Spectral representation of intrinsically stationary fields

A saltwater intrusion study using electrical resistivity distribution was conducted in a coastal aquifer system in the southeastern part of Busan, Korea. This aquifer system is divided into four layers according to the hydrogeologic characteristics, and the horizontal extent of intruded saltwater is determined at each layer through the geostatistical interpretation of electrical resistivity data. Intrinsic random function of order k (IRF-k) kriging is performed with covariance models to produce the plane of spatial mean resistivities. The kriged estimates are evaluated by cross validation, and they showed a good agreement with the true values. The statistics of cross validation represented low errors for the estimates. In the resistivity contour maps more than 5 m below the surface, a dominant direction of saltwater intrusion was observed beginning from the east side. These results led the authors to conclude that the application of geostatistical technique to electrical resistivity data is useful for assessing saltwater intrusion in a coastal aquifer system.

IRFK2D: a computer program for simulating intrinsic random functions of order k

High-quality contour maps can be computer drawn from single track echo-sounding data by combining Universal Kriging and the theory of intrinsic random function of order K (IRFK). These methods interpolate values among the closely spaced points that lie along relatively widely spaced lines. The technique provides a variance which can be contoured as a quantitative measure of map precision. The technique can be used to evaluate alternative survey trackline configurations and data collection intervals, and can be applied to other types of oceanographic data.

The benefits of terrestrial radar interferometry (TRI) for deformation monitoring are restricted by the influence of changing meteorological conditions contaminating the potentially highly precise measurements with spurious deformations. This is especially the case when the measurement setup includes long distances between instrument and objects of interest and the topography affecting atmospheric refraction is complex. These situations are typically encountered with geo-monitoring in mountainous regions, e.g. with glaciers, landslides or volcanoes. We propose and explain an approach for the mitigation of atmospheric influences based on the theory of intrinsic random functions of order k (IRF-k) generalizing existing approaches based on ordinary least squares estimation of trend functions. This class of random functions retains convenient computational properties allowing for rigorous statistical inference while still permitting to model stochastic spatial phenomena which are non-stationary in mean and variance. We explore the correspondence between the properties of the IRF-k and the properties of the measurement process. In an exemplary case study, we find that our method reduces the time needed to obtain reliable estimates of glacial movements from 12 h down to 0.5 h compared to simple temporal averaging procedures.

Numerical modeling of space dependent and variant reservoir-rock properties such as porosity, permeability, etc., are routinely used in the oil industry. The techniques commonly applied are simplistic resulting in poor estimates. Alternatively, a stochastic framework in terms of the theory of Intrinsic Random Functions of order k may be used. Then, "best" estimators in the well established stochastic sense and their associated estimation variance can be derived. In addition, reservoir-rock properties may be simulated in the form of stochastic random fields and subsequently conditioned to reproduce the data measurements and their statistical properties. The advantages of stochastic approach are bared in the use of patterns of variation within the existing data in terms of generalized covariance or variograms. Stochastic models can be used differently depending on their properties. Estimated models may be used in describing and assessing the performance of the reservoir or the estimation of reserves. Simulated ones can provide alternatives to test the sensitivity of flow simulation or to tackle estimation of effective block permeabilities. Introduction The actual performance of a reservoir during the exploitation time is reproduced-predicted from reservoir simulation studies. A controlling factor in the quality of predictions is the numerical description of reservoir-rock properties such as porosity, permeability and fluid saturations(1,2,3). Furthermore, while the sophistication of fluid transport formulations, numerical methods and numerical models continues to increase, quantitative geological modeling remains significantly less advanced. As a result, prediction problems arise from unreliable reservoir description(4). For these reasons, it has already been suggested(3,4) that more emphasis be given to increasing the sophistication and improving the certainty of quantitative reservoir characterization. During the last few years, attempts at reservoir description(5) have had two major aspects. The first is a qualitative-geological and the second quantitative-numerical. Although attention was given and progress has been made in the former(6,7), there is much room for improvement in the latter, so that qualitative information provided by the first aspect can be expressed quantitatively. The present study is a natural step in the continuing effort to improve geological reservoir description. It attempts to provide basic elements of a generalized stochastic framework(8,9) which allows the effective transformation of geological descriptions to computer processable numerical equivalents. The reasons why stochastic models, as opposed to deterministic, chosen are: (1) geological variables exhibit a locally random behavior (Fig. I), but on average there is a structural aspect in spatial variability (correlation structure) which is expressed with the similarity of neighbouring values (Fig. 2); the stochastic approach (2) enables the estimation of large volumes from core samples using best (optimal) estimators; (3) provides a measurement of uncertainty in the above estimation; and (4) allows the generation (simulation) of fields of values which "look like" reality in terms of their variation pattern. The stochastic framework in modeling geological variables was originally introduced in the mining industry(10,11) and found extensive use in ground water hydrology(12,13). In reservoir engineering, few authors have attempted to use stochastic methods for parameter estimation(14,15) or simulation(16,17,18) using restricting

The problem of sequentially scanning and predicting data arranged in a multidimensional array is considered. We introduce the notion of a scandictor, which is any scheme for the sequential scanning and prediction of such multidimensional data. The scandictability of any finite (probabilistic) data array is defined as the best achievable expected performance on that array. The scandictability of any (spatially) stationary random field on /spl Zopf//sup m/ is defined as the limit of its scandictability on finite boxes (subsets of /spl Zopf//sup m/), as their edges become large. The limit is shown to exist for any stationary field, and essentially be independent of the ratios between the box dimensions. Fundamental limitations on scandiction performance in both the probabilistic and the deterministic settings are characterized for the family of difference loss functions. We find that any stochastic process or random field that can be generated autoregressively with a maximum-entropy innovation process is optimally scandicted the way it was generated. These results are specialized for cases of particular interest. The scandictability of any stationary Gaussian field under the squared-error loss function is given a single-letter expression in terms of its spectral measure and is shown to be attained by the raster scan. For a family of binary Markov random fields (MRFs), the scandictability under the Hamming distortion measure is fully characterized.

Two kinds of estimation variance functions for estimating a local average (LA) of a stationary (homogeneous) random field (RF) are derived. One is local estimation (LE) and the other is general estimation (GE) of LA. The former is for estimating LA at the observation location, and the latter is for obtaining LA at any arbitrary location within the RF. The geotechnical implications of these two estimations are that LE is for estimating LA of geotechnical parameters at the spot where the investigations are made, whereas GE is for estimating LA at any arbitrary location within the same layer. The behavior of the two estimation variance functions differs greatly, controlled by the number of observations (i.e. sample size) and the normalized layer thickness (layer thickness divided by autocorrelation distance of RF). Based on the derived estimation variance functions, methods for determining reliable characteristic values of geotechnical parameters and necessary sample size are proposed. The methods are based on the same framework as that of the traditional statistical theory, i.e. confidence interval of estimated parameters. However, the assumption of independently and identically distributed (i.i.d.) samples in the traditional statistical theory is replaced by the assumption of correlated samples from a stationary RF. The results obtained from the proposed methods for LE and GE differ from each other as well as from the traditional results, which has significant implications for geotechnical parameter estimation in geotechnical engineering practice.

A central limit theorem for functions of stationary max-stable random fields on [formula omitted

Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data.

In this article, an atmospheric phase screen (APS) compensation algorithm for a near real-time ground-based interferometry synthetic aperture radar (GB-InSAR) over a mountainous area is investigated. A novel APS compensation scheme is proposed to compensate the fluctuated APS caused by a spatial 3-D inhomogeneous refractivity index distribution without any a priori knowledge of moving location. The proposed method simultaneously addresses to identify moving pixels by a criterion of absolute velocity estimated by the coherent pixels technique (CPT). The proposed method consists mainly of three steps: 1) the stratified APS compensation; 2) identification of moving pixel candidate; and 3) the residual APS [remained APS after 1)] compensation by Kriging interpolation. The steps mentioned above are iteratively applied in order to increase the accuracy of the whole process. In this framework, we develop the 2-D quadratic polynomial model of the refractivity index with respect to slant range and topographic height for modeling the stratified APS. Furthermore, a prediction of the residual APS is achieved by applying the intrinsic random function of order $k$ (IRF- $k$ ) Kriging interpolation, taking into account the nonstationarity of the residual APS. We evaluate the proposed method using zero-baseline GB-differential InSAR (GB-DInSAR) data over a mountainous area located in Minami-Aso, Kumamoto, Japan, through the near real-time post-landslide measurement campaign.

We define multi-self-similar random fields, that is, random fields that are self-similar component-wise. We characterize them, relate them to stationary random fields using a Lamperti-type transformation and study these stationary fields. We also extend the notions of local stationarity and local stationarity reducibility to random fields. Our work is motivated by applications arising from climatological and environmental sciences. We illustrate these new concepts with the fractional Brownian sheet and the Lévy fractional Brownian random field.

Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the "treelike" random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line "winding through every region of the plane." If the random field has the FKG property, then only bounded level lines are possible. For stationary $C^2$ Gaussian random fields with covariance function decaying to 0 at $\infty$, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

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.