ISSN:
1573-8868
Keywords:
variogram modeling
;
positive definite function
;
matrix diagonalization
;
algorithm
Source:
Springer Online Journal Archives 1860-2000
Topics:
Geosciences
,
Mathematics
Notes:
Abstract Suppose that ¯(x1),...,¯Z(xn). are observations of vector-valued random function ¯(x). In the isotropic situation, the sample variogram γ*(h) for a given lag h is $$\bar \gamma ^ * (h) = \frac{1}{{2N(h)}}\mathop \sum \limits_{s(h)} (\overline Z (x_1 ) - \overline Z (x_1 )) \overline {(Z} (x_1 ) - \overline Z (x_1 ))^T $$ where s(h) is a set of paired points with distance h and N(h) is the number of pairs in s(h).. For a selection of lags h1, h2, .... hk such that N (h1) 〉 O. we obtain a ktuple of (semi) positive definite matrices $$\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$$ . We want to determine an orthonormal matrix B which simultaneously diagonalizes the $$\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$$ or nearly diagonalizes them in the sense that the sum of squares of offdiagonal elements is small compared to the sum of squares of diagonal elements. If such a B exists, we linearly transform $$\overline Z (x)$$ by $$\overline Y (x) = B\overline Z (x)$$ . Then, the resulting vector function $$\overline Y (x)$$ has less spatial correlation among its components than $$\overline Z (x)$$ does. The components of $$\overline Y (x)$$ with little contribution to the variogram structure may be dropped, and small crossvariograms fitted by straightlines. Variogram models obtained by this scheme preserve the negative definiteness property of variograms (in the matrix-valued function sense). A simplified analysis and computation in cokriging can be carried out. The principles of this scheme arc presented in this paper.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02087100
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