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2) s 2 / n is an unbiased estimator of the sampling error variance. 9) to each stratum. The average within A)). Howstratum variance is then an unbiased estimator of n var( ever, we have seen that the latter is smallest when the strata are chosen as small and compact as possible. Ideally, we would choose n strata with k = 1. In this case, or with systematic sampling, we have no information left with which to estimate the sampling error. Clearly, with systematic sampling we will never be able to assess the variability due to the positioning of the grid, so we must assume that this is small; that is, that there is no periodicity in the process at the sampling wavelength.

It produces a continuous but non-differentiable predicted surface. Zubrzycki (1957) introduced a process found by counting the number of points of a Poisson process of intensity h within distance ( R / 2 ) of the sample point. Then C(h) =Xmeas( b(0, R ) n b(h, R ) ) where b(h, R ) denotes the ball of radius R centered at h. This gives lo r>R (b) Fig. 12 Zubrzycki's correlation function in two dimensions ( a ) and three dimensions ( b ) . 33) in three dimensions. 12 illustrates these functions. Their behavior near the origin is very similar to an exponential function.

X N are usually, but not always, inside D. They might be on a regular grid, as suggested by the theory of Chapter 3, or they might be those points at which data is available, chosen for other reasons. For instance, networks of rain gauges are set up where observers are available, and data on oil and mineral fields are available where drilling occurred (at spots thought to be fruitful) and from those parts of the field that are being exploited. Under these circumstances, the sample mean may be seriously biased as an estimator of the mean level of the surface within D.