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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