Variogram Model Parameters Research Articles

Application of Dueling DQN and DECGA for Parameter Estimation in Variogram Models

Parameter estimation of variogram models is an important problem in geostatistics and environmental engineering. Most of existing works aim to estimate parameters of variogram single models, while neglecting the parameter estimation of variogram nested model. Most recently, the evolutionary algorithms(EA), including genetic algorithm(GA), are exploited to calculate the parameters of variogram model, which can obtain a more accurate solution. These methods have some hyper-parameters to set and suffer from the well-recognized premature convergence and slow global convergence problem of EA. In this paper, a double elite co-evolutionary genetic algorithm(DECGA) and deep reinforcement learning(dueling DQN) was introduced to estimate the parameters of variogram single or nested models so as to achieve better generalization performance. The DECGA can get the global optimal solution faster than GA with the help of dueling DQN, which can set the hyper-parameters according to the state of DECGA. To verify the effectiveness of the proposed method(DDQNGA), we conduct experiments on the agricultural heavy metal database. Experimental results demonstrate that our method can obtain parameter estimation more accurately. The method proposed in this paper have a certain practical value in the field of geostatistics and environmental engineering.

Practical considerations for kriging groundwater surfaces

AbstractKriging is an accepted method of characterizing the groundwater elevation surface at sites where the water level data are available but where there may be insufficient additional data necessary for groundwater flow modeling. Groundwater surface interpolation via kriging is readily performed using commercial, state of the practice software, but some practitioners may not be able to justify such efforts because the process is not validated within studies documented in the peer‐reviewed literature. This paper describes the available kriging software and literature studies and then uses a case study to compare practical groundwater surface modeling to the studies available in the scientific literature. The literature review shows that the state of the practice as represented by the commercial software approach is consistent with the literature. Specifically, cokriging with groundwater elevation as the primary variable with trend removal and ground‐surface elevation as the secondary variable is an appropriate point of departure in practice. The literature review‐based summary of variogram model parameters (model type, nugget, sill, range, and trend model) was not useful as a quality‐control step to assess the reasonableness of variogram parameters identified by the standard practice of software‐assisted iteration when applied to the case‐study data set. The literature review indicated that groundwater elevation kriging has been performed using as few as 10 data points but a comparison of the case‐study simulated groundwater elevations and groundwater gradient magnitudes and directions indicated that the 30‐well threshold more commonly found in the literature was an appropriate minimum at the study site.

Fuzzy tonnage-average grade model based on extension principle

Due to the nature of the geological and mining activities, different input parameters in the grade estimation and mineral resource evaluation are always tainted with uncertainties. It is possible to investigate the uncertainties related to the measurements and parameters of the variogram model using the fuzzy kriging method instead of the kriging method. The fuzzy kriging theory has already been the subject of relatively various research studies but the main weak point in such studies is that the results of the fuzzy estimations are not used in decision-making and planning. A very common, but key, tool of decision-making for mining engineers is the tonnage-average grade models. Under conditions where measurements or/and variogram model parameters are tainted with uncertainties, the tonnage-average grade model will be uncertain as well. Therefore, it is necessary to use the fuzzy tonnage-grade model instead of the crisp ones, and the next analysis steps and decision-makings are done accordingly. In this paper, the computational principles of the fuzzy tonnage-average grade curve and a case study regarding its usage are presented.

An Improved Workflow for Permeability Estimation from Image Logs with Uncertainty Quantification

Production forecasts are extremely sensitive to the spatial distribution of certain reservoir parameters, especially permeability, which controls fluid flow and efficiency of many recovery methods. Modeling the heterogeneity in permeability distribution is challenging due to various reasons. First, proper sampling is difficult due to biased sampling, dilation effects and sample disturbance. Second, calibration of core-plug measurements with well test data is problematic because of their vast differences in scale; in the case of heavy oil reservoirs, conventional well tests are not feasible due to bitumen’s ultra-low mobility at initial conditions. Therefore, in addition to flow-based measurements, alternative static data including image logs and core photographs, which capture detailed high-resolution information regarding heterogeneities at the millimeter scale, may be integrated for reliable permeability estimation along the wellbore. This paper examines the existing technique for permeability prediction from image logs (or core photographs) and proposes a set of improvements. For example, automatic variogram modeling is performed over a set of moving windows to simulate the sand–shale indicators along the well trajectory, capturing the locally varying anisotropy in variogram model parameters. In addition, it presents a unified approach to properly quantify the uncertainty in the histogram of core measurements and well log data: Joint histogram uncertainty considering the correlations between variables is incorporated into the existing methodology. The millimeter-scale model is constructed and upscaled to the centimeter scale via steady-state flow-based upscaling. This step is repeated with the centimeter-scale model until the final flow-simulation scale is attained. The entire methodology is applied within individual facies with distinct spatial correlations. A field case study involving data from the McMurray Formation (Alberta, Canada) is presented, where the permeability distributions in five different facies are modeled. The results indicate that considering automatic variogram modeling may lead to more realistic representation of the orientation of certain heterogeneous features that are strongly influencing the permeability to flow; furthermore, incorporating histogram uncertainty enables the modeling of porosity and permeability with improved uncertainty.

A Simulated Annealing based Optimization Algorithm for Automatic Variogram Model Fitting

AbstractFitting a theoretical model to an experimental variogram is an important issue in geostatistical studies because if the variogram model parameters are tainted with uncertainty, the latter will spread in the results of estimations and simulations. Although the most popular fitting method is fitting by eye, in some cases use is made of the automatic fitting method on the basis of putting together the geostatistical principles and optimization techniques to: 1) provide a basic model to improve fitting by eye, 2) fit a model to a large number of experimental variograms in a short time, and 3) incorporate the variogram related uncertainty in the model fitting. Effort has been made in this paper to improve the quality of the fitted model by improving the popular objective function (weighted least squares) in the automatic fitting. Also, since the variogram model function (£) and number of structures (m) too affect the model quality, a program has been provided in the MATLAB software that can present optimum nested variogram models using the simulated annealing method. Finally, to select the most desirable model from among the single/multi-structured fitted models, use has been made of the cross-validation method, and the best model has been introduced to the user as the output. In order to check the capability of the proposed objective function and the procedure, 3 case studies have been presented.

Comparison of particle swarm optimization and simulated annealing for locating additional boreholes considering combined variance minimization

One of the most important stages in complementary exploration is optimal designing the additional drilling pattern or defining the optimum number and location of additional boreholes. Quite a lot research has been carried out in this regard in which for most of the proposed algorithms, kriging variance minimization as a criterion for uncertainty assessment is defined as objective function and the problem could be solved through optimization methods. Although kriging variance implementation is known to have many advantages in objective function definition, it is not sensitive to local variability. As a result, the only factors evaluated for locating the additional boreholes are initial data configuration and variogram model parameters and the effects of local variability are omitted. In this paper, with the goal of considering the local variability in boundaries uncertainty assessment, the application of combined variance is investigated to define the objective function. Thus in order to verify the applicability of the proposed objective function, it is used to locate the additional boreholes in Esfordi phosphate mine through the implementation of metaheuristic optimization methods such as simulated annealing and particle swarm optimization. Comparison of results from the proposed objective function and conventional methods indicates that the new changes imposed on the objective function has caused the algorithm output to be sensitive to the variations of grade, domain's boundaries and the thickness of mineralization domain. The comparison between the results of different optimization algorithms proved that for the presented case the application of particle swarm optimization is more appropriate than simulated annealing.

An Integrated Approach to Simulate and Validate Orebody Realizations with Complex Trends: A Case Study in Heavy Mineral Sands

Characterization of spatial variability in earth science commonly requires random fields which are stationary within delineated domains. This contribution presents an alternative approach for simulating attributes in combination with a non-stationary first-order moment. A new procedure is presented to unambiguously decompose the observed behaviour into a deterministic trend and a stochastic residual, while explicitly controlling the modelled uncertainty. The practicality of the approach resides in a straightforward and objective inference of the variogram model and neighborhood parameters. This method does not require a prior removal of the trend. The inference principle is based on minimizing the deviation between empirical and theoretical errors calculated for increasingly distant neighborhood shells. Further, the inference is integrated into a systematic simulation framework and accompanying validation guidelines are formulated. The effort results in a characterization of the resource uncertainty of an existing heavy mineral sand deposit.

FuzzyKrig: a comprehensive matlab toolbox for geostatistical estimation of imprecise information

Using kriging has been accepted today as the most common method of estimating spatial data in such different fields as the geosciences. To be able to apply kriging methods, it is necessary that the data and variogram model parameters be precise. To utilize the imprecise (fuzzy) data and parameters, use is made of fuzzy kriging methods. Although it has been 30 years since different fuzzy kriging algorithms were proposed, its use has not become as common as other kriging methods (ordinary, simple, log, universal, etc.); lack of a comprehensive software that can perform, based on different fuzzy kriging algorithms, the related calculations in a 3D space can be the main reason. This paper describes an open-source software toolbox (developed in Matlab) for running different algorithms proposed for fuzzy kriging. It also presents, besides a short presentation of the fuzzy kriging method and introduction of the functions provided by the FuzzyKrig toolbox, 3 cases of the software application under the conditions where: 1) data are hard and variogram model parameters are fuzzy, 2) data are fuzzy and variogram model parameters are hard, and 3) both data and variogram model parameters are fuzzy.

Considering the Epistemic Uncertainties of the Variogram Model in Locating Additional Exploratory Drillholes

To enhance the certainty of the grade block model, it is necessary to increase the number of exploratory drillholes and collect more data from the deposit. The inputs of the process of locating these additional drillholes include the variogram model parameters, locations of the samples taken from the initial drillholes, and the geological block model. The uncertainties of these inputs will lead to uncertainties in the optimal locations of additional drillholes. Meanwhile, the locations of the initial data are crisp, but the variogram model parameters and the geological model have uncertainties due to the limitation of the number of initial data. In this paper, effort has been made to consider the effects of variogram uncertainties on the optimal location of additional drillholes using the fuzzy kriging and solve the locating problem with the genetic algorithm (GA) optimization method.A bauxite deposit case study has shown the efficiency of the proposed model.

Variogram models reveal habitat gradients predicting patterns of territory occupancy and nest survival among vesper sparrows

Characterizing landscapes as gradients may help illuminate animal—habitat relationships that are either 1) masked by or 2) impractical to investigate using a purely patch‐based perspective. Among other methods, variogram models may reveal these gradients in the environment by quantifying spatial dependence among point samples, yet few analyses of animal habitat relationships employing variograms have been undertaken. Using vegetation volume measurements from 4‐m2plots within breeding vesper sparrowPooecetes gramineusterritories, we calculated four territory‐scale gradients: 1) mean volume, 2) standard deviation of volume, 3) nugget (a measure of fine‐scale variation), and 4) range (an index of patch size). The first two gradients are more commonly employed in animal ecology while the second two were derived using variogram models and are infrequently employed. We next used these gradients in generalized linear models predicting territory occupancy and daily nest survival. We found overwhelming support for employing the range parameter and models indicated 1) birds selected areas with lower average vegetation volume and smaller patch sizes and 2) had lower rates of nest predation in areas with larger patch sizes. While these results indicate a pattern of non‐ideal habitat selection, there was no indication that territories which experienced nest predation were selected disproportionately. Our results underscore the utility of 1) variograms among other methods for quantifying gradients in animal habitat and 2) variogram model parameters in investigating the habitat ecology of animals.

Bayesian area-to-point kriging using expert knowledge as informative priors

Area-to-point (ATP) kriging is a common geostatistical framework to address the problem of spatial disaggregation or downscaling from block support observations (BSO) to point support (PoS) predictions for continuous variables. This approach requires that the PoS variogram is known. Without PoS observations, the parameters of the PoS variogram cannot be deterministically estimated from BSO, and as a result, the PoS variogram parameters are uncertain. In this research, we used Bayesian ATP conditional simulation to estimate the PoS variogram parameters from expert knowledge and BSO, and quantify uncertainty of the PoS variogram parameters and disaggregation outcomes. We first clarified that the nugget parameter of the PoS variogram cannot be estimated from only BSO. Next, we used statistical expert elicitation techniques to elicit the PoS variogram parameters from expert knowledge. These were used as informative priors in a Bayesian inference of the PoS variogram from BSO and implemented using a Markov chain Monte Carlo algorithm. ATP conditional simulation was done to obtain stochastic simulations at point support. MODIS (Moderate Resolution Imaging Spectroradiometer) atmospheric temperature profile data were used in an illustrative example. The outcomes from the Bayesian ATP inference for the Matérn variogram model parameters confirmed that the posterior distribution of the nugget parameter was effectively the same as its prior distribution; for the other parameters, the uncertainty was substantially decreased when BSO were introduced to the Bayesian ATP estimator. This confirmed that expert knowledge brought new information to infer the nugget effect at PoS while BSO only brought new information to infer the other parameters. Bayesian ATP conditional simulations provided a satisfactory way to quantify parameters and model uncertainty propagation through spatial disaggregation.

Tests of Significance for Structural Correlations in the Linear Model of Coregionalization

In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures (one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed tests are generally valid.

Assessing the performance of the ensemble Kalman filter for subsurface flow data integration under variogram uncertainty

The ensemble Kalman filter (EnKF) has recently been proposed as a promising parameter estimation approach for constraining the description of rock flow properties, such as permeability and porosity, to reproduce flow measurements that are modeled as nonlinear functions of these properties. One of the key factors that strongly affect the performance of the EnKF is the quality or representativeness of the prior ensemble of property fields used to initialize the EnKF assimilation procedure. The initial ensemble is commonly constructed by assuming a known geological continuity model such as a variogram. However, geologic continuity models are derived from incomplete information and imperfect modeling assumptions, which can introduce a significant level of uncertainty into the produced models. Neglecting this important source of uncertainty can lead to systematic errors and questionable estimation results. In this paper, we investigate the performance of the EnKF under varying levels of uncertainty in the variogram model parameters. We first attempt to directly estimate variogram model parameters from flow data and show that the complex and nonunique relation they have with the flow data provides little sensitivity for an effective inversion with the EnKF. We then assess the performance of the EnKF for estimation of permeability values under uncertain and incorrect initial variogram parameters and show that any bias in specifying variogram parameters tends to persist throughout the EnKF analysis even though locally reasonable permeability updates may be obtained near observation points. More importantly, we show that when variogram parameters are specified probabilistically to account for the full range of structural variability in the initial permeability ensemble, the EnKF update results are quite promising. The results suggest that under uncertain geologic continuity, the EnKF tends to perform better if a very diverse set of property fields is used to form the initial ensemble than when a deterministic and potentially erroneous variogram model is used. Therefore, in applying the EnKF to model calibration problems, it is preferable to overestimate the uncertainty in geologic continuity and to initialize the EnKF procedure with a wide range of variability in property description than to overlook the variogram uncertainty at the risk of introducing systematic bias that cannot be corrected by the EnKF updates. The practical implications of the results in this paper are significant for designing the EnKF for realistic ensemble model calibration problems where the level of uncertainty in the initial ensemble is usually not known a priori.

Assessing the coupling between surface albedo derived from MODIS and the fraction of diffuse skylight over spatially-characterized landscapes

In this effort, the MODerate Resolution Imaging Spectroradiometer (MODIS) (Collection V005) Bidirectional Reflectance Distribution Function (BRDF)/Albedo algorithm is used to retrieve instantaneous surface albedo at a point in time and under specific atmospheric conditions. These retrievals are then used to study the role that the fraction of diffuse skylight plays under realistic scenarios of anisotropic diffuse illumination and multiple scattering between the surface and atmosphere. Simulations of the sky radiance using the MODTRAN®5.1 radiative transfer model were performed under different aerosol optical properties, illumination conditions, and surface characteristics to describe these effects on surface albedo retrievals from MODIS. This technique was examined using a validation scheme over four measurement sites with varied aerosol levels and landscapes, ranging from croplands to tundra ecosystems, and over extended time periods. Furthermore, a series of geostatistical analyses were performed to examine the types of spatial patterns observed at each measurement site. In particular, Enhanced Thematic Mapper Plus (ETM+) retrievals of surface albedo were acquired to analyze the change in variogram model parameters as a function of increased window-size. Results were then used to assess the degree to which a given point measurement is able to capture the intrinsic variability at the scale of MODIS observations. Assessments of MODIS instantaneous albedos that account for anisotropic multiple scattering, over snow-free and snow-covered lands and at all diurnal solar zenith angles, show a slight improvement over the albedo formulations that treat the downwelling diffuse radiation as isotropic. Comparisons with field measurements show biases improving by 0.004–0.013 absolute units (root-mean-squared error) or 0.1%–2.0% relative error.

Geostatistical analysis of the spatial variation of the berry borer and leaf miner in a coffee agroecosystem

The advent of geostatistics and geographical information systems has made it possible to analyze complex spatial patterns of ecological phenomena over large areas in applied insect ecology and pest management. The objective of this study was to use geostatistics to characterize the spatial structure and map the spatial variation of damage caused by the berry borer (Hypothenemus hampei) and leaf miner (Leucoptera coffeella) in a coffee agroecosystem planted with the cultivar Catuai Vermelho IAC-99. Infestations of berry borer and leaf miner were evaluated in fruits and leaves, respectively. The pests were monitored at 67 georeferenced points in an area of 6.6 ha in 2005, 2006 and 2007. Variograms estimated by the method of moments (MoM) and residual maximum likelihood REML were compared. The latter were generally better in terms of the kriging error coefficients. Spherical variograms estimated by REML for berry borer infestation in fruits had ranges of spatial dependence of 34.62–118.4 m and for the leaf miner they were 53.93–133.7 m. For models fitted by weighted ordinary least squares (OLS) to the MoM experimental variogram, the ranges varied between 37.22 and 68.67 m for the berry borer and 100 and 155.4 m for leaf miner infestation. The variogram model parameters were used with the data for ordinary kriging to map the spatial variation of coffee pests for different monitoring periods. If there was no suitable variogram, inverse distance weighting was used to map the variation. The maps enabled visualization of the intensity of infestation of the insect pests for the different periods evaluated.

Stochastic Simulations of Spatial Variability Based on Multifractal Characteristics

Using multifractal characteristics in the stochastic simulation of environmental and agronomic variables has the potential for producing simulations that better represent features of interest, including distribution properties of the high and low values, than simulations based on geostatistical characteristics. The objective of this study was to compare the performance of stochastic simulations that reproduce certain multifractal characteristics with simulations that reproduce variogram model parameters. Data on corn (Zea mays L.) and wheat (Triticum aestivum L.) yields collected at 1100 and 880 sampling locations, respectively, were used in the simulations. Wheat data represented an example of a variable exhibiting a multifractal scaling and corn an example of a variable exhibiting a monofractal scaling. The conditional simulations were conducted using a simulated annealing procedure. Two simulation scenarios were compared. The first scenario consisted of using the order of the structure function q = 2, which is equivalent to simulating a data set with a certain variogram, that is, a monofractal simulation. The second scenario used the histogram and the structure function with exponent values for q ranging from 1 to 5, a multifractal simulation. Based on 50 simulations, the performances of the two scenarios in representing the highest and lowest values were compared. There was no significant difference in performance of monofractal and multifractal simulations for the variable exhibiting monofractal scaling (corn). For the variable with multifractal scaling (wheat), however, multifractal simulations performed better in predicting low values. For soil or agronomic variables exhibiting multifractal scaling, stochastic simulations based on multifractal characteristics could be preferable to those based on variogram parameters.

Point and Interval Estimation of Variogram Models Using Spatial Empirical Likelihood

We present a spatial blockwise empirical likelihood method for estimating variogram model parameters in the analysis of spatial data on a grid. The method produces point estimators that require no spatial variance estimates to compute, unlike least squares methods for variogram fitting, but are as efficient as the best least squares estimator in large samples. Our approach also produces confidence regions for the variogram, without requiring knowledge of the full joint distribution of the spatial data. In addition, the empirical likelihood formulation extends to spatial regression problems and allows simultaneous inference on both spatial trend and variogram parameters. We examine the asymptotic behavior of the estimator analytically, and investigate its behavior in finite samples through simulation studies.

The analysis of ranked observations of soil structure using indicator geostatistics

Structure is an important physical feature of the soil that is associated with water movement, the soil atmosphere, microorganism activity and nutrient uptake. A soil without any obvious organisation of its components is known as apedal and this state can have marked effects on several soil processes. Accurate maps of topsoil and subsoil structure are desirable for a wide range of models that aim to predict erosion, solute transport, or flow of water through the soil. Also such maps would be useful to precision farmers when deciding how to apply nutrients and pesticides in a site-specific way, and to target subsoiling and soil structure stabilization procedures. Typically, soil structure is inferred from bulk density or penetrometer resistance measurements and more recently from soil resistivity and conductivity surveys. To measure the former is both time-consuming and costly, whereas observations made by the latter methods can be made automatically and swiftly using a vehicle-mounted penetrometer or resistivity and conductivity sensors. The results of each of these methods, however, are affected by other soil properties, in particular moisture content at the time of sampling, texture, and the presence of stones. Traditional methods of observing soil structure identify the type of ped and its degree of development. Methods of ranking such observations from good to poor for different soil textures have been developed. Indicator variograms can be computed for each category or rank of structure and these can be summed to give the sum of indicator variograms (SIV). Observations of the topsoil and subsoil structure were made at four field sites where the soil had developed on different parent materials. The observations were ranked by four methods and indicator and the sum of indicator variograms were computed and modelled for each method of ranking. The individual indicators were then kriged with the parameters of the appropriate indicator variogram model to map the probability of encountering soil with the structure represented by that indicator. The model parameters of the SIVs for each ranking system were used with the data to krige the soil structure classes, and the results are compared with those for the individual indicators. The relations between maps of soil structure and selected wavebands from aerial photographs are examined as basis for planning surveys of soil structure.

Assessing the performance of linear geostatistical tools applied to artificial fisheries data

Most marine species are spatially clustered and thus the analysis of their distribution with geostatistical techniques is becoming increasingly common in fisheries studies, as a way of producing continuous distribution maps, reducing estimated error and analysing species spatial behaviour. The use of geostatistics is coupled with an increased computation power and development of statistical methods, leading to a greater availability of those to fisheries scientists. Thus, after the application of parametric classical geostatistical models to fisheries, a range of methods is now available that possibly are more adequate to fisheries data, typically non-normal, highly skewed and restricted to a low number of samples. However, the use of such recently available methods is still minimal and had not been widely applied or tested for its potential in fisheries studies. In the current work, we use simulated catch data, generated with an underlying spherical covariance function, considering two types of distribution function (Gaussian and log-Gaussian) and three levels of nugget variability (0, 30 and 50%). Twenty to 200 samples were randomly redrawn from the simulated data. Thus, for each case (sampling number, nugget and distribution) first, we test the efficiency of Mantel statistic in detecting significant autocorrelation. Second, we study the extent to which the originally defined covariance structure (variogram model parameters: nugget, sill and range) is recovered by different fitting methods: based on least squares (weighted least squares and Cressie) and on maximum likelihood (ML and relative ML) methods and the performance of the classical and robust variogram estimators. Third, we determine the reliability of several goodness-of-fit criteria (Akaike's, cross-validation, ‘gof’ and minimising function) in detecting the original model type (between spherical, exponential, Gaussian or linear). Finally, the bias of the fitting methods was modelled, so it can be corrected in future studies. Mantel test was able to detect significant autocorrelation in 20–85% of the cases, with Pearson r correlation coefficient varying between 0.04 and 0.12. The test statistic gave the best results with Gaussian data, increasing number of samples (more than 100) and decreasing nugget. Still, overall its behaviour was poor in identifying spatial autocorrelation. All fitting methods analysed gave significantly correct estimates for more than 50–70 samples, even when using log-Gaussian data with 50% nugget. Both estimator types behaved similarly in the Gaussian case, although the robust estimator performed considerably worse for log-Gaussian data. Using the least square methods, both weighted least squares (WLS) and Cressie behaved similarly well, for more than 50 samples, even with 50% nugget. As expected, the precision and accuracy of the estimates increased with the sample size, except for the range, which even when only 20 samples where used the estimates were correct. While the nugget was underestimated for any of the methods tested, the sill was generally overestimated, particularly in the presence of nugget. With increasing nugget variability (from 0 to 50%), more samples were required to accurately estimate the parameters. The range of variograms was in general overestimated by all methods studied. No goodness-of-fit criterion was able to detect the correct spatial model in more than 60% of the cases. Still, Akaike's, ‘gof’ using weights as pairs and the minimising function gave the best results. In this assessment, there was little influence of sample size and amount of nugget. A geostatistical analysis in a fisheries context is often carried out at the border of acceptability from a theoretical point of view; however, the results of this simulation study show that the reliability of a structural analysis can be improved when appropriate tools are employed.

Monitoring Temporal Development of Spatial Soil Water Content Variation: Comparison of Ground Penetrating Radar and Time Domain Reflectometry

We compare the capability of ground penetrating radar (GPR) and time domain reflectometry (TDR) to assess the temporal development of spatial variation of surface volumetric water content. In the case of GPR, we measured surface water content with the ground wave, which is a direct wave between the sender and receiver through the upper centimeters of the soil. Spatial water content variation was measured on 18 d with GPR and TDR during a 30‐d monitoring period. To ensure large fluctuations in the spatial water content variation, we created a heterogeneous pattern of water content by irrigation on 2 d. The temporal development of the spatial variation was studied by means of the variogram and interpolated water content maps. To compare GPR and TDR variograms, we estimated confidence intervals of the experimental variograms and the variogram model parameters with a jackknife approach and a first‐order approximation of model parameter uncertainty. The results showed that the 95% confidence intervals of the GPR experimental variogram were one to two orders of magnitude smaller than the 95% confidence intervals of the TDR experimental variogram because of the larger number of GPR measurements. Consequently, the uncertainty in the variogram model parameters was also much lower for GPR, which meant that the temporal development of the fitted GPR variogram model parameters was easier to interpret. Furthermore, the larger GPR measurement volume resulted in a low spatial nugget variance of 1 × 10−6to 1 × 10−9(m3m−3)2because short distance variation was averaged. This meant that GPR accurately measured spatial correlation lengths, even in the case of low water content variation. Interpolated maps showing the increase of water content due to irrigation and the subsequent gradual drying of the soil were more accurate and reproducible for GPR. It was concluded that the noninvasive GPR measurements provide the means to accurately and consistently monitor the development of spatial water content variation in time.