1. Statistical and Dynamical Downscaling of Atmospheric Model Outputs : A Hydrologic Modeler’s Perspective George Leavesley 1, Lauren Hay 1, Martyn Clark 2, William Gutowski, Jr. 3, and Robert Wilby 4 1 U.S. Geological Survey, Denver, CO 2 University of Colorado, Boulder, CO 3 Iowa State University, Ames, IA 4 King’s College London, London, UK

2. Topics Basic Issues Statistical Downscaling Dynamical Downscaling Bias Spatial and Temporal Distribution Issues

3. Hydrologic Modeling Issues Limited spatial and temporal observations of meteorological variables for use in applying hydrological models Atmospheric models provide spatially and temporally distributed meteorological variables Spatial resolution of hydrometeorological output produced by general circulation models (GCMs) is too coarse to resolve important catchment-scale processes in hydrological models

4. SSDS Statistical Downscaling (SDS) DDDS Dynamical Downscaling (DDS) Empirical relations between features reliably simulated by a global-scale model and surface predictands Regional Climate Model nested within a global-scale model A Solution: Downscaling

5. Downscaling Issues Choice of method –Statistical –Dynamical Application –Scenario generation –Real-time forecasting Bias Spatial and temporal distribution –Point –Areal …

6. Statistical Downscaling Methods Weather generators Weather typing Transfer functions

7. Simple Weather Generator Two state (wet- or dry-day), first order Markov process Three parameters –wet day probability –dry day probability –mean wet day amount or intensity Parameters can be conditioned on climate indices, circulation patterns, season, …

8. Simple Weather Generator p dd = Pr{no precip on day t | no precip on day t-1} p ww = Pr{precip on day t | precip on day t-1} f(x) = 1 / u exp(-x/u) Conditional dry-day probability Conditional wet-day probability On wet day, precip total is generated stochastically using an exponential distribution ( u is the average non-zero precip amount) Wilby, et al., 2002, Hydro Processes

9. Weather Generator Conditioned on Air Flow Indices (z, u, v)  and  are estimated parameters Precipitation occurrence process Mean intensity φ is a random scaling factor used to inflate the variance c is correction factor for bias in ln(u) to u transformation Precip occurs if RV < p w where 0 < RV < 1 Wilby, et al., 2002, Hydro Processes

10. Mean normalized distributions of 500 hPa geopotential height anomolies, averaged over 1970-79 Circulation Pattern Classifications Stehlik and Bardossy, 2002, J. Hydrol. Patterns defined by a set of fuzzy rules CP 1 dry CP 8 wet Weather Typing

11. Circulation Pattern Approach Multivariate stochastic model to compute space- time distribution of daily precipitation Rainfall is modeled as a process coupled to atmospheric circulation patterns using conditional probabilities The annual cycle of model parameters including autocorrelation. Spatial correlation is modeled using a Fourier series Generates daily multi-point precipitation time- series with properties of the observed ones Stehlik and Bardossy, 2002, J. Hydrol.

12. Annual Precipitation Totals and Statistics at Selected Stations

13. Average Annual Cycle at Selected Stations and Fourier Approximation

14. Transfer Function Approach Utilizes empirically determined transfer functions that relate variables of regional or local interest to atmospheric-model simulated large-scale variables Simulated variables may include individual grid values and/or model output statistics (MOS)

15. Transfer Function Methods Linear and nonlinear regression Canonical Correlation Analysis Principal Components Analysis Singular Value Decomposition Artificial Neural Networks Nonhomogeneous hidden Markov models Empirical Orthogonal Functions Expanded Downscaling (EDS) …

16. NCE P National Centers for Environmental Prediction/National Center for Atmospheric Research ReanalysisNCEP Statistical Downscaling from a Global-scale model

17. NCEP 210 km grid spacing Retroactive 51 year record Every 5 days there is an 8-day forecast

18. Statistical Downscaling of the NCEP Output Using a Model Output Statistics (MOS) Technique Multiple linear regression equations developed for 11,000 stations Predictors chosen from over 300 NCEP variables (< 8 chosen for given equation) Predictands are maximum and minimum temperature, precipitation occurrence, and precipitation amounts Stochastic modeling of the residuals in the regression equations to provide ensemble time series 11,000 Climate Station Locations

19. Multiple Linear Regression Transform time series of precipitation –Convert wet and dry days binary time series of 1 and 0 –Wet day amounts are transformed to a normal distribution using a non-parametric probability transform Multiple linear regression with forward selection Over 300 MOS variables evaluated, < 8 selected –These include geopotential height, temp, wind, humidity, surface fluxes, zonal and meridianal moisture fluxes, … Regression conducted using a cross validation procedure

20. Stochastic Modeling of Residuals Temperature –Random number N from gaussian distribution (0, 1) –N times std dev of regression residuals = R –Add R to forecast daily temperature Precipitation Occurrence –Random number N from uniform distribution (0-1) –If N < forecasted probability of preicpitation, then precipitation occurs Precipitation Amount –Residuals modeled like Temperature –Forecasted (normal dist) are transformed back to gamma- type distribution using a non-parametric technique

21. Raw NCEP MOS - based Accuracy of precipitation forecasts Spearman Rank Correlations

22. Raw NCEPMOS - based Accuracy of 2-m maximum temperature forecasts Pearson correlations (r 2 )

23. Dynamical Downscaling 52 km grid node spacing Regional Climate Model – RegCM2 nested within NCEP

24. Dynamical Downscaling East Fork of the Carson Cle Elum Animas Alapaha

25. Dynamical Downscaling Hydrologic Model Input Data Sets

26. Percent days with no precipitation RegCM2 vs Observations OBS RegCM2 ANIMASCLE ELUM E FK CARSONALAPAHA Precipitation Threshold to match OBS precipitation frequency MONTH

27. Dynamic Downscaling of MM5

28. Climate Station Locations

29. Daily Precipitation Mean by Month Percent Rain Days by Month Observed vs MM5 20 km 5 km1.7 km

30. Daily Basin Maximum and Minimum Temperature Mean by Month Observed vs MM5 20 km5 km1.7 km

31. Bias Arise from systematic biases in models plus the effect of unresolved local forcings If the biases are systematic, statistical downscaling can incorporate the correction in the transfer function Dynamically downscaled values are corrected using historic climatology

32. Animas River Basin Colorado, USA 3 selected climate stations 1,805 km 2

33. Representative Elevation of Atmospheric Model Output based on Regional Station Observations Elevation-based Bias Correction

34. Downscaled mean-daily Precipitation by month NCEP 14 NCEPadj 15 RegCM2 26 RegCM2adj 26 SDS 18 % explained variance in observed daily

35. Downscaled mean-daily T MIN by month NCEP 75 NCEPadj 74 RegCM2 66 RegCM2adj 67 SDS 88 % explained variance in observed daily

36. Downscaled mean-daily T MAX by month NCEP 81 NCEPadj 79 RegCM2 72 RegCM2adj 72 SDS 90 % explained variance in observed daily

37. Observed vs Modeled Climatology (Model grid cell in the Ohio River Basin, USA) Wood, et al., 2002, JGR

38. A Probability-Swap Bias Correction Method A quantile-based mapping is constructed of model climatology to observed climatology, by month, for each variable (e.g. the empirical transformation of Panofsky and Brier, 1968) Model simulated values are replaced with values having the same percentiles (nonexceedence proabilities) with respect to the observed climatology that the model values have with respect to the model climatology Wood, et al., 2002, JGR

39. Temporal Distribution Issues Short term forecasts (1-14 days) are available from the Medium Range Forecast (MRF) Model –(daily values) NCEP provides 20 member ensemble of 6- month lead climate forecasts using its global spectral model (GSM) –(monthly values)

40. Disaggregation of Monthly Values to Daily Values For each month in each ensemble, randomly select one year from the observed climatology (e.g. 1995) Select the corresponding month to the ensemble month (e.g. January 1995) The observed daily values of precipitation and temperature for the selected year and month (Janurary 1995) are scaled to match the simulated months precipitation and average temperature Wood et al., 2002, JGR

41. Disaggregation of Monthly Values to Daily Values The randomly selected month provides the daily sequence of precipitation and temperature The random sampling of a climatology year for selection of daily sequences is repeated for each month in each forecast ensemble member Wood et al., 2002, JGR

42. Spatial Distribution Issues Downscaling is done to an observation point or the mean value of a set of observation points When downscaling to multiple points or sets, need to maintain spatial and temporal distributional relations within and across a basin or region (e.g. covariance among observation stations or regions)

43. Statistical Downscaling of Medium Range Forecast Models in the Upper Gunnison River Basin ~10,000 km 2 Forecast Streamflow Nodes Selected set of precipitation gages

44. Schaake Shuffle A method for reconstructing space-time variability in forecasted precipitation and temperature fields Clark et al., 2004, J. Hydrometeorology

45. Schaake Shuffle Feb 1, 2003 Forecast for Day 0 ENS# STN 1 STN2 1 8 8 2 9 6 3 4 5 4 7 7 5 6 2 Historical Data Random Date STN 1 STN2 Feb 3, 2001 7 8 Feb 5, 1996 3 7 Feb 2, 1997 9 2 Feb 1, 1992 5 3 Feb 4, 1989 6 4 4513245132 5324153241 4152341523 5412354123 Rank all data from low to high Keep ensemble number with historical value E1 E2 E3 E4 E5 Assign Ensemble number to forecast day with same rank February 1, 2003, Forecast Day 0, have 2 stations with 5 ensembles forecasted for each station E1 E2 E3 E4 E5 E1 E2 E3 E4 E5

46. Schaake Shuffle Feb 1, 2003 Forecast for Day 0 ENS# STN 1 STN2 1 8 8 2 9 6 3 4 5 4 7 7 5 6 2 E1 E2 E3 E4 E5 Forecasts are shuffled by reordering the ranked forecast data according to the assigned Ensemble # ENS# STN 1 STN2 1 8 8 2 4 7 3 9 2 4 6 5 5 7 6

47. Schaake Shuffle Forecast Day: ENS# 0 (random) 1 2 3 4 1 Feb 3 Feb 4 Feb 5 Feb 6 Feb 7 2001 2 Feb 5 Feb 6 Feb 7 Feb 8 Feb 9 1996 3 Feb 2 Feb 3 Feb 4 Feb 5 Feb 6 1997 4 Feb 1 Feb 2 Feb 3 Feb 4 Feb 5 1992 5 Feb 4 Feb 5 Feb 6 Feb 7 Feb 8 1989 Forecasts days 1- 4 are shuffled similarly, except the random historical date is no longer random, rather it is the next day in the historical time series

48. Upper Gunnison River Basin 15 real-time climate stations Look at correlation between stations calculated using observed daily values. Compare with correlation calculated using observed and SDS and Shuffled- SDS output

49. 0.6 0.7 0.8 0.9 1.0 1.0 0.9 0.8 0.7 0.6 0.7 0.8 0.9 1.0-0.1 0.1 0.3 0.5 0.7 1.0 0.9 0.8 0.7 1.0 0.9 0.8 0.7 0.5 0.3 0.1 0.7 0.5 0.3 0.1 0.6 0.7 0.8 0.9 1.0 1.0 0.9 0.8 0.7 0.6 0.7 0.8 0.9 1.0 Correlation: Observed vs. Observed Correlation Observed vs. Simulated Precipitation Minimum Temperature Maximum Temperature SDS Shuffled-SDS FCday 8 FCday 1

50. Summary Provided an overview of statistical and dynamical downscaling methods Statistical downscaling supports decadal to century timescales Dynamical downscaling currently limited to 10’s of years applications by computer speeds Statistical downscaling assumptions may not be correct for future climate scenario generation. Dynamical downscaling may provide a more physically correct approach but questions of bias need resolved Comparisons to date show no consistent choice of one method over the other

51. Summary Demonstrated higher degree skill of one method over the over varies with the climatic and physiographic region of the world and the performance measures used Question not discussed is how do these methods compare in terms of their use in hydrological models This is the topic of my next presentation TO BE CONTINUED