The task

1. LandCaRe 2020Temporal downscaling of heavy precipitationandsome general thoughts about downscalingRalf Lindau Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

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

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

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

5. Determination of the variance to be added The original (1h) data variance is 4.379 mm2/h2 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 mm2/h2 Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

6. Effects on semi-variogram Thetotal 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) Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

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

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

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

10. Kriging of Rain Beispiel: Regen vom 01.01.1996 bis 07.01.1996 Varianzeigenschaften DWD Original Ergebnis DWD Original Ergebnis Konstante Varianz- reduktion um den BeobFehler BeobFehler: 0.037 mm2/d2 Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

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

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

13. = min Kriging approach Suppose three available observations x1, x2, x3 (old) Kriged new value is l1x1 + l2x2 + l3x3 Its covariance to the old data point x1 is: [x1 (l1x1 + l2x2 + l3x3)] = l1 [x1x1] + l2 [x1x2] + l3 [x1x3] This covariance should be equal to the covariance between prediction point P0 and observation point P1 which is: [x0x1] Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

14. 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: • Predict only a single point • Correct its variance by adding noise • Consider in the next step the predicted value as an old one. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

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

16. 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 x1, x2, x3, ... for the production of xa, xb, xc, ... Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

17. Construction Recipe (1) 1. Time series at data point a: xa = a1x1 a1 = 1 2. Time series at data point b: xb = b1x1 + b2x2 Correlation to a: rab= [xaxb] = [a1x1 (b1x1 + b2x2)] = a1b1 [x1x1] + a2b2 [x1x2] = a1b1 Variance at b: 1 = [xbxb] = [(b1x1 + b2x2)2] = b12 [x1x1] + 2b1b2[x1x2] + b22[x2x2] = b12+ b22 Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

18. Construction Recipe (2) 3. Time series at data point c: xc = c1x1 + c2x2 + c3x3 Correlation to a: rac = [xaxc] = [a1x1 (c1x1 + c2x2 + c3x3)] = a1c1 Correlation to b: rbc = [xbxc] = [(b1x1+ b2x2)(c1x1 + c2x2 + c3x3)] = b1c1 + b2c2 Variance at c: 1 = [xcxc] = [(c1x1 + c2x2 + c3x3)2] = c12+ c22+ c32 Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

19. time=1 .... time=10000 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. Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

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

21. 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 (*). Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

22. 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) Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

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