spatial architectural pattern. Then, the 2D multi-point probability is fused to generate three-dimensional (3D) multi-point statistical probability by the probabilistic fusion. Finally, Monte Carlo sampling is used to predict the spatial distribution of architectures, and an elaborate geological model of a tide-dominated estuarine reservoir is established. Compared with the traditional sequential indication modeling method, the point-to-point error of the model section based on the 2D section reconstruction method is only 25.92%, while the sequential indication modeling method is as high as 58.52%. Even far from the TI, the point-to-point error of the 2D section model is 33.13%. From the cross-validation, the average error of the 2D section is 11%, while the sequential indicator modeling error is 23.1%, which indicates that the accuracy of 2D reconstruction of the estuarine reservoir model is high, and this method is suitable for the establishment of the tide-dominated estuarine reservoir model.

Study on Facies Modeling of Tight Sandstone Reservoir Using Multi-Point Geostatistics Method Based on 2D Training Image—Case Study of Longdong Area, Ordos Basin, China

The Longdong area in the Ordos basin is a typical fluvial reservoir with strong heterogeneity. In order to clarify the distribution law of underground reservoirs in the Longdong area, it is necessary to establish and optimize a 3D geological model to characterize the heterogeneity of reservoirs. This is of great significance for accelerating the exploitation of tight sandstone gas in the southwest of the Ordos basin. This study takes the P2h8 member of the Ct3 research area in the Longdong area as an example, analyzes the core and logging curve shape to divide the sedimentary microfacies, and establishes the facies model. In particular, in view of the difficulty in obtaining 3D training images under the existing conditions in the study area, we use the multi-point geostatistics method combining sequential two-dimensional condition simulation and the direct sampling method to establish the facies model. This method can simulate the 3D geological model by using the 2D training images composed of the digital plane facies diagrams and the well-connection facies diagrams. In addition, we choose the object-based method and sequential indicator method for comparative experiments to verify the feasibility of this method (sequential two-dimensional condition simulation combined with the direct sampling method) from many aspects. The results show that the multi-point geostatistics method based on 2D training images can not only match the well data, but also show the geometric characteristics and contact relationship of the simulation object. The distribution characteristics of sandbody thickness and modeling results are consistent with the actual geological conditions in the study area. This study explores the feasibility of this method in the 3D geological simulation of large-scale fluvial facies tight sandstone reservoirs. Additionally, it also provides a new idea and scheme for the modeling method of geologists in similar geological environments.

A Novel Method of 3D Multipoint Geostatistical Inversion Using 2D Training Images

Abstract The seismic inversion method combined with multipoint geostatistics theory has begun to receive attention, but the acquisition accuracy and calculation efficiency of 3D training image still need more optimization. This paper presents a novel method of 3D multipoint geostatistical inversion based on 2D training images directly. The 2D training image was scanned by the data template to acquire the multipoint statistical probability in 2D direction. The probability fusion method is used to fuse the 2D multipoint probability into 3D multipoint probability. The rock facies types and patterns of the simulated points are obtained by random sampling. On this basis, the elastic parameters are extracted from the statistical rock physics model, and the seismic records are convoluted. Then, the synthetic records and the actual records were compared under a given threshold. If the error exceeds the given threshold, the iterative adaptive spatial sampling method will be used to repeat the process above-mentioned, so as to ensure that the error is below the threshold. Because the 2D training image is easy to obtain and evaluate, the demand problem of 3D training image is solved. The 2D training image scanning, probability storage and access are more convenient, and the adaptive spatial sampling method is more efficient than the reject sampling, so as to ensure the operation efficiency. The model from the Stanford Center for Reservoir Forecasting is selected to test the effectiveness of this newly designed method.

Data-driven construction of Three-dimensional subsurface geological models from limited Site-specific boreholes and prior geological knowledge for underground digital twin

A digital twin is a digital replica or virtual representations of 3D physical entities in the real world. In practice, it is challenging for 3D modelling of subsurface stratigraphy in an underground digital twin due to insufficient site-specific measurements and a lack of efficient 3D spatial prediction tools. In this study, a data-driven and deep learning method, called IC-XGBoost3D, is proposed to build a 3D geological model from limited site-specific boreholes and 2D training images reflecting prior geological knowledge. Anisotropic stratigraphic relationships are learned from two perpendicular 2D training images, and the extracted stratigraphic statistics serve as the input for pre-training a 2D simulation slice. A sequence of 2D simulation slices is then simulated with constraints by site-specific boreholes and subsequently reassembled into a 3D geological model. Note that the parameters of the deep learning method are calibrated with site-specific data and prior training images before being applied to develop the geological model. The model performance is demonstrated and validated using an illustrative example. The proposed method can efficiently generate an anisotropic 3D geological model as a point cloud from two perpendicular training images and sparse boreholes with a high prediction accuracy. More importantly, the proposed method not only infers the most probable 3D geological domain, but also provides a quantitative evaluation of associated 3D stratigraphic uncertainty. Effects of irregular borehole spacing and single training image on the simulation performance are also investigated.

A MPS-based novel method of reconstructing 3D reservoir models from 2D images using seismic constraints

While multi-point geostatistics (MPS) be used to model complex underground reservoirs, this method relies heavily on high-quality three-dimensional (3D) training images that may be difficult to acquire in real field studies. Furthermore, the original MPS technique does not include seismic data, which may noticeably improve underground reservoir predictions. Our proposed model, Se3DRCS, is a novel method to reconstruct 3D geological models from two-dimensional (2D) images and seismic information. In Se3DRCS, we are able to perform MPS-based modeling using 2D training images, rather than 3D images. These 2D images are produced by analyzing well profiles and sedimentary facies planes. After using the Se3DRCS method to obtain an initial geological model, we then generate elastic parameters and synthetic seismograms via direct sampling and convolution with the extracted seismic wavelets, respectively. By comparing the observed and synthetic seismic records, we refine the simulation results using the adaptive sampling method. The initial geological model is then updated with conditional data after each subsequent simulation iteration. When the error between the observed and synthetic seismograms falls below a reasonable threshold value, we have arrived at the best-fitting geological model. Our results indicate that the Se3DRCS method is capable of predicting the data elicited from wells with an accuracy of up to 82.88%, providing constraints on the distribution of branch channels in a delta reservoir, and reproducing local complex non-stationary geological features.

Machine-Learning Techniques Characterize Source-Rock Images at the Pore Scale

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 208632, “Reconstruction and Synthesis of Source-Rock Images at the Pore Scale,” by Timothy Anderson, SPE, Stanford University. The paper has not been peer reviewed. Nanoimaging techniques for characterizing pore-scale structure of shales trade off between high-resolution/high-contrast sample-destructive imaging modalities and low-contrast/low-resolution sample-preserving modalities. Acquisition of nanoscale images also is often time-consuming and expensive. In the complete paper, the author introduces methods for overcoming these challenges in image-based characterization of the fabric of shale source rocks using deep-learning models. Introduction A promising application of data-driven scale-translation techniques is nanoscale imaging. This application is important for studying shales because of the importance of nanoporosity in shale gas storage. Nanoimaging techniques, however, present specific challenges and can result in small image data sets that do not allow for accurate quantification of petrophysical or morphological properties. Consequently, data translation and generation both offer many opportunities to assimilate multiple nano- and microscale modalities and to overcome limitations of nanoimaging systems. The author proposes an image-based characterization work flow (Fig. 1) for data-driven scale translation that uses deep-learning image synthesis and translation models to assimilate multimodal, multiscale, and data-scarce source-rock images for predicting petrophysical and morphological properties. A central challenge in source-rock characterization addressed in this work is the reconstruction of 3D volumes when only 2D training images are available. Image-translation models are presented for reconstruction of 3D image volumes from 2D training data and a porous media image-synthesis algorithm that generalizes to 3D grayscale and multimodal volume generation from 2D training data. The complete paper describes the translation and synthesis models, applies these models to source-rock image data sets, and discusses extensions and future directions for the introduced work flow.

Training Image Optimization Method Based on Convolutional Neural Network and Its Application in Discrete Fracture Network Model Selection

Abstract Training image (TI) is important for multipoint statistics simulation method (MPS), since it captures the spatial geological pattern of target reservoir to be modeled. Generally, one optimal TI is selected before applying MPS by evaluating the similarities between many TIs and the well interpretations of target reservoir. In this paper, we propose a new training image optimization approach based on the convolutional neural network (CNN). First, candidate TIs were randomly sampled several times to obtain the sample dataset. Then, the CNN was used to conduct transfer learning for all samples, and finally, the optimal TI of the conditioning well data is selected through the trained CNN model. By taking advantage of the strong learning ability of CNN in image feature recognition, the proposed method can automatically identify differences in spatial features between the conditioning well data and the samples of the training image. Hence, it effectively resolves the difficulty of spatial matching between discrete datapoints and grid structures. We demonstrated the applicability of our model via 2D and 3D training image selection examples. The proposed methods effectively selected the appropriate TI, and then the pretreatment techniques for improving the accuracy of continuous TI selection were achieved. Moreover, the proposed method was successfully applied to training image selection of a discrete fracture network model. Finally, sensitivity analysis was carried out to show that sufficient conditioning data volume can reduce the uncertainty of the optimization results. By comparing with the improved MDevD method, the advantages of the new method are verified in terms of efficiency and reliability.

A method of reconstructing 3D model from 2D geological cross-section based on self-adaptive spatial sampling: A case study of Cretaceous McMurray reservoirs in a block of Canada

An orthogonal 2D training image is constructed from the geological analysis results of well logs and sedimentary facies; the 2D probabilities in three directions are obtained through linear pooling method and then aggregated by the logarithmic linear pooling to determine the 3D multi-point pattern probabilities at the unknown points, to realize the reconstruction of a 3D model from 2D cross-section. To solve the problems of reducing pattern variability in the 2D training image and increasing sampling uncertainty, an adaptive spatial sampling method is introduced, and an iterative simulation strategy is adopted, in which sample points from the region with higher reliability of the previous simulation results are extracted to be additional condition points in the following simulation to improve the pattern probability sampling stability. The comparison of lateral accretion layer conceptual models shows that the reconstructing algorithm using self-adaptive spatial sampling can improve the accuracy of pattern sampling and rationality of spatial structure characteristics, and accurately reflect the morphology and distribution pattern of the lateral accretion layer. Application of the method in reconstructing the meandering river reservoir of the Cretaceous McMurray Formation in Canada shows that the new method can accurately reproduce the shape, spatial distribution pattern and development features of complex lateral accretion layers in the meandering river reservoir under tide effect. The test by sparse wells shows that the simulation accuracy is above 85%, and the coincidence rate of interpretation and prediction results of newly drilled horizontal wells is up to 80%.

RockFlow: Fast Generation of Synthetic Source Rock Images Using Generative Flow Models

Image-based evaluation methods are a valuable tool for source rock characterization. The time and resources needed to obtain images has spurred development of machine-learning generative models to create synthetic images of pore structure and rock fabric from limited image data. While generative models have shown success, existing methods for generating 3D volumes from 2D training images are restricted to binary images and grayscale volume generation requires 3D training data. Shale characterization relies on 2D imaging techniques such as scanning electron microscopy (SEM), and grayscale values carry important information about porosity, kerogen content, and mineral composition of the shale. Here, we introduce RockFlow, a method based on generative flow models that creates grayscale volumes from 2D training data. We apply RockFlow to baseline binary micro-CT image volumes and compare performance to a previously proposed model. We also show the extension of our model to 2D grayscale data by generating grayscale image volumes from 2D SEM and dual modality nanoscale shale images. The results show that our method underestimates the porosity and surface area on the binary baseline datasets but is able to generate realistic grayscale image volumes for shales. With improved binary data preprocessing, we believe that our model is capable of generating synthetic porous media volumes for a very broad class of rocks from shale to carbonates to sandstone.

Comparison of various 3D pore space reconstruction methods and implications on transport properties of nanoporous rocks

Understanding fluid flow and transport within clay rock is essential for predicting caprock integrity in underground gas storage or as host rock in deep radioactive waste storage. The connectivity and topology of the nanopore space, which drive the transfer mechanisms of these materials, are still poorly known and direct 3D imaging is particularly challenging. In this work, we investigate and compare different stochastic reconstruction approaches based on two-point and multiple-point statistics (MPS) methods and using information from 2D training images for 3D volume rendering at submicron scale. A particular emphasis is given to the maximal critical distance of sampling between two consecutive 2D images which is necessary to obtain a coherent 3D reconstruction of the nanopore structure. We assess how these realizations honour various crucial transport properties of material, namely permeability, effective diffusion and longitudinal dispersion. Morphological features such as pore volume, specific surface, Euler characteristic and tortuosity are used to analyze the results. The methods are employed on a synthetic clay of nanometric porosity for which FIB-SEM images are available. Results indicate that the 3DA-MPS and weighted-3DA-MPS approaches are the most suited for preserving pore space features and transport properties, the choice depending on the level of conditioning data available.

Accelerating simulation for the multiple-point statistics algorithm using vector quantization.

Multiple-point statistics (MPS) is a prominent algorithm to simulate categorical variables based on a sequential simulation procedure. Assuming training images (TIs) as prior conceptual models, MPS extracts patterns from TIs using a template and records their occurrences in a database. However, complex patterns increase the size of the database and require considerable time to retrieve the desired elements. In order to speed up simulation and improve simulation quality over state-of-the-art MPS methods, we propose an accelerating simulation for MPS using vector quantization (VQ), called VQ-MPS. First, a variable representation is presented to make categorical variables applicable for vector quantization. Second, we adopt a tree-structured VQ to compress the database so that stationary simulations are realized. Finally, a transformed template and classified VQ are used to address nonstationarity. A two-dimensional (2D) stationary channelized reservoir image is used to validate the proposed VQ-MPS. In comparison with several existing MPS programs, our method exhibits significantly better performance in terms of computational time, pattern reproductions, and spatial uncertainty. Further demonstrations consist of a 2D four facies simulation, two 2D nonstationary channel simulations, and a three-dimensional (3D) rock simulation. The results reveal that our proposed method is also capable of solving multifacies, nonstationarity, and 3D simulations based on 2D TIs.

Evaluating the morphological completeness of a training image.

Understanding the three-dimensional (3D) stochastic structure of a porous medium is helpful for studying its physical properties. A 3D stochastic structure can be reconstructed from a two-dimensional (2D) training image (TI) using mathematical modeling. In order to predict what specific morphology belonging to a TI can be reconstructed at the 3D orthogonal slices by the method of 3D reconstruction, this paper begins by introducing the concept of orthogonal chords. After analyzing the relationship among TI morphology, orthogonal chords, and the 3D morphology of orthogonal slices, a theory for evaluating the morphological completeness of a TI is proposed for the cases of three orthogonal slices and of two orthogonal slices. The proposed theory is evaluated using four TIs of porous media that represent typical but distinct morphological types. The significance of this theoretical evaluation lies in two aspects: It allows special morphologies, for which the attributes of a TI can be reconstructed at a special orthogonal slice of a 3D structure, to be located and quantified, and it can guide the selection of an appropriate reconstruction method for a special TI.

TiConverter: A training image converting tool for multiple-point geostatistics

TiConverter is a tool developed to ease the application of multiple-point geostatistics whether by the open source Stanford Geostatistical Modeling Software (SGeMS) or other available commercial software. TiConverter has a user-friendly interface and it allows the conversion of 2D training images into numerical representations in four different file formats without the need for additional code writing. These are the ASCII (.txt), the geostatistical software library (GSLIB) (.txt), the Isatis (.dat), and the VTK formats. It performs the conversion based on the RGB color system. In addition, TiConverter offers several useful tools including image resizing, smoothing, and segmenting tools. The purpose of this study is to introduce the TiConverter, and to demonstrate its application and advantages with several examples from the literature.

Pattern density function for reconstruction of three-dimensional porous media from a single two-dimensional image.

Three-dimensional (3D) structures are useful for studying the spatial structures and physical properties of porous media. A 3D structure can be reconstructed from a single two-dimensional (2D) training image (TI) by using mathematical modeling methods. Among many reconstruction algorithms, an optimal-based algorithm was developed and has strong stability. However, this type of algorithm generally uses an autocorrelation function (which is unable to accurately describe the morphological features of porous media) as its objective function. This has negatively affected further research on porous media. To accurately reconstruct 3D porous media, a pattern density function is proposed in this paper, which is based on a random variable employed to characterize image patterns. In addition, the paper proposes an original optimal-based algorithm called the pattern density function simulation; this algorithm uses a pattern density function as its objective function, and adopts a multiple-grid system. Meanwhile, to address the key point of algorithm reconstruction speed, we propose the use of neighborhood statistics, the adjacent grid and reversed phase method, and a simplified temperature-controlled mechanism. The pattern density function is a high-order statistical function; thus, when all grids in the reconstruction results converge in the objective functions, the morphological features and statistical properties of the reconstruction results will be consistent with those of the TI. The experiments include 2D reconstruction using one artificial structure, and 3D reconstruction using battery materials and cores. Hierarchical simulated annealing and single normal equation simulation are employed as the comparison algorithms. The autocorrelation function, linear path function, and pore network model are used as the quantitative measures. Comprehensive tests show that 3D porous media can be reconstructed accurately from a single 2D training image by using the method proposed in this paper.

Mapping directional variations in seismic character using gray-level co-occurrence matrix-based attributes

Texture attributes describe the spatial arrangement of neighboring amplitudes values within a given analysis window. We chose a statistical texture classification method, the gray-level co-occurrence matrix (GLCM), and its derived attributes, to produce a semiautomated description of the spatial arrangement of seismic facies. The GLCM is a measure of how often different combinations of neighboring pixel values occur. We tested the application of directional GLCM-based attributes for the detection of seismic variability within paleoriver features. Calculation of 3D GLCM-based attributes can be done in 13 space directions. The results of GLCM-based attribute calculation differed depending on the chosen GLCM parameters (number of gray levels, analysis window, and direction of calculation). We specifically focused on how the direction of calculation influenced the computation of attributes, while keeping other parameters constant. We first tested the workflow on a 2D training image and later ran on a real seismic amplitude volume from the Vienna Basin. Based on the GLCM-based attributes, we could map the channel features and extract them as geobodies. Additionally, we generated a new set of directional GLCM-based attributes to detect spatial changes in the seismic facies. By comparing these directional attributes, we could determine areas within the channel features having higher directional variability. Areas with higher tendency to directional variations might be associated with changes in lithology, seismic facies, or with seismic anisotropy.

Improvement and Kernelization of T-2DPCA with Application to Face Recognition

T-2DPCA, a novel approach considering the third-order tensors as linear operators on the space of oriented matrices, benefits from treating a 2D image as an inherently integrated object, has been proposed recently and showed better performance than traditional matrix PCA in image analysis and recognition. In T-2DPCA, a reconstructing tubal coefficient is obtained from the defined tensor product, called T-product, of a 2D image and a 2D basis element. In this study, by assuming that an eigenvector of the covariance tensor of the 2D training images is the tensor linear combination, called T-linear combination, of the training images, the T-2DPCA is improved to a new version with better performance. The improved method is further extended to a nonlinear version by using the general kernel trick in machine learning field, but with a new inner product called inside product defined with the T-product in the third-order tensor spaces, and simultaneously the general inner product defended in vector spaces. The effectiveness of the proposed algorithms is tested by face recognition experiment results.

RECONSTRUCTION OF NON-STATIONARY COMPLEX SPATIAL STRUCTURES BY A NOVEL FILTER-BASED MULTI SCALE MPS ALGORITHM

Geologic modeling of complex structures needs realistic algorithms that can extract and reproduce position-dependent statistical relations in these phenomena. For this purpose, methods based on Multi Point Statistics (MPS) should be developed because spatial patterns such as curvilinearity cannot be recognized via two-point statistical concepts. MPS extracts high order spatial relations from a Training Image (TI) as a visually and statistically explicit model and some regional images in non-stationary cases. In this work a novel pattern to pattern filter-based algorithm is first presented for modeling of stationary spatial structures. Then for nonstationary images, in an automatic preprocessing step, one scale and azimuth for each node in final realization map is inferred from regional/auxiliary images. These values are used in patterns similarity measurements and pasting steps in a multi-scale simulation process. Performance of proposed algorithms is investigated by applying the methods on several binary 2D training images. In comparison, lower running time and better visual quality in all cases is clear. In the final non-stationary images, spatial statistics of training images and physical features of regional images are reproduced and traditional methods related concepts such as search tree and realization zoning and their unintended effects are eliminated.