1. Using Synthetic Data to Test Downscaling Methods John Lanzante (GFDL/NOAA)

2. CONCEPTS Testing Downscaling: Like Product Testing My Product 

3. STEP1: Recruit Test Subjects (Gather Data) CONCEPTS STEP2: Feed Cereal For Several Decades (Apply Downscaling Method)

4. CONCEPTS STEP3: How are subjects affected? How well did downscaling do? Not so clear – Need more subjects? Need more data? Real-world data may be limited? Can we generate synthetic data to fill the void?

5. CONCEPTS STEP 4a: Snowmen most affected? Generate a new sample.

6. CONCEPTS STEP 4b: Snow-women affected differently? Generate a new sample.

7. REALISTIC EXAMPLES CASE 1 – Linearity: Simplest downscaling – linear regression

8. REALISTIC EXAMPLES CASE 1 – Strong Nonlinearity: Simplest downscaling – linear regression

9. REALISTIC EXAMPLES SUMMARY CASE 1 – Nonlinearity: Hard to test nonlinearity in real-world data ? (if we are just entering “non-linear regime”) Simulate various degrees of nonlinearity Compare linear & nonlinear downscaling methods Determine amount of degradation Determine time in future when degradation becomes “too large”

10. REALISTIC EXAMPLES CASE 2 – Coastal Error: Downscaling error maximizes along coastline

11. REALISTIC EXAMPLES CASE 2 – Coastal Error: Obs gridpoint  Entirely land Model gridpoint  Partly land, partly ocean

12. REALISTIC EXAMPLES CASE 2 – Coastal Error: Land more detail (extremes) than Ocean (damped) Missing peaks & troughs unrecoverable

13. REALISTIC EXAMPLES SUMMARY CASE 2 – Costal Error: Simulate land & ocean points Downscale land from mixture (land + sea) Vary the proportions of the mixture Is coastal effect due to mixture/mismatch?

14. SYNTHETIC DATA MODEL One Particular Synthetic Data Model: O= Observations M= Model y= year d= day Red = free parameter (user selects the value) O y d = Ō y + O ’ y d  Yearly mean + AR1 O ’ y d = rlag1 * O ’ y d-1 + a y d  AR1 fvar = var Ō / var O [ var O = var Ō + var O ] M y d = O y d + b y d corr = correlation(O,M) a ~ N(0,var a ) Proper choice of a & b b ~ N(0,var b ) yields desired rlag1 & corr

15. SYNTHETIC DATA MODEL STEP 1: Generate Base Time Series rlag1 day-to-day persistence fvar interannual vs. day-to-day variability corr strength of relation: model vs. obs STEP 2: Historical Adjustment mean OBS characteristics of the distribution mean MODEL var OBS var MODEL STEP 3: Future Adjustment mean OBS characteristics of the distribution mean MODEL var OBS var MODEL

16. SYNTHETIC DATA MODEL OUR APPLICATIONS OF THIS MODEL: Downscaling (just getting started) No results yet Applied successfully to several related issues (cross-validation, exceedance statistics, testing two distributions)

17. SUMMARY REAL-WORLD COMPLICATIONS: Results may not be clear-cut: Sample size too small? Multiple factors may contribute? Some conditions more interesting? SOLUTION – GENERATE SYNTHETIC DATA: Advantages of Synthetic Data: Unlimited sample size (enhance signal/noise) Change one factor at a time Prescribe exact conditions Vary factor over a wide range (“turn the knob”) Can extend outside the range of historical data Turn knob “all the way” for unambiguous results

18. A CAUTIONARY NOTE No “One Size Fits All”: No single “best” synthetic data model Must possess appropriate real-world characteristics Ability to vary the relevant factors Possible Models For Future Development: Skewed data (transform Gaussian data nonlinearly?) Precipitation (discrete Markov + bounded distribution?) Model occurrence & amount separately? Multivariate model?

19. THE END

20. REALISTIC EXAMPLES CASE 1 – Weak Nonlinearity: Simplest downscaling – linear regression

21. SUPPLEMENTAL Causes of Nonlinearity? At highest T – model soil becomes excessively dry – T becomes excessive Other possibilities: Water Vapor, Clouds, Sea-Ice, etc.

22. REALISTIC EXAMPLES CASE 2 – Coastal Error: Land  More extremes Ocean  Damped

23. REALISTIC EXAMPLES CASE 2 – Coastal Error: X/Y Plot: Land (model) vs. Land (obs)

24. REALISTIC EXAMPLES CASE 2 – Coastal Error: X/Y Plot: Ocean (model) vs. Land (obs)

25. SYNTHETIC DATA MODEL STEP 4: Fit downscaling model to historical sample STEP 5: Test downscaling in historical & future samples OUR APPLICATIONS OF THIS MODEL: No results to show today Downscaling (just getting started) Guidance in the use of cross-validation Biases in exceedance statistics Testing difference between 2 distributions