Using semi-variograms instead of covariances is only justified under the unbiasedness conditions for the weights: S w_{ i} = 1 and S h _{j} = 0 (see below), and with the Ordinary Kriging method (no trend, second order stationarity) (Deutsch & Journel 1992).
The direct use of variogram values (given the semi-variogram models g_{ A} , g_{ B} and cross-variogram model g_{ AB}), leads in case of m observations of predictand A_{i} and n observations of covariable B_{j} to the following system of CoKriging equations:
(1) |
If we call map A the collection of points with predictand values A_{i}, and map B the points with covariable values B_{j} and if A È B is the 'glued' combination of A and B it means that:
G_{AA} |
has m x m values of g_{A}(h) from the lag vectors h_{ij } found in map A, h = || h_{ij} || |
G_{BA} = G'_{AB} |
containing m x n values of g_{AB}(h) from the vectors h_{ij } found in map A È B. More precisely: the vectors defined by the set A x B (product set consisting of ordered pairs of points taken from A and B). |
G_{BB} |
has n x n values of g_{B}(h) from the lag vectors h_{ij} found in map B, h = || h_{ij} || |
1_{m} and 1'_{m} |
are a column and a row vector both of order m; all elements in the vector are equal to 1 |
1_{n} and 1'_{n} |
are a column and a row vector both of order n; all elements in the vector are equal to 1 |
w |
is a column with m weights w_{ i} assigned to predictand A, and S w_{ i} = 1 |
h |
is a column with n weights h_{ i } assigned to covariable B, and S h_{ j} = 0 |
g_{ pA} |
is a column with m semi-variogram values g_{ A }(h), where h = || h_{ pi }||, the distance from the estimated output pixel p to all visited (sampled) points in map A (predictand map). |
g_{ pAB} |
is a column with n cross-variogram values g_{ AB }(h), where h = || h_{ pi }||, the distance from the estimated output pixel p to all visited (sampled) points in map B (covariable map). |
m_{1} |
is the Lagrange parameter introduced to formalize the unbiasedness and used to find the error variance of the prediction. |
m_{2} |
is the Lagrange parameter for the covariable. |
The solution of the above system gives optimal values for w, h and m_{1} and m_{2}.
These solutions (weights) do not depend on the sampled values in map A or B, but solely on the variogram models used and on the geometric distribution of the measurements, the so-called sampling scheme. They are used to get the prediction (Formula 2) as a linear combination of predictand measurements A_{i} and covariable measurements and B_{j}:
(2) |
The variance of the prediction error (s^{2} , Formula 3) is obtained from inner vector products and m_{1}:
s^{2} = S w_{i} g_{A}(h_{i}) + S h_{j} g_{ AB}(h_{j}) + m_{1} |
(3) |
This expression depends solely on the variogram models g_{ A} , g_{ B} and g_{ AB} and on the geometric layout of the sampled points (the sampling scheme) in both map A and map B.
Note: When the spherical distance option is used, the length of all lag vectors (h) are calculated over the sphere using the projection of the coordinate system that is used by the georeference of the output raster map.
References:
See also: