Multigaussian kriging for point-support estimation: incorporating constraints on the sum of the kriging weights

Abstract

In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations. The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when the mean is unknown.

Appendices

Appendix A

This appendix analyses the conditional bias of the ordinary multigaussian kriging estimator (Eq. 11). To quantify this bias, let us calculate the regression of ϕ(Y _x ) upon [ϕ(Y _x )]^oMK. Since the latter is a function of the ordinary kriging estimator (to simplify, we suppose that this is a one-to-one function), conditioning to [ϕ(Y _x )]^oMK amounts to conditioning to Y ^OK _x :

$$E\{\varphi (Y_{{\mathbf{x}}})|[\varphi (Y_{{\mathbf{x}}})]^{{\rm oMK}} \} = E\{\varphi (Y_{{\mathbf{x}}})|Y^{{\rm OK}}_{{\mathbf{x}}} \}.$$

(43)

The pair ${\left\{{Y_{{\mathbf{x}}} - m,({{Y^{{\rm OK}}_{{\mathbf{x}}} - m}})/{{s^{{\rm OK}}_{{\mathbf{x}}}}}} \right\}}$ has a standard bigaussian distribution with correlation coefficient (Eqs. 8, 10)

$$r^{{\rm OK}}_{{\mathbf{x}}} = \frac{1}{{s^{{\rm OK}}_{{\mathbf{x}}}}}{\sum\limits_{\alpha = 1}^n {\lambda^{{\rm OK}}_{\alpha}\,\rho ({\mathbf{x}}_{\alpha}, {\mathbf{x}})}} = \frac{{1 - (\sigma^{{\rm OK}}_{{\mathbf{x}}})^{2} - \mu _{{\mathbf{x}}}}}{{s^{{\rm OK}}_{{\mathbf{x}}}}}.$$

(44)

Hence, one can write (Rivoirard 1994, p. 50):

$$Y_{{\mathbf{x}}} = m + \frac{{r^{{\rm OK}}_{{\mathbf{x}}}}}{{s^{{\rm OK}}_{{\mathbf{x}}}}}(Y^{{\rm OK}}_{{\mathbf{x}}} - m) + {\sqrt{1 - (r^{{\rm OK}}_{{\mathbf{x}}})^{2}}}\,U,$$

(45)

where U is a standard Gaussian variable independent of Y ^OK _x . This identity allows one to calculate the regression curve between the true and estimated values of ϕ(Y _x ) (Eq. 43):

$$\begin{aligned} & E\{\varphi (Y_{{\mathbf{x}}})|[\varphi (Y_{{\mathbf{x}}})]^{{\rm oMK}} \} \\ & \quad = {\int {\varphi \left(m + \frac{{r^{{\rm OK}}_{{\mathbf{x}}}}}{{s^{{\rm OK}}_{{\mathbf{x}}}}}\left(Y^{{\rm OK}}_{{\mathbf{x}}} - m\right) + {\sqrt {1 - (r^{{\rm OK}}_{{\mathbf{x}}})^{2}}}\,u\right)\,g(u)\,\hbox{d}u.}} \\ \end{aligned}$$

(46)

A comparison with Eq. 11 shows that, in general, a conditional bias is present (Eq. 15), unless r ^OK _x  = s ^OK _x : in this case, the Lagrange multiplier μ _x is equal to zero because of Eqs. 13 and 44, and ordinary multigaussian kriging coincides with simple multigaussian kriging. However, this is a very specific circumstance as it implies that the weight of the mean is equal to zero in the simple kriging system (Eq. 17).

Appendix B

In this appendix, we establish the analytical expressions of the estimation variances associated with multigaussian kriging. To make the demonstration more general, let us consider a weighted average of the normal scores data and mean value that estimates Y _x without bias:

$$Y^{*}_{{\mathbf{x}}} = {\sum\limits_{\alpha = 1}^n {\lambda _{\alpha}\,Y_{{{\mathbf{x}}_{\alpha}}}}} + \left(1 - {\sum\limits_{\alpha = 1}^n {\,\lambda _{\alpha}}}\right) m.$$

(47)

The variance of Y ^* _x and its covariance with Y _x are:

$$ \left\{ {\begin{array}{*{20}l} {{{\text{var}}(Y^{*}_{{\mathbf{x}}} ) = (s^{*}_{{\mathbf{x}}} )^{2} = {\sum\limits_{\alpha = 1}^n {{\sum\limits_{\beta = 1}^n {\lambda _{\alpha } \,\lambda _{\beta } \,\rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}}_{\beta } )} }} }} \hfill} \\ {{{\text{cov}}(Y_{{\mathbf{x}}} ,Y^{*}_{{\mathbf{x}}} ) = r^{*}_{{\mathbf{x}}} s^{*}_{{\mathbf{x}}} = {\sum\limits_{\alpha = 1}^n {\,\lambda _{\alpha } \,\rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}})} }} \hfill} \\ \end{array} } \right.$$

(48)

so that ${\left\{{Y_{{\mathbf{x}}} - m,({{Y^{*}_{{\mathbf{x}}} - m}})/{{s^{*}_{{\mathbf{x}}}}}} \right\}}$ is a standard bigaussian pair with correlation coefficient r ^* _x .

Let us now define an unbiased estimator of ϕ(Y _x ) as:

$$\begin{aligned} \left[\varphi (Y_{{\mathbf{x}}})\right]^{*} & = {\int {\varphi (Y^{*}_{{\mathbf{x}}} + {\sqrt {1 - (s^{*}_{{\mathbf{x}}})^{2}}}\,t)\,g(t)\,{\rm d}t}} \\ & \quad= {\sum\limits_{p = 0}^{+ \infty} {\varphi _{p} (m)\,(s^{*}_{{\mathbf{x}}})^{p} H_{p} {\left({\frac{{Y^{*}_{{\mathbf{x}}} - m}}{{s^{*}_{{\mathbf{x}}}}}} \right)}}} \\ \end{aligned}.$$

(49)

The estimation variance of the estimator defined in Eq. 49 is found by considering the Hermitian expansions of the transfer function (Eq. 27) and by using the orthonormality of the Hermite polynomials for the bigaussian distribution (Eq. 26):

$$\begin{aligned} & {\text{var}}\{\varphi (Y_{{\mathbf{x}}}) - [\varphi (Y_{{\mathbf{x}}})]^{*} \} \\ & \quad= {\text{var}}\{\varphi (Y_{{\mathbf{x}}})\} + {\text{var}}\{[\varphi (Y_{{\mathbf{x}}})]^{*} \} - 2\,{\text{cov}}\{\varphi (Y_{{\mathbf{x}}}),[\varphi (Y_{{\mathbf{x}}})]^{*} \} \\ & \quad= {\sum\limits_{p = 1}^{+ \infty} {\,\varphi ^{2}_{p} (m)\,\{1 + (s^{*}_{{\mathbf{x}}})^{{2p}} - 2\,(r^{*}_{{\mathbf{x}}} s^{*}_{{\mathbf{x}}})^{p} \}.}} \\ \end{aligned}$$

(50)

The estimation variances of the simple and ordinary multigaussian kriging estimators (Eqs. 32, 33) are particular cases of this formula, obtained by using the simple and ordinary kriging weights in Eq. 48.

Appendix C

This appendix aims at obtaining an “optimal” unbiased estimator of ϕ(Y _x ) in case of an unknown mean. By convention, the criterion for optimality is the minimization of the estimation variance. Equation 49 provides a general form of an unbiased estimator of ϕ(Y _x ). Suppose one does not use ordinary kriging but another weighted average of the normal scores data, such that the weights add to one:

$$Y^{*}_{{\mathbf{x}}} = {\sum\limits_{\alpha = 1}^n {\lambda _{\alpha}\, Y_{{{\mathbf{x}}_{\alpha}}}}}\quad{\text{with}}\quad{\sum\limits_{\alpha = 1}^n {\lambda _{\alpha}}} = 1.$$

(51)

Since Y ^* _x does no longer depend on the mean value m, the estimator in Eq. 49 can also be written as follows:

$$[\varphi (Y_{{\mathbf{x}}})]^{*} = {\sum\limits_{p = 0}^{+ \infty} {\varphi _{p} (0)(s^{*}_{{\mathbf{x}}})^{p} H_{p} {\left({\frac{{Y^{*}_{{\mathbf{x}}}}}{{s^{*}_{{\mathbf{x}}}}}} \right)}}}.$$

(52)

It is not obvious that ordinary kriging minimizes the estimation variance (Eq. 50) among all the weighted averages of the normal scores data with total weight equal to one. Actually, this is the case for linear functions of Y _x , but not for any function ϕ. Matheron (1974, p. 30) examined the case of an exponential function (lognormal model) and proposed an algorithm to obtain the optimal weighting for Y ^* _x , i.e. to find the weighted average that minimizes the estimation variance of the lognormal estimator.

A closer look at Eq. 50 shows that the best choice of the weighted average depends on the coefficients {ϕ _p (m), p ∈ N} and that one can construct a more general estimator which minimizes each term of the estimation variance by putting (Eq. 38):

$$[\varphi (Y_{{\mathbf{x}}})]^{*} = {\sum\limits_{p = 0}^{+ \infty} {\varphi _{p} (0)(s^{*}_{{{\mathbf{x}},p}})^{p} H_{p} {\left({\frac{{Y^{*}_{{{\mathbf{x}},p}}}}{{s^{*}_{{{\mathbf{x}},p}}}}} \right)}}}$$

(53)

for a set of weighted averages $\{Y^{*}_{{{\mathbf{x}},p}} = {\sum\nolimits_{\alpha = 1}^n {\lambda _{{\alpha, p}}\,Y_{{{\mathbf{x}}_{\alpha}}}}},p \in N\}$ to be determined.

The unbiasedness of this estimator can be established by recalling that, for any Gaussian random variable Y with mean m and variance s ², one has (Emery 2005b, p. 319):

$$E{\left\{{s^{p} H_{p} {\left({\frac{Y}{s}} \right)}} \right\}} = \frac{{(- m)^{p}}}{{{\sqrt {p\,!}}}}.$$

(54)

Consequently, the disjunctive-type estimator in Eq. 53 has the same expected value as ordinary multigaussian kriging (Eq. 31), which is the same as the expected value of the transfer function to estimate:

$$E\{[\varphi (Y_{{\mathbf{x}}})]^{*} \} = E\{[\varphi (Y_{{\mathbf{x}}})]^{{\rm oMK}} \} = E\{[\varphi (Y_{{\mathbf{x}}})]\}.$$

(55)

The calculation of the estimation variance is more tricky, as the exact expression of this variance depends on the unknown mean value m. To break the deadlock, one has to choose a model (with a specified mean value) and calculate the estimation variance in the framework of this model. Note that, in contrast, the unbiasedness condition (Eq. 55) is independent of this model since the expected value of the estimation error is zero, regardless of the mean value. Here, we will use the “ideal” multigaussian model, for which the normal scores data have a zero mean. In this case, the estimation variance of the Hermite polynomial of degree p, which contributes to the p-th term of Eq. 50, is:

$${\text{var}}\{H_{p} (Y_{{\mathbf{x}}}) - [H_{p} (Y_{{\mathbf{x}}})]^{*} \} = 1 + (s^{*}_{{{\mathbf{x}},p}})^{{2p}} - 2\,(r^{*}_{{{\mathbf{x}},p}}\,s^{*}_{{{\mathbf{x}},p}})^{p}$$

(56)

and the minimization under the restriction on the sum of the weights (Eq. 51) leads to the following system of nonlinear equations:

$$ \left\{ {\begin{array}{*{20}l} {{{\left[ {{\sum\limits_{\beta ,\delta = 1}^n {\lambda _{{\beta ,p}} \,\lambda _{{\delta ,p}} \,\rho ({\mathbf{x}}_{\beta } ,{\mathbf{x}}_{\delta } )} }} \right]}^{{p - 1}} {\sum\limits_{\beta = 1}^n {\lambda _{{\beta ,p}} \,\rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}}_{\beta } )} } + \mu _{p} } \hfill} \\ {{ = {\left[ {{\sum\limits_{\beta = 1}^n {\lambda _{{\beta ,p}} \,\rho ({\mathbf{x}}_{\beta } ,{\mathbf{x}})} }} \right]}^{{p - 1}} \rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}})\quad\forall \alpha = 1,..., n} \hfill} \\ {{{\sum\limits_{\beta = 1}^n {\lambda _{{\beta ,p}} } } = 1.} \hfill} \\ \end{array} } \right.$$

(57)

In practice, this system can be solved by iterations. The initial guess ${\text{\{ $ \lambda $ }}^{{(0)}}_{{{\text{ $ \beta $ ,p}}}} {\text{,}}\;{\text{ $ \beta $ = }}1{\text{,}}...{\text{,}}\;{\text{n\} }}$ may be the solution of the ordinary kriging system (Eq. 9). At step k, one solves the linear system

$$\left\{ {\begin{array}{*{20}l} {{{\sum\limits_{\beta = 1}^n {\lambda ^{{(k)}}_{{\beta ,p}} \,\rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}}_{\beta } )} } + \nu ^{{(k)}}_{p} } \hfill} \\ {{ = {\left[ {\frac{{{\sum\limits_{\beta = 1}^n {\lambda ^{{(k - 1)}}_{{\beta ,p}} \,\rho ({\mathbf{x}}_{\beta } ,{\mathbf{x}})} }}} {{{\sum\limits_{\beta ,\delta = 1}^n {\lambda ^{{(k - 1)}}_{{\beta ,p}} \,\lambda ^{{(k - 1)}}_{{\delta ,p}} \,\rho ({\mathbf{x}}_{\beta } ,{\mathbf{x}}_{\delta } )} }}}} \right]}^{{p - 1}} \rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}})\quad\forall \alpha = 1,..., n} \hfill} \\ {{{\sum\limits_{\beta = 1}^n {\lambda ^{{(k)}}_{{\beta ,p}} } } = 1.} \hfill} \\ \end{array} } \right.$$

(58)

This iterative algorithm is based on a fixed-point method. It is quite easy to implement but may not necessarily converge, or converge slowly. Alternative iterative algorithms can be used to solve system Eq. 57, see for instance Ortega and Rheinboldt (1970) and Dennis and Schnabel (1983).

As a particular case, if the distance from x to any of the data locations {x _α, α=1,..., n} is greater than the range of the correlogram model, then system Eq. 58 is the same as the ordinary kriging system of the unknown mean (Matheron 1971, p. 128), or equivalently the ordinary kriging system of a location distant from all the data:

$$\left\{ {\begin{array}{*{20}l} {{{\sum\limits_{\beta = 1}^n {\lambda ^{{(k)}}_{{\beta ,p}} \,\rho ({\mathbf{x}}_{\alpha } ,{\mathbf{x}}_{\beta } )} } + \nu ^{{(k)}}_{p} = 0\,\forall\alpha = 1,...,\,n} \hfill} \\ {{{\sum\limits_{\beta = 1}^n {\lambda ^{{(k)}}_{{\beta ,p}} } } = 1.} \hfill} \\ \end{array} } \right.$$

(59)

This implies that, far from the data, the disjunctive-type estimator proposed in Eq. 53 coincides with the ordinary multigaussian kriging estimator. Note that both estimators are also the same at the data locations as they honor the values of the transfer function to estimate. As a consequence, the disjunctive estimator (Eq. 53) is a worthy alternative to ordinary multigaussian kriging only for “intermediate” locations, not too close nor too distant from the data.