Analysis of Radar-Rainfall Uncertainties and effects on

Analysis of Radar-Rainfall Uncertainties and effects on

Hydrologic Applications

Emad Habib, Ph.D., P.E.Emad Habib, Ph.D., P.E.University of Louisiana at Lafayette University of Louisiana at Lafayette

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 / droughtWater management

Agricultural

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

Rain Gauge Data: spatial coverage problem

Gauge separation distance (km )

Cor

rela

tion

coef

ficie

nt

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1adjusted

light rain

heavy rain

Central FloridaCentral Florida

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, NCDCSource: Brian Nelson, NCDC

Weather Radar

• 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

• Good temporal and spatial resolution:(6 minute, 1 km x 1o)

Multi-Sensor PrecipitationEstimator (MPE)

WSR-88DRain GaugesSatellite

Solution: Multi-Sensor Rainfall Estimates

• CONUS-wide rainfall data

• Developed for Quantitative Hydrologic Forecasting

• Have their own unknown uncertainties!

Multi-Sensor PrecipitationEstimator (MPE)

WSR-88D

Rain Gauges

Satellite

4 Km

4x4 km2

MPE radar pixels

Experimental Site:Goodwin Creek, Mississippi

•Area ~ 21 km2

•annual rainfall ~ 1440 mm •annual runoff ~ 145 mm•pasture, forest, cultivated land•silt loam and silt clay

Hydrologic Model: GSSHAGridded 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

0

5

10

15

20

25

2/20/19870:00

2/22/19870:00

2/24/19870:00

2/26/19870:00

2/28/19870:00

3/2/1987 0:00

Dis

char

ge (m

3 /sec

)

Observed Predicted (calibration)

Hydrologic Model Calibration

Hydrologic Model Validation

0

10

20

30

40

1/17/2002 1/19/2002 1/21/2002 1/23/2002 1/25/2002

Dis

char

ge (c

ms)

Observed Predicted (validation)

0

4

8

12

16

20P

reci

pita

tion

inte

nsity

(mm

/hr)

MPE

Pixel gauge-avergae

0

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60

5/2/2002 12:00 5/3/2002 0:00 5/3/2002 12:00 5/4/2002 0:00

Dis

char

ge (C

ms)

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30

Pre

cipi

tatio

n in

tens

ity (m

m/h

r) Unadjusted MPE

Gages_avg_middlepixel

0

10

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40

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80

3/20/2002 1:00 3/20/2002 7:00 3/20/2002 13:00 3/20/2002 19:00 3/21/2002 1:00

Dis

char

ge (C

ms)

Unadjusted MPE

Pixel gauge-average

Objective

-- Characterize uncertainties in radar-based rainfall estimates

and …..

-- Develop a methodology for propagating radar uncertainties into hydrologic applications

Approach1. 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

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

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

4 Km

4x4 km2

MPE radar pixels

Experimental Site:Goodwin Creek, Mississippi

•Area ~ 21 km2

•annual rainfall ~ 1440 mm •annual runoff ~ 145 mm•pasture, forest, cultivated land•silt loam and silt clay

ÊÚ

ÊÚ

0 100 200 Kilo m eters

KPOE

KLCH New Orleans

KPOE

KLCH New Orleans

Independent Rain Gauge Network

724 -200

724 -201

723 -201

Rain Gauge StationDischarge Gauge Station

723 -200Vincent Rd.

Gulf South

LaNeuville Rd.

Carriage Light Loop

Millcreek Rd.

STM

FenstermakerCommission Blvd

Lafayette Vineyard

Covenant Church

722 -201

Rain Gauge Sites

Monthly Comparisons

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0

100

300

500Rainfall 2004D

epth

(mm

)


100

200

300Rainfall 2005

Dep

th (m

m)


50

150

250Rainfall 2006

Dep

th (m

m)

GaugeMPE

2004

2005

2006

Validation at hourly scale

Gauge (mm)

MPE

(mm

)

10-1 100 10110-1

100

101

Gauge (mm)

MPE

(mm

)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

Mean (MPE) (mm) 1.53

Mean (gauge) (mm) 1.59

Relative Bias -0.04

Standard Deviation -- MPE (mm) 4.03

Standard Deviation -- Gauge (mm) 4.76

Relative RMSE 1.16

Correlation Coefficient 0.93

Hydrologic predictions driven by MPE

Gauge (mm)

MPE

(mm

)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

Time in days since 1/1/2002 00:00

Dis

char

ge(m

3 /sec

)

122 122.5 1230

10

20

30

40

50

60

70 reference rainfallunadjusted MPEunconditional adjustmentconditional adjustment

storm 12

Now what !

• What should we do with these uncertainties?

• How can we incorporate them into an application of interest?

• We need a realistic yet practical model for the radar uncertainties

Gauge (mm)

MPE

(mm

)

10-1 100 10110-1

100

101

Gauge (mm)

MPE

(mm

)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

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

• 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

r

s

RR

=ε

Radar Error

724 -200

724 -201

723 -201

Rain Gauge StationDischarge Gauge Station

723 -200Vincent Rd.

Gulf South

LaNeuville Rd.

Carriage Light Loop

Millcreek Rd.

STM

Fenstermaker

Commission Blvd

Lafayette Vineyard

Covenant Church

722 -201

r

s

RR

=ε

Radar Error

r

s

RR

=ε

Radar Error

r

s

RR

=ε

Systematic Systematic componentcomponent

Random Random componentcomponent

Systematic Component

• Overall Bias:

• Conditional Bias:

-0.4-0.2

00.20.40.60.8

11.21.4

0.13-0.30

0.30-0.50

0.50-0.80

0.80-1.25

1.25-2.0

2-4 4-7 7-10 > 10

Gauge Intensity (mm/h)

Rel

ativ

e B

ias

2004 2005 2006

][][

r

sREREB =

r

rrsr r

rRRErCB ]|[)( ==

Surface rainfall

radar rainfall

Spatial variability of radar bias

00.20.40.60.8

11.21.41.61.8

2

1 2 3 4 5 6 7 8 9 10 11Storm Number

Bia

s Fa

ctor


Dis

char

ge(m

3 /sec

)

22 22.2 22.4 22.6 22.80

5

10

15

20reference rainfallunadjusted MPEunconditional adjustmentconditional adjustment

storm 2


Dis

char

ge(m

3 /sec

)

22 22.2 22.4 22.6 22.80

5

10

15


storm 2


Dis

char

ge(m

3 /sec

)

22 22.2 22.4 22.6 22.80

5

10

15


storm 2


Dis

char

ge(m

3 /sec

)

22 22.2 22.4 22.6 22.80

5

10

15


storm 2


Dis

char

ge(m

3 /sec

)

122 122.5 1230

10

20

30

40

50

60


storm 12


Dis

char

ge(m

3 /sec

)

122 122.5 1230

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20

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60


storm 12


Dis

char

ge(m

3 /sec

)

122 122.5 1230

10

20

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60


storm 12


Dis

char

ge(m

3 /sec

)

122 122.5 1230

10

20

30

40

50

60


storm 12

Random Component: Error Probability Distribution

• Marginal Distribution:

– Bias and Variance: Eε; VARε

– Conditional dependence: ε|R• Joint Distribution:

– Spatial auto-correlation: ρε(s)

– Temporal auto-correlation: ρε(t)

r

s

RR

=εsurface rainfall

radar rainfall

Error Marginal Distribution

Radar random error

Prob

abili

ty

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

(b)

Radar random error

Prob

abili

ty

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

(b)

normal

Radar random error

Prob

abili

ty

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

(b)

lognormal

r

s

RR

=ε

surface rainfall

radar rainfall

VARε ≠ constant

VARε = f (Rr)

VARε|Rr=r

0

1

2

3

4

5

6

7

8

0.13-0.30

0.30-0.50

0.50-0.80

0.80-1.25

1.25-2.0

2-4 4-7 7-10 > 10

Gauge Intensity (mm/h)

Rel

ativ

e S

TD(e

rror

) 2004 2005 2006

Error Spatial Dependencies

Radar Rainfall Corr. Matrix

Spatial dependencies of radar error fields are NOT negligible

Radar Error Corr. Matrix

Error Temporal Dependencies

Temporal Auto-correlation MatrixAuto-correlation of each pixel with itself and all other pixels

Time Lag: 15min Time Lag: 30min Time Lag: 45min

Temporal dependencies of radar error fields seem to be negligible

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, Rr)

• Substitute in to generate realizations of probable surface rainfall fields that reflect radar uncertainties

r

sRR

=ε

Simulation Scenarios

CASE 1:- Variance of radar error is independent of radar magnitude. -No spatial correlation in simulated error fields.

Var(Var(εε) ) = Const.= Const.ρρεε= 0= 0

CASE 2:- Variance of radar error is NOT independent of radar magnitude. - No spatial correlation in simulated error fields.

Var(Var(εε) ) = f (= f (RRrr))ρρεε= 0= 0

CASE 3:-Variance of radar error is NOT independent of radar magnitude. - Spatial correlation in simulated error fields is preserved.

Var(Var(εε) ) = f (= f (RRrr))ρρεε≠≠ 00

Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette

Simulation of Radar Error - Case 1

Simulated errors (yellow dots) do NOT follow the trend of observed errors (blue plus sign)




No conditioning on error

large errors may imposed on a large magnitudes of radar and result in unrealistically large simulated radar data.




y = 0.8104e-0.0398x

R2 = 0.8927

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Radar (mm/h)

Stan

dar D

evia

tion

of L

og E

rror

for a

Mov

ing

Win

dow






Simulation of Radar Error - Case 2Correlation of Observed Error Field

Correlation of Observed Radar Field

Correlation of Simulated Error Field

Correlation of Simulated Radar Field




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=

*i i iLε µ= + Ω

Where L and L* are lower triangular matrix and its transpose respectively.

Simulated vector of error

Mean error vector

Decomposed var-covar matrixRandom number generator

iε =

iµ =*L =

iΩ =






Simulation of Radar Error - Case 3Correlation of Simulated Error Field Correlation of Observed Error Field

Correlation of Simulated Radar Field Correlation of Observed Radar Field


Time in days since 1/1/2002

Dis

char

ge(m

3 /sec

)

122 122.5 1230

20

40

60


Dis

char

ge(m

3 /sec

)

122 122.5 1230

20

40

60


Dis

char

ge(m

3 /sec

)

122 122.5 1230

20

40

60


Dis

char

ge(m

3 /sec

)

122 122.5 1230

20

40

60

Conclusions

• Radar error has complex structure in terms of its marginal and joint distribution

• It follows a log-normal distribution with non-constant variance

• Error auto-correlations are non-negligible and sometimes significant (especially spatial correlations)

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)

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 spatio-temporal 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

Thank You!

Questions….

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Analysis of Radar-Rainfall Uncertainties and effects on