Cokriging Technique Research Articles

Geostatistical techniques for approximate location of pipe burst events in water distribution systems

This paper focusses on the customisation and further enhancement of the recently developed data-driven methodology for the automated near real-time detection of pipe bursts and other (e.g. sensor faults) events at the district metered area (DMA) level. Assuming the availability of pressure/flow data from an increased number of sensors deployed in a DMA, the aim is to: (i) overcome the limitations of the probabilistic inference engine when dealing with the increased data availability; and (ii) exploit the event information resulting from the analysis of the larger number of DMA signals for determining the approximate location of the pipe burst events within the DMA. This is achieved by making use of a multivariate Gaussian mixtures-based graphical model and geostatistical techniques. The novel detection and location methodology is demonstrated and tested on a series of simulated pipe burst events that were performed by opening hydrants in a real-life DMA in the UK. The results obtained illustrate that the new methodology can successfully determine the approximate location of pipe bursts within a DMA (in addition to detecting them in a fast and reliable manner). The performance comparison of several geostatistical techniques shows that the Ordinary Cokriging technique outperforms all other techniques tested.

Geostatistical improvements of evapotranspiration spatial information using satellite land surface and weather stations data

The objective of the present study was to use the simple cokriging methodology to characterize the spatial variability of Penman–Monteith reference evapotranspiration and Thornthwaite potential evapotranspiration methods based on Moderate Resolution Imaging Spetroradiometer (MODIS) global evapotranspiration products and high-resolution surfaces of WordClim temperature and precipitation data. The climatic element data referred to 39 National Institute of Meteorology climatic stations located in Minas Gerais state, Brazil and surrounding states. The use of geostatistics and simple cokriging technique enabled the characterization of the spatial variability of the evapotranspiration providing uncertainty information on the spatial prediction pattern. Evapotranspiration and precipitation surfaces were implemented for the climatic classification in Minas Gerais. Multivariate geostatistical determined improvements of evapotranspiration spatial information. The regions in the south of Minas Gerais derived from the moisture index estimated with the MODIS evapotranspiration (2000–2010), presented divergence of humid conditions when compared to the moisture index derived from the simple kriged and cokriged evapotranspiration (1961–1990), indicating climate change in this region. There was stronger pattern of crossed covariance between evapotranspiration and precipitation rather than temperature, indicating that trends in precipitation could be one of the main external drivers of the evapotranspiration in Minas Gerais state, Brazil.

Assessment of soil property spatial variation based on the geostatistical simulation

The main objective in the present study was to assess the spatial variation of chemical and physical soil properties and then use this information to select an appropriate area to install a pasture rehabilitation experiment in the Zereshkin region, Iran. A regular 250 m grid was used for collecting a total of 150 soil samples (from 985 georeferenced soil pits) at 0 to 30, and 30 to 60 cm layers. Soil samples were analyzed for pH, EC, N, K, P, Na, Ca, Mg and SAR. Conventional statistical methods and geostatistics were performed in order to analyze soil properties spatial dependence. Mean, standard deviation, skewness, and kurtosis for all measured variables were evaluated. All variograms generally were well structured with a relatively large nugget effect. Soil properties such as pH, P semivariograms were best fitted by spherical models, while SAR, Na were best fitted by spherical models. In the beginning kriging were performed in order to analyze spatial variation of chemical and physical soil properties, then for enhancing estimation accuracy and comparing results we used cokriging technique. Comparison of the results using statistical techniques showed that kriging technique has acceptable accuracy in characterizing the spatial variability. Also results showed that although kriging technique has acceptable accuracy in characterizing the spatial variability of soil properties but if higher accuracy is needed, cokriging is preferred to kriging particularly when the extra variable has been used.

Comparison of Different Univariate and Multivariate Geostatistical Methods by Porosity Modeling of an Iranian Oil Field

Geostatistical methods are grouped in two main divisions: univariate and multivariate. When there is adequate amount of primary data, univariate methods such as kriging and SGS give a good representation of property distribution in the reservoir, but practical difficulties appear when there is no sufficient data. In such a case it is necessary to choose multivariate geostatistical methods in which some covariables are contributed to model the primary variable. Multivariate geostatistics is a broad term that encompasses all geostatistical methods that utilize more than one variable to predict some physical property of the earth. Bivariate geostatistics is obviously the simplest subset of the multivariate techniques and thus the standard cokriging technique can be called multivariate geostatistics. Cokriging estimates the unsampled value using the primary and secondary variable values from all primary variable sampled locations and also the covariable at an unsampled location, thus there is a need to construct the secondary and cross covariograms. Hence, it is a time-consuming approach. Collocated cokriging is a valuable alternative to full cokriging, which overcomes these problems. It only requires the covariable sample to be available at every location where the principal variable must be estimated. The authors used this in the framework of sequential simulation to produce various realizations. They compare moving average, kriging, and SGS as univariate methods to collocated cokriging and collocated cosimulation as multivariate methods by porosity modeling of an Iranian oil field, although there were only 7 wells available.

A cokriging based approach to reconstruct air pollution maps, processing measurement station concentrations and deterministic model simulations

One of the aims of regional Environmental Authorities is to provide citizens information about the quality of the atmosphere over a certain region. To reach this objective Environmental Authorities need suitable tools to interpolate the data coming from monitoring networks to domain locations where no measures are available. In this work a spatial interpolation system has been developed to estimate 8-h mean daily maximum ozone concentrations and daily mean PM10 concentrations over a domain, starting from measured concentration values. The presented approach is based on a cokriging technique, using the results of a deterministic Chemical Transport Model (CTM) simulation as secondary variable. The developed methodology has been tested over a 60 × 60 km 2 domain located in Northern Italy, including Milan metropolitan area, one of the most polluted areas in Europe.

An integrated air quality forecast system for a metropolitan area

Air quality forecasting is an important issue in environmental research, due to the effects that air pollutants have on population health. To deal with this topic, in this work an integrated modelling system has been developed to forecast daily maximum eight hours ozone concentrations and daily mean PM10 concentrations, up to two days in advance, over an urban area. The presented approach involves two steps. In the first step, artificial neural networks are identified and applied to get point-wise forecasting. In the second step, the forecasts obtained at the monitoring station locations are spatially interpolated all over the domain using the cokriging technique, which allows to improve the spatial interpolation in the absence of densely sampled data. The integrated modelling system has been then applied to a case study over Northern Italy, performing a validation over space and time for the year 2004 and analyzing if the limit values for the protection of human health set by the European Commission are respected. The presented approach represents a fast and reliable way to provide decision makers and the general public with air quality forecasting, and to support prevention and precautionary measures.

Structural reliability analysis based on the cokriging technique

Approximation methods are widely used in structural reliability analysis because they are simple to create and provide explicit functional relationships between the responses and variables in stead of the implicit limit state function. Recently, the kriging method which is a semi-parameter interpolation technique that can be used for deterministic optimization and structural reliability has gained popularity. However, to fully exploit the kriging method, especially in high-dimensional problems, a large number of sample points should be generated to fill the design space and this can be very expensive and even impractical in practical engineering analysis. Therefore, in this paper, a new method—the cokriging method, which is an extension of kriging, is proposed to calculate the structural reliability. cokriging approximation incorporates secondary information such as the values of the gradients of the function being approximated. This paper explores the use of the cokriging method for structural reliability problems by comparing it with the Kriging method based on some numerical examples. The results indicate that the cokriging procedure described in this work can generate approximation models to improve on the accuracy and efficiency for structural reliability problems and is a viable alternative to the kriging.

Quantification of uncertainty in pedotransfer function‐based parameter estimation for unsaturated flow modeling

While pedotransfer functions (PTFs) have long been applied to estimate soil hydraulic parameters for unsaturated flow and solute transport modeling, the uncertainty associated with the estimates is often ignored. The objective of this study is to evaluate uncertainty of the PTF‐estimated soil hydraulic parameters and its effect on numerical simulation of moisture flow. Contributing to the parameter estimation uncertainty are (1) the PTF intrinsic uncertainty caused by limited data used for PTF training and (2) the PTF input uncertainty in pedotransfer variables (i.e., PTF inputs). The PTF intrinsic uncertainty is assessed using the bootstrap method by generating multiple bootstrap realizations of the soil hydraulic parameters; the realizations follow normal or lognormal distributions. The PTF input variables (i.e., bulk density and soil texture) are obtained using the cokriging technique. The PTF input uncertainty is quantified by assuming that the cokriging estimates follow a normal distribution. Our results show that the PTF input uncertainty dominates over the PTF intrinsic uncertainty and determines the spatial distribution of the PTF parameter estimation uncertainty. When the parameter estimation uncertainty is included, the spatial variability of the measured soil hydraulic parameters is better captured. This is also the case for the observed moisture contents, whose spatial variability is well bracketed by the prediction intervals. However, this is only possible after the PTF input uncertainty is considered. These results suggest that additional sample acquisition for the PTF input variables would have a more favorable impact on reduction of the parameter estimation uncertainty than collecting additional soil hydraulic parameter measurements for PTF development.

레이더 정량강우와 연계한 홍수유출 및 범람해석 시스템 확립 II. Cokriging을 이용한 2차원 정량강우 산정

수문모형의 많은 입력자료 가운데 강우 관측자료가 모의결과에 가장 주요한 원인을 가진다. 과거부터 수문모형은 유역에서의 평균강우량 자료를 한 지점에서의 강우량 자료에 의존해 왔다. 그러나, 지점에서 측정된 강우량 자료를 유역을 대표하는 평균강우량 산정에 이용할 경우 부적절한 경우가 발생할 수 있으며, 특히 그 계측망이 조밀하지 못할 경우 더욱 그러하다. 레이더의 경우 보다 넓은 유역 전체를 관측하기 때문에 유역에서의 평균강우량을 제공함에 있어 좋은 관측장비라 할 수 있다. 레이더를 이용하여 강우를 관측할 경우 직접적인 강우관측이 아니기 때문에 QPE에서 몇몇 한계가 있을 수 있으나, 강우자료의 공간변화도에 대한 좋은 정보를 제공해 준다. 또한 지상의 강우관측소에서의 강우측정자료는 지점에서의 강우량에 대한 정확한 정보를 제공해 준다. 따라서, 본 연구의 목적은 지상의 강우관측소에서 관측된 강우량 자료와 레이더를 이용하여 관측된 자료, 즉 두 가지 관측자료를 Ordinary cokriging 보간을 이용하여 통합함으로써 개선된 강우량 추정방법을 개발한다. Among various input data to hydrologic models, rainfall measurements arguably have the most critical influence on the performance of hydrologic model. Traditionally, hydrologic models have relied on point gauge measurements to provide the area-averaged rainfall information. However, rainfall estimates from gauges become inadequate due to their poor representation of areal rainfall, especially in situations with sparse gauge network. Alternatively, radar that covers much larger areas has become an attractive instrument for providing area- averaged precipitation information. Despite of the limitation of the QPE(Quantitative Precipitation Estimation) using radar, we can get the better information of spatial variability of rainfall fields. Also, rain-gauges give us the better quantitative information of rainfall field. Therefore, in this study, we developed improved methodologies tu estimate rainfall fields using an ordinary cokriging technique which optimally merges radar reflectivity data into rain-gauges data.

Stochastically conditioned flow paths and travel time distribution in highly heterogeneous synthesized aquifers / Chemins d’écoulement et distribution des temps de parcours au sein d’aquifères synthétiques hautement hétérogènes sous conditions stochastiques

Characterization of heterogeneity at the field scale generally requires detailed aquifer properties such as transmissivity and hydraulic head. An accurate delineation of these properties is expensive and time consuming, and for many if not most groundwater systems, is not practical. As an alternative approach, stochastic representation of random fields is used and presented in this paper. Specifically, an iterative stochastic conditional simulation approach was applied to a hypothetical and highly heterogeneous pre-designed aquifer system. The approach is similar to the classical co-kriging technique; it uses a linear estimator that depends on the covariance functions of transmissivity (T), and hydraulic head (h), as well as their cross-covariances. A linearized flow equation along with a conditional random field generator constitutes the iterative process of the conditional simulation. One hundred equally likely realizations of transmissivity fields with pre-specified geostatistical parameters were generated, and conditioned to both limited transmissivity and head data. The successful implementation of the approach resulted in conditioned flow paths and travel-time distribution under different degrees of aquifer heterogeneity. This approach worked well for fields exhibiting small variances. However, for random fields exhibiting large variances (greater than 1.0), an iterative procedure was used. The results show that, as the variance of the ln[T] increases, the flow paths tend to diverge, resulting in a wide spectrum of flow conditions, with no direct discernable relationship between the degree of heterogeneity and travel time. The applied approach indicates that high errors may result when estimation of particle travel times in a heterogeneous medium is approximated by an equivalent homogeneous medium.

PA—Precision AgricultureSoil Tillage Resistance as a Tool to map Soil Type Differences

Abstract Precision agriculture is based on spatial knowledge of soil and crop conditions in the management decisions. In this paper, a method to improve determination of soil physical properties is proposed. Current practice is to analyse soil samples, taken at several locations in a field, for physical properties. To obtain a sound coverage, many soil samples have to be analysed. When soil properties are correlated, it is possible to reduce the number of soil samples of one property and enhance its spatial resolution with the information of another, more densely sampled, property. In this study, information gathered automatically during the major soil tillage operation, ploughing, is used to improve the spatial resolution of sampled topsoil clay content. Plough draught was measured during two seasons on a 6 ha field. The specific plough draught varied between 30 and 50 kNm −2 . Clay content varied between 6 and 22% and topsoil moisture content range was 120–240 g kg −1 during the first year and 180–300 g kg −1 in the second year. The specific plough draught maps of both years showed a similar spatial pattern with a cross correlation coefficient at zero distance of 0·6. The use of specific plough draught as co-variable in the co-kriging technique made it possible to decrease the number of topsoil clay content samples from 60 to 18 ha −1 with only 20% increase in prediction error. On this field, the spatial variation in top soil clay content was correlated with the spatial variation of the crop yield.

PA—Precision Agriculture: Soil Tillage Resistance as a Tool to map Soil Type Differences

Precision agriculture is based on spatial knowledge of soil and crop conditions in the management decisions. In this paper, a method to improve determination of soil physical properties is proposed. Current practice is to analyse soil samples, taken at several locations in a field, for physical properties. To obtain a sound coverage, many soil samples have to be analysed. When soil properties are correlated, it is possible to reduce the number of soil samples of one property and enhance its spatial resolution with the information of another, more densely sampled, property. In this study, information gathered automatically during the major soil tillage operation, ploughing, is used to improve the spatial resolution of sampled topsoil clay content.Plough draught was measured during two seasons on a 6 ha field. The specific plough draught varied between 30 and 50 kNm−2. Clay content varied between 6 and 22% and topsoil moisture content range was 120–240 g kg−1during the first year and 180–300 g kg−1in the second year. The specific plough draught maps of both years showed a similar spatial pattern with a cross correlation coefficient at zero distance of 0·6.The use of specific plough draught as co-variable in the co-kriging technique made it possible to decrease the number of topsoil clay content samples from 60 to 18 ha−1with only 20% increase in prediction error. On this field, the spatial variation in top soil clay content was correlated with the spatial variation of the crop yield.

Assessment of the method used to construct clearness index maps for the new European Solar Radiation Atlas (ESRA)

We present an assessment of the methods used to construct maps for a new solar radiation atlas for Europe. For this atlas, station data and satellite-derived data are used in an interpolation/merging process to derive maps of the long-term monthly global radiation that cover an area ranging from 30° W to 70° E and from 25° to 75° N. Our focus is on the discussion of the accuracy of the method applied: a co-kriging technique. Special emphasis is put on a discussion as to whether the use of satellite-derived radiation maps with a low spatial resolution brings benefits.

An iterative geostatistical inverse method for steady flow in the vadose zone

An iterative geostatistical inverse approach is developed to estimate conditional effective unsaturated hydraulic conductivity parameters, soil‐water pressure head, and degree of saturation in heterogeneous vadose zones. This approach is similar to the classical cokriging technique, and it uses a linear estimator that depends on covariances and cross covariances of unsaturated hydraulic parameters, soil‐water pressure head, and degree of saturation. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the residual covariance and cross‐covariance functions in an iterative manner. As a result, the nonlinear relationship between unsaturated hydraulic conductivity parameters and head is incorporated in the estimation and the estimated fields are approximate conditional means. The ability of the iterative approach is demonstrated through some numerical examples.

An Iterative Stochastic Inverse Method: Conditional Effective Transmissivity and Hydraulic Head Fields

An iterative stochastic approach is developed to estimate transmissivity and head distributions in heterogeneous aquifers. This approach is similar to the classical cokriging technique; it uses a linear estimator that depends on the covariances of transmissivity and hydraulic head and their cross covariance. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the covariances and cross‐covariance function of transmissivity and hydraulic head fields in an iterative manner. As a result the nonlinear relationship between transmissivity and head is incorporated in the estimation, and the estimated fields are approximate conditional means. The ability of the iterative approach is tested with some deterministic and stochastic inverse problems. The results show that the estimated transmissivity and hydraulic head fields have smaller mean square errors than those obtained by classical cokriging even in the aquifer with variance of transmissivity up to 3.

GEOSTATISTICAL ANALYSES OF TRACE ELEMENTS IN SOILS AND PLANTS

Statistical and geostatistical analyses were conducted to estimate both correlation and spatial distributions of trace elements in soils and plants within a corn field. Statistical analysis of AB-DTPA-extractable trace elements in soils and the total elemental content of plants indicated that Mo in corn leaves was negatively correlated with soil Cu and Fe. Copper aggravates Mo deficiency in plants because Cu interferes with the role of Mo in the enzymatic reduction of NO3. Geostatistical analyses of the soil trace elements, Cu, Zn, Mn, Fe, and Mo, showed that these elements were spatially interdependent. Iron, Mn, and Mo in corn leaves were spatial variables characterized by linear, spherical, and exponential variogram models, respectively. However, Cu and Zn contents in corn leaves were randomly distributed in the field. Using the relationship between soil Cu and plant Mo, and a cokriging technique, plant Mo estimation was significantly improved by incorporating the soil Cu information. Compared with kriging, cokriging reduced the mean error of the estimates by about 5 times, reduced the mean square error and the mean kriging variance up to 48%, and increased the correlation of estimates and measurements from 0.49 to 0.61.

An Iterative Cokriging‐Like Technique for Ground‐Water Flow Modeling

AbstractThe importance of field heterogeneities in ground‐water pollution problems has been widely recognized during the last few decades. To address the impact of field heterogeneities on ground‐water flow and solute transport, many different stochastic methods have been developed. Among all these stochastic methods kriging is the most popular one used by many practitioners to interpolate and extrapolate measured transmissivity data. However, hydraulic head measurements are generally more abundant than transmissivity data. Therefore, the cokriging technique which utilizes both the head and transmissivity measurements to estimate transmissivity and/or hydraulic head distributions has also received much attention in recent years.Classical cokriging relies on a linear predictor approach and uses covariance and cross covariance functions derived from a first‐order approximation. Consequently, it often results in head and transmissivity fields that can produce unacceptable velocity distributions. In this paper, we develop an iterative method which combines classical cokriging and a numerical flow model to obtain optimum estimates of transmissivity and head distributions and to alleviate the limitations of classical cokriging. Through several numerical examples, we demonstrate that this method is superior to the classical cokriging method in terms of producing mass conservative velocity fields. In addition, results of the study also indicate that hydraulic head measurements can improve for our prediction of ground‐water flow directions and paths in aquifers significantly.

Practice of geostatistics in the estimation of ion deposition from acid precipitation in the presence of sparse data networks

The estimation of the spatial variability of ion deposition from acid precipitation is commonly faced with the problem of sparse data networks sampled over a short period of time. In such cases, geostatistical techniques are pertinent as long as they are properly applied. According to some practical examples, the use of ordinary kriging 1 given an experimental semi-variogram function based on scattered data values can be misleading in terms of predicted estimation variances; however, the choice of additional sampling stations based on such a semi-variogram remains valid. It is also preferable to work with regularly spaced data values that allow the identification of preferential directional variabilities even from a small number of data points. In order to predict the performance of kriging, the use of semi-variogram cross-validation techniques in the presence of small data sets can be misleading and is not recommended. Finally, the integration of additional information from denser precipitation networks through the cokriging technique is questionable when based on a very small number of concomitant deposition and precipitation data values.

COSIM: a FORTRAN IV program for coconditional simulation

Recently, a computer algorithm was presented for joint estimation of random functions; this cokriging technique demonstrated the utility and increased accuracy obtained through best linear unbiased estimation based on auto- and cross-correlation. A worthwhile extension of cokriging is coconditional simulation, a technique whereby several nonconditionally simulated random functions are conditioned using cokriging. Although coconditional simulation can be performed, one random function at a time, using kriging, this can result in an incorrect portrayal of the cross-correlation between the simulated random functions. This is particularly true if one, or several variables is sampled sparsely. Coconditional simulation based on cokriging correctly reproduces variable cross-correlation independent of variable sampling density.

Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity

This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.