Universal Kriging.pdf

University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Universal kriging The Ordinary Kriging (OK) that was discussed earlier is based on the constant mean model given by Z(s) = μ+ δ(s) where δ(·) has mean zero and variogram 2γ(·). Many times this is a too simple model to use. The mean can be a function of the coordinates X, Y , in some linear, quadratic, or higher form. For example the value of Z at location s can be expressed now as Z(si) = β0 + β1Xi + β2Yi + δ(si), linear Or Z(si) = β0 + β1Xi + β2Yi + β3X 2 i + β4XiYi + β5Y 2 i + δ(si), quadratic, etc. If this is the case then we say that there is a trend of the polynomial type. We need to take this into account when we find the kriging weights. The predicted value Z(s0) at location s0 will be again a linear combination of the observed Z(si), i = 1, · · · , n values: Z?(s0) = w1Z(s1) + w2Z(s2) + · · · + wnZ(sn) = n∑ i=1 wiZi where w1 + w2 + · · · + wn = 1 Suppose now that a trend of the linear form is present. Then the value Z?(s0) can be expressed as Z?(s0) = n∑ i=1 wiZi = n∑ i=1 wiβ0 + n∑ i=1 wiβ1Xi + n∑ i=1 wiβ2Yi + n∑ i=1 wiδ(si) or Z?(s0) = n∑ i=1 wiZi = β0 + β1 n∑ i=1 wiXi + β2 n∑ i=1 wiYi + n∑ i=1 wiδ(si) (1) But also the value of Z(s0) can be expressed (based on the linear trend) as Z(s0) = β0 + β1X0 + β2Y0 + δ(s0) (2) Compare (1) and (2). In order to ensure that we have an unbiased estimator we will need the following conditions: 1 n∑ i=1 wiXi = X0 n∑ i=1 wiYi = Y0 and n∑ i=1 wi = 1 As with ordinary kriging, to find the weights when a trend is present we need to minimize the mean square error (MSE) min ( Z(s0) ? n∑ i=1 wiZ(si) )2 subject to the above constraints. This minimization will be unconstrained if we incorporate the 3 constraints in the objective function. The result is a system of n+ 3 equations for the linear trend. If the trend is quadratic we will need n + 6 equations, and n + 10 equations for a three- dimentional quadratic trend, etc.