Analysis of Radar-Rainfall Uncertainties and effects on Hydrologic Applications. Emad Habib, Ph.D., P.E. University of Louisiana at Lafayette

Transcription

1 Analysis of Radar-Rainfall Uncertainties and effects on Hydrologic Applications Emad Habib, Ph.D., P.E. University of Louisiana at Lafayette

2 Motivation Rainfall is a process with significant variability across a wide-range of scales Such variability has implications for variety of engineering/environmental applications A main challenge is how to measure rainfall and characterize its variability Flash Floods / drought Water management Agricultural

3 Rain Gauges Most representative of how much rainfall reaches the surface at a certain point Known sources of errors: Wind under-catch Evaporation Losses Wetting Losses Calibration Errors

4 Rain Gauge Data: spatial coverage problem Correlation coefficient Central Florida adjusted light rain heavy rain Gauge separation distance (km)

5 More importantly! Questionable Data Quality There are always issues with rain gauge data. Missing data, Zero reports, transmission errors, tipping bucket errors, poorly maintained equipment, staggered reporting times etc. --- OHRFC Source: Brian Nelson, NCDC

6 Weather Radar Good temporal and spatial resolution: (6 minute, 1 km x 1 o ) Limitations Calibration problems Z-R relationship Vertical profile variations Brightband and hail contamination beam blocking Improper beam filling and overshooting Evaporative, condensational, and wind effects Range degradation

7 Satellite Rain Gauges WSR-88D Multi-Sensor Precipitation Estimator (MPE)

8 Solution: Multi-Sensor Rainfall Estimates Rain Gauges CONUS-wide rainfall data Developed for Quantitative Hydrologic Forecasting Have their own unknown uncertainties! Satellite WSR-88D Multi-Sensor Precipitation Estimator (MPE)

9 4x4 km 2 MPE radar pixels Experimental Site: Goodwin Creek, Mississippi 4 Km Area ~ 21 km 2 annual rainfall ~ 1440 mm annual runoff ~ 145 mm pasture, forest, cultivated land silt loam and silt clay

10 Hydrologic Model: GSSHA Gridded Surface Subsurface Hydrologic Analysis A physically-based fully distributed hydrologic model (Ogden and Downer, 2002). Uses finite difference and finite volume methods to simulate different hydrologic processes: Precipitation; plant interception; infiltration; unsaturated soil water movement; ET; overland flow; channel routing; lateral groundwater flow Model setup in this study: two-dimensional diffusive wave for overland flow one-dimensional explicit diffusive wave method for channel flow Penman-Monteith equation for ET. Green&Ampt infiltration with redistribution for the unsaturated zone Parameters assigned based on land-use & soil types Overland and channel roughness Soil hydraulic parameters (Ks, porosity, etc.) ET parameters

11 Hydrologic Model Calibration 25 Observed Predicted (calibration) 20 Discharge (m 3 /sec) /20/1987 0:00 2/22/1987 0:00 2/24/1987 0:00 2/26/1987 0:00 2/28/1987 0:00 3/2/1987 0:00

12 Hydrologic Model Validation 40 Observed Predicted (validation) Discharge (cms) /17/2002 1/19/2002 1/21/2002 1/23/2002 1/25/2002

13 20 Precipitation intensity (mm/hr) MPE Pixel gauge-avergae Discharge (Cms) /2/ :00 5/3/2002 0:00 5/3/ :00 5/4/2002 0:00

14 Precipitation intensity (mm/hr) Discharge (Cms) Unadjusted MPE Gages_avg_middlepixel Unadjusted MPE Pixel gauge-average 0 3/20/2002 1:00 3/20/2002 7:00 3/20/ :00 3/20/ :00 3/21/2002 1:00

15 Objective -- Characterize uncertainties in radarbased rainfall estimates and.. -- Develop a methodology for propagating radar uncertainties into hydrologic applications

16 Approach 1. Perform validation/verification Analysis of radar-based MPE products: Quantify and characterize uncertainty in radar-rainfall estimates 2. Develop a radar error model Stochastic simulation of ensembles of error fields 3. Propagation of radar uncertainties into hydrologic prediction application Runoff prediction uncertainty

17 Radar Rainfall Validation Issues Rain gauges are the only reasonable ground reference standard but Acute problem in the sampling area difference: some 8 orders of magnitude! Temporal differences Difference in sample location Appropriate statistical methodologies are not well established

18 Hypothesis Validation of MPE products is possible only if we have a network of gauges that is: Independent High quality of data High density within scale of MPE product

19 4x4 km 2 MPE radar pixels Experimental Site: Goodwin Creek, Mississippi 4 Km Area ~ 21 km 2 annual rainfall ~ 1440 mm annual runoff ~ 145 mm pasture, forest, cultivated land silt loam and silt clay

20 Independent Rain Gauge Network KPOE ÊÚ Rain Gauge Station Discharge Gauge Station Fenstermaker Commission Blvd KLCH New Orleans Covenant Church ÊÚ Gulf South Lafayette Vineyard STM Carriage Light Loop Kilometers Vincent Rd. LaNeuville Rd Millcreek Rd

21 Rain Gauge Sites

22 Monthly Comparisons Depth (mm) Depth (mm) Depth (mm) Rainfall 2004 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2005 Rainfall 2005 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2006 Rainfall 2006 Gauge MPE Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

23 Validation at hourly scale Mean (MPE) (mm) 1.53 Mean (gauge) (mm) 1.59 Relative Bias MPE (mm) Gauge (mm) Standard Deviation -- MPE (mm) 4.03 Standard Deviation -- Gauge (mm) 4.76 Relative RMSE 1.16 Correlation Coefficient 0.93 MPE (mm) Gauge (mm)

24 80 Hydrologic predictions driven by MPE MPE (mm) Gauge (mm) Discharge (m 3 /sec) 70 storm 12 reference rainfall unadjusted MPE 60 unconditional adjustment conditional adjustment Time in days since 1/1/ :00

25 Now what! What should we do with these uncertainties? How can we incorporate them into an application of interest? MPE (mm) Gauge (mm) We need a realistic yet practical model for the radar uncertainties MPE (mm) Gauge (mm)

26 Approach It is difficult and unpractical to model each error source independently The idea is to treat different sources of radar uncertainties as a combined error Rs ε = R r Characterize probability distribution and dependence structure Develop an ensemble generator of surface-rainfall fields It has to stay faithful to what we learned about the error characteristics Propagate radar errors and generate ensemble of hydrologic simulations and quantify the end uncertainties

27 Radar Error ε = R s R r Rain Gauge Station Discharge Gauge Station Fenstermaker Commission Blvd Covenant Church Gulf South Lafayette Vineyard STM Carriage Light Loop Vincent Rd Millcreek Rd. LaNeuville Rd

28 Radar Error ε = R R s r

29 Radar Error ε = R R s r Systematic component Random component

30 Overall Bias: Systematic Component Conditional Bias: Bias Factor Relative Bias B = CB( r E[ R E[ R Spatial variability of radar bias > Gauge Intensity (mm/h) Storm Number r ) = s r ] ] E[ R s R r Surface rainfall radar rainfall r r = r r ]

31 Discharge (m 3 /sec) storm 2 reference rainfall unadjusted MPE unconditional adjustment conditional adjustment Time in days since 1/1/ :00 Discharge (m 3 /sec) 70 storm 12 reference rainfall unadjusted MPE unconditional adjustment conditional adjustment Time in days since 1/1/ :00

32 Random Component: Error Probability Distribution Marginal Distribution: Bias and Variance: E{ε}; VAR{ε} ε = R R s r surface rainfall radar rainfall Conditional dependence: ε R Joint Distribution: Spatial auto-correlation: ρ ε (s) Temporal auto-correlation: ρ ε (t)

33 Error Marginal Distribution 1 Probability normal lognormal ε = R R s r surface rainfall radar rainfall Radar random error (b) (b)

34 VAR{ε} constant VAR{ε} = f (R r ) VAR{ε R r =r} 8 Relative STD(error) > Gauge Intensity (mm/h)

35 Error Spatial Dependencies Radar Rainfall Corr. Matrix Radar Error Corr. Matrix Spatial dependencies of radar error fields are NOT negligible

36 Error Temporal Dependencies Time Lag: 15min Time Lag: 30min Time Lag: 45min Temporal Auto-correlation Matrix Auto-correlation of each pixel with itself and all other pixels Temporal dependencies of radar error fields seem to be negligible

37 Ensemble generator Main error (ε) characteristics: Lognormal distribution Conditional bias Conditional variance Correlation in space Generate random-fields of radar errors: ε = f (s, t, R r ) R Substitute in ε = s to generate realizations of probable Rr surface rainfall fields that reflect radar uncertainties

38 Simulation Scenarios CASE 1: - Variance of radar error is independent of radar magnitude. -No spatial correlation in simulated error fields. Var(ε) = Const. ρ ε = 0 CASE 2: - Variance of radar error is NOT independent of radar magnitude. - No spatial correlation in simulated error fields. Var(ε) = f (R( r ) ρ ε = 0 CASE 3: -Variance of radar error is NOT independent of radar magnitude. - Spatial correlation in simulated error fields is preserved. Var(ε) = f (R( r ) ρ ε 0

39 Simulation of Radar Error - Case 1 Var(ε) = Const. ρ ε = 0 Simulated errors (yellow dots) do NOT follow the trend of observed errors (blue plus sign) Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

40 Simulation of Radar Error - Case 1 Var(ε) = Const. ρ ε = 0 No conditioning on error large errors may imposed on a large magnitudes of radar and result in unrealistically large simulated radar data. Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

41 Simulation of Radar Error - Case 2 Var(ε) = f (R( r ) ρ ε = 0 Standar Deviation of Log Error for a Moving Window y = e x R 2 = Radar (m m /h) Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

42 Simulation of Radar Error - Case 2 Var(ε) = f (R( r ) ρ ε = 0 Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

43 Simulation of Radar Error - Case 2 Var(ε) = f (R( r ) ρ ε = 0 Correlation of Simulated Error Field Correlation of Observed Error Field Correlation of Simulated Radar Field Correlation of Observed Radar Field Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

44 Simulation of Radar Error - Case 3 Var(ε) = f (R( r ) ρ ε 0 The variance-covariance matrix can be decomposed using the Cholesky decomposition and used with Monte Carlo method to simulate random fields with similar variance-covariance matrices. C = LL * Where L and L * are lower triangular matrix and its transpose respectively. ε = µ + L Ω * i i i ε = i µ = i * L = Ω = i Simulated vector of error Mean error vector Decomposed var-covar matrix Random number generator Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

45 Simulation of Radar Error - Case 3 Var(ε) = f (R( r ) ρ ε 0 Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

46 Simulation of Radar Error - Case 3 Var(ε) = f (R( r ) ρ ε 0 Correlation of Simulated Error Field Correlation of Observed Error Field Correlation of Simulated Radar Field Correlation of Observed Radar Field Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

47 60 Discharge (m 3 /sec) Time in days since 1/1/2002

48 Conclusions Radar error has complex structure in terms of its marginal and joint distribution It follows a log-normal distribution with nonconstant variance Error auto-correlations are non-negligible and sometimes significant (especially spatial correlations)

49 Conclusions Using oversimplified models of radar error, or ignoring its dependency structure, lead to unrealistic representation of hydrologic model uncertainties (too narrow or too wide) Runoff uncertainty bounds were sensitive to the assumed error spatial correlation (especially during the rising parts of the hydrographs)

50 Closing Remarks. Radar-based estimates are valuable resource for hydrologic application ---their uncertainties have to be quantified and modeled We tried to provide insight into the complex spatiotemporal characteristics of radar error and how they impact our interpretation of uncertainties in hydrologic predictions A radar-error model has been tested in terms of ability to re-produce plausible surface rainfall fields This model can be applied to various hydrological /environmental /ecological applications that rely on radar-rainfall estimates produce prediction uncertainties

51 Thank You! Questions.