Intrinsic Random Functions Of Order Research Articles

Tangent fields, intrinsic stationarity, and self similarity

This paper studies the local structure of continuous random fields on Rd taking values in a complete separable linear metric space V. Extending seminal work of Falconer, we show that the generalized (1+k)-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of V-valued intrinsic random functions of order k (IRFk, k=0,1,…). To this end, we focus on the special case where V is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order V-valued IRFk’s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar V-valued IRFk’s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valued versions of the general k-th order operator fractional Brownian fields and are characterized by their self-similarity operator exponent as well as a finite trace class operator valued spectral measure. We conclude with several examples motivating future applications to probability and statistics.

Iterative Atmospheric Phase Screen Compensation for Near-Real-Time Ground-Based InSAR Measurements Over a Mountainous Slope

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.

Quantification of methane emissions in a Mediterranean landfill (Southern Spain). A combination of flux chambers and geostatistical methods

In this study, landfill gas emissions from a landfill located in southern Spain were estimated using static surface flux chambers and applying and comparing four geostatistical methods; ordinary kriging, lognormal kriging, intrinsic random functions of order K and indicator kriging. This paper presents the methodology used to calculate methane (and carbon dioxide) emissions from a landfill in southern Spain. Static flux chambers were used to estimate emissions through the sealing layer of a landfill assuming that the geospatial mean best expresses the average value of these emissions. This study considers several geostatistical methods for obtaining the corresponding spatial estimation, using measurements obtained from static flux chambers and finding the best proven results. The most appropriate geostatistical analysis method was found to be indicator kriging and lognormal kriging because of the simplicity of its implementation and the transformation of the flux measurements. Methane surface emissions (100 g·m−2·d−1) and visualization of the hotspots were significant enough to result in the placement of a new cover across the entire landfill. This additional cover had an immediate impact on the effectiveness of the recovery system and increased LFG collection flow rates by 15% with an increase in CH4 concentration in the collected gas from 50% to 60%.

Intrinsic random functions for mitigation of atmospheric effects in terrestrial radar interferometry

Abstract 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.

Spectral representation of intrinsically stationary fields

Transferring the concept of processes with weakly stationary increments to arbitrary locally compact Abelian groups two closely related notions arise: while intrinsically stationary random fields can be seen as a direct analog of intrinsic random functions of order k applied by G. Matheron in geostatistics, stationarizable random fields arise as a natural analog of definitizable functions in harmonic analysis. We concentrate on intrinsically stationary random fields related to finite-dimensional, translation-invariant function spaces, establish an orthogonal decomposition of random fields of this type, and present spectral representations for intrinsically stationary as well as stationarizable random fields using orthogonal vector measures.

Interpolating datasets with trends: A modified median polish approach

This article investigates an alternative method to deal with data sets in the presence of trends. Median polish kriging (MPK) was introduced as an alternative solution to universal kriging or intrinsic random functions of order k (IRF-k) for estimation in the presence of trends. The maps obtained using the original MPK algorithm show banding artefacts which do not appear in the reference data set. A modified version of MPK was introduced to attempt to remove the banding artefacts. The results confirm the improvement in quality of estimate using the modified version of MPK (called MPKm), which takes into account the problems of clustered samples and boundary effect associated with the re-addition of the trend along bands. The variation introduced in the median polish algorithm proved to be satisfactory in eliminating the artefacts.

Study of the spatio-temporal variation of soil moisture under forest using intrinsic random functions of order k

The aim of this study is to describe the spatial pattern and temporal evolution of soil water content observed at the plot scale during the soil drying and wetting cycle. The intrinsic random function of order k (IRF- k) approach was used which decomposes the drift and covariance structure to define models of spatial covariance through increments of a sufficiently high order, so that the drift can be filtered out and stationarity attained. The raw directional variograms of soil water content of the study site in southern Italy showed a clear parabolic increase and zonal anisotropy in the northern and eastern directions. The order of the drift ( k) was 2, whereas the generalized covariance model varied during the soil drying and wetting cycle. The maps of soil water content showed the dynamics of soil water redistribution as a function of the temporal rainfall pattern. The spatial variation of soil water was affected by two main factors: topography and texture. Also the spatial distribution of the kriging standard errors changed over time as a function of the date of soil water content measurement. Finally, a critical analysis on the use of the generalized covariance approach is developed and the results compared with the ones of universal kriging.

Intrinsic random function of order k kriging of electrical resistivity data for estimating the extent of saltwater intrusion in a coastal aquifer system

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

IRFK2D is an ANSI Fortran-77 program that generates realizations of an intrinsic function of order k (with k equal to 0, 1 or 2) with a permissible polynomial generalized covariance model. The realizations may be non-conditional or conditioned to the experimental data. The turning bands method is used to generate realizations in 2D and 3D from simulations of an intrinsic random function of order k along lines that span the 2D or 3D space. The program generates two output files, the first containing the simulated values and the second containing the theoretical generalized variogram for different directions together with the theoretical model. The experimental variogram is calculated from the simulated values while the theoretical variogram is the specified generalized covariance model. The generalized variogram is used to assess the quality of the simulation as measured by the extent to which the generalized covariance is reproduced by the simulation. The examples given in this paper indicate that IRFK2D is an efficient implementation of the methodology.

Sole egg distributions in space and time characterised by a geostatistical model and its estimation variance

Geostatistics is used to model the spatio-temporal variability in egg volumetric density measured by monitoring surveys performed on a major spawning ground of sole in the Bay of Biscay (France). An approximation formula is proposed for computing the estimation variance of the mean egg density estimated over space and time. The major structural feature is the time invariance in the spatial distribution of the relative density (density(x,t)/mean(t)). This leads to interpretation of the different samplings as different realisations of the same random function in space. Because of the time-repeated samplings at each station, drift (or trend) and residuals can be separated. The drift is defined as the spatial structural component that is constant over time. The spatio-temporal variability was described by a multiplicative model with two effects: a time effect corresponding to a proportionality to the spatial average at each time and a space effect corresponding to the time-invariant spatial distribution of the relative density. Residuals showed structure in space but not in time and no space-time interaction. Geostatistical aggregation curves were used to characterise invariance over time of the aggregation pattern, the drift was mapped using non-stationary IRF-k (Intrinsic Random Functions of order k) kriging and variograms and crossvariograms were used to study the structure in the residuals.

GCINFE: A computer program for inference of polynomial generalized covariance functions

GCINFE, an ANSI FORTRAN-77 coded computer program for statistical inference of polynomial generalized covariance parameters, is examined in this paper. The generalized covariance function describes a nonstationary random field within the framework of the theory of intrinsic random functions of order k. The program utilizes four different estimation methods: minimum norm, minimum variance unbiased quadratic, minimum variance unbiased quadratic iterative and maximum likelihood. An important feature of these methods is that they provide a measure of uncertainty of the estimated parameters as defined by the standard deviation of the estimation error. The sampling distribution statistics of the four different methods evaluated through the Monte Carlo simulation technique are also presented for an example of the generalized covariance model.

Stochastic Modeling Of Space Dependent Reservoir-Rock Properties

Abstract 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

Spatial Variability of Saturated Hydraulic Conductivity of the Subsoil of Two Forested Watersheds

AbstractSubsurface flow has been shown to be greater in soils with spatially dependent hydraulic properties. The objective of this study was to quantify the spatial dependency in saturated hydraulic conductivity, Ksat, of two potential shallow waste disposal sites in eastern Tennessee. The Guelph Permeameter was used to determine Ksat of the subsoil of two forested watersheds, Walker Branch and Melton Branch. The simultaneous equations approach resulted in more than 50% negative Ksat values and was concluded to be invalid for these forested subsoils. As an alternative, flux density was used as an index of the saturated hydraulic conductivity for spatial correlation analysis and for determination of scaling factors. the spatial realizations of hydraulic conductivity for both watersheds were found to be intrinsic random functions of order zero, i.e., stationary, with a generalized covariance function for Walker Branch consisting of a pure nugget and for Melton Branch a nugget with a linear term. Semivariograms revealed no spatial dependency for separation distances >4.2 m at Walker Branch. Spatial dependency at Melton Branch for separation distances <30 m was described by an exponential model. The hydraulic conductivity of Walker Branch subsoils had a greater arithmetic mean (1.50 vs. 0.76 × 10−5 m s−1) and greater random variability than Melton Branch subsoils. These findings suggest that soils common to Walker Branch Watershed are hydrologically preferable for shallow waste disposal.

A nonparametric view of generalized covariances for kriging

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.

Automated mapping of the ocean floor using the theory of 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.

On the estimation of the generalized covariance function

The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables.

Geostatistical Approach in Coal Resource Estimation: ABSTRACT

The conventional reserve/resource estimation methods (triangular, polygonal, and isopach) are briefly described and then compared against a new geostatistical approach. The comparison is based on estimates using real-world data from southern Illinois. The geostatistical procedure employed is a kriging technique called the intrinsic random functions of order k, using spherical covariance models. While the intrinsic random function approach is usually considered to be a local estimation procedure, the introduction of sampling methodology allows its extension to global estimation problems without excessive cost for computation. The major advantages of the geostatistical approach are that it provides a built-in measure of the precision of its estimates, thus allowing the dete mination of confidence limits, and also gives additional insight into the spatial variability of coal seams. The estimates are improved as more geological information of the coal deposits becomes available. End_of_Article - Last_Page 705------------

Automatic Kriging And Contouring In The Presence Of Trends (Universal Kriging Made Simple)

Abstract The main purpose of this paper is to present a new automatic interpolation and contouring technique which automatically recognizes the spatial structure of the dam to be contoured even when, trends are present and data are not regularly spaced. A brief relief of contouring problems is presented the idea of spatial structure and its characterization by a generalized covariance is then introduced and the POLYPAC program is described. Applications in various areas (oil shales, sulphide mines, coal field, cement industry, geophysical data) are illustrated and a detailed appendix covers the mathematical theory of the intrinsic random functions of order k, which is the basis for the program. References to other similar programs are included. Conceptual Background Introduction Attempting to survey everything that has been written on the subject of automatic contouring is a challenging task as ideas are generated in various fields with little communication between each other, such as coal reserves, ore reserve estimation or geophysical mapping, not to mention geography and meteorology. Also, the concern of contouring packages for high cosmetic quality like G.P.C.P., CSPL. Stampede, etc... tends to push people away from the fundamentals of the problem and its conceptualization, Most of the time the interpolation method used in a program is unknown to the map user, or he is left with the choice of up to 32 different interpolation subroutines in the same package, with no further hint at which method is adapted to his problem. It is our opinion that most mapping problems can be encased in the frame of a unifying theory which will allow the generation of the most adequate map for a given purpose. This theory has been known as kriging or simply geostatistics and was developed by Matheron, The full name of the most complete version of it is "theory of intrinsic random functions of order k", and all its intricacies have now been turned into another black box, which produces the best theoretical maps within the framework of distance weighted and trend-surface interpolation. It is one of the purposes of this paper to show which concepts are needed to understand why a given interpolation algorithm is good in certain cases and unacceptable in others. These concepts of spatial structure, drift, trend, high-frequency noise, estimation variance have been known for a long time, but we must admit that map users should be presented with something else than a truckload of equations. The contouring problem is easily split in two; the first problem is that of the generation of II. regular grid of points, the second being that of aesthetically contouring this fine grid. So far only the first problem retained the attention of geostatisticians, with few exceptions. Broken contours are not acceptable even if the theory behind the grid generation technique is faultless (Olea, 1972). There now exists a program matching the most sophisticated theory and superb contours.