Power Variogram Research Articles

Research on wind farm participating in AGC based on wind power variogram characteristics.

The increasing integration of large-scale wind power aggravates the difficulty of maintaining system frequency deviations in a certain range. The frequency regulation pressure of conventional generators increases, which requires wind farms to participate in system frequency regulation. In this paper, a multi-area interconnected power system frequency response model with wind power is established. Based on the frequency response model, the state space model of regional interconnected power system is presented. Then, the wind power variogram characteristics are introduced for estimating wind power variations in different time-scales. By predicting the wind power variations in AGC time-scale, a strategy of wind farm participating in AGC system is proposed and performed based on model predictive control (MPC). The control strategy makes the conventional units and wind farms to participate in AGC system coordinately. Simulation results are provided which verifies the feasibility and validity of the proposed strategy.

Anticipatory AGC control strategy based on wind power variogram characteristic

Increasing penetration of wind power has a negative effect on system frequency deviation due to the fast variability and uncertainty characteristics of wind. In this study, a wind power variation prediction-based automatic generation control (AGC) feedforward control method is applied within the conventional AGC framework to minimise system frequency and tie-line deviations. Firstly, the variogram function is introduced for analysing the characteristics of wind power variations. A relationship between the variogram of wind power and the trend component of wind power is found and then a three-parameter power-law model is established. Then, the generation rate constrains of AGC generators is considered as the main constraint factor that influences wind power smoothing. By predicting the wind power variations in AGC control time-scale, a coordination AGC feedforward control strategy is proposed. The control strategy makes different types of AGC generators act in advance to improve the ability of the power system, corresponding to the forecasted ramp variations in wind power generation. Finally, the performance of the proposed coordination control strategy is verified and tested using the actual data from China. Simulation results show that with this new strategy, frequency deviation under wind power variations can be effectively decreased.

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

Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on Rd. Then, we show that appropriate functions of the Brown–Resnick random field with a power variogram and of the Smith random field satisfy the central limit theorem. Another strong motivation for our work lies in the fact that central limit theorems for random fields on Rd have been barely considered in the literature. As an application, we briefly show the usefulness of our results in a risk assessment context.

ABC model selection for spatial extremes models applied to South Australian maximum temperature data

Max-stable processes are a common choice for modelling spatial extreme data as they arise naturally as the infinite-dimensional generalisation of multivariate extreme value theory. Statistical inference for such models is complicated by the intractability of the multivariate density function. Nonparametric, composite likelihood-based, and Bayesian approaches have been proposed to address this difficulty. More recently, a simulation-based approach using approximate Bayesian computation (ABC) has been employed for estimating parameters of max-stable models. ABC algorithms rely on the evaluation of discrepancies between model simulations and the observed data rather than explicit evaluations of computationally expensive or intractable likelihood functions. The use of an ABC method to perform model selection for max-stable models is explored. Three max-stable models are regarded: the extremal-t model with either a Whittle–Matérn or a powered exponential covariance function, and the Brown–Resnick model with power variogram. In addition, the non-extremal Student-t copula model with a Whittle–Matérn or a powered exponential covariance function is also considered. The method is applied to annual maximum temperature data from 25 weather stations dispersed around South Australia.

Predictive Assessment of Groundwater Flow Uncertainty in Multiscale Porous Media by Using Truncated Power Variogram Model

The spatial distribution of hydrogeological properties is essentially heterogeneous. Heterogeneity can be characterized quantitatively using geostatistics, which conventionally assumes that the stochastic process is stationary. However, growing evidence indicates that the spatial variability has the multiscale self-similarity characteristics and can be better characterized using non-stationary model but with statistically homogeneous increments. A general framework is developed in this work to conduct the uncertainty quantification analysis by using truncated power variogram model, which can explicitly account for measurement scale, observation scale, and window scale. The effect of the multiscale characteristics of the hydrogeological properties on the uncertainty and the consequential risk associated with the groundwater flow process is investigated. A synthetic two-dimensional saturated steady-state groundwater flow problem is used to evaluate the performance to predict the flow field distribution. For comparative purposes, the evaluation is based on both the truncated power and the traditional variogram models when the underlying porous medium is a random fractal field. The results show that the truncated power variogram model can perform the uncertainty quantification more accurately, and the adoption of traditional variogram model tends to result in a smoother estimation on the flow field and underestimate the uncertainty associated with the hydraulic head prediction. Upscaling is generally inevitable to avoid predictive uncertainty underestimation when the underlying random field exhibits multiscale characteristics.

A parametric model bridging between bounded and unbounded variograms

A simple variogram model with two parameters is presented that includes the power variogram for fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadric model. One parameter controls the sample path roughness of the process. The other parameter allows for a smooth transition between bounded and unbounded variograms, that is, between stationary and intrinsically stationary processes in a Gaussian framework, or between mixing and non‐ergodic Brown–Resnick processes when modeling spatial extremes. Copyright © 2017 John Wiley & Sons, Ltd.

Characterization of Aquifer Multiscale Properties by Generating Random Fractal Field with Truncated Power Variogram Model Using Karhunen–Loève Expansion

The traditional geostatistics to describe the spatial variation of hydrogeological properties is based on the assumption of stationarity or statistical homogeneity. However, growing evidences show and it has been widely recognized that the spatial distribution of many hydrogeological properties can be characterized as random fractals with multiscale feature, and spatial variation can be described by power variogram model. It is difficult to generate a multiscale random fractal field by directly using nonstationary power variogram model due to the lack of explicit covariance function. Here we adopt the stationary truncated power variogram model to avoid this difficulty and generate the multiscale random fractal field using Karhunen–Loève (KL) expansion. The results show that either the unconditional or conditional (on measurements) multiscale random fractal field can be generated by using truncated power variogram model and KL expansion when the upper limit of the integral scale is sufficiently large, and the main structure of the spatial variation can be described by using only the first few dominant KL expansion terms associated with large eigenvalues. The latter provides a foundation to perform dimensionality reduction and saves computational effort when analyzing the stochastic flow and transport problems.

On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere

Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential models, as well as power variograms with 0<α≤1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and power variograms with 1<α≤2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on the sphere is provided.

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.

Effects of evolving scales of heterogeneity on hydraulic head predictions under convergent flow conditions

Two-dimensional steady-state flow towards a well that fully penetrates a randomly heterogeneous aquifer with a deterministically prescribed constant head boundary is considered. Flow occurs over an infinite hierarchy of mutually uncorrelated, statistically homogeneous and isotropic random fields (modes) of natural log-transmissivity, Y, each of which is associated with a Gaussian variogram. A lower and upper cut-off of the hierarchy, respectively related to the length scale of the domain and data support (sample volume), is introduced. This allows the direct incorporation of the scale dependence of the integral scale of Y into groundwater flow (ensemble) moments and leads to a geostatistical description of the system in terms of a (stationary) truncated power variogram (TPV). An analytical solution for the hydraulic head mean and variance is developed based on recursive approximations of exact nonlocal moment equations. The solution allows assessment of the functional dependences of the distribution of the leading (statistical) moments of hydraulic head on parameters of the variogram associated with the hierarchy of log-transmissivity modes.

Stochastic analysis of groundwater flow in a two‐dimensional generalized fractal field

Stochastic analysis of groundwater flow in a generalized fractal field is performed in this study. The random field is described by fractional Levy motion (fLm), which is a generalized version of traditional fractional Brownian motion (fBm) and is superior to describe a field with a high degree of variability. A truncated power variogram of the fLm is derived using the weighted superposition of mutually uncorrelated exponential variograms. When the Levy index of fLm α equals 2, the fBm is recovered. When the upper and lower cutoffs of the truncated power variogram are close, the stationary exponential model can be well approximated. First‐order perturbation analyses of flow in a two‐dimensional fLm field are performed and results are compared to those in the stationary exponential and fractal fBm fields. Since the proposed general fractal model has broader applications than the stationary and fBm models, it is versatile enough to simulate flow in different scenarios and provide more accurate modeling results.

On the geostatistical characterization of hierarchical media

The subsurface consists of porous and fractured materials exhibiting a hierarchical geologic structure, which gives rise to systematic and random spatial and directional variations in hydraulic and transport properties on a multiplicity of scales. Traditional geostatistical moment analysis allows one to infer the spatial covariance structure of such hierarchical, multiscale geologic materials on the basis of numerous measurements on a given support scale across a domain or “window” of a given length scale. The resultant sample variogram often appears to fit a stationary variogram model with constant variance (sill) and integral (spatial correlation) scale. In fact, some authors, who recognize that hierarchical sedimentary architecture and associated log hydraulic conductivity fields tend to be nonstationary, nevertheless associate them with stationary “exponential‐like” transition probabilities and variograms, respectively, the latter being a consequence of the former. We propose that (1) the apparent ability of stationary spatial statistics to characterize the covariance structure of nonstationary hierarchical media is an artifact stemming from the finite size of the windows within which geologic and hydrologic variables are ubiquitously sampled, and (2) the artifact is eliminated upon characterizing the covariance structure of such media with the aid of truncated power variograms, which represent stationary random fields obtained upon sampling a nonstationary fractal over finite windows. To support our opinion, we note that truncated power variograms arise formally when a hierarchical medium is sampled jointly across all geologic categories and scales within a window; cite direct evidence that geostatistical parameters (variance and integral scale) inferred on the basis of traditional variograms vary systematically with support and window scales; demonstrate the ability of truncated power models to capture these variations in terms of a few scaling parameters; show that exponential and truncated power variograms are often difficult to distinguish from each other, which helps explain why hierarchical data may appear to fit the former; note that truncated power models are unique in their ability to represent multiscale random fields having either Gaussian or heavy‐tailed symmetric Levy stable probability distributions; detail the way in which these models allow conditioning the spatial statistics of such fields on multiscale measurements via cokriging; and illustrate these capabilities on multiscale hydraulic data from an unconfined aquifer near Tübingen, Germany.

A general fractal model of flow and solute transport in randomly heterogeneous porous media

One of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous. This paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media. We describe the random field using fractional Levy motion (fLm), which more effectively represents a field with a high degree of variability. Understanding flow and solute transport in the fLm field expands the applicability of stochastic theory. Following the work of V. Di Federico and S. P. Neuman (1997) and V. Di Federico et al. (1999), we derive truncated isotropic and anisotropic power variograms for the fLm fields. The results are analogous to but more general than those of fractional Brownian motion (fBm) (V. Di Federico and S. P. Neuman (1997) and V. Di Federico et al. (1999)). For a random field, when the Levy index α in fLm is replaced by a constant of 2, fBm is recovered. This study employs first‐order perturbation analyses of flow and solute transport in two‐dimensional fLm fields. For the stationary, fBm, and fLm fields, macrodispersion coefficients are compared. The plume is found to grow slowest in the fLm field but will show highest concentration. The proposed fractal model is general in the sense that when the Levy index of fLm α equals 2, the results of flow and solute transport for the commonly used fBm model are recovered. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the results for the stationary exponential or Gaussian model can be well approximated. Since the proposed general fractal model has broader applications than the stationary and fBm models, it simulates flow and solute transport in more different scenarios to yield more accurate modeling results.

Blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition

This paper offers a blueprint for extending perturbative ensemble moment (and potentially other, e.g., Gaussian or perturbative Karhunen‐Loéve) solutions of stochastic flow and transport problems in composite, randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large. The proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition. It considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram, i.e., a truncated fractal. This consideration is supported either exactly or approximately by a growing amount of observational data. It allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales. Combining this fractal decomposition with a variational multiscale decomposition formalism, motivated by the multiscale nature of turbulence, allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was. The problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales. The proposed approach does not require introducing effective parameters on any scale; instead, it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution. The approach nevertheless allows subgrid closure (another idea inspired by the literature on turbulence) through subgrid refinement or the use of effective subgrid parameters. It additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements. Details are presented in the context of steady state single‐phase flow in a composite medium.

Generating and scaling fractional Brownian motion on finite domains

Power variograms of statistically isotropic or anisotropic fractal fields (common in earth science) are weighted integrals of variograms representing statistically homogeneous fields (modes) having mutually uncorrelated increments. Large‐ and small‐scale cutoffs were previously assumed proportional to length scales of the sampling window and data support. We verify this assumption numerically for two‐dimensional isotropic fractional Brownian motion (fBm). It was previously concluded semi‐empirically that, for Hurst coefficient H = 0.25, the constant of proportionality is μ = 1/3. We confirm this but find μ to vary with mode type and H. We find that due to lack of ergodicity, sample fBm variograms generated on finite windows exhibit directional dependence and differ sharply between realizations. Many realizations are required to obtain an average sample variogram resembling the theoretical power model, especially for persistent fields. We propose generating fBm on finite windows using truncated power variograms and provide guidance for doing so effectively.

Relationship between juxtaposed, overlapping, and fractal representations of multimodal spatial variability

It has been shown by Neuman [1990], Di Federico and Neuman [1997], and Di Federico et al. [1999] that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or Gaussian variograms of overlapping, homogeneous random fields (modes) having mutually uncorrelated increments and variance proportional to a power 2H of the integral (spatial correlation) scale where H is the Hurst coefficient. Low‐ and high‐frequency cutoffs are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy, created by a hiatus of modes associated with discrete ranges of scales. Alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed (mutually exclusive) materials or categories, each having its own architecture and attributes. The spatial distribution of categories is characterized by indicator random variables and their attributes by random fields. This paper focuses on expressions developed by Lu and Zhang [2002] in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories. Upon rewriting their expressions for statistically homogeneous and anisotropic media, it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space (overlap) in a way that preserves their local architecture, their attributes become mutually uncorrelated. Categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space; the idea is analogous to the well‐known dual continuum concept in which two categories, most commonly fractures and porous blocks, are considered to overlap. In reality, the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space. The variogram of an attribute, sampled jointly over overlapping statistically homogeneous and anisotropic categories, is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories, weighted by their volumetric proportions. It is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or Gaussian variograms, such that the product of category probability density function (pdf) and attribute variance is proportional to a power 2H + 1 of the attribute integral scale, they represent a fractal field characterized by a power or truncated power variogram as in the work of Di Federico and Neuman. Any category pdf that satisfies this condition, and any attribute pdf having a variance that satisfies the same condition, are admissible; neither of them need to be Gaussian (to form fractional Brownian motion) or Levy‐like (to approximate fractional Levy motion) for the attribute to form a random fractal. Intermediate cutoffs render the latter model discrete, as is that of Lu and Zhang. Hence the two multimodal models are mutually consistent and complementary.

Surface Roughness and Fractal Dimension

Surface roughness is one of the main characteristics of fabric responsible for hand feeling. By using KES, the surface height variation (SHV) trace is obtained. For characterization of roughness the mean absolute deviation (MAD; denoted by Kawabata as SMD) is usually used. The main aim of this work is quantitative description of complexity of roughness based on the SHV. The procedure of surface complexity parameters evaluation from SHV traces is described. The core is inspection of power spectral density and variogram (or autocorrelation function) behavior and classification of SHV signal to some groups. For individual groups, fractal dimension and surface roughness characteristics are computed by suitable methods. Proposed procedure was checked on the simulated SHV profiles and on practical examples.

Anisotropy, lacunarity, and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms

It has been shown by Di Federico and Neuman [1997, 1998a, b] that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework, which views hydraulic conductivity as a random fractal characterized by a power variogram. Such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments. Di Federico and Neuman [1997] have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually uncorrelated fields (modes) that are homogeneous and isotropic. We show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic, provided the ratios between principal integral (spatial correlation) scales are the same for all modes. We then analyze the effect of filtering out (truncating) modes of low, high, and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains. A low‐frequency cutoff renders the truncated hierarchy homogeneous. The integral scales of the lowest‐ and highest‐frequency cutoff modes are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale (in a manner previously shown to be consistent with observations in the isotropic case). It also allows (in principle) bridging across scales at a specific locale, as well as among locales, by adopting either site‐specific or generalized variogram parameters. The introduction of intermediate cutoffs allows us to account, in a straightforward manner, for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales (for example, where textural and structural features are associated with distinct ranges of scale, such as fractures having discrete ranges of trace length and density, which dissect the rock into matrix blocks having corresponding ranges of sizes). We explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale, variance, covariance, and spectra of a truncated fractal field. We then develop an expression for the equivalent hydraulic conductivity of a box‐shaped porous block, embedded within a multiscale log hydraulic conductivity field, under mean‐uniform flow. The block is larger than the support scale of the field but is smaller than a surrounding sampling window. Consequently, its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function, conditioned on a known mean value of support‐scale conductivity across the window, are given explicitly by our multiscale theory.

VARFIT: a fortran-77 program for fitting variogram models by weighted least squares

The inference of spatial variogram parameters by the classical non-parametric approach (method of moments) requires the fitting of a model to the experimental variogram (the variogram calculated from the observational data). The usual approach is to perform this fitting `by eye' in a trial-and-error process that continues until the fitting is considered satisfactory. This procedure allows the user more freedom to include his/her own knowledge about the geology, geophysics, etc., of the spatial variable. Nevertheless a program of automatic fitting may be of considerable help for a number of different reasons: (a) as a preliminary estimation of the parameters, (b) when a high number of variograms must be modelled on a routine basis, (c) in order to make a more objective modelling (so that different practitioners will obtain the same results) and (d) in order to perform a more statistically sound fitting (each experimental variogram point should be weighed according to its own statistical uncertainty). The VARFIT program performs a nonlinear minimization of the weighed squared differences between the experimental variogram and the model. Five different weighing functions as well as four different variogram models (spherical, exponential, Gaussian and power models) are provided as options. It is possible to fit a nugget effect and a geometric anisotropy (in 2D). The fitting of an anisotropic power variogram model to rainfall data is shown to illustrate the versatility of the program. Results from simulation experiments are provided in order to make some general conclusions about the performance of the different weighing functions. An experimental variogram data set is given in order to check the program implementation.

Reply [to “Comment on ‘Scaling of random fields by means of truncated power variograms and associated spectra’ by Vittorio Di Federico and Shlomo P. Neuman”

Reply [to “Comment on ‘Scaling of random fields by means of truncated power variograms and associated spectra’ by Vittorio Di Federico and Shlomo P. Neuman”