Uncertainty Representation and Quantification in Precipitation Data Records Yudong Tian

1. Uncertainty Representation and Quantification in Precipitation Data Records Yudong Tian Collaborators: Ling Tang, Bob Adler, George Huffman, Xin Lin, Fang Yan, Viviana Maggioni and Matt Sapiano University of Maryland & NASA/GSFC http://sigma.umd.edu Sponsored by NASA ESDR-ERR Program

2. Outline • What is uncertainty • 2. Uncertainty quantification relies on error modeling • Finding a good error model • Uncertainties in precipitation data records • 5. Conclusions

3. Information Uncertainty “There are known knowns. These are things we know that we know.”There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know.” -- Donald Rumsfeld “There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know.” -- Donald Rumsfeld But how much? Uncertainty quantification is to know how much we do not know “There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know.” -- Donald Rumsfeld

4. What we do not know affects what we know Information Knowns Knowledge Signal Deterministic Systematic errors Uncertainty Unknowns Ignorance Noise Stochastic Random errors Uncertainty determines reliability of information

5. For ESDRs, uncertainty quantification amounts to determining systematic and random errors Information Knowns Knowledge Signal Deterministic Systematic errors Uncertainty Unknowns Ignorance Noise Stochastic Random errors 5

6. Xi Xi Ti Ti Systematic and random error are defined by the error model Error model determines the uncertainty definition and representation 6

7. Two types of error models can be used for precipitation data records • The additive error model: • The multiplicative error model: • or • Xi: measurements in data records • Ti: truth, error free. • a, b: systematic error -- knowledge • ε: random error -- uncertainty

8. Xi Xi Ti Ti Different error models produce incompatible definition of uncertainty ε Which one is better? 8

9. What is a good error model? • It cleanly separates signal and noise • 2. It has good predictive skills

10. A bad error model: • Mixes signal and noise • 2. Lack of predictive skills Under-fitted model: systematic leaking into random errors Over-fitted model: random leaking into systematic errors Xi Xi 10 Ti Ti

11. Test with NASA Precipitation Data • Data: TMPA 3B42RT [ Tropical Rainfall Measuring Mission (TRMM) Multi-satellite Precipitation Analysis (TMPA) Version 6 real-time product, 3B42RT ] • Reference data: CPC-UNI [ Climate Prediction Center (CPC) Daily Gauge Analysis for the contiguous United Sates ] • Study period: three years [ 09/2005-08/2008 ] • Resolution: daily, 0.25-degree 11

12. 3B42RT Mean Daily Rainrate Additive error model: under-fitting makes systematic errors leak into random errors Additive Model Multiplicative Model Uncertainty will be inflated due to the leakage

13. Error leakage produces random errors with a complex dependency and distribution Additive Model Multiplicative Model 13

14. Model-predicted measurements Comparison of data distributions Actual measurements The multiplicative error model predicts better Additive Model Multiplicative Model

15. Testing multiplicative model on more data records σ(amplitude of random error -- uncertainty) TMPA 3B42 TMPA 3B42RT NOAA Radar 15

16. Spatial distribution of the model parameters b a and b (systematic error) TMPA 3B42 TMPA 3B42RT NOAA Radar a 16

17. Uncertainty quantification in sensor data • Time period: 3 years, 2009 ~ 2011 • Reference: Q2 [ NOAA NSSL Next Generation QPE, bias-corrected with NOAA NCEP Stage IV (hourly, 4-km) ] • Satellite sensor ESDRs: TMI and AMSR-E [ TMI: TRMM Microwave Imager; AMSR-E: Advanced Microwave Scanning Radiometer for EOS onboard Aqua ] • Resolution: 5-minute, 0.25-degree • Error Model: 17

18. Uncertainty in satellite sensor data σ(random error - uncertainty) TMI AMSR-E 18

19. Systematic error in satellite sensor data a b TMI AMSR-E

20. Summary • 1. Uncertainty in data record is defined by error model • 2. A good error model • -- simplifies uncertainty quantification [ σ vs. σ=f(Ti) ] • -- produces accurate and consistent uncertainty info • -- has predictive skills • 3. Multiplicative model is recommended for high resolution precipitation data records • 4. A standard error model unifies uncertainty definition and quantification, helps end users. 20

21. References • Tian et al., 2012: Error modeling for daily precipitation measurements: additive or multiplicative? submitted to Geophys. Rev. Lett. • Monday: • M. R. Sapiano; R. Adler; G. Gu; G. Huffman: Estimating bias errors in the GPCP monthly precipitation product, IN14A-04, 4:45 • Wednesday: • Ling Tang; Y. Tian; X. Lin:Measurement uncertainty of satellite-based precipitation sensors.  H33C-1314, 1:40 PM (poster). • Viviana Maggioni; R. Adler; Y. Tian; G. Huffman; M. R. Sapiano; L. Tang: Uncertainty analysis in high-time resolution precipitation products. H33C-1316, 1:40 PM (poster). • Thursday: • Uncertainties in Precipitation Measurements and Their Hydrological Impact • Conveners: Yudong Tian and Ali Behrangi • Posters (H41H), 8:00 AM -12:20 PM • Oral (H44E), 4:00 PM – 6:00 PM, Room 3018 • Website: • http://sigma.umd.edu 21

22. Extra slides 22

23. What we do not know hurts what we know Knowns | Unknowns Knowledge | Ignorance Signal | Noise --------------------------------------------------------- Information | Uncertainty Uncertainty determines the information content 23

24. The multiplicative error model • A nonlinear multiplicative measurement error model: • Ti: truth, error free. Xi: measurements • With a logarithm transformation, • the model is now a linear, additive error model, with three parameters: • A=log(α), B=β, xi=log(Xi), ti=log(Ti)

25. Additive model does not have a constant variance 25

26. For ESDR, uncertainty quantification amounts to determining systematic and random errors Knowns | Unknowns Knowledge | Ignorance Signal | Noise Deterministic | Stochastic Predictable | Unpredictable Systematic errors | random errors --------------------------------------------------------- Uncertainty determines the information content 26

27. The multiplicative error model has clear advantages • Clean separation of systematic and random errors • More appropriate for measurements with several orders of magnitude variability • Good predictive skills • Tian et al., 2012: Error modeling for daily precipitation measurements: additive or multiplicative? to be submitted to Geophys. Rev. Lett.

28. Probability distribution of the model parameters A B σ TMI AMSR-E F16 F17

29. Spatial distribution of the model parameters A B σ(random error) TMI AMSR-E F16 F17 29

30. Spatial distribution of the model parameters A B σ(random error) TMI AMSR-E 30

31. Correct error model is critical in quantifying uncertainty Xi Xi Xi Ti Ti Ti

32. Optimal combination of independent observations (or how human knowledge grows) Information content 32

33. “Conservation of Information Content”

34. Why uncertainty quantification is always needed Information content

35. Summary and Conclusions Created bias-corrected radar data for validation Evaluated biases in PMW imagers: AMSR-E, TMI and SSMIS Constructed an error model to quantify both systematic and random errors 35