On a continuous spectral algorithm for simulating non-stationary Gaussian random fields

This paper presents an algorithm for simulating Gaussian random fields with zero mean and non-stationary covariance functions. The simulated field is obtained as a weighted sum of cosine waves with random frequencies and random phases, with weights that depend on the location-specific spectral density associated with the target non-stationary covariance. The applicability and accuracy of the algorithm are illustrated through synthetic examples, in which scalar and vector random fields with non-stationary Gaussian, exponential, Matérn or compactly-supported covariance models are simulated.     

Empirical Orthogonal Function (EOF) analysis of spatial random fields: Theory,accuracy of the numerical approximations and sampling effects

Empirical Orthogonal Function (EOF) analysis of spatial random fields involves calculation of the eigenfunctions of the covariance kernel of the field. For real-world applications, a numerical approximation is necessary because the process is spatially discretized. An approximation for two-dimensional fields is proposed and then, analytical solutions of the integral problem are derived and used to study the accuracy of the numerical approximations. Sampling effects are also considered.     

Simulation of intrinsic random fields of order <Emphasis Type="Italic">k</Emphasis> with a continuous spectral algorithm

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matérn generalized cross-covariance.     

Covariance functions and models for complex-valued random fields

In Geostatistics, primary interest often lies in the study of the spatial, or spatial-temporal, correlation of real-valued random fields, anyway complex-valued random field theory is surely a natural extension of the real domain. In such a case, it is useful to consider complex covariance functions which are composed of an even real part and an odd imaginary part. Generating complex covariance functions is not simple at all, but the procedure, developed in this paper, allows generating permissible covariance functions for complex-valued random fields in a straightforward way. In particular, by recalling the spectral representation of the covariance and translating the spectral density function by using a shifting factor, complex covariances are obtained. Some general aspects and properties of complex-valued random fields and their moments are pointed out and some examples are given.     

HYDRO_GEN: A spatially distributed random field generator for correlated properties

This paper describes a new method for generating spatially-correlated random fields. Such fields are often encountered in hydrology and hydrogeology and in the earth sciences. The method is based on two observations: (i) spatially distributed attributes usually display a stationary correlation structure, and (ii) the screening effect of measurements leads to the sufficiency of a small search neighborhood when it comes to projecting measurements and data in space. The algorithm which was developed based on these principles is called HYDRO_GEN, and its features and properties are discussed in depth. HYDRO_GEN is found to be accurate and extremely fast. It is also versatile: it can simulate fields of different nature, starting from weakly stationary fields with a prescribed covariance and ending with fractal fields. The simulated fields can display statistical isotropy or anisotropy.     

Gamma random field simulation by a covariance matrix transformation method

In studies involving environmental risk assessment, Gaussian random field generators are often used to yield realizations of a Gaussian random field, and then realizations of the non-Gaussian target random field are obtained by an inverse-normal transformation. Such simulation process requires a set of observed data for estimation of the empirical cumulative distribution function (ECDF) and covariance function of the random field under investigation. However, if realizations of a non-Gaussian random field with specific probability density and covariance function are needed, such observed-data-based simulation process will not work when no observed data are available. In this paper we present details of a gamma random field simulation approach which does not require a set of observed data. A key element of the approach lies on the theoretical relationship between the covariance functions of a gamma random field and its corresponding standard normal random field. Through a set of devised simulation scenarios, the proposed technique is shown to be capable of generating realizations of the given gamma random fields.     

Statistical Travel-time Tomography in Terms of Stacking Velocity

— Velocity evaluation is a key step in seismic analysis. The covariance of the true velocity field must be known when interpolating or simulating velocities from well measurements using geostatistical methods. In addition, inversion procedures often require information pertaining to this covariance. Traditionally it has been taken to be the covariance of stacking velocities. We present a simple example to show that this approximation can lead to significant errors. Better methods, such as those of Touati (1996) and Iooss (1998), use the variance of prestack picked travel times as a function of offset to infer that of the velocities. In this paper we extend their results on the estimation of the covariance of the reflected traveltimes, and obtain an explicit expression for the covariance of the square of the stacking slowness as a function of the covariance of the velocities. Although we are not able to invert the formula analytically to yield an explicit estimator for these parameters, the results obtained using it furnish a good and quick estimation of the velocity's covariance. This is illustrated with synthetic examples.     

用EOF分析方法研究强震前中国大陆水平位移场时空分布特征

利用大气科学常用的经验正交函数(EOF)分解方法, 对中国大陆连续观测的基准站的水平位移每周解时间序列进行分析。 首先对时间序列中不连续的数据进行内插处理, 并通过线性拟合从时间序列中去掉长期滑动速率的影响。 结合震例研究强震前水平位移场在NS向和EW向的时间和空间的分布特征。     

Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging

Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.     

Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging

Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.     

A stochastic model reduction method for nonlinear unconfined flow with multiple random input fields

In this paper we present a stochastic model reduction method for efficiently solving nonlinear unconfined flow problems in heterogeneous random porous media. The input random fields of flow model are parameterized in a stochastic space for simulation. This often results in high stochastic dimensionality due to small correlation length of the covariance functions of the input fields. To efficiently treat the high-dimensional stochastic problem, we extend a recently proposed hybrid high-dimensional model representation (HDMR) technique to high-dimensional problems with multiple random input fields and integrate it with a sparse grid stochastic collocation method (SGSCM). Hybrid HDMR can decompose the high-dimensional model into a moderate M-dimensional model and a few one-dimensional models. The moderate dimensional model only depends on the most M important random dimensions, which are identified from the full stochastic space by sensitivity analysis. To extend the hybrid HDMR, we consider two different criteria for sensitivity test. Each of the derived low-dimensional stochastic models is solved by the SGSCM. This leads to a set of uncoupled deterministic problems at the collocation points, which can be solved by a deterministic solver. To demonstrate the efficiency and accuracy of the proposed method, a few numerical experiments are carried out for the unconfined flow problems in heterogeneous porous media with different correlation lengths. The results show that a good trade-off between computational complexity and approximation accuracy can be achieved for stochastic unconfined flow problems by selecting a suitable number of the most important dimensions in the M-dimensional model of hybrid HDMR.     

Conditional components for simulation of vector random fields

Conditional component random fields (CC) based on Cholesky decomposition of the multivariate spectra are introduced in this study to develop a new method for conditional simulation of vector attributes in environmental and geological phenomena. The CC are independent random fields with covariance models obtained from projections and conditioning in the frequency domain. The approach is to simulate one attribute in the physical space and use the results to estimate the other attributes in the frequency domain. Then, a CC for the next attribute is simulated and projected on the other attributes. In general, any attribute is built as the sum of inverse Fourier transform of the orthogonal projection of previous simulated CC plus a last CC simulated in the physical space. This simulation approach continues in this fashion for several attributes and the order of them may be changed for different realizations. This method allows for data conditioning and simulation. A simplified version for intrinsically correlated random fields allows for an approach that avoids the frequency domain.     

Norm-dependent covariance permissibility of weakly homogeneous spatial random fields and its consequences in spatial statistics

 Permissibility of a covariance function (in the sense of Bochner) depends on the norm (or metric) that determines spatial distance in several dimensions. A covariance function that is permissible for one norm may not be so for another. We prove that for a certain class of covariances of weakly homogeneous random fields, the spatial distance can be defined only in terms of the Euclidean norm. This class includes commonly used covariance functions. Functions that do not belong to this class may be permissible covariances for some non-Euclidean metric. Thus, a different class of covariances, for which non-Euclidean norms are valid spatial distances, is also discussed. The choice of a coordinate system and associated norm to describe a physical phenomenon depends on the nature of the properties being described. Norm-dependent permissibility analysis has important consequences in spatial statistics applications (e.g., spatial estimation or mapping), in which one is concerned about the validity of covariance functions associated with a physically meaningful norm (Euclidean or non-Euclidean).     

An efficient multivariate random field generator using the fast Fourier transform

It is often convenient to use synthetically generated random fields to study the hydrologic effects of spatial heterogeneity. Although there are many ways to produce such fields, spectral techniques are particularly attractive because they are fast and conceptually straightforward. This paper describes a spectral algorithm for generating sets of random fields which are correlated with one another. The algorithm is based on a discrete version of the Fourier-Stieltjes representation for multidimensional random fields. The Fourier increment used in this representation depends on a random phase angle process and a complex-valued spectral factor matrix which can be readily derived from a specified set of cross-spectral densities (or cross-covariances). The inverse Fourier transform of the Fourier increment is a complex random field with real and imaginary parts which each have the desired coveriance structure. Our complex-valued spectral formulation provides an especially convenient way to generate a set of random fields which all depend on a single underlying (independent) field, provided that the fields in question can be related by space-invariant linear transformations. We illustrate this by generating multi-dimensional mass conservative groundwater velocity fields which can be used to simulate solute transport through heterogeneous anisotropic porous media.     

Borehole radar velocity inversion using cokriging and cosimulation

A new radar velocity tomography method is presented based on slowness covariance modeling and cokriging of the slowness field using only measured travel time data. The proposed approach is compared to the classical LSQR algorithm using various synthetic models and a real data set. In each case, the proposed method provides comparable to or better results than LSQR. One advantage of this approach is that it is self-regularized and requires less a priori information. The covariance model also allows stochastic imaging of slowness fields by geostatistical simulations. Stable characteristics and uncertain features of the inverted models can then be easily identified.     

Spatial sampling design and covariance-robust minimax prediction based on convex design ideas

This paper presents new ideas on sampling design and minimax prediction in a geostatistical model setting. Both presented methodologies are based on regression design ideas. For this reason the appendix of this paper gives an introduction to optimum Bayesian experimental design theory for linear regression models with uncorrelated errors. The presented methodologies and algorithms are then applied to the spatial setting of correlated random fields. To be specific, in Sect. 1 we will approximate an isotropic random field by means of a regression model with a large number of regression functions with random amplitudes, similarly to Fedorov and Flanagan (J Combat Inf Syst Sci: 23, 1997). These authors make use of the Karhunen Loeve approximation of the isotropic random field. We use the so-called polar spectral approximation instead; i.e. we approximate the isotropic random field by means of a regression model with sine-cosine-Bessel surface harmonics with random amplitudes and then, in accordance with Fedorov and Flanagan (J Combat Inf Syst Sci: 23, 1997), apply standard Bayesian experimental design algorithms to the resulting Bayesian regression model. Section 2 deals with minimax prediction when the covariance function is known to vary in some set of a priori plausible covariance functions. Using a minimax theorem due to Sion (Pac J Math 8:171–176, 1958) we are able to formulate the minimax problem as being equivalent to an optimum experimental design problem, too. This makes the whole experimental design apparatus available for finding minimax kriging predictors. Furthermore some hints are given, how the approach to spatial sampling design with one a priori fixed covariance function may be extended by means of minimax kriging to a whole set of a priori plausible covariance functions such that the resulting designs are robust. The theoretical developments are illustrated with two examples taken from radiological monitoring and soil science.     

Multiscale flow and transport model in three-dimensional fractal porous media

This paper proposes a multiscale flow and transport model which can be used in three-dimensional fractal random fields. The fractal random field effectively describes a field with a high degree of variability to satisfy the one-point statistics of Levy-stable distribution and the two-point statistics of fractional Levy motion (fLm). To overcome the difficulty of using infinite variance of Levy-stable distribution and to provide the physical meaning of a finite domain in real space, truncated power variograms are utilized for the fLm fields. The fLm model is general in the sense that both stationary and commonly used fractional Brownian motion (fBm) models are its special cases. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the stationary model is well approximated. The commonly used fBm model is recovered when the Levy index of fLm is 2. Flow and solute transport were analyzed using the first-order perturbation method. Mean velocity, velocity covariance, and effective hydraulic conductivity in a three-dimensional fractal random field were derived. Analytical results for particle displacement covariance and macrodispersion coefficients are also presented. The results show that the plume in an fLm field moves slower at early time and has more significant long-tailing behavior at late time than in fBm or stationary exponential fields. The proposed fractal transport model has broader applications than those of stationary and fBm models. Flow and solute transport can be simulated for various scenarios by adjusting the Levy index and cutoffs of fLm to yield more accurate modeling results.     

Wavelet analysis of characteristic length scales and orientation of two-dimensional heterogeneous porous media

The global wavelet energy spectrum (GWES) has been used on one-dimensional transects of porous media to identify the characteristic length scale of a material property. The characteristic length scale is a measure of the distance over which property values are correlated. We extend the wavelet analysis technique to two-dimensional stationary porous media for identifying both the characteristic length scales and orientation of heterogeneity. We theoretically develop and evaluate the GWES for isotropic and anisotropic random fields using the two-dimensional Mexican hat wavelet. The relationship between the wavelet scale and the characteristic length scale of a two-dimensional random field is investigated for different covariance structures. The ability of the GWES to identify multiple characteristic length scales and orientations in a random field with nested covariance structures is also investigated. We use the technique to identify the characteristic length scales of a log hydraulic conductivity field used in a laboratory experiment.     

Southern hemisphere low level wind circulation statisticsfrom the Seasat scatterometer

Analyses of remotely sensed low-level wind vector data over the Southern Ocean are performed. Five-day averages and monthly means are created and the month-to-month variability during the winter (July-September) of 1978 is investigated. The remotely sensed winds are compared to the Australian Bureau of Meteorology (ABM) and the National Meteorological Center (NMC) surface analyses. In southern latitudes the remotely sensed winds are stronger than what the weather services’ analyses suggest, indicating underestimation by ABM and NMC in these regions. The evolution of the low-level jet and the major stormtracks during the season are studied and different flow regimes are identified. The large-scale variability of the meridional flow is studied with the aid of empirical orthogonal function (EOF) analysis. The dominance of quasi-stationary wave numbers 3, 4, and 5 in the winter flow is evident in both the EOF analysis and the mean flow. The signature of an exceptionally strong blocking situation is evident in July and the special conditions leading to it are discussed. A very large intraseasonal variability with different flow regimes at different months is documented.     

A note on smoothness measures for space–time surfaces

The differentiability of a random field has a direct relationship with the differentiability of its covariance function. We review the concept of differentiability of space–time covariance models and random fields, and its implications on predictions. We analyze the change of behavior of the covariance function at the origin and at different space–time lags away from the origin, by using the concept of smoothness which can be considered the geometrical view of the differentiability. We propose a way to measure the smoothness of any covariance function, and apply it to purely spatial and space–time covariance functions.