Standard Article

# Kriging

Spatial/Temporal Modeling and Analysis

Published Online: 15 SEP 2006

DOI: 10.1002/9780470057339.vak003

Copyright © 2002 John Wiley & Sons, Ltd

Book Title

## Encyclopedia of Environmetrics

Additional Information

#### How to Cite

Switzer, P. 2006. Kriging. Encyclopedia of Environmetrics. 2.

#### Publication History

- Published Online: 15 SEP 2006

This is not the most recent version of the article. View current version (15 JAN 2013)

Suppose a spatial field is partially observed at selected sites and the goal is to infer the field at unobserved sites. This is the problem of spatial estimation, sometimes called spatial prediction. Examples of spatial random fields are soil nutrient concentrations over an agricultural domain observed on a survey grid (*see* Soil Surveys), hydrologic variables over an aquifer observed at well locations, and air quality measurements over an air basin observed at monitoring sites. Kriging is a method for converting the data into an estimate of the field together with a measure of error or uncertainty.

In its simplest form, a kriging estimate of the field at an unobserved location is an optimized linear combination of the data at the observed locations. The coefficients of the kriging estimate and the associated error measure both depend on the spatial configuration of the data, the unobserved location relative to the data locations, and spatial correlation or the degree to which one location can be predicted from a second location as a function of spatial separation.

Kriging is named for D. Krige who published an early account [7] with applications to estimation of a mineral ore body. The method has close links to Wiener optimal linear filtering in the theory of random functions, Gandin objective analysis in meteorology [4], spatial splines, (*see* Splines in Nonparametric Regression) and generalized least squares estimation (*see* Least Squares, General) in a spatial context. Kriging methods have been studied and applied extensively since 1970 and have been adapted, extended, and generalized. For example, kriging has been generalized to classes of nonlinear functions of the observations, modified to increase robustness (*see* Bayesian Methods and Modeling), extended to take advantage of covariate information, adapted for fields whose statistical properties are spatially evolving, and placed into a formal Bayesian framework.

Matheron [8] is the most influential early systematic exposition of kriging. Book length accounts published in the 1990s include [1]–[3], [5], [6], [9]–[11] of which [2] is the most comprehensive. Related entries in this encyclopedia are Multivariate kriging; Space–time covariance models; Space–time Kalman filter; Spatial design, optimal; Variogram.

### Error Models

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

Kriging is a procedure for spatial prediction at an unobserved location, using data at observed locations, optimized with reference to a specific error criterion. The criterion is the squared prediction error at the unobserved location – averaged over a conceptual class of spatial prediction problems that have the same configuration of observed and unobserved locations. The specification of this averaging class is the model under which the optimization is carried out and the estimation error is reported.

The usual model under which kriging calculations are done is that of a spatial stochastic process that generates spatial fields over the geographical region of interest. A stochastic process model is selected with characteristics that reflect characteristics of the available data. With this averaging model, the stated kriging properties are purely conceptual – they refer to *average* prediction errors that would be seen if the same kriging procedure were applied to the same prediction problem on spatial fields generated repeatedly by the selected stochastic process. Locations of the observed sites within the geographical domain are fixed under this averaging model, but not the values of the observations themselves. The fact that the kriging averaging model does not fix the values of the observations can be seen as a limitation.

An alternative to this stochastic process averaging model treats the whole spatial field as fixed and considers the configuration of observed and unobserved sites as one configuration from a specified class of possible configurations. The error associated with spatial estimation is then the average error associated with the entire class of specified configurations. For example, the class of configurations might consist of all possible shifts of a fixed configuration. In some important cases there is an approximate operational equivalence between an averaging model based on a fixed spatial field with movable site configurations and an averaging model with a random spatial field and a fixed-site configuration, particularly when the number of observation locations is large.

When randomness is deliberately introduced into observation sites, such as stratified random sampling for example (*see* Sampling, Environmental), it is common to use the fixed-field averaging model with movable site configurations. If the goal is to estimate an area average value, then the assigned estimation errors will be operationally similar for the fixed field and stochastic field averaging models, even though the stochastic field model will treat the results of the randomized observation sites as fixed locations. However, for estimating (predicting) field values at specified sites, as in interpolation and mapping, an averaging model that uses only randomization of the observation sites would not be meaningful for the computation of estimation error.

### Notation

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

*R* denotes the spatial region of interest. Commonly *R* is a two-dimensional region of the earth's surface in geostatistical applications. But *R* may be the topsoil layer of a geographical area considered as a three-dimensional region. *R* may be a stratigraphic column or a transect along the ocean bottom, considered as a one-dimensional region. *R* may be a time period during which measurements are made, also considered as a one-dimensional region. The formalisms of kriging are the same regardless of the dimension of *R*.

*Z*(*x*) denotes the value of the field at location *x*, where *x* is any site in the domain *R*. For example, *Z*(*x*) may be the concentration of a pollutant at a surface location *x* in a specified air basin *R*. We may generalize *Z*(*x*) to be a multivariate field, i.e. *Z*(*x*) represents a suite of variables at each location *x*, such as a suite of multiple-pollutant concentrations and weather variables at each site *x* in the air basin *R* (*see* Multivariate Kriging).

Let *x*_{i}, *i* = 1, …, *n*, denote *n* locations in the region *R* where the field *Z*(*x*) has been observed. *x*_{0} denotes a specified but arbitrary unobserved location in *R*.

*Z*_{n} is the data vector, i.e. the *n*-vector of observations *Z*(*x*_{1}), …, *Z*(*x*_{n}). *Z*_{0} = *Z*(*x*_{0}) is the value of the field at the unobserved location *x*_{0}.

Observations are not always in the form of discrete point observations. Measuring devices may produce locally averaged values, typical of remote sensing data. Or, data may be measured continuously along one-dimensional trajectories such as data from moving tracking devices. Kriging methods can be generalized to accommodate such data.

For spatial stochastic process models the following further notation is used:

**μ**(*x*) is the model expected value of the random variable *Z*(*x*) at a location *x* in the domain *R*. **μ**_{n} is the *n*-vector, **μ**(*x*_{i}), *i* = 1, …, *n*, of expected values at the *n* data locations. **μ**_{0} is the expected value, **μ**(*x*_{0}), at the unobserved location *x*_{0}.

*c*(*x*, *x*′) is the model covariance between the pair of random variables *Z*(*x*) and *Z*(*x*′) at two locations *x* and *x*′ in the domain *R*. *c*_{n} is the *n* × *n* matrix, {*c*(*x*_{i}, *x*_{j})}, of variances and covariances between pairs of random variables at data locations. *c*_{0} is the *n*-vector of model covariances, {*c*(*x*_{i}, *x*_{0})}, between the random variable *Z*(*x*_{0}) at an unobserved location *x*_{0} and the random variables *Z*(*x*_{i}) at each of the *n* observed locations, and *c*_{0, 0} is the model variance for the random variable at the unobserved location *x*_{0}.

γ(*x*, *x*′) is the model semivariogram, defined as one-half the expected value of the squared difference between random variables *Z*(*x*), *Z*(*x*′) at two different locations *x*, *x*′ (*see* Variogram). Equivalently, 2γ(*x*, *x*′) = *c*(*x*, *x*) + *c*(*x*′, *x*′) − 2*c*(*x*, *x*′). **γ**_{0} is the *n*-vector [γ(*x*_{i}, *x*_{0})] whose components are the modeled expected squared differences between random variables at the observation locations paired with the random variable at the estimation location. γ_{n} is the *n* × *n* matrix [γ(*x*_{i}, *x*_{j})] whose components are the modeled expected squared differences between the random variables at pairs of observation locations.

### Unbiased Linear Kriging

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

A linear kriging estimator, denoted *Z*_{0}*, estimates the value of the field at an unobserved site *x*_{0} using a specific linear function of data values *Z*_{n} at observed sites. We write *Z*_{0}* = λ_{0} + *Z*′_{n} **λ**_{n, 0}, where **λ**_{n, 0} is an *n*-vector of coefficients that multiply the values at observed locations. The notation emphasizes that kriging coefficients depend on the estimation location *x*_{0}. All coefficients λ_{0} and **λ**_{n, 0} are chosen to minimize the expected squared error under the selected model subject to an unbiasedness requirement. Equivalently, unbiased kriging minimizes the variance of the estimation error. Because the estimator is linear, the error variance can be expressed in terms of the model variances and covariances, namely

- (1)

Unbiasedness, as used here, requires that the *average* value of the kriging estimator, *Z*_{0}*, be the same as the average value of what is being estimated, *Z*_{0}, where the averages are taken over all putative realizations from the selected stochastic model. Thus, a stochastic model specification serves to determine both the class of model-unbiased linear estimators and the model variance of an unbiased linear estimator. As remarked earlier, the observation locations are kept fixed under the averaging model, but not the values of the observations themselves; this point sometimes raises questions regarding the meaning and appropriateness of the ‘average’ unbiasedness condition. The average unbiasedness restriction can sometimes have awkward consequences.

### Modeling the Spatial Mean Function

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

Kriging uses a parameterized mean function μ(*x*) that is linear in the unknown parameters. Specifically, the model mean at an arbitrary location *x* in the region *R* can be expressed as μ(*x*) = *b*′ *f*(*x*), where *b* is a *p*-vector of unknown mean function parameters and *f*(*x*) is a specified known *p*-vector of covariates specific to the location *x*. For example, a covariate might be a specified function of elevation above sea level for two-dimensional regions, or it might simply be latitude to reflect a north–south linear trend. Denote by *f*_{0} = *f*(*x*_{0}) the model mean *p*-vector of covariates at the estimation location *x*_{0}, and denote by *f*_{n} the *p* × *n* matrix of covariates at the *n* observation locations.

This parameterization of the mean function implies that the unbiasedness constraint for kriging is equivalent to *b*′ *f*_{0} = λ_{0} + *b*′ *f*_{n}**λ**_{n, 0}. This relation represents *p* separate linear constraints on the kriging coefficients that must be satisfied regardless of the value of *b*, the unknown mean function parameter vector. Optimized linear estimation with this kind of parametric mean function and unbiasedness constraint has been called *universal kriging*. The kriging coefficient optimization problem is solved by minimizing the estimation error variance, a quadratic function of the coefficients, subject to *p* linear constraints on the coefficients that render the estimator unbiased. The mean function parameters are not explicitly estimated nor is this the goal; the goal remains the estimation of the spatial field *Z*(*x*) at unobserved locations.

The simplest case is a single parameter mean, *p* = 1, with the scalar covariate *f*(*x*) having the same value at all observed and unobserved locations *x* in the estimation neighborhood. This is a mean-stationary model. Without loss of generality we can take *f*(*x*) = 1 and the scalar parameter becomes the unknown, regionally constant, expected value. It follows that there are two unbiasedness constraints on the kriging coefficients: the sum of all kriging coefficients is one and λ_{0} = 0. Optimized linear estimation in the context of this stationary model with an unspecified constant model mean has been called *ordinary kriging*. The unbiasedness constraints of ordinary kriging permit the estimation error variance to be expressed in terms of the variogram, namely

- (2)

While 1 for the error variance applies to any unbiased linear estimator, 2 applies only to suitably constrained estimators. Its advantage lies in the fact that spatial data can be used more readily to fit a variogram model than a covariance model, and the class of models for variograms is larger than the class of models for covariances.

A different mean-stationary model, with *f*(*x*) = 1 at every location *x*, assigns a specific value *m* to the regionally constant mean, rather than have it be an unknown parameter as in ordinary kriging. There is now a single unbiasedness constraint on the kriging coefficients, namely λ_{0} = *m*(1 − 1_{n}′**λ**_{n, 0}), where 1_{n} is the *n*-vector of ones and 1_{n}′**λ**_{n, 0} is the sum of the *n* kriging coefficients that multiply the respective *n* observations. So λ_{0} is typically not zero and the kriging coefficients do not add up to one. This kriging scenario has been called *simple kriging*. A distinction between simple and ordinary kriging can be seen when all *n* observations have the same value *z*, say: with ordinary kriging the estimate at the unobserved location is also *z*, whereas with simple kriging the estimate at the unobserved location is a linear combination of *z* and the specified model mean *m*.

The parameterization of the model mean as a function of spatial location has two important implications. First, we have seen that it restricts the class of permissible unbiased estimators. Second, the mean function will impact the spatial variances and covariances. Roughly, a more complex mean function will imply that the spatial field has smaller residuals, hence smaller variances and covariances which are computed using residuals. The effect of a more complex mean parameterization on kriging estimation error is not always clear: smaller errors are associated with smaller variances and covariances, but larger errors are associated with a more restrictive unbiasedness constraint.

Suppose that the stationary model mean *m* is uncertain and is represented by a distribution with mean **μ**_{m} and variance *v*_{m} that are subjectively specified. Then the single unbiasedness constraint on the kriging coefficients is the same as that for simple kriging with **μ**_{m} replacing *m*. However, the model error variance for estimating the spatial field at location *x*_{0}, as given by 1, is now augmented by the term *v*_{m}(1 − 1_{n}′**λ**_{n, 0})^{2}, which will affect the optimization of the kriging coefficients and yield a larger estimation error variance compared with simple kriging. The resulting kriging optimization and associated model errors are sometimes called ‘Bayesian’ kriging, although a posterior distribution for the unknown *Z*_{0} is not involved. Simple kriging becomes the special case *v*_{m} = 0.

In comparing Bayesian kriging with ordinary kriging, the augmented error variance term vanishes for ordinary kriging. But the two unbiasedness constraints of ordinary kriging will cause the minimized value of 1 to be larger than the error associated with Bayesian kriging under the Bayesian model hypothesis. The apparent precision gains of Bayesian kriging over ordinary kriging depend on the reasonableness of the Bayes model specification. The error-averaging model implied by Bayesian kriging is over putative realizations from a spatial stochastic process, with each realization selected from a model with a *different* mean level.

### Modeling Spatial Covariances

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

As seen in 1, the error variance for unbiased linear kriging estimates of the random field at an unobserved location can be expressed in terms of covariances between pairs of random variables at different geographical locations. Typically, these covariances are taken from a parametric covariance function of two spatial arguments. This function is necessarily positive definite over the range of its arguments, which restricts the class of potential covariance models. In kriging practice, all parameters of the covariance function are explicitly specified, unlike the case of a parametric mean. For example, covariances may be specified by an explicit function of the geographical separation between the pair of locations. In this case the model is said to be *covariance stationary*, i.e. invariant to geographical shifting or translation. Furthermore, if the covariance model depends on the geographical separation only through geographical distance, then the covariance model is said to be *isotropic*, i.e. invariant to geographical rotation. An example is a covariance that is a negative exponential function of distance between location pairs.

This stationarity implies that, on average, variability is the same in all parts of the spatial field of interest. In particular, calculated kriging error variances will depend only on the relative spatial configuration of the observed and unobserved locations. Imposing covariance stationarity is like extending the error-averaging model to include all translations of this spatial configuration. Stationarity is imposed mainly as a model of convenience because it makes it possible to estimate covariances from the observed data; observation pairs with approximately the same separation are considered as statistical replicates, from which a covariance can be estimated. Nonstationary covariance models would allow for model-estimated kriging errors to be different in different parts of the region *R*, even if the data configuration remained the same.

Kriging optimization produces kriging coefficients that reflect the geometry of the observation locations in relation to the unobserved location *x*_{0}. Typically, observations closer to *x*_{0} will have larger coefficients; clustered observations will tend to each have smaller coefficients, thus kriging is said to decluster the data; observations that line up with *x*_{0} along directions with slowly decreasing covariance will tend to have larger coefficients; if two observations are approximately collinear with *x*_{0}, then the one farthest from *x*_{0} will tend to have a small coefficient.

In the case of ordinary kriging, with its constraints on the kriging coefficients, it was pointed out in 2 that kriging error variances may be expressed in terms of expected squared differences between pairs of random variables. For stationary models these quantities are derived from a variogram function that depends only on geographical separation. An example is a variogram that is a linear function of distance between location pairs – a model that does not have a corresponding covariance model.

Parameters of a spatial covariance or variogram model are typically inferred from the available observations on the field. For example, we may use squared linear *contrasts* of the observations for this purpose. A linear contrast is a linear combination of the observations with zero expected value. What is a contrast and what is not a contrast depends on the model specification of the mean field. For example, with a mean stationary model any difference between a pair of observations is a linear contrast. Since a contrast has zero expectation, the squared contrast will be an unbiased estimator of the variance of that linear combination which, in turn, is a known function of the covariance parameters. By equating squared contrasts to their expectations we get estimating equations for the covariance parameters. If there are *n* observation locations and *p* linear constraints on the mean function, then there will be *n* − *p* linearly independent contrasts that can be used for estimation of covariance parameters.

Error variances, reported for kriging estimates at unobserved locations, are computed from the chosen covariance model and the model parameter estimates derived from the data at the observed locations. Since the covariance or variogram parameter estimates are themselves random quantities under the model assumptions, the reported estimation error variance is itself an estimate, even if one fully accepts the parametric form of the covariance model. Statistical properties of the error estimate depend on a fuller specification of the random field model.

### Nugget Effect and Measurement Error

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

Models for spatial covariance usually involve two or three spatial scales. The first represents spatial continuity on very short scales, less than the separation between observations. The second scale is that comparable to the separation between nearby observations, called the neighborhood scale. The third scale is a regional scale comparable to separation between nonproximate observations. Short-scale variability is commonly inferred by extrapolation of neighborhood scale variability, although a better method is to have the sampling design incorporate closely spaced observations. When extrapolation of covariances to short separations suggests a discontinuity at zero separation, this discontinuity is called the *nugget effect*. The nugget effect is a contribution to variability without spatial continuity. Parametric variance–covariance models commonly include a parameter for the discontinuity at zero separation.

Measurement error, if appreciable, can be regarded as a spatial field superposed on the underlying field of interest. Then the data at the observed locations include the contributions of measurement error. Measurement error models typically have zero mean over the whole field and have completely specified spatial covariance structure. This complete specification of measurement error statistics permits the estimation of covariance parameters of the underlying spatial field of interest, using the available corrupted observations.

Frequently, measurement errors are taken to be spatially uncorrelated on all spatial scales. In this case, the measurement error variance acts statistically like an addition to the short-scale nugget effect of the underlying field. Both the nugget effect and the zero-mean measurement error have no effect on unbiasedness constraints for kriging coefficients. However, these short-scale contributions to spatial variability can substantially affect the optimization of kriging coefficients and the reported kriging error variance at an unobserved location. For example, with a stationary model only the diagonal elements of the covariance matrix *C*_{n} involve the nugget and measurement error, and these diagonal elements can be large relative to the off-diagonal elements that involve only neighborhood scale variability. The effect is that the *n* optimized kriging coefficients associated with the *n* observations become more nearly equal and less dependent on their proximity to the unobserved location. If the measurement error variance is not specified and cannot be distinguished from the nugget variability, then the kriging estimates at unobserved locations are unaffected, but the reported kriging error variance will be too large.

### Neighborhood Kriging

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

Commonly, kriging imposes a further ‘neighborhood’ constraint. This further constraint forces most kriging coefficients to be zero, i.e. those coefficients corresponding to observations that are not within a restricted geographical neighborhood of the estimation location *x*_{0}. In practice, neighborhoods may be defined by specifying both a minimum and maximum number of observations to be included. Neighborhood kriging does not arise as an automatic consequence to the modeling exercise. Rather it is a subjective but often useful constraint. Its principal advantage is that the specification of spatial models for unbiasedness constraints and for estimation errors need not involve spatial scales larger than neighborhoods.

With neighborhood kriging the mean function parameter, *b*, could be neighborhood-specific, and thus the mean function viewed over the whole region *R* becomes effectively nonparametric and possibly quite complex. A common approach is to take the mean function to be a locally linear function of the geographical coordinates. For example, for a two-dimensional region, the model mean is described by an unknown planar surface over the neighborhood that includes the unobserved estimation location and the associated observation locations. The planar mean surface would vary as we vary the estimation location and its associated neighborhood. The unbiasedness constraints on the kriging coefficients then involve only the observations within the neighborhood of the estimation location. The associated model error estimation variance only involves covariances for interpoint separation on the neighborhood scale. However, the simpler, more robust model specifications used for neighborhood kriging could, in principle, still yield larger kriging estimation errors because most kriging coefficients are arbitrarily set to zero.

### Co-Kriging and Multivariate Spatial Estimation

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

A spatial field is a multivariate field if, at each location *x* in the region *R*, *Z*(*x*) is a vector of attributes. For example, *Z*(*x*) could be a suite of pollutant concentrations, surface temperature, humidity, and wind speed. (For a discussion of multivariate spatial statistics see the entry on multivariate kriging.) One might consider each of these spatial attributes as a separate field with separate modeling as described above. However, spatial estimation of ozone concentration, say, could have smaller error variance if the available ozone data were supplemented by the available covariate data of surface air temperature.

Incorporation of the surface temperature data for ozone estimation can take different forms depending on the error-averaging model. The spatial variability of ozone and surface temperature could be modeled as a bivariate random field. The ozone estimate at an unobserved location is taken to be a linear function of both the observed ozone data and the observed covariate temperature data. If the squared error of the estimate is averaged over the modeled random variations in *both* variables, keeping only the location information fixed, then the optimized estimate is called a co-kriging estimate.

The observation locations may be different for different variables. For example, a common situation is one where a covariate's datum is available at every location in the region *R*. In this case co-kriging is sometimes further constrained, for simplicity, to assign zero weight to all covariate locations that are not coincident with an ozone observation location or the current ozone estimation location. Unbiasedness constraints that ensure a zero average error, for the selected averaging model, will depend on assumed parametric mean structures for each of the variables. For example, if each variable is taken to be mean stationary with an unknown mean parameter, then the unbiasedness constraints will be that the co-kriging estimator has no constant term, that the coefficients that multiply the ozone observations sum to one, and that the coefficients that multiply the covariate temperature observations sum to zero. The optimization of the linear co-kriging estimate requires modeling of the cross-covariance between variable pairs at location pairs.

If only the ozone variation figures in the error-averaging model, while the covariate information is held fixed, then the modeling requirements are different. For example, suppose that the mean ozone field at location *x*, conditional on the covariate field, is taken to be an unspecified linear function of the covariate at location *x*. Then we are in the universal kriging modeling situation described above. With neighborhood kriging, the unknown parameters of the conditional mean ozone field are different for different neighborhoods, leading to a nonparametric conditional mean ozone field. With this approach there is no need to model cross-covariances.

### Estimation of Spatial Averages

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

A spatial average is simply the actual average value of the spatial field over a specified region or block, possibly the entire region *R*. The error is the difference between this spatial average and its estimate. As before, we minimize the squared error averaged over the putative ensemble of all realizations of the spatial field generated by a modeled spatial stochastic process. For a stationary model it is important to note that the error is not the difference between the estimate and the expected value of the stochastic process, because under the error-averaging model the true spatial average will be different for each realization.

The same spatial mean and spatial covariance models used for optimized unbiased linear estimation of the field at unobserved locations can also be used for optimized linear unbiased estimation of spatial averages, again using a linear function of the available data at the *n* observation locations. Note that not all observation locations need be inside the averaging block. The resulting linear kriging estimate will be the same as the spatial average of the kriging estimates for all locations within the specified region, provided all *n* data locations are used to estimate every location. With neighborhood kriging, where different subsets of the data are used to estimate different locations, the correspondence will not be exact. The error variance of an unbiased linear estimate of a spatial average is formally like 1, where **λ** is the *n*-vector of coefficients that multiplies the *n* observed values of the field, *C*_{0} is the *n*-vector consisting of *n* covariances each averaged over all locations within the estimation block, and *C*_{0, 0} is the average covariance between all pairs of points in the specified block.

It is informative to compare this kriging model error variance, which uses a stochastic field model and fixed sampling locations, with the error variance obtained for simple random sampling where only randomization of location is used and the spatial field is fixed but unknown. Suppose that in the latter case the estimate is always taken to be an equally weighted average of the *n* observations. Then, by averaging over the randomization of sample locations, the error variance is **σ**^{2}/*n*, where **σ**^{2} is the variance of the fixed field *Z*(*x*) over *x* in *R*. However, once the sample locations are selected, one could treat them as fixed and compute the error variance as in 1 using the spatial covariance model applied to the observed configuration of sample locations. If this calculation were repeated for repeated randomizations of the sample locations, then the average of these repeated error variance calculations should also be approximately **σ**^{2}/*n*, provided that the spatial covariance model is consistent with the empirical **σ**^{2}.

Thus, the advantage of the spatial covariance model approach is that it provides estimation error variances specific to the actual data location configuration, while its disadvantage is that one needs to model spatial covariances.

### Nonlinear Kriging

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

Nonlinear spatial estimation can have different meanings. For example, the area fraction of the spatial region *R* where a pollutant concentration exceeds a threshold, *z*, is a spatial average of a nonlinear function of the concentration. This nonlinear function is the dichotomous indicator function *I*(*x*, *z*) that is either one or zero, as the concentration at location *x* does or does not exceed the threshold. This problem can be addressed, though not necessarily efficiently, by transforming the *n* observations on the concentration field to ones and zeros. Then one uses a linear function of the transformed data to estimate the spatial average of the transformed field. This becomes a standard kriging problem for the transformed field. The spatial covariances *c*(*x*, *x*′) for the transformed dichotomous field can be expressed in terms of the joint probabilities that the concentrations at two locations, *x* and *x*′, both exceed the threshold. Without further constraints, linear kriging estimates of the area fraction might not fall between 0% and 100% since kriging coefficients can be negative.

If we choose multiple threshold values, then the estimated area fractions, taken together, can be seen as an estimate of a spatial distribution function. The simplest approach is to treat each threshold as a different kriging estimation problem with separate modeling of the indicator covariances at each threshold value. The possibility arises that estimated area fractions for higher thresholds could turn out larger than the estimates for lower thresholds. At the expense of more complex modeling, covariances and cross-covariances for multiple thresholds could be simultaneously specified and would, in principle, more fully exploit the observations. See [2], for example, for a discussion of coindicator kriging and disjunctive kriging.

The kriging estimate of an area fraction is in effect the spatially integrated value of pointwise estimates of the dichotomous variable at every location *x* in *R*. The value being estimated at location *x* is either one or zero (*see* Binary Data). The error is the difference between the true zero/one and an estimate that is a linear function of the neighboring observed zeros and ones. The indicator kriging estimate minimizes the variance of this error. The estimate, which is commonly (but not always) a number between zero and one, is sometimes interpreted as an estimated exceedance probability appropriate to the location *x*, conditional on the neighboring data. However, considered as an estimate of this conditional probability, the local indicator kriging estimate will be biased with an unknown bias that depends on the joint distribution of the indicators at the observed and unobserved locations.

Nonlinear kriging is also used to refer to a generalization of linear kriging estimates to encompass nonlinear functions of the data at the observed locations, but where the goal remains to estimate the value of the spatial field *Z*(*x*) at unobserved locations. Thus one might consider additive rather than strictly linear functions of the observations, i.e. the estimate is the sum of individually transformed observations. Selection of optimal transformations requires modeling of bivariate distributions for pairs of observations, rather than just the covariances needed for linear kriging.

### Spatial Simulation

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

Kriging procedures, used to estimate a spatial field at unobserved locations, provide a context for assessing error variance and for optimizing the estimator within a prescribed class of estimators. Kriging minimizes the estimation variance separately at each estimation location, and uses only covariance information for location pairs. If the spatial field of such kriging estimates over the whole region *R* is represented by a map, this map will not have the same spatial properties as the underlying true field. For example, it is inappropriate to use the kriging map to infer properties related to the variability or spatial correlation of the true field or to the area fraction exceeding a threshold.

A pure Bayes approach would use a complete random field probabilistic specification such as a Gaussian field (*see* Random Field, Gaussian) specification or a pointwise transformation of a Gaussian field such as a lognormal field. This random field model becomes the prior distribution of possible field realizations, of which the field of interest is assumed to be one example. What is different here is that the imposition of a Gaussian specification makes it possible to readily characterize the posterior joint distribution of the unobserved location values, given the observed location values. The derived posterior random field model will again be a Gaussian field with posterior means and covariances computed directly from the specified prior model and the observed data.

An estimate of the field at an unobserved location can now be represented by a marginal posterior normal distribution with posterior mean and variance that reflect the observation locations and their values. This posterior mean and variance are equivalent to the ‘simple’ kriging estimate and its corresponding error variance. However, the Gaussian prior model specification also leads to probabilistic statements regarding the magnitude of the estimation error such as the frequency of large errors, taken over putative realizations of the random field model. In a different example of the versatility of the Bayes framework, one can readily compute the probability of exceeding a concentration threshold at an unobserved location, given concentration data at the observed locations, as the tail probability of a normal distribution with the location-specific computed conditional mean and conditional variance. The spatial field of these posterior tail probabilities can then be used as a risk map.

To estimate more complicated functionals of the spatial field that involve multiple locations simultaneously, one could use multiple realizations of the posterior spatial field. The functional is then computed for each realization and a sample frequency distribution of estimates is thus generated that approximates the posterior distribution of the functional, given the observations. Sampling from the posterior distribution requires discretization of the region *R*, typically using a fine grid of *N* nodes, and drawing repeatedly from an *N*-variate normal distribution with the calculated conditional mean *N*-vector and conditional *N* × *N* covariance matrix. Note that no information is gained from a single field simulation regarding estimation uncertainty. Simulations can be constrained to globally mimic statistics of the sample data, but such constraints may not be consistent with exhibiting full posterior sampling variability.

Multiple simulations will exhibit, in aggregate, all statistical characteristics consistent with the selected prior Gaussian field model, conditional on the observations. For example, the area fraction exceeding a concentration threshold (*see* Exceedance Over Threshold) will have a posterior distribution that can be estimated by the empirical distribution of the area fraction computed from each of the multiple realizations. The estimated posterior distribution, which may be plotted as a histogram, provides an interpretable uncertainty characterization for the unknown area fraction given the available observations, with the assumed Gaussian prior model specification. In principle, these multiple simulations may be used in the same way to represent the uncertainty of more complicated functionals of the spatial field such as flow characteristics in a porosity field, by repeating the flow calculation for each of the sampled fields.

Apart from the imposition of a Gaussian-like prior model that can be difficult to validate from the available observations, the multiple simulation approach typically requires substantial computing (*see* Simulation and Monte Carlo Methods). Careful attention to questions of sampling design can help to mitigate some computational issues. What are needed are richer classes of prior models that are amenable to conditioning on the observations and are amenable to posterior sampling. In some situations it may be reasonable to specify properties of a prior spatial model by using examples of plausible spatial patterns derived from exogenous knowledge or experience that are then regarded as a samples from the prior model. Statistics of these sample patterns can then be used in building prior probabilities.

If the prior model specification has adjustable parameters, for example a Gaussian random field model with a parametric mean function and a parametric spatial covariance function, then the Bayes formulation also needs a completely specified joint prior distribution for the parameters. The posterior distribution for the random field, given the observations, requires integration over the parameter prior distribution, which typically yields a non-Gaussian field posterior distribution. This approach allows for the propagation of parameter uncertainty to the final statements of uncertainty about unobserved values of the field. The implied error-averaging model incorporates a different parameter selection for each sampled realization of the field. In principle, it would also be possible to produce posterior distributions for the model parameters, but the goal is still the characterization of the uncertainty of the field *Z*(*x*) given the available data.

### References

- Top of page
- Error Models
- Notation
- Unbiased Linear Kriging
- Modeling the Spatial Mean Function
- Modeling Spatial Covariances
- Nugget Effect and Measurement Error
- Neighborhood Kriging
- Co-Kriging and Multivariate Spatial Estimation
- Estimation of Spatial Averages
- Nonlinear Kriging
- Spatial Simulation
- References

- 1 (
- 2 & (
- 31991). Statistics for Spatial Data, Wiley, New York.(
- 41965). Objective Analysis of Meteorological Fields, Israel Program for Scientific Translations, Jerusalem.(
- 51997). Geostatistics for Natural Resources Evaluation, Oxford University Press, Oxford.(
- 61997). Introduction to Geostatistics: Applications to Hydrogeology, Cambridge University Press, Cambridge.(
- 71951). A statistical approach to some basic mine valuation problems on the Witwatersrand, Journal of the Chemical, Metallurgical and Mining Society of South Africa 52, 119–139.(
- 81965). Les Variables Regionalisees et leur Estimation, Masson, Paris.(
- 9 (
- 10 (
- 11 (