uncertainty representation and quantification in precipitation data records yudong tian
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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. - PowerPoint PPT PresentationTRANSCRIPT
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
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1. What is uncertainty
2. Uncertainty quantification relies on error modeling
3. Finding a good error model
4. Uncertainties in precipitation data records
5. Conclusions
Outline
Uncertainty quantification is to know how much we do not know
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“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
Information
“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
Uncertainty
But how much?
Uncertainty determines reliability of information
What we do not know affects what we know
Information
KnownsKnowledge
Signal Deterministic
Systematic errors
yUncertaint1~yReliabilit
Uncertainty
UnknownsIgnorance Noise StochasticRandom errors
Uncertainty
UnknownsIgnorance Noise StochasticRandom errors
Information
KnownsKnowledge
Signal Deterministic
Systematic errors
For ESDRs, uncertainty quantification amounts to determining systematic and random errors
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Systematic and random error are defined by the error model
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Error model determines the uncertainty definition and representation
Ti
Xi
Ti
Xi
Xi: measurements in data records
Ti: truth, error free.
a, b: systematic error -- knowledge
ε: random error -- uncertainty
The multiplicative error model:
or
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The additive error model:
Two types of error models can be used for precipitation data records
ii bTaX eTX ii
)ln()ln( ii TbaX
8Which one is better?
Ti
Xi
Ti
Xi
Different error models produce incompatible definition of uncertainty ε
ii bTaX eTX ii
1. It cleanly separates signal and noise
2. It has good predictive skills
What is a good error model?
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1. Mixes signal and noise2. Lack of predictive skills
Ti
Xi
Ti
Xi
A bad error model:
Under-fitted model: systematic leaking into random errors
Over-fitted model: random leaking into
systematic errors
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
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Additive error model: under-fitting makessystematic errors leak into random errorsAdditive Model Multiplicative Model
3B42RT Mean Daily Rainrate
Uncertainty will be inflated due to the leakage
)ln()ln( ii XbaY ii XbaY
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Error leakage produces random errors with a complex dependency and distribution
Additive Model Multiplicative Model
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The multiplicative error model predicts better
Additive Model Multiplicative Model
Model-predicted measurements
Actual measurementsComparison of data distributions
Testing multiplicative model on more data records
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)()ln()ln( stdevXbaY ii
σ(amplitude of random error -- uncertainty)
TMPA 3B42 TMPA 3B42RT NOAA Radar
b
Spatial distribution of the model parameters
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)()ln()ln( stdevXbaY ii
a and b (systematic error)
TMPA 3B42 TMPA 3B42RT NOAA Radar
a
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
eTX ii
Uncertainty in satellite sensor data
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TMI
AMSR-E
)()ln()ln( stdevXbaY ii
σ(random error - uncertainty)
a b
TMI
AMSR-E
)()ln()ln( stdevXbaY ii Systematic error in satellite sensor data
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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.
Summary
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• 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, 4:45Wednesday: • 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
References
Extra slides
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What we do not know hurts what we know
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Knowns | UnknownsKnowledge | Ignorance
Signal | Noise---------------------------------------------------------
Information | Uncertainty
Uncertainty determines the information content
yUncertaintContentnInformatio 1
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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)
The multiplicative error model
),0(~ 2 N
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Additive model does not have a constant variance
For ESDR, uncertainty quantification amounts to determining systematic and random errors
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Knowns | UnknownsKnowledge | Ignorance
Signal | NoiseDeterministic | Stochastic
Predictable | UnpredictableSystematic errors | random errors
---------------------------------------------------------
Uncertainty determines the information content
yUncertaintContentnInformatio 1
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• 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.
The multiplicative error model has clear advantages
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Probability distribution of the model parameters
A B σ
TMI
AMSR-E
F16
F17
)()log()log( stdevXBAY ii
Spatial distribution of the model parameters
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TMI
AMSR-E
F16
F17
)()log()log( stdevXBAY ii A B σ(random error)
Spatial distribution of the model parameters
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TMI
AMSR-E
)()log()log( stdevXBAY ii A B σ(random error)
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Correct error model is critical in quantifying uncertainty
Ti
Xi
Ti
Xi
Ti
Xi
Optimal combination of independent observations(or how human knowledge grows)
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Information content
“Conservation of Information Content”
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Why uncertainty quantification is always needed
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Information content
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
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