Kernel Principal Component Analysis for Efficient, Differentiable Parameterization of Multipoint Geostatistics

The purpose of history matching is to achieve geological realizations calibrated to the historical performance of the reservoir. For complex geological structures it is usually intractable to run tens of thousands of full reservoir simulation to trace the most probable geological model. Hence the inadequacy of the history-matching results frequently leads to poor estimation of the true model and high uncertainty in production forecasting. Reduced-order modeling procedures, which have been applied in many application areas including reservoir simulation, represent a promising means for constructing efficient surrogate models. Nonlinear dimensionality reduction techniques allow for encapsulating the high-resolution complex geological description of reservoir into a low-dimensional subspace, which significantly reduces number of unknowns and provides an efficient way to construct a proxy model based on the the reduced-dimension parameters. Polynomial Chaos Expansions (PCE) is a powerful tool to quantify uncertainty in dynamical system when there is probabilistic uncertainty in the system parameters. In reservoir simulation it has been shown to be more accurate and efficient compared to traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution is proved when the order of the PCE is increased. Accordingly PCE proxy can be used as the pseudo-simulator to represent the surface responses of the uncertain variables. When the objective and constraints of a reservoir model is described by multivariate polynomial functions, there are very efficient algorithms to compute the global solutions. We have developed a workflow at which incorporates PCE to find the global minimum of the misfit surface and assess the uncertainty associated with. The accuracy of the PCE proxy increases with the additional trial runs of the reservoir simulator. We conduct a two dimensional synthetic case study of a fluvial channel as well as a real field example to demonstrate the effectiveness of this approach. Kernel Principal Component Analysis (KPCA) is used to parameterize the complex geological structure. The study has revealed useful reservoir information and delivered more reliable production forecasts. PCE-based history match enhances the quality and efficiency of the estimation of the most probable geological model and improve the confidence interval of production forecasts.

The Late Carboniferous strata exposed in the Sacramento Mountains in Southern New Mexico, USA, have long been considered classic exposures documenting reciprocal high-frequency mixed carbonate–siliciclastic cyclicity and shelf-edge algal-mound growth. The growth style and internal architecture of these phylloid algae mounds depend on their position on the shelf and are controlled by potential accommodation space, depth of the photic zone and hydrodynamic energy. The combination of these parameters results in a laterally variable amount of reworked phylloid algae debris and in-situ mound core facies along the depositional profile. This variable architecture can be observed on the outcrop and results in a complex distribution of these two lithofacies in three dimensions that is challenging to reproduce in a 3D geocellular model. Two geostatistical estimation algorithms are used to stochastically model carbonate buildups: surface-based and multipoint statistics (MPS)-based. The surface-based model uses two-point statistics and is built by first recreating the overall geometry of the mound and then reproducing the internal architecture using indicator Gaussian simulation, but requires strong secondary trend data to reproduce the correct facies architecture. The MPS model successfully recreates both the geometry and internal architecture of the mound, but requires a complicated training image and complex multigrid simulation that would be hard to implement in subsurface. This comparison demonstrates that modelling carbonate buildup geometry and internal architecture is not trivial and requires complex workflow with secondary trends. These secondary trends require a significant amount of prior knowledge that is easily extracted from outcrop observations, but would be difficult to assess in subsurface data.

Kernel principal component analysis (KPCA) has been effectively applied as an unsupervised non-linear feature extractor in many machine learning applications and suggested for various data stream classification tasks requiring a nonlinear transformation scheme to reduce dimensions. However, the dimensionality reduction ability is restricted because of KPCA’s high time complexity. So the practicality of KPCA on large datasets is rare. Therefore in this paper, we proposes a novel kind of incremental kernel principal component analysis algorithm: Data characteristic extraction based on IPCA algorithm –DCEIPCA, which allows efficient processing of large datasets and overcome the insufficient of KPCA. On the basis of DCEIPCA, we propose Classification expert system (CES) for intrusion detection system. Extensive experiments on KDDcup99 datasets confirm the superiority of Intrusion detection system based on CES(IDSCES) over other recent Intrusion detection system[4-11].

Two-point or multiple-point statistics? A comparison between the ensemble Kalman filtering and the ensemble pattern matching inverse methods

A multiobjective dominance and decomposition algorithm for reservoir model history matching

Summary Accurate characterization of sub-surface oil reservoirs is an essential prerequisite to the design and implementation of enhanced oil recovery (EOR) scenarios. Specifically, in reservoir characterization, integrating static and dynamic data into reservoir models to construct accurate and realistic models has received considerable attention. Unlike most of the conventional geostatistical approaches of integrating data into reservoir models that are based on semi-variograms (two-point statistics) as a measure of spatial connectivity, a complete multiple-point (MP) statistic framework is presented in this paper. In contrast to two-point statistic methods, MP statistics-based methods are capable of reproducing curvilinear geological structures. The algorithm starts with extracting MP statistics from training images (TI) using an optimal spatial template. After collecting different patterns and building the MP histogram, the pattern reproduction process commences. This process begins from data locations and then grows to fill the whole reservoir domain. The algorithm accounts for three main practical issues: uncertainty in geological scenarios, scanning template and non-stationarity. The MP statistics algorithm (growthsim) is capable of integrating data from multiple data sources. Among these data types is dynamic data or flow history. The conventional approach to integrate production information into reservoir models is by iterative perturbation of the reservoir model until the production history of the reservoir is matched. Iterative methods have been applied till date to random fields that are completely characterized by a two-point co-variance function. In contrast, this paper presents a forward modelling approach that investigates history matching within a MP modelling framework. A novel technique implemented in this research is based on the merging of MPs inferred from history matched and geological models. Pattern growth is performed subsequently by sampling from the merged MP histograms. History matched models using the presented approach show an excellent agreement with underlying geological descriptions and match production history.

We propose a new kernel entropy component analysis approach to object recognition with image set by fusing the information entropy. Since the geometry of Symmetric Positive Definite (SPD) matrices, we model the image set with its covariance matrix (nonsingular). Thus the object recognition with image set can be model as classifying problem on the riemannian manifold space. Given the proper kernel function derived from efficient metric for SPD matrices, points lie on manifold space represented by covariance matrices can be cast into high dimensional euclidean space. In this euclidean space, traditional methods used for dimensional reduction can be applied directly. Accounting for the measurability to information of the Renyi entropy, we fuse the information entropy to the dimensional reduction progress of our method. Similar to Kernel Principal Component Analysis (KPCA), the method accomplish data dimensionality reduction by projection onto a subset of entropy preserving KPCA axes. But this subset does not need to correspond to the top eigenvalues of the kernel matrix. The positive results of experiment on datasets demonstrated effectiveness of our method.

In the evaluation of Lagrangian particle tracking (LPT) measurement data the use of spatially binned flow statistics in the form of one, two or multi-point statistics is often an essential step towards better understanding of the measured flow fields. Increasingly there is a focus towards uncertainty quantification of the measurement system however these evaluations are seldom used to directly improve the statistics by directly involving them into the calculation. We present our Functional Binning approach which makes use of such uncertainty information as a core component for the calculation of improved statistics. The improvements towards prior approaches are shown utilizing synthetic data as well as data from a real-world subsonic jet experiment. Beyond the initial formulation for one-point statistics, we show that this approach is readily extended towards two-point statistics and explore more advanced utilizations of uncertainty information for the optimal selection of particle pairs. Furthermore, the benefits of more individualized particle error estimations are investigated and some strategies for archiving such information are investigated.

As an important modeling parameter in multi-point geostatistics, training images determine the modeling effect to a great extent. It is necessary to evaluate and optimize the applicability of candidate training images before modeling by multi-point geostatistics. Conventional two-dimensional training images can’t describe the overlapping relation of sedimentary facies in space, and there is a deficiency in describing the event relation of single data. This paper puts forward a new training image optimization method. The basic idea is to arrange and analyze sand bodies filled with sedimentary sand bodies in point dams and river channels in different periods. The method of obtaining three-dimensional training images is to use sand thickness maps and sedimentary facies maps for spatial constraints. The simulation test shows that compared with the sedimentary facies model obtained from two-dimensional training images, the sedimentary facies model obtained from three-dimensional training images through multi-point geostatistics has high compatibility and is more in line with geological understanding. On the basis of fully understanding the development characteristics of the sedimentary system and quantitative geometry of sedimentary body in the study area, sedimentary microfacies models based on sequential indicator simulation method and target simulation method are established, respectively. By comparing and analyzing the sedimentary microfacies models established by three different methods, the results show that the multi-point geostatistics can stably present the planar distribution characteristics and overlapping spatial relationship of sedimentary microfacies and reproduce the complex spatial structure and geometric shape of fluvial facies. The model established by this method is more in line with the geological sedimentary model.

A workflow for modelling fluvial meander-belt successions: Combining forward stratigraphic modelling and multi-point geostatistics

High resolution sequence stratigraphical methods have been used in the detailed geological modeling of an element of the Ness formation of a North Sea field. This updated geological description has been used as input to the stochastic geological model MOHERES. 14 realizations of facies architecture and petrophysical properties have been generated. The stochastically generated realizations have been scaled up to a refined element of an existing deterministic, history matched reservoir simulation model for the full Upper Brent reservoir. The upscaled versions of the realizations have been connected to the full field model, and reservoir simulations have been performed to compare twelve years of production history with simulated results for well production, RFT- and PLT-data. Six of the realizations gave good to very good results when compared to measured production data. These realizations were used for simulation of the future production performance to the year 2010, making it possible to estimate its uncertainty.

This paper discusses the effects of introducing nonlinear interactions and noise-filtering to the covariance matrix used in Markowitz’s portfolio allocation model, evaluating the technique’s performances for daily data from seven financial markets between January 2000 and August 2018. We estimated the covariance matrix by applying Kernel functions, and applied filtering following the theoretical distribution of the eigenvalues based on the Random Matrix Theory. The results were compared with the traditional linear Pearson estimator and robust estimation methods for covariance matrices. The results showed that noise-filtering yielded portfolios with significantly larger risk-adjusted profitability than its non-filtered counterpart for almost half of the tested cases. Moreover, we analyzed the improvements and setbacks of the nonlinear approaches over linear ones, discussing in which circumstances the additional complexity of nonlinear features seemed to predominantly add more noise or predictive performance.

Slope reliability analysis has always been a research focus for geotechnical engineers. Currently, most studies use stationary random fields to simulate the spatial variability of soil parameters. However, many field test data show that soil parameters such as cohesion and friction angle often exhibit obvious nonstationary characteristics along the burial depth direction. Moreover, the slope reliability analysis method based on Monte Carlo simulation requires a combination of the FEM and a large number of computational resources, resulting in extremely low computational efficiency. Therefore, this paper proposes a convolutional neural network (CNN)-based surrogate model for slope reliability analysis, emphasizing the nonstationary spatial variability of cohesion and friction angle. First, stationary and nonstationary random fields considering the spatial variability of shear strength parameters are generated, and sample data sets are collected using an automated computational program. Next, a CNN surrogate model is established to predict the factor of safety (FOS), which is trained using nine sample data sets containing stationary and nonstationary random fields. Finally, the model performance is validated using six slope models with different shear strength parameters. The results show that the CNN surrogate model can adapt to the data characteristics of different random fields and also shows good predictive performance in handling unseen sample data sets. The predicted FOS values show a high degree of consistency with the results calculated by the FEM, with the computation time being significantly reduced.

For an ensemble-based history matching of a channelized reservoir, loss of geological plausibility is challenging because of pixel-based manipulation of channel shape and connectivity despite sufficient conditioning to dynamic observations. Regarding the loss as artificial noise, this study designs a serial denoising autoencoder (SDAE) composed of two neural network filters, utilizes this machine learning algorithm for relieving noise effects in the process of ensemble smoother with multiple data assimilation (ES-MDA), and improves the overall history matching performance. As a training dataset of the SDAE, the static reservoir models are realized based on multipoint geostatistics and contaminated with two types of noise: salt and pepper noise and Gaussian noise. The SDAE learns how to eliminate the noise and restore the clean reservoir models. It does this through encoding and decoding processes using the noise realizations as inputs and the original realizations as outputs of the SDAE. The trained SDAE is embedded in the ES-MDA. The posterior reservoir models updated using Kalman gain are imported to the SDAE which then exports the purified prior models of the next assimilation. In this manner, a clear contrast among rock facies parameters during multiple data assimilations is maintained. A case study at a gas reservoir indicates that ES-MDA coupled with the noise remover outperforms a conventional ES-MDA. Improvement in the history matching performance resulting from denoising is also observed for ES-MDA algorithms combined with dimension reduction approaches such as discrete cosine transform, K-singular vector decomposition, and a stacked autoencoder. The results of this study imply that a well-trained SDAE has the potential to be a reliable auxiliary method for enhancing the performance of data assimilation algorithms if the computational cost required for machine learning is affordable.

We propose a nonlinear covariance region descriptor for target tracking. The target object appearance and spatial information is represented using a covariance matrix in a target derived Hilbert space using kernel principal component analysis. A similarity measure is derived, which computes the similarity of a candidate image region to the learned covariance matrix. A variational technique is provided to maximize the similarity measure, which iteratively finds the best matched object region. Tracking performance is demonstrated on a variety of sequences containing noise, occlusions, illumination changes, background clutter, etc.