1. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Kriging Connection between Stepwise Kriging and Data Construction and Stepwise Kriging of Victorian LWP Ralf Lindau

2. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Two cases of Downscaling Two principle cases: Data consists of averages (1 h rain sum  30 min rain sum). Downscaling should produce averages of smaller scale. The variance of each scale should be increased by a certain amount. The pdf should contain more extremes. Data consists of point measurements (DWD rain stations  rain map of Germany) Downscling should produce synthetic data in observation gaps. The variance and pdf should remain constant.

3. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Kriging approach = min Suppose three available observations x 1, x 2, x 3 (old) Kriged new value is 1 x 1 +  2 x 2 + 3 x 3 Its covariance to the old data point x 1 is: [x 1 ( 1 x 1 + 2 x 2 +  3 x 3 )] = 1 [x 1 x 1 ] + 2 [x 1 x 2 ] + 3 [x 1 x 3 ] This covariance should be equal to the covariance between prediction point P 0 and observation point P 1 which is: [x 0 x 1 ]

4. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 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 – 18. Januar 2010 Data Construction Stepwise data construction with correct mutual correlations. Construct n time series at n locations so that the spatial correlation between all locations are „correct“ (known covariance matrix as input needed (as usual)). Use weighted averages of uncorrelated normalized time series x a, x b, x c,... for the production of x 1, x 2, x 3,...

6. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Construction Recipe Time series at data point 1: Time series at data point 2: Correlation between data point 1 and 2: Variance at data point 2 Time series at data point 3: Correlation between data point 1 and 3: Correlation between data point 2 and 3: Variance at data point 3

7. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Construction vs. Kriging ConstructionKriging n random time series x a, x b, x c,... are used. Coefficients a ij determine the weights of each random time series. The correct variance is finally achieved by adding noise from an additional random time series. Both methods produce successively data points x 1, x 2, x 3,... n random time series x a, x b, x c,... are used. Coefficients c ij determine the weights of each already produced data point. The correct variance is finally achieved by adding noise from an additional random time series.

8. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 The transformation Thus, the only difference is: Data Construction uses weighted averages of n random time series, Stepwise Kriging uses weighted averages of n already produced data points. But each already produced data point is in the end itself a weighted average of the used random time series. Does a fixed transformation between Stepwise Kriging and Data Construction exist? Yes:

9. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Kriging is faster? So, Data Construction and Stepwise Kriging are essentially equal, leading to identical results. However, Construction needs information from all already included random time series. This is time consuming (modern talking: expensive). Kriging takes most information from the immediate surrounding. We might disregard far away data points. This might save computing time. (modtalk: cheap) Data Construction Stepwise Kriging

10. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Is NNW Kriging applicable? Far away data points will have negligible weights. This allows us to stop the calculation at a certain radius and save computer time. An interruption is not only desirable, but necessary (if larger fields should be constructed) Why? To determine the last of 200 x 200 = 40000 data points, the covariance matrix is as large as 39999 x 39999 (nearly 1,600,000,000 numbers). This is difficult to solve. When to stop? Normal:If no data point with a positive weight is able to reduce the error further. Stepwise:Small radius  correlations to larger distances become wrong. So, allow negative weights again.

11. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 NNW Kriging vs DS Kriging Kriging error Potential change of kriging error (aasumption: all old wights remain constant) So far: No-Negative-Weights Kriging 1.Choose that data point with the best potential error reduction (above equation). 2.Calculate weights and error exactly (solve matrix). 3.Include more and more data points by repeating the procedure. 4.If no error reduction with a positive weight is possible, one new data point is ready. 5.Repeat everything for all grid points. Now: Deep-Search Kriging 1.Choose those 15 data points with the best potential error reduction (above equation). 2.Calculate weights and error exactly (solve 15 matrices) and determine in this way the actually best. 3.Include more and more data points by repeating the procedure. 4.Ensure that all weights remain between -1 and 1. 5.If no further error reduction is possible, or 15 data points are included, one new data point is ready. 6.Repeat everything for all grid points.

12. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Weight constraint Instable solution (huge positive next to huge negative weights, if negative weights are allowed.  Constraint (-1;1) necessary.

13. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Three versions No-negative-weights Krigingconventional Lindau Kriging Stepwise Krigingnoise is added to each new point each new point is considered as old search radius enlarged Deep-search Krigingconventional, but with an enlarged search radius

14. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Conventional Kriging Original Data Used Data Pseudo- measurements NoNegWeights Kriging Input DeepSearch Kriging Output

15. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Stepwise Kriging Original Data Difference to Original Step-kriged Step-kriged pdf-corrected (by cumulative freq.)

16. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Auto-correlation Not pdf-corrected 0 : Original correlations : Input to kriging routine + : Output of kriging routine

17. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Cumulus Clouds

18. Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Summary The two methods Data Construction and Stepwise Kriging produce identical results. From only a few meaurements Stepwise kriging is able to produce full clouds fields having the correct autocorrelation. Pdf can be corrected afterwards, without distroying the nice structure.