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Sustainable and economical farming needs precise adaptation to the varying soil- and plant properties within fields. Consequently, farming operations have to be adjusted to this in a site-specific way.

Agricultural productivity may be constrained by many factors such as water scarcity, soil degradation, and use of marginal quality water. In Morocco, the main degradation processes occurring for irrigated areas are on-site impact (soil salinization and/or alkalinization) and off-site impacts (pollution of groundwater by salts and nitrates). Since 1995, the Moroccan Public Irrigation Agency has installed and maintained a network of soil and groundwater monitoring stations. For the present study, we selected the soil and water sampling sites on spatial representativeness in the perimeter, the main soil types, and the hydrogeological variants. Soil salinity, alkalinity, and sodicity as well as groundwater salinity, nitrates, and water table level were recorded to determine spatiotemporal variability and dynamics of groundwater salinity. A good understanding of its evolution in space and time will make possible to obtain reliable models for spatiotemporal prediction, estimation of the missing data, cartography, and over the long term for the delineation of risky zones. The spatiotemporal analysis of the groundwater salinity shows the presence of a strong spatial dependence and a weak temporal dependence. The spatiotemporal dependence of the residuals is very weak and primarily consists in random fluctuations. Consequently, a simple model was adopted, containing two components: a spatial component explaining more than 50% of the total variability of groundwater salinity and a temporal component that explains almost 77% of the remaining variability. Overall, this model explains more than 90% of total observed variability. Cartography of the average groundwater salinity was also established by kriging, by computing mean spatial variograms on the basis of per site data. The spatial variogram of the northern area was adjusted by the Gaussian model characterized by a sill of 3 dS2/m2 and a range of 12,526 m, while the southern area was adjusted by a Gaussian model with a sill of 0.2 dS2/m2 and a range of 9,674 m, with a nugget effect of 0.06 dS2/m2.

The knowledge about the magnitude, the spatial extent, the distribution and the evolution of salinity over a period of time is essential for the better management of salt-affected soils. Soil salinity is determined, conventionally, by measuring the electrical conductivity of a saturated past extract (ECe). However, given the spatio-temporal variability of salinity, numerous samples are necessary, which makes the conventional procedure laborious and expensive. As an alternative, the apparent electrical conductivity of soil (ECa) can be measured in the field by the use of the electromagnetic induction (EMI) method. This method is fast and allows making extensive ECa determination in space and monitoring. In the present study, an area of 2,060 ha has been investigated in the irrigation district of Tadla, central Morocco. Twelve soil samples were collected for ECe measurement, while 92 ECa measurements were realized with EM38. The pairs of ECe-ECa values allowed establishing the calibration equation permitting to convert the ECa into ECe values and for other ECa values for which ECe was not accomplished. The geostatistics was used to develop maps for the risk of soil salinization. Initially, a threshold for the risk of soil salinization was determined, and indicators were built. Later, the spatial variability of these indicators was described and modelled using the variogram. Finally, the maps were generated based on a non-parametric method of geostatistical interpolation, that is, indicator kriging. The results showed that the study area presents various degrees of soil salinization risks. The south-eastern part and small areas in central west and east of the study area have a low risk of salinization. In contrast, the south-western, the north-western and the central parts have a high risk of salinization. All the remaining parts of the study area have a moderate risk of salinization. It is concluded that the combined use of ECe and ECa-EM38 values and geostatistics allowed establishing a reliable soil salinization risk map and help to develop rehabilitation plan for the salt-affected soils.

Orebody modeling is critical for the evaluation and engineering of mineral deposits. The building of the 3D geometry is conventionally based on vertical and horizontal sections interpreted by a mine geologist. In more advanced cases, geostatistical methods are used such as indicator kriging and/or simulations, truncated gaussian or plurigaussian simulations, which allows to automate the modeling process. These methods are probabilistic and use the variogram to represent the geological heterogeneity. Multiple-point geostatistics (MPG) is an alternative to traditional variogram-based geostatistical modeling, whereas a fully explicit representation of the geological patterns (a training image) is used in place of variograms. Although it is now routinely used in modeling of oil and gas reservoirs, there are few studies showing application of this technique in mineral deposits. The advantages of the MPG approach are to provide a more realistic representation of the geology through a more accessible parameterization (the visual training image instead of the analytic variogram). This paper presents initial results of MPG with the SNESIM algorithm applied to multiple lithological domains at a Brazilian iron ore deposit. Additionally, the steps involved in dataset preparation for adequate use of the algorithm are discussed.

The problem of edge effects in sequential simulation is widely acknowledged but usually overlooked in geostatistical simulations. They can, however, have significant effects on simulations, especially for situations in which the variogram range is relatively large compared to the size of the simulation area (volume). Failure to account properly for the issue can bias simulations by generating values with ranges of correlation and statistical characteristics that differ from those specified. In this paper, we investigate edge effects in detail using sequential Gaussian simulations and derive the critical threshold at which an edge effect becomes significant. Edge correction techniques, such as guard areas, are discussed as a means of mitigating the effects.

Reproduction of complex 3D patterns is not possible using algorithms that are constrained to two-point (covariance or variogram) statistics. A unique pattern-growth algorithm (GrowthSim) is presented in this paper that performs multiple point spatial simulation of patterns conditioned to multiple point data. Starting from conditioning data locations, patterns are grown constrained to the pattern statistics inferred from a training image. This is in contrast to traditional multiple-point statistics based-algorithms where the simulation progresses one node at a time. In order to render this pattern growth algorithm computationally efficient, two strategies are employed—(i) computation of an optimal spatial template for pattern retrieval, and (ii) pattern classification using filters. To accurately represent the spatial continuity of large-scale features, a multi-level simulation scheme is implemented. In addition, a scheme for applying affine transformation to spatial patterns is presented to account for local variation in spatial patterns in a target reservoir. The GrowthSim algorithm is demonstrated for developing the reservoir model for a deepwater turbidite system. Lobes and channels that exhibit spatial variations in orientation, density and meandering characteristics characterize the reservoir. The capability of GrowthSim to represent such non-stationary features is demonstrated.

Simulation results depend on many parameters such as: modeling assumptions (e.g. Gaussian or Indicator simulation); implementation (e.g. point or block simulation); and case-specific parameters (e.g. top-cut values). The user is often in the dark when it comes to the impact on the results of modeling assumptions and software implementation. Not much literature is available and checks are difficult to complete. This paper is a comparative study of two point and block Gaussian related simulations. The study shows that careful calibration/validation is necessary in both cases to avoid significant biases. Point simulation is easy to validate against primary data because the support does not change; block simulation is more difficult to validate. Both software/algorithms were able to provide what they have been designed for, i.e. conditional simulated values that reproduce the required grade point or block distribution and variogram. Differences, however, were noted between the re-blocked point simulation and direct block simulation results. Differences in application and results of the methods together with advantages and disadvantages are discussed.

This chapter reviews methods for selecting sampling locations in contaminated soils for three situations. In the first situation a global estimate of the soil contamination in an area is required. The result of the surey is a number or a series of numbers per contaminant, e.g. the estimated mean concentration, median, 90th percentile, or the cumulative frequency distribution for the area as a whole. In the second case we want more spatial detail, and interest is in the mean or median concentration for several delineated blocks. Finally, in the third case the aim is to construct a high resolution map of the concentrations, for instance by geostatistical interpolation. For the first aim, design-based sampling methods, in which locations are selected by probability sampling, are most appropriate. Several basic sampling designs are described. Laboratory costs can be saved by bulking soil samples. The precision of estimates can be increased by exploiting ancillary information on variables correlated with the contaminants. For mapping purposes, model-based sampling methods, in which locations typically are selected by purposive sampling, are the best option. Examples are sampling on a centred grid, spatial coverage sampling, and geostatistical sampling. A simple method, based on the k-means clustering algorithm, is described for computing spatial coverage samples. For geostatistical interpolation a variogram is required. Variogram estimation is enhanced by adding several tens of locations within short distance of the locations of a grid or spatial coverage sample. A separate section describes sampling methods for detecting and for delineating hot spots.

Many environmental variables that are relevant to precision agriculture, such as crop and soil properties and climate, vary both in time and space. Farmers can often benefit greatly from accurate information about the status of these variables at any particular point in time and space to aid their management decisions on irrigation, fertilizer and pesticide applications, and so on. Practically, however, it is not feasible to measure a variable exhaustively in space and time. Space–time geostatistics can be useful to fill in the gaps. This chapter explains the basic elements of space–time geostatistics and uses a case study on space–time interpolation of the normalized difference vegetation index (NDVI) as an indicator of biomass in a Dutch potato field. Space–time geostatistics proves to be a useful extension to spatial geostatistics for precision agriculture, although theoretical as well as practical advances are required to mature this subject area and make it ready to be used for within-season, within-field decision making by farmers.

Preface 1 An Overview of Geostatistics and Precision Agriculture M. A. Oliver Abstract 1.1 Introduction 1.2 The Theory of Geostatistics 1.3 Case study: Football Field 2 Sampling in Precision Agriculture: Part I R. Kerry, M. A. Oliver and Z. L. Frogbrook Abstract 2.1 Introduction 2.2 Variograms to guide sampling 2.3 Use of the variogram to guide sampling for bulking 2.4 The variogram to guide grid-based sampling 2.5 Variograms to improve predictions from sparse sampling 2.6 Conclusions References 3 Sampling in Precision Agriculture, Optimal Designs from Uncertain Models B. P. Marchant and R. M. Lark Abstract 3.1 Introduction 3.2 The linear mixed model: estimation, predictions and uncertainty 3.3 Optimizing sampling schemes by spatial simulated annealing 3.4 Conclusions References 4 The Spatial Analysis of Yield Data T. W. Griffin Abstract 4.1 Introduction 4.2 Background of site-specific yield monitors 4.3 Managing Yield Monitor Data 4.4 Spatial statistical analysis of yield monitor data 4.5 Case study: Spatial analysis of yield monitor data from a field-scale experiment 4.6 Conclusion References 5 Space-time Geostatistics for Precision Agriculture: A Case Study of NDVI Mapping for a Dutch Potato Field G. B. M. Heuvelink and F. M. van Egmond Abstract 5.1 Introduction 5.2 Description of the Lauwersmeer study site and positional correction of NDVI data 5.3 Exploratory data analysis of Lauwersmeer data 5.4 Space-time geostatistics 5.5 Application of space-time geostatistics to the Lauwersmeer farm data 5.6 Discussion and Conclusions References 6 Delineating Site-specific Management Units with Proximal Sensors D. L. Corwin and S. M. Lesch Abstract 6.1 Introduction 6.2 Directed Sampling with a Proximal Sensor 6.3 Delineation of SSMUs with a Proximal Sensor 6.4 Case Study Using Apparent Soil Electrical Conductivity (ECa) - San Joaquin Valley, CA 6.5 Conclusion References 7 Using Ancillary Data to Improve Prediction of Soil and Crop Attributes in Precision Agriculture P. Goovaerts and R. Kerry Abstract 7.1 Introduction 7.2 Theory 7.3 Case study 1: the Yattendon site 7.4 Case Study 2: the Wallingford site 7.5 Conclusions References 8 Spatial Variation and Site-specific Management Zones R. Khosla, D. G. Westfall, R. M. Reich, J.S. Mahal and W. J. Gangloff Abstract 8.1 Introduction 8.2 Quantifying spatial variation in soil and crop properties 8.3 Site-specific management zones 8.4 Statistical evaluation of management zone delineation techniques: A case study 8.5 Conclusions References 9 Weeds, Worms and Geostatistics R. Webster Abstract 9.1 Introduction 9.2 Weeds 9.3 Nematodes 9.4 The future of geostatistics in precise pest control References 10 The Analysis of Spatial Experiments M.J. Pringle, T.F.A. Bishop, R.M. Lark, B.M. Whelan and A.B. McBratney Abstract 10.1 Introduction 10.2 Background 10.3 Management-class experiments 10.4 Local-response experiments 10.5 Alternative approaches to experimentation 10.6 Issues for the future 10.7 Conclusions References 11 Application of Geostatistical Simulation in Precision Agriculture R. Gebbers and S. de Bruin Abstract 11.1 Introduction 11.2 Case study I: uncertainty of a pH map 11.3 Case study II: uncertainty in the position of geographic objects 11.4 Case study III: uncertainty propagation in soil mapping 11.5 Application of geostatistical simulation in precision agriculture: summary References 12 Geostatistics and Precision Agriculture: A Way Forward J. K. Schueller Abstract 12.1 Introduction 12.2 Weather, time, and space 12.3 Farmers, advisors and researchers 12.4 Issues, ideas and questions 12.5 Past, present, and future References Appendix: Software Index

It is common practice in geostatistics to use the variogram to describe the spatial dependence structure of the underlying random field. However, the variogram is sensitive to outlying observations and strongly influenced by the marginal distribution of the random field. As an alternative to spatial modeling using the variogram we consider describing the spatial correlation by means of copula functions. We present three methods for performing spatial interpolation using copulas. By exploiting the relationship between bivariate copulas and indicator covariances, the first method performs indicator kriging and disjunctive kriging. As a second method we propose a simple kriging of the rank-transformed data. The third method is a plug-in Bayes predictor, where the predictive distribution is calculated using the conditional copula given the observed data and the model parameters. We show that the latter approach generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Joker data set from the spatial interpolation comparison SIC2004.

Many different interpolation procedures have been used to generate maps of precipitation amount from point data. Several case analyses have shown that making use of covariates such as elevation may increase the accuracy of predictions. Kriging-based approaches, as often employed for mapping precipitation amount, are usually based on a global variogram model and the assumption is made that spatial variation is the same at all locations. This chapter assesses the impact on prediction accuracy of using (i) local variogram models as against a global variogram model and (ii) multivariate approaches as against univariate approaches. Various kriging-based interpolation procedures are applied along with inverse distance weighting and regularised splines with tension. The results suggest that multivariate approaches such as kriging with an external drift may provide more accurate predictions than standard univariate approaches such as ordinary kriging. In addition, kriging based on local variogram models, rather than a global variogram model, is shown to provide smaller prediction errors.

The main purpose of this study was to examine the applicability of geostatistical modeling to obtain valuable information for assessing the environmental impact of sewage outfall discharges. The data set used was obtained in a monitoring campaign to S. Jacinto outfall, located off the Portuguese west coast near Aveiro region, using an AUV. The Matheron’s classical estimator was used the compute the experimental semivariogram, which was fitted to three theoretical models: spherical, exponential and Gaussian. The cross-validation procedure suggested the best semivariogram model and ordinary kriging was used to obtain the predictions of salinity at unknown locations. The generated map shows clearly the plume dispersion in the studied area, indicating that the effluent does not reach the nearby beaches. Our study suggests that an optimal design for the AUV sampling trajectory from a geostatistical prediction point of view, can help to compute more precise predictions and hence to quantify more accurately dilution. Moreover, since accurate measurements of plume’s dilution are rare, these studies might be very helpful in the future for validation of dispersion models.

This chapter sets the scene for the two main topics of this book, namely geostatistics and precision agriculture. The aim is to provide readers with a foundation for what is to come in the other chapters and an understanding of why the subjects make suitable companions. The history and basic theory of geostatistics are covered, together with the history of precision agriculture and of geostatistics in precision agriculture. The two core techniques of geostatistics, variography and kriging, are described, together with examples of how they can be applied. Methods of estimating the variogram and fitting an authorized model to the experimental values are explained and illustrated. There are many types of kriging; ordinary and disjunctive kriging are described briefly in this chapter, and others types are portrayed in subsequent chapters. The application of the variogram and kriging are illustrated with a case study of an arable field in England.

Very large spatially-referenced datasets, for example, those derived from satellite-based sensors which sample across the globe or large monitoring networks of individual sensors, are becoming increasingly common and more widely available for use in environmental decision making. In large or dense sensor networks, huge quantities of data can be collected over small time periods. In many applications the generation of maps, or predictions at specific locations, from the data in (near) real-time is crucial. Geostatistical operations such as interpolation are vital in this map-generation process and in emergency situations, the resulting predictions need to be available almost instantly, so that decision makers can make informed decisions and define risk and evacuation zones. It is also helpful when analysing data in less time critical applications, for example when interacting directly with the data for exploratory analysis, that the algorithms are responsive within a reasonable time frame. Performing geostatistical analysis on such large spatial datasets can present a number of problems, particularly in the case where maximum likelihood. Although the storage requirements only scale linearly with the number of observations in the dataset, the computational complexity in terms of memory and speed, scale quadratically and cubically respectively. Most modern commodity hardware has at least 2 processor cores if not more. Other mechanisms for allowing parallel computation such as Grid based systems are also becoming increasingly commonly available. However, currently there seems to be little interest in exploiting this extra processing power within the context of geostatistics. In this paper we review the existing parallel approaches for geostatistics. By recognising that diffeerent natural parallelisms exist and can be exploited depending on whether the dataset is sparsely or densely sampled with respect to the range of variation, we introduce two contrasting novel implementations of parallel algorithms based on approximating the data likelihood extending the methods of Vecchia [1988] and Tresp [2000]. Using parallel maximum likelihood variogram estimation and parallel prediction algorithms we show that computational time can be significantly reduced. We demonstrate this with both sparsely sampled data and densely sampled data on a variety of architectures ranging from the common dual core processor, found in many modern desktop computers, to large multi-node super computers. To highlight the strengths and weaknesses of the diffeerent methods we employ synthetic data sets and go on to show how the methods allow maximum likelihood based inference on the exhaustive Walker Lake data set.

We propose a hierarchical model coupled to geostatistics to deal with a non-gaussian data distribution and take explicitly into account complex spatial structures (i.e. trends, patchiness and random fluctuations). A common characteristic of animal count data is a distribution that is both zero-inflated and heavy tailed. In such cases, empirical variograms are no more robust and most structural analyses result in poor and noisy estimated spatial variogram structures. Thus kriged maps feature a broad variance of prediction. Moreover, due to the heterogeneity of wildlife population habitats, a nonstationary model is often required. To avoid these difficulties, we propose a hierarchical model that assumes that the count data follow a Poisson distribution given a theoretical sighting density which is a latent variable to be estimate. This density is modelled as the product of a positive long range trend by a positive stationary random field, characterized by a unit mean and a variogram function. A first estimate of the drift is used to obtain an estimate of the variogram of residuals including a correction term for variance coming from the Poisson distribution and weights due to the non-constant spatial mean. Then a kriging procedure similar to a modified universal kriging is implemented to directly map the latent density from raw count data. An application on fin whale data illustrates the effectiveness of the method in mapping animal density in a context that is presumably non-stationary. E. Bellier and P. Monestiez Biostatistique et Processus Spatiaux, INRA, Domaine Saint-Paul, Site Agroparc, 84914 Avignon cedex 9, France E. Bellier ( ) Norwegian Institute for Nature Research NINA, NO-7485 Trondheim, NORWAY e-mail: edwige.bellier@nina.no C. Guinet Centre d’Etudes Biologiques de Chize, CNRS, 79360 Villiers-en-Bois, France P.M. Atkinson and C.D. Lloyd (eds.), geoENV VII – Geostatistics for Environmental Applications, Quantitative Geology and Geostatistics 16, DOI 10.1007/978-90-481-2322-3 1, c Springer Science+Business Media B.V. 2010 1

Automatically generating maps of a measured variable of interest can be problematic. In this work we focus on the monitoring network context where observations are collected and reported by a network of sensors, and are then transformed into interpolated maps for use in decision making. Using traditional geostatistical methods, estimating the covariance structure of data collected in an emergency situation can be difficult. Variogram determination, whether by method-of-moment estimators or by maximum likelihood, is very sensitive to extreme values. Even when a monitoring network is in a routine mode of operation, sensors can sporadically malfunction and report extreme values. If this extreme data destabilises the model, causing the covariance structure of the observed data to be incorrectly estimated, the generated maps will be of little value, and the uncertainty estimates in particular will be misleading. Marchant and Lark [2007] propose a REML estimator for the covariance, which is shown to work on small data sets with a manual selection of the damping parameter in the robust likelihood. We show how this can be extended to allow treatment of large data sets together with an automated approach to all parameter estimation. The projected process kriging framework of Ingram et al. [2007] is extended to allow the use of robust likelihood functions, including the two component Gaussian and the Huber function. We show how our algorithm is further refined to reduce the computational complexity while at the same time minimising any loss of information. To show the benefits of this method, we use data collected from radiation monitoring networks across Europe. We compare our results to those obtained from traditional kriging methodologies and include comparisons with Box-Cox transformations of the data. We discuss the issue of whether to treat or ignore extreme values, making the distinction between the robust methods which ignore outliers and transformation methods which treat them as part of the (transformed) process. Using a case study, based on an extreme radiological events over a large area, we show how radiation data collected from monitoring networks can be analysed automatically and then used to generate reliable maps to inform decision making. We show the limitations of the methods and discuss potential extensions to remedy these.

This chapter describes three geostatistical methods to incorporate secondary information into the mapping of soil and crop attributes to improve the accuracy of their predictions. The application of the methods is illustrated in two case studies. Cokriging is the multivariate extension of the well known ordinary kriging. It does not require ancillary data to be available at all nodes of the interpolation grid, whereas kriging with external drift and simple kriging with local means do. Cokriging, however, is more demanding in terms of variogram inference and modelling. The other two methods use ancillary data to model the spatial trend of the primary variable. Kriging with an external drift can account for local changes in the linear correlation between primary and secondary variables. Simple kriging with local means, which applies kriging to regression residuals and adds the kriged residual to the regression estimate, is the most straightforward of these methods to implement. The prediction performance of each technique was evaluated by cross-validation. As the results are site-specific, the choice of technique for a given site should be guided by the results of cross-validation.

The spatial structures revealed in remotely sensed imagery are essential information characterizing the nature and the scale of spatial variation of sea ice processes. This study evaluates the potential capability of using semi-variogram of intrinsic regionalization model for change detection of sea ice. Up to now, the second-order variogram has been widely used to describe the spatial variations within an image, but it demonstrates the limitation to discriminate distinct image spatial structures. This study introduces a different geo-statistic metric, in which spatial structures of sea ice are considered a combination of two stochastic second-order stationary models. Firstly, the multi-gamma model is used to characterize continuous variations corresponding to water or the background of sea ice. The second model is a tessellation model, in which the image domain is randomly separated into non-overlapping cells. In each cell, a random value is independently assigned. It is called the mosaic model. In this paper, the mosaic model is constructed by a Poisson tessellation. The linear combination of these two stochastic models defines the mixture model to represent spatial structures of sea ice presented in SAR intensity imagery. This algorithm is applied to Radarsat-1 images acquired different days to identify the change of sea ice.

International audience; We propose a hierarchical model coupled to geostatistics to deal with a non-gaussian data distribution and take explicitly into account complex spatial structures (i.e. trends, patchiness and random fluctuations). A common characteristic of animal count data is a distribution that is both zero-inflated and heavy tailed. In such cases, empirical variograms are no more robust and most structural analyses result in poor and noisy estimated spatial variogram structures. Thus kriged maps feature a broad variance of prediction. Moreover, due to the heterogeneity of wildlife population habitats, a nonstationary model is often required. To avoid these difficulties, we propose a hierarchical model that assumes that the count data follow a Poisson distribution given a theoretical sighting density which is a latent variable to be estimate. This density is modelled as the product of a positive long range trend by a positive stationary random field, characterized by a unit mean and a variogram function. A first estimate of the drift is used to obtain an estimate of the variogram of residuals including a correction term for variance coming from the Poisson distribution and weights due to the non-constant spatial mean. Then a kriging procedure similar to a modified universal kriging is implemented to directly map the latent density from raw count data. An application on fin whale data illustrates the effectiveness of the method in mapping animal density in a context that is presumably non-stationary

Three methods were introduced for estimating the local semivariogram for use in procedures such as super-resolution pattern prediction. The first is simply to use a training image to estimate the global semivariogram required. The second method employs a deconvolution–convolution procedure to estimate the local semivariogram. The estimated semivariogram represents proportions and so a further step is required to convert the proportions semivariogram to represent a binary field. The third method is an integration of the first two methods obtained by weighted linear combination across the lags of the semivariograms. The results are evaluated using the known target local semivariogram. The integrated method provides some advantages. The discussion points to problems and potential future improvements on the method.

This study examines the effect of considering data from rain gauges nearby the boundaries of Galicia (NW Spain) in order to minimize the border effect. Two datasets were considered: the first one comprised 232 climatic stations within Galicia and the second one consisted of 322 rain gauges including the former 232 from Galicia and adding 90 stations from boundary provinces (42 from Asturias, 31 from Leon and 17 from Zamora). Total monthly rainfall data from 2006 was analyzed and descriptive statistics demonstrated slight differences between both datasets. Theoretical structures were described for all the studied monthly datasets. Spatial dependence analysis showed that the best-fitting semivariogram model structure was the same for both datasets in most of the cases, even though the model parameters showed great differences. Similarly, cross-validation parameter values were clearly distinct among datasets; mostly, the ones corresponding to the 322 stations dataset were closer to the ideal values. Ordinary kriging was performed for both datasets and resulting variance maps showed improvements when the information from boundary regions was taken into account. These improvements can reach up to 25% of the maximum variance value and they were observed in wet months such as January whereas, in dry months such as July, no improvement was observed. Minimum error values were usually lower when extra information was used in the interpolations. In conclusion, a better mapping of the rainfall within a region can be achieved using data from boundary areas, reducing the variance of the estimates.

We compare different approaches to fine scale simulation of aquifer heterogeneity of meandering river depositional elements, based on the study of a 3-D quarry exposure of historical point bar-channel sediments of the Lambro River (Po plain, Northern Italy). The starting point is a sedimentological and hydrostratigraphic hierarchic model obtained after mapping of five quarry faces with centimeter-scale detail. The vertical facies maps show the shape and size of two superimposed composite bars, of their component unit bars and channel fills and the distribution of the individual facies within them. Textural and poro-perm analyses allowed the definition of the properties of four basic hydrofacies (Open Framework Gravels, Gravelly Sands and Sandy Gravels, Clean Sands, Sandy Silts and Clays), with permeability contrasts by at least one order of magnitude \((1{0}^{-9} < \mathrm{K} < 1{0}^{-1})\). The correlation of hydrofacies has been quantified after discretization of the maps with square cells (side 0.05 m), by both transition-probability geostatistics and variographic analysis, to support 3-D pixel-oriented simulation of the volume. We found a high level of correspondence between the semivariogram ranges and the experimental transition probabilities computed on the entire dataset. Several realizations of 3-D conditioned simulations, that honour the vertical facies maps, were computed using Sequential Indicator Simulation (SIS) and T-Progs (transition-probability geostatistics software). Both methods yield more realistic results if the highest rank depositional elements are simulated separately than if the sedimentary volume is simulated on the whole. Image analyses on random sections through selected realizations shows that, in this specific case, SIS yields the most realistic simulations. However, both techniques are not capable of accounting for trends of depositional features that determine a non-stationary behaviour at the facies scale.

This chapter considers the importance of spatial scale in sampling and investigates various methods by which the variogram can be used to determine an appropriate sampling scheme or interval for grid sampling. When no prior information is available on the scale of variation, and the variable of interest is unlikely to be strongly correlated to available ancillary data, a nested survey and analysis provides a first approximation to the variogram and the approximate spatial scale. If the variable of interest appears related to ancillary data such as aerial photographs or elevation, variograms of these data can provide an indication of the likely scale of variation in the soil or crop. Existing variograms of soil or crop properties can be used to determine how many cores of soil or samples from plants should be taken to form a composite (bulked) sample to reduce the local noise. Such variograms can also be used with the kriging equations to determine a grid sampling interval with a specific tolerable error, or an interval of less than half the variogram range can be used to ensure a spatially dependent sample. Finally, if the scale of variation is large in relation to the field size, a variogram estimated by residual maximum likelihood (REML) or standardized variograms from ancillary data can be used to krige data from a small, but spatially dependent sample. Each of the methods investigated is illustrated with a case study.