Aggregation in Geostatistical Problems

A random process Z(·) defined on point support has quite different characteristics to those of aggregations of Z(·). For example, \(Z\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{s} } \right)\) has larger variance than \(Z\left( B \right) \equiv \int_B {Z\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{u} } \right)} d\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{u} / \int_B {d\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{u} } \), where \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{s} }\) is a point chosen at random within the block B; B is often referred to as the support of Z(B). Suppose that a resource Z(·) is sampled, yielding data \(Z \equiv {\left( {Z\left( {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{s} }_1}} \right), \ldots ,Z\left( {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{s} }_n}} \right)} \right)^\prime }\). However, the resource is extracted in blocks B1,…, BN. Let B denote a generic block and suppose that it is desired to predict g(Z(B)) based on the data \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Z} }\). The conditional expectation, \(E\left\{ {g\left( {Z\left( B \right)} \right)\left| {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Z} } \right.} \right\}\), minimizes the mean-squared prediction error, but it is impossible to estimate it without making over-utopian parametric assumptions. Current approaches to the problem require knowledge of “block-to-block”, and “block-to-sample”, parameters that cannot be estimated from the data. This paper proposes alternatively to make the kriging predictor more variable; the result is an optimal predictor for g(Z(B)) that is unbiased for a Gaussian process and approximately unbiased for a non-Gaussian process and sufficiently smooth g.