Spatial prediction of river channel topography by kriging

Topographic information is fundamental to geomorphic inquiry, and spatial prediction of bed elevation from irregular survey data is an important component of many reach‐scale studies. Kriging is a geostatistical technique for obtaining these predictions along with measures of their reliability, and this paper outlines a specialized framework intended for application to river channels. Our modular approach includes an algorithm for transforming the coordinates of data and prediction locations to a channel‐centered coordinate system, several different methods of representing the trend component of topographic variation and search strategies that incorporate geomorphic information to determine which survey data are used to make a prediction at a specific location. For example, a relationship between curvature and the lateral position of maximum depth can be used to include cross‐sectional asymmetry in a two‐dimensional trend surface model, and topographic breaklines can be used to restrict which data are retained in a local neighborhood around each prediction location. Using survey data from a restored gravel‐bed river, we demonstrate how transformation to the channel‐centered coordinate system facilitates interpretation of the variogram, a statistical model of reach‐scale spatial structure used in kriging, and how the choice of a trend model affects the variogram of the residuals from that trend. Similarly, we show how decomposing kriging predictions into their trend and residual components can yield useful information on channel morphology. Cross‐validation analyses involving different data configurations and kriging variants indicate that kriging is quite robust and that survey density is the primary control on the accuracy of bed elevation predictions. The root mean‐square error of these predictions is directly proportional to the spacing between surveyed cross‐sections, even in a reconfigured channel with a relatively simple morphology; sophisticated methods of spatial prediction are no substitute for field data. Copyright © 2007 John Wiley & Sons, Ltd.     

Copula-based geostatistical modeling of continuous and discrete data including covariates

It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.     

Geostatistical traveltime tomography in elliptically anisotropic media

In geological materials, anisotropy may arise due to different mechanisms and can be found at different scales. Neglecting anisotropy in traveltime tomographic reconstruction leads to artefacts that can obscure important subsurface features. In this paper, a geostatistical tomography algorithm to invert cross‐hole traveltime data in elliptically anisotropic media is presented. The advantages of geostatistical tomography are that the solution is regularized by the covariance of the model parameters, that known model parameters can be used as constraints and fitted exactly or within a prescribed variance and that stochastic simulations can be performed to appraise the variability of the solution space. The benefits of the algorithm to image anisotropic media are illustrated by two examples using synthetic georadar data and real seismic data. The first example confirms suspected electromagnetic anisotropy in the vadose zone caused by relatively rapid water content variations with respect to wavelength at georadar frequencies. The second presents how sonic log data can be used to constrain the inversion of cross‐well seismic data and how geostatistical simulations can be used to infer parameter uncertainty. Results of both examples show that considering anisotropy yields a better fit to the data at high ray angles and reduces reconstruction artefacts.     

基于地质统计先验信息的储层物性参数同步反演

本文提出的储层物性参数同步反演是一种高分辨率的非线性反演方法,该方法综合利用岩石物理和地质统计先验信息,在贝叶斯理论框架下,首先通过变差结构分析得到合理的变差函数,进而利用快速傅里叶滑动平均模拟算法（Fast Fourier TransformMoving Average,FFT-MA）和逐渐变形算法（Gradual Deformation Method,GDM）得到基于地质统计学的储层物性参数先验信息,然后根据统计岩石物理模型建立弹性参数与储层物性参数之间的关系,构建似然函数,最终利用Metropolis算法实现后验概率密度的抽样,得到物性参数反演结果。并将此方法处理了中国陆上探区的一块实际资料,本方法的反演结果具有较高的分辨率,与测井数据吻合度较高;由于可以直接反演储层物性参数,避免了误差的累积,大大减少了不确定性的传递,且计算效率较高。     

Declustering experimental variograms by global estimation with fourth order moments

The variogram is a key parameter for geostatistical estimation and simulation. Preferential sampling may bias the spatial structure and often leads to noisy and unreliable variograms. A novel technique is proposed to weight variogram pairs in order to compensate for preferential or clustered sampling . Weighting the variogram pairs by global kriging of the quadratic differences between the tail and head values gives each pair the appropriate weight, removes noise and minimizes artifacts in the experimental variogram. Moreover, variogram uncertainty could be computed by this technique. The required covariance between the pairs going into variogram calculation, is a fourth order covariance that must be calculated by second order moments. This introduces some circularity in the calculation whereby an initial variogram must be assumed before calculating how the pairs should be weighted for the experimental variogram. The methodology is assessed by synthetic and realistic examples. For synthetic example, a comparison between the traditional and declustered variograms shows that the declustered variograms are better estimates of the true underlying variograms. The realistic example also shows that the declustered sample variogram is closer to the true variogram.     

Geostatistical simulation of random fields with bivariate isofactorial distributions by adding mosaic models

This work deals with the geostatistical simulation of a family of stationary random field models with bivariate isofactorial distributions. Such models are defined as the sum of independent random fields with mosaic-type bivariate distributions and infinitely divisible univariate distributions. For practical applications, dead leaf tessellations are used since they provide a wide range of models and allow conditioning the realizations to a set of data via an iterative procedure (simulated annealing). The model parameters can be determined by comparing the data variogram and madogram, and enable to control the spatial connectivity of the extreme values in the realizations. An illustration to a forest dataset is presented, for which a negative binomial model is used to characterize the distribution of coniferous trees over a wooded area.     

Spatial prediction on river networks: comparison of top‐kriging with regional regression

Top‐kriging is a method for estimating stream flow‐related variables on a river network. Top‐kriging treats these variables as emerging from a two‐dimensional spatially continuous process in the landscape. The top‐kriging weights are estimated by regularising the point variogram over the catchment area (kriging support), which accounts for the nested nature of the catchments. We test the top‐kriging method for a comprehensive Austrian data set of low stream flows. We compare it with the regional regression approach where linear regression models between low stream flow and catchment characteristics are fitted independently for sub‐regions of the study area that are deemed to be homogeneous in terms of flow processes. Leave‐one‐out cross‐validation results indicate that top‐kriging outperforms the regional regression on average over the entire study domain. The coefficients of determination (cross‐validation) of specific low stream flows are 0.75 and 0.68 for the top‐kriging and regional regression methods, respectively. For locations without upstream data points, the performances of the two methods are similar. For locations with upstream data points, top‐kriging performs much better than regional regression as it exploits the low flow information of the neighbouring locations. Copyright © 2012 John Wiley & Sons, Ltd.     

Impact of log-transmissivity variogram structure on groundwater flow and transport predictions

We analyze the impact of the choice of the variogram model adopted to characterize the spatial variability of natural log-transmissivity on the evaluation of leading (statistical) moments of hydraulic heads and contaminant travel times and trajectories within mildly (randomly) heterogeneous two-dimensional porous systems. The study is motivated by the fact that in several practical situations the differences between various variogram types and a typical noisy sample variogram are small enough to suggest that one would often have a hard time deciding which of the tested models provides the best fit. Likewise, choosing amongst a set of seemingly likely variogram models estimated by means of geostatistical inverse models of flow equations can be difficult due to lack of sensitivity of available model discrimination criteria. We tackle the problem within the framework of numerical Monte Carlo simulations for mean uniform and radial flow scenarios. The effect of three commonly used isotropic variogram models, i.e., Gaussian, Exponential and Spherical, is analyzed. Our analysis clearly shows that (ensemble) mean values of the quantities of interest are not considerably influenced by the variogram shape for the range of parameters examined. Contrariwise, prediction variances of the quantities examined are significantly affected by the choice of the variogram model of the log-transmissivity field. The spatial distribution of the largest/lowest values of the relative differences observed amongst the tested models depends on a combination of variogram shape and parameters and relative distance from internal sources and the outer domain boundary. Our findings suggest the need of developing robust techniques to discriminate amongst a set of seemingly equally likely alternative variogram models in order to provide reliable uncertainty estimates of state variables.     

On the geostatistical approach to the inverse problem

The geostatistical approach to the inverse problem is discussed with emphasis on the importance of structural analysis. Although the geostatistical approach is occasionally misconstrued as mere cokriging, in fact it consists of two steps: estimation of statistical parameters (“structural analysis”) followed by estimation of the distributed parameter conditional on the observations (“cokriging” or “weighted least squares”). It is argued that in inverse problems, which are algebraically undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical approach is that instead of adjusting a grid-dependent and potentially large number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once this fitting is accomplished, the estimation of block conductivities ensues in a predetermined fashion without fitting of additional parameters. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundamental differences between the two approaches are: (a) they use different principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statistically unbiased estimates of the parameters that are not strongly dependent on the discretization, while the other method is biased and its bias becomes worse by refining the discretization into zones with different conductivity.     

Cokriging for enhanced spatial interpolation of rainfall in two Australian catchments

Rainfall data in continuous space provide an essential input for most hydrological and water resources planning studies. Spatial distribution of rainfall is usually estimated using ground‐based point rainfall data from sparsely positioned rain‐gauge stations in a rain‐gauge network. Kriging has become a widely used interpolation method to estimate the spatial distribution of climate variables including rainfall. The objective of this study is to evaluate three geostatistical (ordinary kriging [OK], ordinary cokriging [OCK], kriging with an external drift [KED]), and two deterministic (inverse distance weighting, radial basis function) interpolation methods for enhanced spatial interpolation of monthly rainfall in the Middle Yarra River catchment and the Ovens River catchment in Victoria, Australia. Historical rainfall records from existing rain‐gauge stations of the catchments during 1980–2012 period are used for the analysis. A digital elevation model of each catchment is used as the supplementary information in addition to rainfall for the OCK and kriging with an external drift methods. The prediction performance of the adopted interpolation methods is assessed through cross‐validation. Results indicate that the geostatistical methods outperform the deterministic methods for spatial interpolation of rainfall. Results also indicate that among the geostatistical methods, the OCK method is found to be the best interpolator for estimating spatial rainfall distribution in both the catchments with the lowest prediction error between the observed and estimated monthly rainfall. Thus, this study demonstrates that the use of elevation as an auxiliary variable in addition to rainfall data in the geostatistical framework can significantly enhance the estimation of rainfall over a catchment.     

变差函数中加权系数拟合方法的改善

变差函数拟合效果的优劣是提高求解变差函数精度的关键步骤,结合加权多项式法和线性规划法的优点,提出使用处理滞后距的方法得到权系数的线性规划法进行球状模型的参数估计,对新疆贝克滩水系沉积物化探数据进行空间分析,结果表明:预测值与样品实测值误差较小,提高预测值的精度.     

Relationships between correlation lengths and integral scales for covariance models with more than two parameters

In geostatistical applications, the terms correlation length and range are often used interchangeably and refer to a characteristic covariance length ξ that normalizes the lag distance in the variogram or the covariance model. We present equations that strictly define the correlation length (r _c ) and integral range (ℓ _c ). We derive analytical expressions for r _c and ℓ _c of the Whittle–Matérn, fluctuation gradient curvature and rational quadratic covariances. For these covariances, we show that the correlation length and integral range for a given model are not fully determined by ξ. We define non-trivial covariance functions, and we formulate an ergodicity index based on ℓ _c . We propose using the ergodicity index to compare coarse-grained measures corresponding to non-trivial covariance functions with different parameters. Finally, we discuss potential applications of the proposed covariance models in stochastic subsurface hydrology.     

Stochastic delineation of capture zones: classical versus Bayesian approach

A Bayesian approach to characterize the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations is presented and compared with the classical or non-Bayesian approach. The transmissivity field is modelled as a random space function and conditioned on distributed measurements of the transmissivity. In conventional geostatistical methods the mean value of the log transmissivity and the functional form of the covariance and its parameters are estimated from the available measurements, and then entered into the prediction equations as if they are the true values. However, this classical approach accounts only for the uncertainty that stems from the lack of ability to exactly predict the transmissivity at unmeasured locations. In reality, the number of measurements used to infer the statistical properties of the transmissvity field is often limited, which introduces error in the estimation of the structural parameters. The method presented accounts for the uncertainty that originates from the imperfect knowledge of the parameters by treating them as random variables. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of the parameters. The classical and Bayesian approach to stochastic capture zone delineation are detailed and applied to a hypothetical flow field. Two different sampling densities on a regular grid are considered to evaluate the effect of data density in both methods. Results indicate that the predictions of the Bayesian approach are more conservative.     

Direct forecasting of subsurface flow response from non-linear dynamic data by linear least-squares in canonical functional principal component space

Inverse modeling is widely used to assist with forecasting problems in the subsurface. However, full inverse modeling can be time-consuming requiring iteration over a high dimensional parameter space with computationally expensive forward models and complex spatial priors. In this paper, we investigate a prediction-focused approach (PFA) that aims at building a statistical relationship between data variables and forecast variables, avoiding the inversion of model parameters altogether. The statistical relationship is built by first applying the forward model related to the data variables and the forward model related to the prediction variables on a limited set of spatial prior models realizations, typically generated through geostatistical methods. The relationship observed between data and prediction is highly non-linear for many forecasting problems in the subsurface. In this paper we propose a Canonical Functional Component Analysis (CFCA) to map the data and forecast variables into a low-dimensional space where, if successful, the relationship is linear. CFCA consists of (1) functional principal component analysis (FPCA) for dimension reduction of time-series data and (2) canonical correlation analysis (CCA); the latter aiming to establish a linear relationship between data and forecast components. If such mapping is successful, then we illustrate with several cases that (1) simple regression techniques with a multi-Gaussian framework can be used to directly quantify uncertainty on the forecast without any model inversion and that (2) such uncertainty is a good approximation of uncertainty obtained from full posterior sampling with rejection sampling.     

Histogram and variogram inference in the multigaussian model

Several iterative algorithms are proposed to improve the histogram and variogram inference in the framework of the multigaussian model. The starting point is the variogram obtained after a traditional normal score transform. The subsequent step consists in simulating many sets of gaussian values with this variogram at the data locations, so that the ranking of the original values is honored. The expected gaussian transformation and the expected variogram are computed by an averaging operation over the simulated datasets. The variogram model is then updated and the procedure is repeated until convergence. Such an iterative algorithm can adapt to the case of tied data and despike the histogram. Two additional issues are also examined, referred to the modeling of the empirical transformation function and to the optimal pair weighting when computing the sample variogram.     

Identification of heterogeneous aquifer transmissivity using an AE-based method

Determination of hydraulic head, H, as a function of spatial coordinates and time, in ground water flow is the basis for aquifer management and for prediction of contaminant transport. Several computer codes are available for this purpose. Spatial distribution of the transmissivity, T(x,y), is a required input to these codes. In most aquifers, T varies in an erratic manner, and it can be characterized statistically in terms of a few moments: the expected value, the variance, and the variogram. Knowledge of these moments, combined with a few measurements, permits one to estimate T at any point using geostatistical methods. In a review of transmissivity data from 19 unconsolidated aquifers, Hoeksema and Kitanidis (1985) identified two types of the logtransmissivity Y= ln(T) variations: correlated variations with variance sigma2Yc and correlation scale, I(Y), on the order of kilometers, and uncorrelated variations with variance sigma2Yn. Direct identification of the logtransmissivity variogram, Gamma(Y), from measurements is difficult because T data are generally scarce. However, many head measurements are commonly available. The aim of the paper is to introduce a methodology to identify the transmissivity variogram parameters (sigma2Yc, I(Y), and sigma2Yn) using head data in formations characterized by large logtransmissivity variance. The identification methodology uses a combination of precise numerical simulations (carried out using analytic element method) and a theoretical model. The main objective is to demonstrate the application of the methodology to a regional ground water flow in Eagle Valley basin in west-central Nevada for which abundant transmissivity and head measurements are available.     

Geostatistical mapping of geomorphic variables in the presence of trend

Mapping geomorphic variables geostatistically, specifically by kriging, runs into difficulties when there is trend. The reason is that the variogram required for the kriging must be of residuals from any trend, which in turn cannot be estimated optimally by the usual method of trend surface analysis because the residuals are correlated. The difficulties can be overcome by the use of residual maximum likelihood (REML) to estimate both the trend and the variogram of the residuals simultaneously. We summarize the theory of REML as it applies to kriging in the presence of trend. We present the equations to show how estimates of the trend are combined with kriging of residuals to give empirical best linear unbiased predictions (E‐BLUPs). We then apply the method to estimate the height of the sub‐Upper‐Chalk surface beneath the Chiltern Hills of southeast England from 238 borehole data. The variogram of the REML residuals is substantially different from that computed by ordinary least squares (OLS) analysis. The map of the predicted surface is similar to that made from kriging with the OLS variogram. The variances, however, are substantially larger because (a) they derive from a variogram with a much larger sill and (b) they include the uncertainty of the estimate of the trend. Copyright © 2006 John Wiley & Sons, Ltd.     

Stochastic simulation of regionalized ground motions using wavelet packets and cokriging analysis

Performance‐based earthquake engineering often requires ground‐motion time‐history analyses to be performed, but very often, ground motions are not recorded at the location being analyzed. The present study is among the first attempt to stochastically simulate spatially distributed ground motions over a region using wavelet packets and cokriging analysis. First, we characterize the time and frequency properties of ground motions using the wavelet packet analysis. The spatial cross‐correlations of wavelet packet parameters are determined through geostatistical analysis of regionalized ground‐motion data from the Northridge and Chi‐Chi earthquakes. It is observed that the spatial cross‐correlations of wavelet packet parameters are closely related to regional site conditions. Furthermore, using the developed spatial cross‐correlation model and the cokriging technique, wavelet packet parameters at unmeasured locations can be best estimated, and regionalized ground‐motion time histories can be synthesized. Case studies and blind tests using data from the Northridge and Chi‐Chi earthquakes demonstrate that the simulated ground motions generally agree well with the actual recorded data. The proposed method can be used to stochastically simulate regionalized ground motions for time‐history analyses of distributed infrastructure and has important applications in regional‐scale hazard analysis and loss estimation. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonparametric geostatistical risk mapping

In this work, a fully nonparametric geostatistical approach to estimate threshold exceeding probabilities is proposed. To estimate the large-scale variability (spatial trend) of the process, the nonparametric local linear regression estimator, with the bandwidth selected by a method that takes the spatial dependence into account, is used. A bias-corrected nonparametric estimator of the variogram, obtained from the nonparametric residuals, is proposed to estimate the small-scale variability. Finally, a bootstrap algorithm is designed to estimate the unconditional probabilities of exceeding a threshold value at any location. The behavior of this approach is evaluated through simulation and with an application to a real data set.