Geo479/579: Geostatistics Ch17. Cokriging

Introduction Data sets often contain more than one variable of interest These variables are usually spatially cross-correlated Data sets often contain not only the primary variable but also secondary variables of interest These variables are usually spatially cross-correlated and ths contain useful information about the Primary variable.

We can see from the cross H plots that U values are correlated with V values and also is correlation decreases as the Value of H increases thus these vales are spatially correlated. The scatter plot becomes fatter as H increases.

The Cokriging System A method for estimation that minimizes the variance of the estimation error by exploiting the cross-correlation between several variables Cross-correlated information contained in the secondary variable should help reduce the variance of the estimation errors Now if we can use this variable in addition to the primary variable to estimate the value then this will help reduce the covariance

The Cokriging System When is the secondary variable useful in estimates? Primary variable of interest is under sampled then the only information we have is the cross correlated information For example we can use rainfall data to predict erosion…

The Cokriging System The cokriging estimate is a linear combination of both primary and secondary data values This is the Equation used in Ordinary Kriging pg 279 17,1 ij kriging equation plus a term for the secondary variable.

The Cokriging System The development of the cokriging system is identical to the development of ordinary kriging system Estimation Error R can be defined as This is a modification of the error estimation in Ordinary Kriging( pg 279)

The Cokriging System Using matrix notation we can write w = { a1, a2, a3,…an, b1, b2, b3,…bm} Z = { U1, U2…..Ui, V1,….Vj}

The Cokriging System Using Equation 9.14 (p216), 12.6 (p283), 16.3 (p372) we can write The Expansion of the term is similar to what we did in OK, pg 283, eq 12.7

This is similar to Chapter 16 The Expansion of the term is similar to what we did in OK, pg 283, eq 12.7

The Cokriging System 1) Unbiasedness condition Reference to Chapter 12 E{U0} should be equal to the mean. Note: Other nonbias conditions are also possible

It is similar to unbiasedness in Ordinary Kriging (p281) We set error at as 0:

The Cokriging System 2) Minimizing error variance Lagrangean Relaxation:

The Cokriging System Lagrangean Relaxation: Original Lagrange parameter: (12.9)

The Cokriging System Equating n+m+2 partial derivatives of Var{R} to zero, we get the following system of equations Similar to eq 12.9

This is similar to minimizing the varianves of error in Ordinary Kriging The set of weights that minimize the error variance under the unbiasedness condition satisfies the following n+1 equations - ordinary kriging system: (12.11) (12.12)

Minimization of the Error Variance The ordinary kriging system expressed in matrix (12.13) (12.14)

The Cokriging System Positive definiteness must hold for the set of auto- and cross-variograms (Eq16.44, p391).

The Cokriging System If the primary and secondary variables both exist at all data locations and the auto- and cross-variograms are proportional to the same basic model then the cokriging estimates will be identical to the ordinary kriging estimates

A Cokriging Example

A Case Study Compares cokriging and ordinary kriging Undersampled variable U is estimated using 275 U & 470 V sample data for cokriging and only the 275 U data for ordinary kriging

Ordinary kriging 275 U values Using eq 17.11 for the variogram model

Cokriging 275 U and 470 V values Using eq 17.11 for the variogram model Two non-bias conditions 1) uses the initial conditions 2) uses only one nonbias condition

In the alternate unbiased condition, the unknown U value is now estimated as a weighted linear combination of nearby V values adjusted by a constant so that their mean is equal to the mean of the U values

Negative estimates occur because of the nonbias condition

Cokriging with two nonbias conditions is less than satisfactory A physical process with both negative and positive weighting scheme is difficult to imagine Cokriging with one nonbias condition considerably improved the spread of errors and bias Though we had to calculate global means of U and V

If the spatial continuity is modeled using semivariograms then they can be converted to covariance values for cokriging matrices by following equation:

The Cokriging System If we want an estimate over a local area A, there are two options: 1) Average of point estimations within A

The Cokriging System 2) Replace all the covariance terms and in point cokriging system, with average covariance values and Using block Kriging Pg 325

The Cokriging System With the unbiasedness conditions, we can calculate error variance as follows