Multigaussian kriging for point-support estimation: incorporating constraints on the sum of the kriging weights

In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations. The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when the mean is unknown.     

Conditional bias-penalized kriging (CBPK)

Simple and ordinary kriging, or SK and OK, respectively, represent the best linear unbiased estimator in the unconditional sense in that they minimize the unconditional (on the unknown truth) error variance and are unbiased in the unconditional mean. However, because the above properties hold only in the unconditional sense, kriging estimates are generally subject to conditional biases that, depending on the application, may be unacceptably large. For example, when used for precipitation estimation using rain gauge data, kriging tends to significantly underestimate large precipitation and, albeit less consequentially, overestimate small precipitation. In this work, we describe an extremely simple extension to SK or OK, referred to herein as conditional bias-penalized kriging (CBPK), which minimizes conditional bias in addition to unconditional error variance. For comparative evaluation of CBPK, we carried out numerical experiments in which normal and lognormal random fields of varying spatial correlation scale and rain gauge network density are synthetically generated, and the kriging estimates are cross-validated. For generalization and potential application in other optimal estimation techniques, we also derive CBPK in the framework of classical optimal linear estimation theory.     

Assessment of regional estimation error by canonical residual kriging

Various regional flood frequency analysis procedures are used in hydrology to estimate hydrological variables at ungauged or partially gauged sites. Relatively few studies have been conducted to evaluate the accuracy of these procedures and estimate the error induced in regional flood frequency estimation models. The objective of this paper is to assess the overall error induced in the residual kriging (RK) regional flood frequency estimation model. The two main error sources in specific flood quantile estimation using RK are the error induced in the quantiles local estimation procedure and the error resulting from the regional quantile estimation process. Therefore, for an overall error assessment, the corresponding errors associated with these two steps must be quantified. Results show that the main source of error in RK is the error induced into the regional quantile estimation method. Results also indicate that the accuracy of the regional estimates increases with decreasing return periods. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A brief review of kriging and its application to optimal interpolation and observation well selection

ABSTRACT

The current state of kriging in subsurface hydrology is critically reviewed. In an application to a region where boreholes already exist, methods of optimal location of additional observation wells for geophysical parameter investigation and optimal interpolation for the purpose of solving the inverse problem are investigated. The particular case of the location of wells for the measurements of transmissivity and hydraulic head in the Kennet Valley Chalk aquifer, UK, is examined. Results of interpolation of measured hydraulic conductivity values by kriging are compared with results from a standard graphical package for interpolation. Reference is also made to the distribution obtained by the inverse method (in which the conductivity distribution is obtained from the head distribution). On the basis of the application, the conditional simulation (in which the generated data are both consistent with field values and the field statistical structure) is deemed to be the best. It is also found that different methods of interpolation give widely different distributions in the case of hydraulic conductivity. It is suggested that the kriged map or conditional map of the transmissivity should serve as the basis for regional discretization to which corrections via the inverse model may be made.     

Regional flood frequency analysis using residual kriging in physiographical space

In this article, an approach using residual kriging (RK) in physiographical space is proposed for regional flood frequency analysis. The physiographical space is constructed using physiographical/climatic characteristics of gauging basins by means of canonical correlation analysis (CCA). This approach is a modified version of the original method, based on ordinary kriging (OK). It is intended to handle effectively any possible spatial trends within the hydrological variables over the physiographical space. In this approach, the trend is first quantified and removed from the hydrological variable by a quadratic spatial regression. OK is therefore applied to the regression residual values. The final estimated value of a specific quantile at an ungauged station is the sum of the spatial regression estimate and the kriged residual. To evaluate the performance of the proposed method, a cross‐validation procedure is applied. Results of the proposed method indicate that RK in CCA physiographical space leads to more efficient estimates of regional flood quantiles when compared to the original approach and to a straightforward regression‐based estimator. Copyright © 2010 John Wiley & Sons, Ltd.     

Incorporating elevation in rainfall interpolation in Tunisia using geostatistical methods

Abstract

This paper compares the performance of three geostatistical algorithms, which integrate elevation as an auxiliary variable: kriging with external drift (KED); kriging combined with regression, called regression kriging (RK) or kriging after detrending; and co-kriging (CK). These three methods differ by the way by in which the secondary information is introduced into the prediction procedure. They are applied to improve the prediction of the monthly average rainfall observations measured at 106 climatic stations in Tunisia over an area of 164 150 km² using the elevation as the auxiliary variable. The experimental sample semivariograms, residual semivariograms and cross-variograms are constructed and fitted to estimate the rainfall levels and the estimation variance at the nodes of a square grid of 20 km?×?20 km resolution and to develop corresponding contour maps. Contour diagrams for KED and RK were similar and exhibited a pattern corresponding more closely to local topographic features when (a) the network is sparse and (b) the rainfall–elevation correlation is poor, while CK showed a smooth zonal pattern. Smaller prediction variances are obtained for the RK algorithm. The cross-validation showed that the RMSE obtained for CK gave better results than for KED or RK.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Feki, H., Slimani, M., and Cudennec, C., 2012. Incorporating elevation in rainfall interpolation in Tunisia using geostatistical methods. Hydrological Sciences Journal, 57 (7), 1294–1314.     

Wavelet analysis residual kriging vs. neural network residual kriging

 This paper deals with the problem of spatial data mapping. A new method based on wavelet interpolation and geostatistical prediction (kriging) is proposed. The method – wavelet analysis residual kriging (WARK) – is developed in order to assess the problems rising for highly variable data in presence of spatial trends. In these cases stationary prediction models have very limited application. Wavelet analysis is used to model large-scale structures and kriging of the remaining residuals focuses on small-scale peculiarities. WARK is able to model spatial pattern which features multiscale structure. In the present work WARK is applied to the rainfall data and the results of validation are compared with the ones obtained from neural network residual kriging (NNRK). NNRK is also a residual-based method, which uses artificial neural network to model large-scale non-linear trends. The comparison of the results demonstrates the high quality performance of WARK in predicting hot spots, reproducing global statistical characteristics of the distribution and spatial correlation structure.     

Comparison between the indirect approach and kriging with samples of different support for estimation using samples of different length

In this paper we compare two estimation methods to deal with samples of different support: (1) the indirect approach using accumulation and (2) kriging with samples of different support. These two methods were tested in a simple example. The estimates of the two methods were compared against a benchmark scenario. The benchmark consisted of kriging using a complete set of samples on the same support. The effects of the nugget effect, variogram range and type on the weight of long samples, the estimate, and the error variance were assessed. Kriging with samples of different support led to lower error variance and to estimates closer to the estimates of the benchmark scenario. Furthermore, in the case of spatially continuous attributes (low nugget effect), the indirect approach assigns greater weight to long samples than kriging with samples of different support. A cross validation study comparing the two methods with a database from a bauxite deposit was performed. The results of the cross validation study showed that kriging with samples of different support resulted in more precise estimates.     

Geostatistical interpolation of object counts collected from multiple strip transects: Ordinary kriging versus finite domain kriging

Data collected along transects are becoming more common in environmental studies as indirect measurement devices, such as geophysical sensors, that can be attached to mobile platforms become more prevalent. Because exhaustive sampling is not always possible under constraints of time and costs, geostatistical interpolation techniques are used to estimate unknown values at unsampled locations from transect data. It is known that outlying observations can receive significantly greater ordinary kriging weights than centrally located observations when the data are contiguously aligned along a transect within a finite search window. Deutsch (1994) proposed a kriging algorithm, finite domain kriging, that uses a redundancy measure in place of the covariance function in the data-to-data kriging matrix to address the problem of overweighting the outlying observations. This paper compares the performances of two kriging techniques, ordinary kriging (OK) and finite domain kriging (FDK), on examining unexploded ordnance (UXO) densities by comparing prediction errors at unsampled locations. The impact of sampling design on object count prediction is also investigated using data collected from transects and at random locations. The Poisson process is used to model the spatial distribution of UXO for three 5000 × 5000 m fields; one of which does not have any ordnance target (homogeneous field), while the other two sites have an ordnance target in the center of the site (isotropic and anisotropic fields). In general, for a given sampling transects width, the differences between OK and FDK in terms of the mean error and the mean square error are not significant regardless of the sampled area and the choice of the field. When 20% or more of the site is sampled, the estimation of object counts is unbiased on average for all three fields regardless of the choice of the transect width and the choice of the kriging algorithm. However, for non-homogeneous fields (isotropic and anisotropic fields), the mean error fluctuates considerably when a small number of transects are sampled. The difference between the transect sampling and the random sampling in terms of prediction errors becomes almost negligible if more than 20% of the site is sampled. Overall, FDK is no better than OK in terms of the prediction performances when the transect sampling procedure is used.     

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.     

Rainfall network design through comparative kriging methods

Abstract

The methods of simple and disjunctive kriging are applied and compared in the estimation of optimum locations of recording raingauges as part of a network for the determination of storm characteristics to be used in forecasting and design. Some advantages are shown but problems arise when there are large differences in storm structures and movements. Another source of uncertainty is in the modelling of the semi-variogram. Application is made to the management of an area of the Severn-Trent water basin, UK, with 13 autographic raingauges.     

Distance-based kriging relying on proxy simulations for inverse conditioning

Let us consider a large set of candidate parameter fields, such as hydraulic conductivity maps, on which we can run an accurate forward flow and transport simulation. We address the issue of rapidly identifying a subset of candidates whose response best match a reference response curve. In order to keep the number of calls to the accurate flow simulator computationally tractable, a recent distance-based approach relying on fast proxy simulations is revisited, and turned into a non-stationary kriging method where the covariance kernel is obtained by combining a classical kernel with the proxy. Once the accurate simulator has been run for an initial subset of parameter fields and a kriging metamodel has been inferred, the predictive distributions of misfits for the remaining parameter fields can be used as a guide to select candidate parameter fields in a sequential way. The proposed algorithm, Proxy-based Kriging for Sequential Inversion (ProKSI), relies on a variant of the Expected Improvement, a popular criterion for kriging-based global optimization. A statistical benchmark of ProKSI’s performances illustrates the efficiency and the robustness of the approach when using different kinds of proxies.     

Generation of runoff series at ungauged locations using empirical orthogonal functions in combination with kriging

The method of Empirical Orthogonal Functions (EOF method) is combined with an objective interpolation technique, kriging, to generate runoff series at ungauged locations. In a case study the results are compared to series interpolated by a combination of EOF analysis and regression using catchment characteristics as independent variables. The results are also compared to linear weighting of an existing runoff series, a commonly used method for spatial interpolation. The influence of altitude on the runoff is studied comparing kriging based on 2 and 3 coordinates. The study showed that the capacity of EOF analysis combined with kriging is as good as the traditionally used linear weighting. The results, when altitude is included in the kriging, are improved.     

Statistical interpolation of ozone measurements from satellite data (TOMS, SBUV and SAGE II) using the kriging method

This study demonstrates that ordinary kriging in spherical coordinates using experimental semi-variograms provides highly usable results, especially near the pole in winter and/or where there could be data missing over large areas. In addition, kriging allows display of the spatial variability of daily ozone measurements at different pressure levels. Three satellite data sets were used: Total Ozone Mapping Spectrometer (TOMS) data, Solar Backscattered UltraViolet (SBUV), and the Stratospheric Aerosol and Gas Experiment (SAGE II) ozone profiles. Since SBUV is a nadir-viewing instrument, measurements are only taken along the sun-synchronous polar orbits of the satellite. SAGE II is a limb-viewing solar occultation instrument, and measurements have high vertical resolution but poor daily coverage. TOMS has wider coverage with equidistant distribution of data (resolution 1° × 1.25°) but provides no vertical information. Comparisons of the resulting SBUV-interpolated (column-integrated) ozone field with TOMS data are strongly in agreement, with a global correlation of close to 98%. Comparisons of SBUV-interpolated ozone profiles with daily SAGE II profiles are relatively good, and comparable to those found in the literature. The interpolated ozone layers at different pressure levels are shown.     

BME prediction of continuous geographical properties using auxiliary variables

Using auxiliary information to improve the prediction accuracy of soil properties in a physically meaningful and technically efficient manner has been widely recognized in pedometrics. In this paper, we explored a novel technique to effectively integrate sampling data and auxiliary environmental information, including continuous and categorical variables, within the framework of the Bayesian maximum entropy (BME) theory. Soil samples and observed auxiliary variables were combined to generate probability distributions of the predicted soil variable at unsampled points. These probability distributions served as soft data of the BME theory at the unsampled locations, and, together with the hard data (sample points) were used in spatial BME prediction. To gain practical insight, the proposed approach was implemented in a real-world case study involving a dataset of soil total nitrogen (TN) contents in the Shayang County of the Hubei Province (China). Five terrain indices, soil types, and soil texture were used as auxiliary variables to generate soft data. Spatial distribution of soil total nitrogen was predicted by BME, regression kriging (RK) with auxiliary variables, and ordinary kriging (OK). The results of the prediction techniques were compared in terms of the Pearson correlation coefficient (r), mean error (ME), and root mean squared error (RMSE). These results showed that the BME predictions were less biased and more accurate than those of the kriging techniques. In sum, the present work extended the BME approach to implement certain kinds of auxiliary information in a rigorous and efficient manner. Our findings showed that the BME prediction technique involving the transformation of variables into soft data can improve prediction accuracy considerably, compared to other techniques currently in use, like RK and OK.     

Moving window kriging with geographically weighted variograms

This study adds to our ability to predict the unknown by empirically assessing the performance of a novel geostatistical-nonparametric hybrid technique to provide accurate predictions of the value of an attribute together with locally-relevant measures of prediction confidence, at point locations for a single realisation spatial process. The nonstationary variogram technique employed generalises a moving window kriging (MWK) model where classic variogram (CV) estimators are replaced with information-rich, geographically weighted variogram (GWV) estimators. The GWVs are constructed using kernel smoothing. The resultant and novel MWK–GWV model is compared with a standard MWK model (MWK–CV), a standard nonlinear model (Box–Cox kriging, BCK) and a standard linear model (simple kriging, SK), using four example datasets. Exploratory local analyses suggest that each dataset may benefit from a MWK application. This expectation was broadly confirmed once the models were applied. Model performance results indicate much promise in the MWK–GWV model. Situations where a MWK model is preferred to a BCK model and where a MWK–GWV model is preferred to a MWK–CV model are discussed with respect to model performance, parameterisation and complexity; and with respect to sample scale, information and heterogeneity.     

Geostatistical prediction of a local geometric geoid - kriging and cokriging with the use of EGM2008 geopotential model

We investigate prediction abilities of different variants of kriging and different combinations of data in a local geometric (GNSS/leveling based) geoid modeling. In order to generate local geoid models, we have used GNSS/leveling data and EGM2008 geopotential model. EGM2008 has been used twofold. Firstly, it was used as a basic long wave-length trend to be removed from geoid undulation data to generate a residual field of geoid heights modeled later by kriging (remove-restore technique). Secondly, EGM2008-based undulations were used as a secondary variable in a cokriging prediction procedure (as pseudo-observations). Besides the use of EGM2008, the kriging-based local geometric geoid models were generated only on the basis of raw undulations data. Kriging itself was used in two variants, i.e. ordinary kriging and universal kriging for univariate and bivariate cases (cokriging). The quality of kriging-based prediction for all its variants and all data combinations have been investigated on one fixed validation dataset consisting of 86 points and three training data sets characterized by a different density of sampling. Results of this study indicate that incorporation of EGM08 as a long wave-length trend in kriging prediction procedure outperforms cokriging strategy based on incorporation of EGM08 as a secondary spatially correlated variable.     

Comparison of ordinary kriging and artificial neural network for spatial mapping of arsenic contamination of groundwater

In this technical note, we investigate the hypothesis that ‘non-linearity matters in the spatial mapping of complex patterns of groundwater arsenic contamination’. The spatial mapping pertained to data-driven techniques of spatial interpolation based on sampling data at finite locations. Using the well known example of extensive groundwater contamination by arsenic in Bangladesh, we find that the use of a highly non-linear pattern learning technique in the form of an artificial neural network (ANN) can yield more accurate results under the same set of constraints when compared to the ordinary kriging method. One ANN and a variogram model were used to represent the spatial structure of arsenic contamination for the whole country. The probability for successful detection of a well as safe or unsafe was found to be atleast 15% larger than that by kriging under the country-wide scenario. The probability of false hopes, which is a serious issue in public health monitoring was found to be significantly lower (by more than 10%) than that by kriging.     

Sequential kriging and cokriging: Two powerful geostatistical approaches

A sequential linear estimator is developed in this study to progressively incorporate new or different spatial data sets into the estimation. It begins with a classical linear estimator (i.e., kriging or cokriging) to estimate means conditioned to a given observed data set. When an additional data set becomes available, the sequential estimator improves the previous estimate by using linearly weighted sums of differences between the new data set and previous estimates at sample locations. Like the classical linear estimator, the weights used in the sequential linear estimator are derived from a system of equations that contains covariances and cross-covariances between sample locations and the location where the estimate is to be made. However, the covariances and cross-covariances are conditioned upon the previous data sets. The sequential estimator is shown to produce the best, unbiased linear estimate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used. It also provides a new way to incorporate additional information into a previous estimation.