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R. M. Lark - One of the best experts on this subject based on the ideXlab platform.
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the matern Variogram model implications for uncertainty propagation and sampling in geostatistical surveys
Geoderma, 2007Co-Authors: B P Marchant, R. M. LarkAbstract:Abstract The Matern Variogram model has been advocated because it is flexible and can represent varied behaviour at small lags. We show how the constraints on the spherical and exponential Variogram at short lags ignore a possible source of uncertainty in the Variogram and so in kriging surveys, that the Matern model can describe. Matern, spherical and exponential Variogram models were fitted by maximum likelihood to a set of log 10 (K) observations made on a regular grid at Broom's Barn Farm, Suffolk, England. The likelihood profiles of the Matern parameter estimates were asymmetric. Thus the uncertainty of these estimates could only be adequately assessed by a Bayesian approach. The uncertainty of estimated parameters of the Matern Variogram was larger than for the exponential Variogram. This is an indication that the assumption of an exponential model limits the behaviour that may be described by the Variogram. Thus uncertainty analyses where an exponential Variogram is assumed may underestimate the uncertainty of kriged estimates. Bayesian analysis of the kriged estimates of log 10 (K) at Broom's Barn Farm using the Matern Variogram revealed an observable component of uncertainty due to Variogram uncertainty. When an exponential Variogram model was used, the estimate of this component of uncertainty was negligible. The Matern Variogram should therefore be used rather than the exponential model when assessing the adequacy of a Variogram estimate. A method of designing sample schemes which is suitable for both estimating a Matern Variogram and interpolation is suggested.
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on spatial prediction of soil properties in the presence of a spatial trend the empirical best linear unbiased predictor e blup with reml
European Journal of Soil Science, 2006Co-Authors: R. M. Lark, B R Cullis, S J WelhamAbstract:Geostatistical estimates of a soil property by kriging are equivalent to the best linear unbiased predictions (BLUPs). Universal kriging is BLUP with a fixed-effect model that is some linear function of spatial coordinates, or more generally a linear function of some other secondary predictor variable when it is called kriging with external drift. A problem in universal kriging is to find a spatial variance model for the random variation, since empirical Variograms estimated from the data by method-of-moments will be affected by both the random variation and that variation represented by the fixed effects. The geostatistical model of spatial variation is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatistics). Statisticians use residual maximum likelihood (REML) to estimate variance parameters, i.e. to obtain the Variogram in a geostatistical context. REML estimates are consistent (they converge in probability to the parameters that are estimated) with less bias than both maximum likelihood estimates and method-of-moment estimates obtained from residuals of a fitted trend. If the estimate of the random effects variance model is inserted into the BLUP we have the empirical BLUP or E-BLUP. Despite representing the state of the art for prediction from a linear mixed model in statistics, the REML-E-BLUP has not been widely used in soil science, and in most studies reported in the soils literature the Variogram is estimated with methods that are seriously biased if the fixed-effect structure is more complex than just an unknown constant mean (ordinary kriging). In this paper we describe the REML-E-BLUP and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend.
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estimating Variogram uncertainty
Mathematical Geosciences, 2004Co-Authors: B P Marchant, R. M. LarkAbstract:The Variogram is central to any geostatistical survey, but the precision of a Variogram estimated from sample data by the method of moments is unknown. It is important to be able to quantify Variogram uncertainty to ensure that the Variogram estimate is sufficiently accurate for kriging. In previous studies theoretical expressions have been derived to approximate uncertainty in both estimates of the experimental Variogram and fitted Variogram models. These expressions rely upon various statistical assumptions about the data and are largely untested. They express Variogram uncertainty as functions of the sampling positions and the underlying Variogram. Thus the expressions can be used to design efficient sampling schemes for estimating a particular Variogram. Extensive simulation tests show that for a Gaussian variable with a known Variogram, the expression for the uncertainty of the experimental Variogram estimate is accurate. In practice however, the Variogram of the variable is unknown and the fitted Variogram model must be used instead. For sampling schemes of 100 points or more this has only a small effect on the accuracy of the uncertainty estimate. The theoretical expressions for the uncertainty of fitted Variogram models generally overestimate the precision of fitted parameters. The uncertainty of the fitted parameters can be determined more accurately by simulating multiple experimental Variograms and fitting Variogram models to these. The tests emphasize the importance of distinguishing between the Variogram of the field being surveyed and the Variogram of the random process which generated the field. These Variograms are not necessarily identical. Most studies of Variogram uncertainty describe the uncertainty associated with the Variogram of the random process. Generally however, it is the Variogram of the field being surveyed which is of interest. For intensive sampling schemes, estimates of the field Variogram are significantly more precise than estimates of the random process Variogram. It is important, when designing efficient sampling schemes or fitting Variogram models, that the appropriate expression for Variogram uncertainty is applied.
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Two robust estimators of the cross-Variogram for multivariate geostatistical analysis of soil properties
European Journal of Soil Science, 2003Co-Authors: R. M. LarkAbstract:Summary If we wish to describe the coregionalization of two or more soil properties for estimation by cokriging then we must estimate and model their auto- and cross-Variogram(s). The conventional estimates of these Variograms, obtained by the method-of-moments, are unduly affected by outlying data which inflate the Variograms and so also the estimates of the error variance of cokriging predictions. Robust estimators are less affected. Robust estimators of the auto-Variogram and the pseudo cross-Variogram have previously been proposed and used successfully, but the multivariate problem of estimating the cross-Variogram robustly has not yet been tackled. Two robust estimators of the cross-Variogram are proposed. These use covariance estimators with good robustness properties. The robust estimators of the cross-Variogram proved more resistant to outliers than did the method-of-moments estimator when applied to simulated fields which were then contaminated. Organic carbon and water content of the soil was measured at 256 sites on a transect and the method-of-moments estimator, and the two robust estimators, were used to estimate the auto-Variograms and cross-Variogram from a prediction subset of 156 sites. The data on organic carbon included a few outliers. The method-of-moments estimator returned larger values of the auto- and cross-Variograms than did either robust estimator. The organic carbon content at the 100 validation sites on the transect was estimated by cokriging from the prediction data plus a set of Variograms fitted to the method-of-moments estimates and two sets of Variograms fitted to the robust estimates. The ratio of the actual squared prediction error to the cokriging estimate of the error variance was computed at each validation site. These results showed that cokriging using Variograms obtained by the method-of-moments estimator overestimated the error variance of the predictions. By contrast, cokriging with the robustly estimated Variograms gave reliable estimates of the error variance of the predictions.
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estimating Variograms of soil properties by the method of moments and maximum likelihood
European Journal of Soil Science, 2000Co-Authors: R. M. LarkAbstract:Summary Variograms of soil properties are usually obtained by estimating the Variogram for distinct lag classes by the method-of-moments and fitting an appropriate model to the estimates. An alternative is to fit a model by maximum likelihood to data on the assumption that they are a realization of a multivariate Gaussian process. This paper compares the two using both simulation and real data. The method-of-moments and maximum likelihood were used to estimate the Variograms of data simulated from stationary Gaussian processes. In one example, where the simulated field was sampled at different intensities, maximum likelihood estimation was consistently more efficient than the method-ofmoments, but this result was not general and the relative performance of the methods depends on the form of the Variogram. Where the nugget variance was relatively small and the correlation range of the data was large the method-of-moments was at an advantage and likewise in the presence of data from a contaminating distribution. When fields were simulated with positive skew this affected the results of both the method-of-moments and maximum likelihood. The two methods were used to estimate Variograms from actual metal concentrations in topsoil in the Swiss Jura, and the Variograms were used for kriging. Both estimators were susceptible to sampling problems which resulted in over- or underestimation of the variance of three of the metals by kriging. For four other metals the results for kriging using the Variogram obtained by maximum likelihood were consistently closer to the theoretical expectation than the results for kriging with the Variogram obtained by the method-of-moments, although the differences between the results using the two approaches were not significantly different from each other or from expectation. Soil scientists should use both procedures in their analysis and compare the results.
Margaret A Oliver - One of the best experts on this subject based on the ideXlab platform.
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Determining nugget:sill ratios of standardized Variograms from aerial photographs to krige sparse soil data
Precision Agriculture, 2008Co-Authors: Ruth Kerry, Margaret A OliverAbstract:Maps of kriged soil properties for precision agriculture are often based on a Variogram estimated from too few data because the costs of sampling and analysis are often prohibitive. If the Variogram has been computed by the usual method of moments, it is likely to be unstable when there are fewer than 100 data. The scale of variation in soil properties should be investigated prior to sampling by computing a Variogram from ancillary data, such as an aerial photograph of the bare soil. If the sampling interval suggested by this is large in relation to the size of the field there will be too few data to estimate a reliable Variogram for kriging. Standardized Variograms from aerial photographs can be used with standardized soil data that are sparse, provided the data are spatially structured and the nugget:sill ratio is similar to that of a reliable Variogram of the property. The problem remains of how to set this ratio in the absence of an accurate Variogram. Several methods of estimating the nugget:sill ratio for selected soil properties are proposed and evaluated. Standardized Variograms with nugget:sill ratios set by these methods are more similar to those computed from intensive soil data than are Variograms computed from sparse soil data. The results of cross-validation and mapping show that the standardized Variograms provide more accurate estimates, and preserve the main patterns of variation better than those computed from sparse data.
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determining the effect of asymmetric data on the Variogram i underlying asymmetry
Computers & Geosciences, 2007Co-Authors: Ruth Kerry, Margaret A OliverAbstract:Matheron's usual Variogram estimator can result in unreliable Variograms when data are strongly asymmetric or skewed. Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. This paper examines the effects of underlying asymmetry on the Variogram and on the accuracy of prediction, and the second one examines the effects arising from outliers. Standard geostatistical texts suggest ways of dealing with underlying asymmetry; however, this is based on informed intuition rather than detailed investigation. To determine whether the methods generally used to deal with underlying asymmetry are appropriate, the effects of different coefficients of skewness on the shape of the experimental Variogram and on the model parameters were investigated. Simulated annealing was used to create normally distributed random fields of different size from Variograms with different nugget:sill ratios. These data were then modified to give different degrees of asymmetry and the experimental Variogram was computed in each case. The effects of standard data transformations on the form of the Variogram were also investigated. Cross-validation was used to assess quantitatively the performance of the different Variogram models for kriging. The results showed that the shape of the Variogram was affected by the degree of asymmetry, and that the effect increased as the size of data set decreased. Transformations of the data were more effective in reducing the skewness coefficient in the larger sets of data. Cross-validation confirmed that Variogram models from transformed data were more suitable for kriging than were those from the raw asymmetric data. The results of this study have implications for the 'standard best practice' in dealing with asymmetry in data for geostatistical analyses.
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determining the effect of asymmetric data on the Variogram ii outliers
Computers & Geosciences, 2007Co-Authors: Ruth Kerry, Margaret A OliverAbstract:Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. The first paper of this series examined the effects of the former on the Variogram and this paper examines the effects of asymmetry arising from outliers. Simulated annealing was used to create normally distributed random fields of different size that are realizations of known processes described by Variograms with different nugget:sill ratios. These primary data sets were then contaminated with randomly located and spatially aggregated outliers from a secondary process to produce different degrees of asymmetry. Experimental Variograms were computed from these data by Matheron's estimator and by three robust estimators. The effects of standard data transformations on the coefficient of skewness and on the Variogram were also investigated. Cross-validation was used to assess the performance of models fitted to experimental Variograms computed from a range of data contaminated by outliers for kriging. The results showed that where skewness was caused by outliers the Variograms retained their general shape, but showed an increase in the nugget and sill variances and nugget:sill ratios. This effect was only slightly more for the smallest data set than for the two larger data sets and there was little difference between the results for the latter. Overall, the effect of size of data set was small for all analyses. The nugget:sill ratio showed a consistent decrease after transformation to both square roots and logarithms; the decrease was generally larger for the latter, however. Aggregated outliers had different effects on the Variogram shape from those that were randomly located, and this also depended on whether they were aggregated near to the edge or the centre of the field. The results of cross-validation showed that the robust estimators and the removal of outliers were the most effective ways of dealing with outliers for Variogram estimation and kriging.
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comparing sampling needs for Variograms of soil properties computed by the method of moments and residual maximum likelihood
Geoderma, 2007Co-Authors: Ruth Kerry, Margaret A OliverAbstract:It has been generally accepted that the method of moments (MoM) Variogram, which has been widely applied in soil science, requires about 100 sites at an appropriate interval apart to describe the variation adequately. This sample size is often larger than can be afforded for soil surveys of agricultural fields or contaminated sites. Furthermore, it might be a much larger sample size than is needed where the scale of variation is large. A possible alternative in such situations is the residual maximum likelihood (REML) Variogram because fewer data appear to be required. The REML method is parametric and is considered reliable where there is trend in the data because it is based on generalized increments that filter trend out and only the covariance parameters are estimated. Previous research has suggested that fewer data are needed to compute a reliable Variogram using a maximum likelihood approach such as REML, however, the results can vary according to the nature of the spatial variation. There remain issues to examine: how many fewer data can be used, how should the sampling sites be distributed over the site of interest, and how do different degrees of spatial variation affect the data requirements? The soil of four field sites of different size, physiography, parent material and soil type was sampled intensively, and MoM and REML Variograms were calculated for clay content. The data were then sub-sampled to give different sample sizes and distributions of sites and the Variograms were computed again. The model parameters for the sets of Variograms for each site were used for cross-validation. Predictions based on REML Variograms were generally more accurate than those from MoM Variograms with fewer than 100 sampling sites. A sample size of around 50 sites at an appropriate distance apart, possibly determined from Variograms of ancillary data, appears adequate to compute REML Variograms for kriging soil properties for precision agriculture and contaminated sites.
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geostatistics for environmental scientists
2001Co-Authors: R Webster, Margaret A OliverAbstract:Preface 1 Introduction 2 Basic Statistics 3 Prediction and Interpolation 4 Characterizing Spatial Processes: The Covariance and Variogram 5 Modelling the Variogram 6 Reliability of the Experimental Variogram and Nested Sampling 7 Spectral Analysis 8 Local Estimation or Prediction: Kriging 9 Kriging in the Presence of Trend and Factorial Kriging 10 Cross-Correlation, Coregionalization and Cokriging 11 Disjunctive Kriging 12 Stochastic Simulation (new file) Appendix A Appendix B References Index
Ruth Kerry - One of the best experts on this subject based on the ideXlab platform.
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Determining nugget:sill ratios of standardized Variograms from aerial photographs to krige sparse soil data
Precision Agriculture, 2008Co-Authors: Ruth Kerry, Margaret A OliverAbstract:Maps of kriged soil properties for precision agriculture are often based on a Variogram estimated from too few data because the costs of sampling and analysis are often prohibitive. If the Variogram has been computed by the usual method of moments, it is likely to be unstable when there are fewer than 100 data. The scale of variation in soil properties should be investigated prior to sampling by computing a Variogram from ancillary data, such as an aerial photograph of the bare soil. If the sampling interval suggested by this is large in relation to the size of the field there will be too few data to estimate a reliable Variogram for kriging. Standardized Variograms from aerial photographs can be used with standardized soil data that are sparse, provided the data are spatially structured and the nugget:sill ratio is similar to that of a reliable Variogram of the property. The problem remains of how to set this ratio in the absence of an accurate Variogram. Several methods of estimating the nugget:sill ratio for selected soil properties are proposed and evaluated. Standardized Variograms with nugget:sill ratios set by these methods are more similar to those computed from intensive soil data than are Variograms computed from sparse soil data. The results of cross-validation and mapping show that the standardized Variograms provide more accurate estimates, and preserve the main patterns of variation better than those computed from sparse data.
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determining the effect of asymmetric data on the Variogram i underlying asymmetry
Computers & Geosciences, 2007Co-Authors: Ruth Kerry, Margaret A OliverAbstract:Matheron's usual Variogram estimator can result in unreliable Variograms when data are strongly asymmetric or skewed. Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. This paper examines the effects of underlying asymmetry on the Variogram and on the accuracy of prediction, and the second one examines the effects arising from outliers. Standard geostatistical texts suggest ways of dealing with underlying asymmetry; however, this is based on informed intuition rather than detailed investigation. To determine whether the methods generally used to deal with underlying asymmetry are appropriate, the effects of different coefficients of skewness on the shape of the experimental Variogram and on the model parameters were investigated. Simulated annealing was used to create normally distributed random fields of different size from Variograms with different nugget:sill ratios. These data were then modified to give different degrees of asymmetry and the experimental Variogram was computed in each case. The effects of standard data transformations on the form of the Variogram were also investigated. Cross-validation was used to assess quantitatively the performance of the different Variogram models for kriging. The results showed that the shape of the Variogram was affected by the degree of asymmetry, and that the effect increased as the size of data set decreased. Transformations of the data were more effective in reducing the skewness coefficient in the larger sets of data. Cross-validation confirmed that Variogram models from transformed data were more suitable for kriging than were those from the raw asymmetric data. The results of this study have implications for the 'standard best practice' in dealing with asymmetry in data for geostatistical analyses.
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determining the effect of asymmetric data on the Variogram ii outliers
Computers & Geosciences, 2007Co-Authors: Ruth Kerry, Margaret A OliverAbstract:Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. The first paper of this series examined the effects of the former on the Variogram and this paper examines the effects of asymmetry arising from outliers. Simulated annealing was used to create normally distributed random fields of different size that are realizations of known processes described by Variograms with different nugget:sill ratios. These primary data sets were then contaminated with randomly located and spatially aggregated outliers from a secondary process to produce different degrees of asymmetry. Experimental Variograms were computed from these data by Matheron's estimator and by three robust estimators. The effects of standard data transformations on the coefficient of skewness and on the Variogram were also investigated. Cross-validation was used to assess the performance of models fitted to experimental Variograms computed from a range of data contaminated by outliers for kriging. The results showed that where skewness was caused by outliers the Variograms retained their general shape, but showed an increase in the nugget and sill variances and nugget:sill ratios. This effect was only slightly more for the smallest data set than for the two larger data sets and there was little difference between the results for the latter. Overall, the effect of size of data set was small for all analyses. The nugget:sill ratio showed a consistent decrease after transformation to both square roots and logarithms; the decrease was generally larger for the latter, however. Aggregated outliers had different effects on the Variogram shape from those that were randomly located, and this also depended on whether they were aggregated near to the edge or the centre of the field. The results of cross-validation showed that the robust estimators and the removal of outliers were the most effective ways of dealing with outliers for Variogram estimation and kriging.
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comparing sampling needs for Variograms of soil properties computed by the method of moments and residual maximum likelihood
Geoderma, 2007Co-Authors: Ruth Kerry, Margaret A OliverAbstract:It has been generally accepted that the method of moments (MoM) Variogram, which has been widely applied in soil science, requires about 100 sites at an appropriate interval apart to describe the variation adequately. This sample size is often larger than can be afforded for soil surveys of agricultural fields or contaminated sites. Furthermore, it might be a much larger sample size than is needed where the scale of variation is large. A possible alternative in such situations is the residual maximum likelihood (REML) Variogram because fewer data appear to be required. The REML method is parametric and is considered reliable where there is trend in the data because it is based on generalized increments that filter trend out and only the covariance parameters are estimated. Previous research has suggested that fewer data are needed to compute a reliable Variogram using a maximum likelihood approach such as REML, however, the results can vary according to the nature of the spatial variation. There remain issues to examine: how many fewer data can be used, how should the sampling sites be distributed over the site of interest, and how do different degrees of spatial variation affect the data requirements? The soil of four field sites of different size, physiography, parent material and soil type was sampled intensively, and MoM and REML Variograms were calculated for clay content. The data were then sub-sampled to give different sample sizes and distributions of sites and the Variograms were computed again. The model parameters for the sets of Variograms for each site were used for cross-validation. Predictions based on REML Variograms were generally more accurate than those from MoM Variograms with fewer than 100 sampling sites. A sample size of around 50 sites at an appropriate distance apart, possibly determined from Variograms of ancillary data, appears adequate to compute REML Variograms for kriging soil properties for precision agriculture and contaminated sites.
Sara Focaccia - One of the best experts on this subject based on the ideXlab platform.
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Estimating Thermal Response Test Coefficients: Choosing Coordinate Space of The Random Function
Mathematical Geosciences, 2016Co-Authors: Roberto Bruno, Francesco Tinti, Sara FocacciaAbstract:In shallow geothermal systems, the main equivalent underground thermal properties are commonly calculated with a thermal response test (TRT). This is a borehole heat exchanger production test where the temperature of a heat transfer fluid is recorded over time at constant power heat injection/extraction. The equivalent thermal parameters (thermal conductivity, heat capacity) are simply deduced from temperature data regression analysis that theoretically is a logarithmic function in the time domain, or else a linear function in the log-time domain. By interpreting the recorded temperatures as a regionalized variable whose drift is the regression function, in both cases the formal problem is a linear estimation of the mean. If the autocorrelation function (Variogram, covariance) of residuals is known, coefficient variance can be directly deduced. Coefficient estimates are independent of the drift form adopted, and the residuals are the same in the same points. The random function is different in the time domain, however, and in the log-time domain. In fact, residual Variograms are different due to the transformation of the coordinate space. This paper uses a TRT case study to examine the consequences of coordinate space transformation for a random function, namely its Variogram. The specific question addressed is the choice of coordinate space and Variogram.
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thermal response test for shallow geothermal applications a probabilistic analysis approach
Geothermal Energy, 2015Co-Authors: Francesco Tinti, Sara Focaccia, Roberto BrunoAbstract:Thermal Response Test (TRT) is an onsite test used to characterize the thermal properties of shallow underground, when used as heat storage volume for shallow geothermal application. It is applied by injecting/extracting heat into geothermal closed-loop circuits inserted into the ground. The most common types of closed loop are the borehole heat exchangers (BHE), horizontal ground collectors (HGC), and energy piles (EP). The interpretation method of TRT data is generally based on a regression technique and on the calculation of thermal properties through different models, specific for each closed loop and test conditions. A typical TRT record is a graph joining a series of experimental temperatures of the thermal carrier fluid. The proposed geostatistical approach considers the temperature as a random function non-stationary in time, with a given trend, therefore the record is considered as a ‘realization’, one of the possible results; the random nature of the test results is transferred to the fluctuations and a Variogram modeling can be applied, which may give many information on the TRT behavior. In this paper, a nested probabilistic approach for TRT output interpretation is proposed, which can be applied for interpreting TRT data, independently of the different methodologies and technologies adopted. In the paper, for the sake of simplicity, the probabilistic approach is applied to the ‘infinite line source’ (ILS) methodology, which is the most commonly used for BHE. The probabilistic approach, based on Variogram modeling of temperature residuals, is useful for identifying with robust accuracy the time boundaries (initial time t 0 and the final time t f) inside which makes temperature regression analysis possible. Moreover, Variograms are used into the analysis itself to increase estimation precision of thermal parameter calculation (ground conductivity λ g, ground capacity c g, borehole resistance R b). Finally, the probabilistic approach helps keep under control the effect of any cause of result variability. Typical behaviors of power, flows, and temperatures and of their interaction with the specific closed-loop circuit and geo-hydrological system are deepened by Variogram analysis of fluctuations.
J W Van Groenigen - One of the best experts on this subject based on the ideXlab platform.
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The influence of Variogram parameters on optimal sampling schemes for mapping by kriging
Geoderma, 2000Co-Authors: J W Van GroenigenAbstract:Abstract Using spatial simulated annealing (SSA), spatial sampling schemes can be optimised for minimal kriging variance. Two optimisation criteria are presented in this paper. The first criterion minimises the average kriging variance, the second the maximum kriging variance. In a simple case with 23 observations, performances of a sampling scheme obtained with SSA were compared with performances of a triangular grid. SSA reduced the average kriging variance from 40.64 to 39.99 [unit] 2 . The maximum kriging variance was reduced from 86.83 to 53.36 [unit] 2 . Starting with a preliminary, irregularly data set of 100 observations, an additional sampling scheme of 10 observations was optimised. This reduced the average kriging variance from 21.62 to 15.83 [unit] 2 . The maximum kriging variance was reduced from 70.22 to 34.60 [unit] 2 . As the kriging variance is considerably influenced by Variogram parameters, their influence on the optimised sampling schemes was investigated. A Gaussian Variogram produced a different sampling scheme than an exponential Variogram with the same nugget, sill and (effective) range. Exponential, spherical and linear Variograms without nugget resulted in irregular similar sampling schemes. A very short range resulted in irregular sampling schemes, with observations separated by distances larger than twice the range. For a spherical Variogram, the magnitude of the relative nugget effect did not affect the sampling schemes, except for very high values (0.75).
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constrained optimisation of soil sampling for minimisation of the kriging variance
Geoderma, 1999Co-Authors: J W Van Groenigen, W Siderius, A SteinAbstract:This paper introduces the extended Spatial Simulated Annealing (SSA) method to optimise spatial sampling schemes for obtaining the minimal kriging variance. Sampling schemes are optimised at the point level. Boundaries and previous observations can be taken into account. This procedure extends ordinary SSA which focuses entirely on Variogram estimation and even distribution of observations over the area. We applied it to texture and phosphate content on a river terrace in Thailand. First, sampling was conducted for estimation of the Variogram using ordinary SSA. The Variograms thus obtained were used in extended SSA, yielding a reduction of the mean kriging variance of the sand percentage from 28.2 to 23.7(%)2. The Variograms were used subsequently in a geomorphologically similar area. Optimised sampling schemes for anisotropic variables differed considerably from those for isotropic ones. The schemes were especially efficient in reducing high values of the kriging variance near boundaries of the area.
