Improving estimation and prediction of extremes using CBPKF
Improving Estimation and Prediction of Extremes Using Conditional Biaspenalized Kalman Filter M. M. Saifuddin1, D.-J. Seo1, H. Lee2, S. Noh1, and J. Brown3 1Dept.
of Civil Eng., The Univ. of Texas at Arlington, Arlington, TX 2LEN Technologies, Oak Hill, VA. 3Hydrologic Solutions Limited, Southampton, UK This material is based upon work supported by NSF under Grants No. IIP 1237767 and CyberSEES-1442735, and by NOAA/OAR/OWAQ/JTTI under Grant No. NA16OAR4590232. Jan 23, 2017
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Improving estimation and prediction of extremes using CBPKF
Motivation • “Optimal” state estimation
for highly uncertain systems commonly under- and overestimate large and small predictands, respectively • This is wholly
unsatisfactory if estimation/prediction of extremes is of primary concern, • but is inevitable given the optimality criterion inherent in all least squares-based approaches Jan 23, 2017
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Improving estimation and prediction of extremes using CBPKF
Optimal linear estimation (OLE) Z HX V
• Observation model
• Estimator
X * WZ
• Minimize
EV E X , X * [( X X * )( X X * )T ]
• Optimal estimate • Error variance
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X * [ H T R 1 H ]1 H T R 1Z
[ H T R 1 H ]1
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Fisher solution
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Improving estimation and prediction of extremes using CBPKF
Co-minimize Type-II error
• Type-II error (a miss)
E[ Xˆ | X x] x
• Cannot be reduced by calibration
• Type-I error (a false hit)
E[ X | Xˆ xˆ ] xˆ
• Can be reduced by calibration
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Improving estimation and prediction of extremes using CBPKF
Conditional bias-penalized OLE (Seo 2012) • Minimizes
EV CB
* * T where CB E X [( X E X * [ X | X x])( X E X * [ X | X x])
• Estimate
X * [ Hˆ T 1 H ]1 Hˆ T 1Z
• Estimation variance
T 1 1 ˆ [ H H ]
CBPOLE solution
T T 1 where Hˆ H XX XZ 1 R (1 )ZX XX XZ HXZ ZX H T Hˆ T 1 Hˆ I XX
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Improving estimation and prediction of extremes using CBPKF
CBP-OLE • Streamflow prediction • Brown and Seo (2012, Hydrologic Processes) • Conditional bias-penalized kriging • Seo (2012, Stosch Environ Res Risk Analysis) • Extended Rain gauge-only precipitation analysis • Seo et al. (2014, Journal of Hydrology)
• Radar-gauge multisensor precipitation analysis • Kim et al. (2016, Journal of Hydrology) Jan 23, 2017
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Improving estimation and prediction of extremes using CBPKF
CBPKF solution (Seo and Saifuddin submitted to SERRA) • Filtered estimate
Xˆ k |k Xˆ k |k 1 K k [Z k H k Xˆ k |k 1 ] • CBPK gain
K k [ H kT 11H k 221H k 22 ]1[ H kT 11 21] • Estimation variance
k |k (1 )XX {(1 )[ H kT 11 H k 221 H k 22 ]}1 • Use XX Jan 23, 2017
k |k 1 AMS Meeting, Seattle, WA
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Improving estimation and prediction of extremes using CBPKF
Numerical experiment • Dynamical model (AR(1))
X k k 1 X k 1 Gk 1wk 1 k 1 X k 1 w,k 1wk 1 • Observation eq.
Z k H k X k Vk UX k Vk Jan 23, 2017
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Improving estimation and prediction of extremes using CBPKF
Numerical experiment (cont.) • Stationary • Various levels of predictability, model uncertainty, obs. uncertainty
• Nonstationary • Time-varying predictability
p k 1
k 1
0.5 kp1 0.95
• Time-varying model uncertainty
p w, k 1
w,k 1 w w
wp,k 1 0.01
• Time-varying observational uncertainty
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p v,k
v , k v v
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vp,k 0.01 9
All stationary
Nonstationary model error
All nonstationary
Nonstationary obs. error
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Improving estimation and prediction of extremes using CBPKF
CBPKF very significantly reduces conditional RMSE
CBPKF slightly increases unconditional RMSE Jan 23, 2017
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Improving estimation and prediction of extremes using CBPKF
CBPKF error variance is generally larger and more accurate
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Improving estimation and prediction of extremes using CBPKF
Real-world experiment • Comparatively evaluate EnCBPKF with EnKF for streamflow
prediction • 23 headwater basins in TX • Models used - Sacramento soil moisture accounting, unit hydrograph • Only significant events considered (peak flow > 100 m3/s) • Model soil moisture states augmented with modelsimulated flow to render observation eq. linear (Rafael Nasal et al. 2014) Jan 23, 2017
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0.1
0.2
0.3
All data 7.6 m3/s (Pr=0.25) 17.6 m3/s (Pr=0.5) 32.7 m3/s (Pr=0.75) 102.8 m3/s (Pr=0.9) 135.5 m3/s (Pr=0.95)
0
Mean CRPSS
0.4
Mean continuous ranked probability skill score (CRPSS) of EnCBPKF forecasts over EnKF forecasts for MCKT2 (perfect QPF assumed)
0
1
2
3 (Days)
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0.1
0.2
0.3
All data 2.8 m3/s (Pr=0.25) 18.9 m3/s (Pr=0.5) 69.7 m3/s (Pr=0.75) 125.2 m3/s (Pr=0.9) 164.5 m3/s (Pr=0.95)
0
Mean CRPSS
0.4
Mean CRPSS of EnCBPKF for MDST2 (perfect QPF assumed)
0
1
2
3 (Days)
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0.04
0.08
0.12
All data 2.9 m3/s (Pr=0.25) 11.9 m3/s (Pr=0.5) 72.1 m3/s (Pr=0.75) 131.0 m3/s (Pr=0.9) 172.7 m3/s (Pr=0.95)
0
Mean CRPSS
0.16
Mean CRPSS of EnCBPKF for MTPT2 (perfect QPF assumed)
0
1
2
3 (Days)
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
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Improving estimation and prediction of extremes using CBPKF
Conclusions and future research recommendations • EnCBPKF improves over EnKF for prediction of high flows out to about
24 hrs of lead time for basins less than 1,000 km2 in TX • The improvement is very significant for short lead times but diminishes quickly • EnCBPKF predictions are better able to reflect the observed flow both in rising and falling limbs and provides more realistic uncertainty spread • EnCBPKF is computationally significantly more expensive • Needs to solve additional (nxn) and (mxm) linear systems where n and m are the sizes of the observation and state vectors, respectively • Future research needs include • Computationally efficient formulation, objective specification the weight, α • Dealing with phase errors in streamflow assimilation • Additional experience from diverse applications Jan 23, 2017
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Hydrologic Applications of Weather Radar - An Urban View
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