Universal Kriging

Algorithm

The regionalized variable theory assumes that the spatial variation of any variable Z can be expressed as the sum of two major components. These components are:

a structural component, associated with a constant mean value or a constant trend [m(x)]
a stochastic, spatially correlated component, known as the variation of the regionalized variable [e'(x)].

If x is a position in 2 dimensions in space, then the value of variable Z at x is given by:

Z(x) = m(x) + e'(x)

(1)

While in Ordinary Kriging it is assumed that the mean is constant across the entire region of study (second order stationarity), in Universal Kriging the mean is a function of the site coordinates. Then, m(x) in equation (1) reads:

(2)

where:

a_k	are the local trend or drift coefficients
p_k(x)	are functions of the site coordinates (trend equations)
x	is a two dimensional vector

In ILWIS, the local trend or drift is either represented by a linear expression or by a quadratic expression, so the general equation (2) can be rewritten as:

m(x) = a₀ + a₁x_i + a₂y_i + a₃x_i² + a₄x_iy_i + a₅y_i²

(3)

where:

x_i, y_i	are the XY-coordinates of the i th control point
a₁...a₅	are the unknown trend or drift coefficients.

If the degree = 1, then the local trend is linear and a₃ =a₄ = a₅ = 0.
Equation (3) will then read:

m(x) = a₀ + a₁x_i + a₂y_i

(3a)

If the degree = 2, then the local trend is quadratic.

Remark that the parameters of equation (3) are recomputed for each output pixel.

The expressions for the local trend can be incorporated into the system of simultaneous equations used to find the Kriging weights. For a system of 5 input points and a local linear trend, the set of equations read (in matrix form):

where:

h_ik	is the distance between input point i and input point k
h_pi	is the distance between output pixel p and input point i
g(h_ik)	is the value of the semi-variogram model for distance h_ik, i.e. the semi-variogram value for the distance between input point i and input point k
g(h_pi)	is the value of the semi-variogram model for the distance h_pi , i.e. the semi-variogram value for the distance between output pixel p and input point i
x_i, y_i	are the XY-coordinates of input point i
w_i	is a weight factor for input point i
l	is a Lagrange multiplier, used to minimize possible estimation error
a₁, a₂	are the local trend coefficients of the first order trend
x_p, y_p	are the XY-coordinates of output pixel p

This matrix form has to be solved for each output pixel in the same way as described in Kriging : algorithm, section on Ordinary Kriging. Once the weights of the input point values are known it is possible to calculate an estimate or predicted value for the output map and to calculate the error variance and the standard error.

References:

Burrough, P.A. 1986. Principles of Geographical Information Systems for Land Resources Assessment. Oxford University Press. 194 pp.
Burrough, P.A. and R.A. McDonnell 1998. Principles of Geographical Information Systems. Oxford University Press. 330 pp.
Davis, J.C. 1973. Statistics and data analysis in geology. Wiley, New York. 646 pp.
Pebesma, E.J. 1996. Mapping Groundwater Quality in the Netherlands. KNAG/Faculteit Ruimtelijke Wetenschappen, Utrecht University. 128 pp.