Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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)
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
10
20
30
40
50
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)
0
5
10
15
20
25
30
Pre
cipi
tatio
n in
tens
ity (m
m/h
r) Unadjusted MPE
Gages_avg_middlepixel
0
10
20
30
40
50
60
70
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
Monthly Comparisons
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0
100
300
500Rainfall 2004D
epth
(mm
)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0
100
200
300Rainfall 2005
Dep
th (m
m)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0
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
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
Time in days since 1/1/2002 00:00
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
Time in days since 1/1/2002 00:00
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
Time in days since 1/1/2002 00:00
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
Time in days since 1/1/2002 00:00
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
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
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
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
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
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)
Var(Var(εε) ) = Const.= Const.ρρεε= 0= 0
Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette
Simulation of Radar Error - Case 1
No conditioning on error
large errors may imposed on a large magnitudes of radar and result in unrealistically large simulated radar data.
Var(Var(εε) ) = Const.= Const.ρρεε= 0= 0
Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette
Simulation of Radar Error - Case 2
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
Var(Var(εε) ) = f (= f (RRrr))ρρεε= 0= 0
Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette
Simulation of Radar Error - Case 2
Var(Var(εε) ) = f (= f (RRrr))ρρεε= 0= 0
Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette
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
Var(Var(εε) ) = f (= f (RRrr))ρρεε= 0= 0
Amir Aghakouchak, Emad Habib - Department of Civil Engineering, University of Louisiana at Lafayette
Simulation of Radar Error - Case 3
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 =
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 3
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 3Correlation of Simulated Error Field Correlation of Observed Error Field
Correlation of Simulated Radar Field Correlation of Observed Radar Field
Var(Var(εε) ) = f (= f (RRrr))ρρεε≠≠ 00
Time in days since 1/1/2002
Dis
char
ge(m
3 /sec
)
122 122.5 1230
20
40
60
Time in days since 1/1/2002
Dis
char
ge(m
3 /sec
)
122 122.5 1230
20
40
60
Time in days since 1/1/2002
Dis
char
ge(m
3 /sec
)
122 122.5 1230
20
40
60
Time in days since 1/1/2002
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