Spatial Variation by Bertil Matérn (auth.)

By Bertil Matérn (auth.)

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F. (a, b; c) for the volume of the intersection of the two spheres where the distance tx1 - x 2 1 between tne centers of the spheres equals c. V is the sum of two spherical segments. Thus V,. a~ f .. r r h,. u (w) dw {J (3. 2. 4), and a2 + c2- b2 ac=---2ac It must be remembered that h(w) =O if lwl > r. For some low values of n we find the following cor. 's corresponding to (4) n=I n =2 n=3 n =5 r(2Av)=I-v r(2Av) =I- (2/n) [v VI - v2 +arcsin v] r(2Av)=I-3V/2+v 3 f2 r(2Av) =I- I5vf8 + Sv3 /4- 3V 5 f8 These expressions are valid for lvl

It is clear from (r) that also in this case the cor. f. of the observed process has a discontinuity at the origin. For u > o, r(u) is a weighted average of the values of r 1 in some neighbourhood of u. 's will be discussed further in the next section. We shall also consider rounding off errors. Let the observed z(x) be defined as a multiple, nh, of the length, h, of the rounding-off-interval, with n determined from We thus consider rounding off to the nearest lower multiple of h. If the rounding off is to the nearest multiple of h, the covariance structure is the same.

The statistical properties of sampling schemes (and experimental designs) are mostly dependent on the nature of the short distance variation. See the discussion in Matern (I947, pp. ) and Jowett (I955). Of course "local" and "long-distance" are relative concepts. They must be seen in relation to the size of the sampling strata, the experimental blocks, etc. Furthermore, data (tables, maps) on the spatial variation are often confined to a very limited region, so that inferences cannot be made about the longdistance variation.

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