1. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Surrogates and Kriging Part I: Kriging Ralf Lindau

2. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Contents Stepwise Kriging (the future) Simple Kriging (the past) avoiding negative weights Victorian LWP Stefanian Permeability

3. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Summary of 18.5.2009 Distingish between two types: 1. downscaling of averages (true downscaling) 2. downscaling of point measurements (interpolation) Example for average downscaling: Precipitation from 60 to 30 min Three methods for point downscaling: 1. Linear interpolation plus noise 2. Stepwise kriging 3. Stepwise spatio-temporal data construction

4. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Stepwise Kriging The covariances of a new kriging point to all old observation points are correct by definition. However the explained variance is smaller than 1 (normalized case). This leads to an underestimation of the correlation. Thus: Do not use the kriging technique several times in series for all intermediate points. But: 1.Predict only a single point 2.Correct its variance by adding noise 3.Consider in the next step the predicted value as an old one.

5. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Sequential vs Stepwise J.A. Vargas-Guzman, T.-C. Jim Yeh, 1999: Sequential kriging and cokriging: Two powerful geostatistical approaches Stochastic Environmental Research and Risk Assessment, 13, 416-435. The sequential estimator is shown to produce the best, unbiased linear estimate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used. The expression Sequential Kriging means here: Decomposition of a huge matrix into several smaller matrices, which can be solved easier. The presented method of Stepwise Kriging means something else: Add the information and correct the variance stepwise, data point by data point.

6. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Stepwise Kriging We proudly present: The algorithm is readily programmed and awaits eagerly to work.

7. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 So far the future. Now, the past: two examples for simple kriging. 1. Clouds (Victor) 2. Soil (Stefan) Most important difference to OK: No trend:no gradient, only (normalized) anomalies. No negative weights:dont panic, no more discussion about that.

8. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Negative weights in literature Deutsch, C.V., 1996: Correcting for Negative Weights in Ordinary Kriging, Computers & Geociences, 22, (7), 765-773. Negative weights in ordinary kriging (OK) arise when data close to the location being estimated screen outlying data. Depending on the variogram and the amount of screening, the negative weights can be significant; there is nothing in the OK algorithm that alerts the kriging system about the zero threshold for weights. Negative weights, when interpreted as probabilities for constructing a local conditional distribution, are nonphysical. Also, negative weights when applied to high data values may lead to negative and nonphysical estimates. In these situations the negative weights in ordinary kriging must be corrected. An algorithm is presented to reset negative kriging weights, and compensating positive weights to zero. The sum of the remaining nonzero weights is restandardized to 1.0 to ensure unbiasedness. The situations when this correction is appropriate are described and a number of examples are given. Deutsch presented a steamroller method to avoid negative kriging weights. My method is slightly ;-) more elegant.

9. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Autocorrelation Difficulties arise for cumulus clouds, where only a small number of non-zero observations may occur. In this case the autocorrelation function is not derivable. Solution: Kriging is based on the mean autocorrelation for all cumulus cases. 0.47 0.02 0.211.07

10. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Victorian LWP measurement krigerror krig

11. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Concluding the total variance Lines have a smaller variance than the entire field. Why? Divide a field into several lines. Then the total variance is equal to: internal: the mean variance inside the lines plus external: the variance of the lines means Technically, there are more short distances in lines than in the entire area. Multiply the relative frequencies of distances with their corresponding variance. Then the expected total variances (lines/area) can be concluded.

12. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Stefanian Permeability OriginalKriged (the truth) 0.697 0.378 0.319 Semivariogram Org Krig

13. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Nice effects of kriging Half of the variance is attributed to errors. This error variance can be interpreted in two ways: Pure inaccuracy of measurements. (Observation error) Point measuremnts are less representative for the entire grid box. (Small scale variability) For both cases the reduction of variance ís necessary. The kriged field is superior compared to the the original. It can be called Truth for this spatial resolution

14. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Only the edge OriginalKriged 0.839 0.553 0.286

15. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Random 31% Original Kriged 0.670 0.425 0.245

16. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 True, Edge, 31% True Edge 31% 0.697 – 0.319 = 0.378 0.839 – 0.553 = 0.286 0.670 – 0.425 = 0.245 Total – Error = Remain

17. Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008 Summary It is the difficulty to calculate total and error variance from a reduced data amount, which defines mainly how much variability is remaining in the obtained kriging result. Strong smoothing may occur. But the main reasons are underestimation of the total variance or overestimation of the error variance. Not the kriging method itself. Stepwise kriging is ready to go.