1. CWR 6536 Stochastic Subsurface Hydrology
Optimal Estimation of Hydrologic Parameters using Kriging

2. Types of Kriging
Simple kriging is optimal estimation of a random field, e.g. T(x), with a known mean, m(x), and a known covariance PTT(x,x’).
Ordinary kriging is optimal estimation of a random field, e.g. T(x), with an unknown constant or linearly trending mean, but a known semivariogram gTT(x,x’).
Universal kriging is optimal estimation of a random field, e.g. T(x), with an unknown polynomial trending mean, but a known semivariogram gTT(x,x’).

3. Universal Kriging Assume random field has the form:
Define kriging estimate as:
To ensure unbiasedness:
Therefore must choose li so that:

4. Universal Kriging
Proceed as before minimizing the estimation variance subject to these 4 constraints, i.e.
Use the method of Lagrange multipliers which adjoins the constraints to the objective function.

5. Universal Kriging
This results in a linear system of N+4 equations with N+4 unknowns:
Or in matrix notation:

6. Universal Kriging
Once li and mi are determined construct estimate from
Calculate the kriging variance

7. Properties of Universal Kriging Process
The non-stationary mean is fitted only locally. It does not appear in the estimation of T, only in the kriging variance.
At each new point x0 a new mean is fit. Therefore result is different than fitting a polynomial trend to the system as a whole and kriging the residuals.
Problem is that must estimate variogram around drift. Because drift is unknown cannot easily estimate directly from data. There are software packages that can do this e.g. GSLIB (Deutsch and Journel).
All other previously discussed properties of the kriging process apply

8. Log-Kriging
If data appears skewed and non-normal it may be better to take the log transform of the data, fit a covariance or variogram to the transformed data, and krige the transformed variable.
The question arises..Does a good estimate of Ln T lead to a good estimate of T?

9. Log Kriging
If the variable T is truly multivariate log-normal then the relationship between the moments of the arithmetic and log parameters is:

10. Log Kriging
For simple kriging an unbiased estimate of the un-transformed parameter is obtained using:
For ordinary kriging an unbiased estimate of the un-transformed parameter is obtained by:

11. Log Kriging
In practice these transformations do not always produce unbiased estimates because these expressions are not very robust to errors in the multi-lognormal assumption
A suggested technique to overcome biasedness (i.e. arithmetic mean of estimated values noticeably different from the arithmetic mean of the measured values) is to iteratively use the transformation:

12. Log Kriging
It also has been suggested that when estimating transmissivities it is preferable to obtain an unbiased estimate of Ln T and then to use T=exp(lnT) (i.e. the geometric mean) as a biased estimate of the effective transmissivity (good for uniform 2D flow)
Recall that have minimized the variance of the estimate of the log transformed variable not the arithmetic variable.

13. Normal Score Transform Kriging
If data appears skewed and non-normal can also take the normal score transform of the data, fit a covariance or variogram to the normal score transformed data, and krige the transformed variable.
To get estimate of the original variable backtransform normal score estimate. Estimate of the original variable may be biased
Recall that have minimized the variance of the estimate of the normal score transformed variable not the original variable.