1. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 LandCaRe 2020 Temporal downscaling of heavy precipitation and some general thoughts about downscaling Ralf Lindau

2. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 The task Soil erosion model within the LandCaRe „model chain“ needs rain input with a temporal resolution of 30 min. CLM output is available hourly. Downscaling technique is needed. First step: All model grid boxes with more than 20 mm daily precipitation are extracted from CLM output.

3. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 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.

4. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Principle of average downscaling Two coarse averages x i and x j are altered by a random  x. It results: The original covariance x i x j plus the added variance  x  x This is valid for each scale as x i and x j have an arbitrary time lag.

5. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Determination of the variance to be added The original (1h) data variance is 4.379 mm 2 /h 2 Averaging over 2,4,8 hours reduces the variance. A linear fit enables us to estimate the potential variance for 0.5 h time resolution: 5.457 mm 2 /h 2

6. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Effects on semi-variogram The total variance (horizontal lines) is increased (as desired) from 4.379 to 5.455 (mm/h) 2 This increase (as desired) is added equally to each scale (see dashed line for difference)

7. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Effects on pdf Problem: Additive noise creates negative rain values Original pdfDownscaled pdf

8. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Multiplicative noise Solution: Multiplicative noise instead of additive noise OriginalDownscaled (down – org) / (down + org)

9. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Point downscaling Well done. But what about the second type of downscaling? (Production of synthetic data in observation gaps) Why did kriging perform such a marvelous job? Do you remember? You don‘t.

10. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Kriging of Rain Beispiel: Regen vom 01.01.1996 bis 07.01.1996 DWD OriginalErgebnis Varianzeigenschaften DWD Original Ergebnis BeobFehler: 0.037 mm 2 /d 2 Konstante Varianz- reduktion um den BeobFehler

11. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Linear Interpolation Once Dr Lindau wrote four pages on that topic with the following summary: Linear Interpolation underestimates the variance of each scale by a quarter of the variance found in the smallest resolved scale. Thus, the correct spatial structure can be obtained by just adding a constant, known amount of variance.

12. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Kriging In the considered case kriging worked well because the variance not resolved by DWD stations was small. (Two 10 km separated stations measure a fairly similar daily precipitation.) However, if kriging is used as interpolation tool to estimate many virtuel data points between a few observations, it will underestimate the intermediate spatial variance.

13. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 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 ]

14. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 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.

15. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Recapitulation So far I presented two techniques: 1. Simple linear interpolation plus adding noise (quarter of small scale variance) 2. Stepwise Kriging In the following I will present a third one: 3. Stepwise data construction

16. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Stepwise 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 1, x 2, x 3,... for the production of x a, x b, x c,...

17. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Construction Recipe (1) 1. Time series at data point a: x a = a 1 x 1 a 1 = 1 2. Time series at data point b: x b = b 1 x 1 + b 2 x 2 Correlation to a:r ab = [x a x b ] = [a 1 x 1 (b 1 x 1 + b 2 x 2 )] = a 1 b 1 [x 1 x 1 ] + a 2 b 2 [x 1 x 2 ] = a 1 b 1 Variance at b: 1 = [x b x b ] = [(b 1 x 1 + b 2 x 2 ) 2 ] = b 1 2 [x 1 x 1 ] + 2b 1 b 2 [x 1 x 2 ] + b 2 2 [x 2 x 2 ] = b 1 2 + b 2 2

18. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Construction Recipe (2) 3. Time series at data point c: x c = c 1 x 1 + c 2 x 2 + c 3 x 3 Correlation to a:r ac = [x a x c ] = [a 1 x 1 (c 1 x 1 + c 2 x 2 + c 3 x 3 )] = a 1 c 1 Correlation to b:r bc = [x b x c ] = [(b 1 x 1 + b 2 x 2 )(c 1 x 1 + c 2 x 2 + c 3 x 3 )] = b 1 c 1 + b 2 c 2 Variance at c:1 = [x c x c ] = [(c 1 x 1 + c 2 x 2 + c 3 x 3 ) 2 ] = c 1 2 + c 2 2 + c 3 2

19. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Construction examples 10000 of such fields are produced. Each of them can be considered as time step. Statistical property is: For each pair of locations a given correlation is constructed. time=1....time=10000

20. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Check of correlation properties InputOutput Difference Correlation for one example point (16,11) to all others. Correlation is well reproduced. The remaining 1599 checks will be shown next time ;-)

21. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Spatial correlation of individual fields So far the spatial correlation between two points obtained by averaging in time. For some processes only a single field is available. In such cases spatial correlations are obtained by averaging over data pairs of equal distance. The method produces fields with varying spatial structure; however in average (+) it is correct. If a single strictly correct field is desired, the best of the 10000 produced can be selected (*).

22. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 The future (bright) ;-) Future steps: Use mutually uncorrelated but internal correlated time series, so that a realistic temporal development results. Use others than gaussian distributed time series. The pdf of the used underlying time series determine the pdf of the obtained fields. In this way any desired output-pdf could be created. Allow to include observations by prescribing a few members of the basic time series. In this way the method would be able to reproduce also the Victorian mean and not only the structure (the correct position of highs and lows)

23. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 Summary 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