Average Kriging Variance Research Articles

Optimization of Monitoring Network to the Rainfall Distribution by Using Stochastic Search Algorithms: Lesson from Pakistan

Agricultural production is greatly influenced by environmental parameters such as temperature, rainfall, humidity, and wind speed. The accurate information about environmental parameters plays a vital and useful role when making policies for the agriculture sector as well as for other sectors. Pakistan meteorological department observed these environmental parameters at more than 90 stations. The allocation of these monitoring stations is not made systematically correct. This leads to inaccurate predictions for unobserved locations. The study aims to propose a monitoring network by which these prediction errors of the environmental parameters can be minimized. The well-known prediction techniques named, model-based ordinary kriging and model-based universal kriging (UK) with the known Matheron variogram model are used for prediction purposes. We investigate the monitoring network of Pakistan for rainfall and focus on both the optimal deletion/addition of monitoring stations from/to this network. The two stochastic search algorithms, spatial simulated annealing, and genetic algorithm are used for optimization purposes. Furthermore, the minimization of the Average Kriging Variance (AKV) is taken as the interpolation accuracy measure. The spatial simulated annealing exhibits a lower AKV as compared to the Genetic algorithm when adding/removing the optimal/redundant locations from the monitoring network.

Combining point and regular lattice data in geostatistical interpolation

This work studies how to include both point and areal measurements when estimating gaussian fields by kriging. To achieve this objective, three geostatistical approaches are considered for the areal distributed data: (a) regionalized measurements that are geographically referenced by their centroid as if they were point measurements, (b) regionalized measurements that are explicitly accounted by formally computing all the needed covariances (i.e. area-to-area, area-to-point and point-to-point covariances, respectively) and (c) regionalized measurements that are used as an external drift variable. Results indicate that the measurement error corresponding to the areal data plays a key role to decide when the spatial support of the areal measurements is relevant. For small measurement errors, it is necessary to explicitly consider the spatial support of the areal measurements to avoid large estimation variances. For large measurement errors, the difference between defining areal measurements by using their actual spatial support and defining areal measurements by referencing them by their centroids (i.e. gravity centre) is small. In this situation, it is possible to use the areal measurements as an external drift instead of merging both types of information (i.e. point and areal data) as measurements for kriging. In this case, the cross-validation analysis shows a larger coefficient of determination, similar average kriging variance and smaller mean square error than the obtained in the case of merging point and areal measurements for kriging.

Optimal network design for spatial prediction of soil redistribution (137Cs) based on a minimax approach

Measuring 137 Cs is considered an effective method to study soil redistribution rate and hence needs sampling at a number of sites. The spatial configuration of the network of sites to be sampled has a substantial effect on the soil redistribution assessment. Here, motivated by sampling 137 Cs, we adopted a model-based approach. For this, we chose the average kriging variance (AKV) as a design criterion. In fact, by minimizing the AKV of soil 137 Cs prediction in the paired sub-catchments of Iran's Golestan province, we determined the optimal sampling design in the case that no directly measured prior information of the primary variable of interest ( 137 Cs ) is available. However, the AKV depends on some unknown parameters and preliminary estimates of model parameters are not available. To overcome this problem, we apply the minimax approach which minimizes the maximum value of design criterion over the misspecification of parameters . The method is illustrated taking into account the ancillary information (slope%) from representative Sub-catchments ( Sample and Testifier, each around 190 ha in size ). A simulated annealing algorithm is used to search for an optimal design from among all possible designs. Since, the number of sampling points is often limited by time and budgetary constraints, we use a sequential-based method for selecting the sample size. It is shown that 60 sites are sufficient for the proposed Sample and Testifier sub-catchments.

Optimally Locating Additional Drill Holes in Three Dimensions Using Grade and Simulated Annealing

Abstract Average kriging variance is a standard tool used in optimization of the location of additional drill holes. However, this tool cannot distinguish between areas with different priorities. This limitation could be eliminated by using weighted average kriging variance. This paper extends the problem of optimal location to three dimensional cases, use grade as a weight and search optimum locations by simulated annealing. Weighted average kriging variance is used as objective function. The method is applied to a copper deposit. Results have shown that weighting of the estimation variance with “grade” is effective only when the difference among the grades estimated for different blocks is considerable.

Spatial sampling design based on convex design ideas and using external drift variables for a rainfall monitoring network in Pakistan

Spatial sampling design is concerned with the optimal allocation of samples to spatial coordinates in order to improve in a well-defined sense the estimation and prediction of spatial random fields. Unfortunately, objective functions in spatial sampling design seem to be so complicated so far that most often stochastic search algorithms are used to get these design criteria optimized. Our intention is to show that the minimization of the average kriging variance design criterion shows a mathematically tractable structure when considering the random field as a linear regression model with infinitely many random coefficients. Either the Karhunen–Loeve expansion or the polar spectral representation of the random field may be used to get such a favourable representation. Well-known convex experimental design theory may be applied then to this high dimensional cosine-sine-Bessel surface harmonics random coefficients regression model to calculate spatial sampling designs. We study a monitoring network for rainfall during the monsoon in Pakistan and consider both the optimal deletion and subsequent addition of monitoring stations from/to this network. Only deterministic optimization algorithms and no stochastic search algorithms are used for the task of network optimization. As external drift variables determining the rainfall trend wind, humidity and elevation are considered.

Sampling design optimization for multivariate soil mapping

Much attention has been paid to sampling design optimization over the past decades. Many methods have been developed and applied, but only a few of these deal with simultaneous optimization of the sampling design for multiple soil variables. In this paper we present a method implemented as R-code that minimizes the average kriging variance (AKV) for multiple soil variables simultaneously. The method is illustrated with real soil data from an experimental field in central Czech Republic. The goal of the method is to minimize the sample size while keeping the AKV values of all tested soil variables below given thresholds. We defined and tested two different objective functions, critical AKV optimization and weighted sum of AKV optimization, both based on the AKV minimization with annealing algorithm. The crucial moment for such an optimization is defining the mutual spatial relationship between all soil variables with the Linear Model of Coregionalization and proper modelling of all (cross)variograms which are used in the optimization process. In addition, a separate optimization was made for each of the tested soil characteristics to evaluate a possible gain of the simultaneous approach. The results showed that the final design for multivariate sampling is “fully-optimal” for one soil variable — optimal number of observations and optimal structure of sampling pattern, and “sub-optimal” for the others, while no clear difference between the two optimization criteria was found. We can recommend using the method in situations where periodical soil surveys are planned and where multivariate soil characteristics are determined from the same soil samples at once.

Comparison of Variance‐Reduction and Space‐Filling Approaches for the Design of Environmental Monitoring Networks

Abstract: This article focuses on the design of groundwater monitoring networks to detect contamination with nitrates from agricultural origin. This is a problem that has been in the minds of the general public, scientists, governmental agencies, and legislators for some time now. If one looks at European statistics, despite still incomplete data, in 13% of the regions the 50 mg/l European water quality standard for drinking water (Drinking Water Directive, 98/83/EC) is exceeded in more than 25% of the monitoring stations. A compliance groundwater nitrate‐monitoring network is developed for a case study in the south of Portugal (Gabbro of Beja aquifer system), using both variance‐reduction and space‐filling approaches. In the first the variance of the error of estimation obtained by ordinary kriging is used, after building a covariance model, and the objective is to minimize the average kriging variance. In the second approach a criterion for the quality of spatial coverage is used, usually based on a metric. The objective here is also usually to minimize a function of the metric (criterion). The search for a solution is made in an iterative manner, by replacing one station, analyzing the result, and deciding whether to keep the solution or not. Due to the enormous number of possible combinations even for small networks, a structured search method, the simulated annealing method, is used to obtain the final network.

The influence of variogram parameters on optimal sampling schemes for mapping by kriging

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).

Disjunctive Kriging: 2. Examples

The disjunctive kriging (DK) method described in detail in the previous paper of this series is illustrated by two examples. The first example is for the electrical conductivity (EC) which is approximately lognormally distributed. In the second example the natural logarithm of the EC is used to produce an approximately normally distributed data set. The data set was chosen since the Hermite coefficients for a normal and lognormal distribution could be determined analytically as well as to facilitate a comparison between ordinary and disjunctive kriging. The capability of DK to be a nonlinear estimator and an estimator of the conditional probability is explored. The results indicate that DK is a better estimator than linear estimators (i.e., ordinary kriging) in the sense of reduced variance of errors and average kriging variance even when the data are approximately normally or lognormally distributed. Also, using DK one can obtain an estimate of the conditional probability.