Towards Probabilistic Towards Probabilistic Quantitative Precipitation Quantitative Precipitation
Estimation: Estimation: Modeling Radar-Rainfall Error Modeling Radar-Rainfall Error
StructureStructureWitold F. Krajewski, Grzegorz J. Ciach, Witold F. Krajewski, Grzegorz J. Ciach,
and Gabriele Villariniand Gabriele Villarini
“First I shall make some experiments before I proceed further, because my intention is to consult experience first and then by means of reasoning show why such experiment is bound to work in such a way. And this is the true rule by which those who analyze natural effect must proceed; and although nature begins with the cause and ends with the experience, we must follow the opposite course, namely, […] begin with the experience and by means of it investigate the cause.”
Leonardo da VinciLeonardo da Vinci
InputInput + + UncertaintyUncertainty
Uncertainty PropagationUncertainty PropagationKey Concept:Key Concept:
Output Output + + UncertaintyUncertainty
Transformation:Transformation:(hydrologic (hydrologic
prediction model)prediction model)
DeterministicDeterministicoror
StochasticStochastic
Product-Error Driven Product-Error Driven ApproachApproach
• Collect reliable data on the relation Collect reliable data on the relation between different RR products and the between different RR products and the corresponding corresponding True RainfallTrue Rainfall;;
• Create a flexible model of this relation and Create a flexible model of this relation and apply it to the PQPE product generator;apply it to the PQPE product generator;
• Develop empirically based generalizations Develop empirically based generalizations of the model for different situations.of the model for different situations.
Combined effect of all error sources!Combined effect of all error sources!
DefinitionsDefinitions
• True Rainfall:True Rainfall: Amount of rain-water Amount of rain-water falling on a specified area in a specified falling on a specified area in a specified intervalinterval
• Radar Rainfall (RR):Radar Rainfall (RR): An approximation An approximation of the True Rainfall based on radar dataof the True Rainfall based on radar data
• RR Uncertainties:RR Uncertainties: All discrepancies All discrepancies between RR and the corresponding between RR and the corresponding True RainfallTrue Rainfall
• Ground Reference (GR):Ground Reference (GR): Approximation Approximation of True Rainfall, based on rain-gauge of True Rainfall, based on rain-gauge measurements, used to evaluate RR measurements, used to evaluate RR
Mathematical Mathematical ApparatusApparatus
Describe family of bivariate frequency Describe family of bivariate frequency distributions (“verification distributions (“verification distributions“):distributions“):
((RRr r , R, Raa))A,T,dA,T,d
with A,T,d indexing space, time scales, with A,T,d indexing space, time scales, and radar range, and radar range, RRaa is is True RainfallTrue Rainfall
Bivariate distribution (Bivariate distribution (XX11 , X , X22) can be ) can be expressed in two equivalent ways thorough expressed in two equivalent ways thorough the relationships:the relationships:
RRrr = h = h11 (R (Raa , , εε11)) physical meaningphysical meaning
RRaa = h = h22 (R (Rrr , , εε22)) good for PQPEgood for PQPE
hhii - deterministic factor - deterministic factor
εεii - independent random variable - independent random variable
Mathematical Mathematical ApparatusApparatus
Ground Reference ErrorsGround Reference Errors
• The errors in GR based on single rain-gauge The errors in GR based on single rain-gauge are large. They can dominate the radar-gauge are large. They can dominate the radar-gauge comparisons and lead to confusing resultscomparisons and lead to confusing results
• The GR errors should not be ignoredThe GR errors should not be ignored• Two ways to deal with the problem:Two ways to deal with the problem:
– Building more accurate GR systems;Building more accurate GR systems;– Filtering GR errors from the radar-gauge Filtering GR errors from the radar-gauge
verification samplesverification samples
Oklahoma DataOklahoma DataResultsResults
(after considerable QC/QA)(after considerable QC/QA)
Range Effect AnalysisRange Effect Analysis
Zone IV
Zone I
ARS Micronet
Hourly DataHourly Data Cold Cold (NDJFM)(NDJFM)
Warm Warm (AMO)(AMO)
Hot Hot (JJAS)(JJAS)
Entire Entire datasetdataset
Zone I (<75 km)Zone I (<75 km) 0.95 0.78 0.76 0.82
Zone II (70-105)Zone II (70-105) 0.88 0.76 0.73 0.78
Zone III (100-145) Zone III (100-145) 0.87 0.68 0.65 0.72
Zone IV (140-185)Zone IV (140-185) 1.29 0.78 0.65 0.83
Zone V (>180 km)Zone V (>180 km) 2.33 1.11 0.75 1.12
Overall BiasOverall Bias
Cold (NDJFM) Warm (AMO)
AllHot (JJAS)
Con
ditio
nal G
auge
Mea
n (m
m)
Radar-Rainfall (mm)
Cold (NDJFM) Warm (AMO)
Hot (JJAS)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e (m
m)
All
Radar-Rainfall (mm)
Additive Additive errorerror
Cold (NDJFM) Warm (AMO)
Hot (JJAS)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e
All
Radar-Rainfall (mm)
Multiplicative Multiplicative errorerror
Ran
dom
err
or q
uant
iles
Radar-Rainfall (mm)
Cold (NDJFM)
Ran
dom
err
or q
uant
iles
Radar-Rainfall (mm)
Warm (AMO)
Ran
dom
err
or q
uant
iles
Radar-Rainfall (mm)
Hot (JJAS)
Cold (NDJFM)
Separation lag (km)
Spa
tial c
orre
latio
n of
the
ran
dom
err
or, e
Separation lag (km)
Spa
tial c
orre
latio
n of
the
ran
dom
err
or, e
Warm (AMO)
Separation lag (km)
Spa
tial c
orre
latio
n of
the
ran
dom
err
or, e
Hot (JJAS)
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Time lag (minutes)
Tem
pora
l cor
rela
tion
of t
he r
ando
m e
rror
, e
Modeling ResultsModeling Results
Radar-Rainfall (mm)Radar-Rainfall (mm)
Con
ditio
nal G
auge
Mea
n (m
m)
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IZone I
Radar-Rainfall (mm)Radar-Rainfall (mm)
Con
ditio
nal G
auge
Mea
n (m
m)
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIZone II
Radar-Rainfall (mm)Radar-Rainfall (mm)
Con
ditio
nal G
auge
Mea
n (m
m)
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIIZone III
Radar-Rainfall (mm)Radar-Rainfall (mm)
Con
ditio
nal G
auge
Mea
n (m
m)
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IVZone IV
Radar-Rainfall (mm)Radar-Rainfall (mm)
Con
ditio
nal G
auge
Mea
n (m
m)
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone VZone V
Radar-Rainfall (mm)Radar-Rainfall (mm)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IZone I
Radar-Rainfall (mm)Radar-Rainfall (mm)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIZone II
Radar-Rainfall (mm)Radar-Rainfall (mm)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIIZone III
Radar-Rainfall (mm)Radar-Rainfall (mm)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IVZone IV
Radar-Rainfall (mm)Radar-Rainfall (mm)
Ran
dom
err
or s
tand
ard
devi
atio
n,
e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone VZone V
Separation distance (km)Separation distance (km)
Spa
tial c
orre
latio
n of
the
ran
dom
com
pone
nt, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IZone I
Separation distance (km)Separation distance (km)
Spa
tial c
orre
latio
n of
the
ran
dom
com
pone
nt, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIZone II
Separation distance (km)Separation distance (km)
Spa
tial c
orre
latio
n of
the
ran
dom
com
pone
nt, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIIZone III
Separation distance (km)Separation distance (km)
Spa
tial c
orre
latio
n of
the
ran
dom
com
pone
nt, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IVZone IV
Separation distance (km)Separation distance (km)
Spa
tial c
orre
latio
n of
the
ran
dom
com
pone
nt, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone VZone V
Time lag (minutes)Time lag (minutes)
Tem
pora
l cor
rela
tion
of t
he r
ando
m c
ompo
nent
, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IZone I
Time lag (minutes)Time lag (minutes)
Tem
pora
l cor
rela
tion
of t
he r
ando
m c
ompo
nent
, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIZone II
Time lag (minutes)Time lag (minutes)
Tem
pora
l cor
rela
tion
of t
he r
ando
m c
ompo
nent
, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IIIZone III
Time lag (mintes)Time lag (mintes)
Tem
pora
l cor
rela
tion
of t
he r
ando
m c
ompo
nent
, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone IVZone IV
Time lag (minutes)Time lag (minutes)
Tem
pora
l cor
rela
tion
of t
he r
ando
m c
ompo
nent
, e
Cold (NDJFM) Warm (AMO)
Hot (JJAS) All
Zone VZone V
Time scale (hours)
Coe
ffic
ient
a (
stan
dard
dev
iatio
n)
Cold Warm
Hot All
Time scale (hours)
Coe
ffic
ient
b (
stan
dard
dev
iatio
n)
Cold Warm
Hot All
Time scale (hours)
Exp
onen
t c
(sta
ndar
d de
viat
ion)
Cold Warm
Hot All
GR Error FilteringGR Error Filtering
• Assume that, for given spatio-temporal Assume that, for given spatio-temporal resolution resolution (A,T)(A,T) and radar-range and radar-range (d)(d), , we have available:we have available:– Large sample of corresponding (Large sample of corresponding (RRrr ,R ,Rgg) )
pairs;pairs;– Detailed information about spatial rainfall Detailed information about spatial rainfall
variability in this sample.variability in this sample.
• Can we retrieve a good estimate of the Can we retrieve a good estimate of the verification distribution (verification distribution (RRrr , R , Raa)?)?
PicoNePicoNe
tt
Oklahoma PicoNetOklahoma PicoNet
One HourOne Hour
ConclusionsConclusions• Extensive empirical analysisExtensive empirical analysis• Confirmation of strong range effects and seasonal Confirmation of strong range effects and seasonal
dependencedependence• Strong dependence on radar-rainfallStrong dependence on radar-rainfall• Non-negligible space-time dependence of the Non-negligible space-time dependence of the
random error componentrandom error component• Temporal scale invariance (some)Temporal scale invariance (some)• Scarce empirical information limits hypothesis Scarce empirical information limits hypothesis
testing on point vs. area differencetesting on point vs. area difference• Fairly simple structure of the ensemble generator Fairly simple structure of the ensemble generator
(Gaussian random errors)(Gaussian random errors)
Remaining WorkRemaining Work
• Analysis of OK Piconet: point vs. area difference;Analysis of OK Piconet: point vs. area difference;• Analysis of Micronet with Vance AFB WSR-88D data: range Analysis of Micronet with Vance AFB WSR-88D data: range
effect;effect;• Analysis of other radars in the region: calibration;Analysis of other radars in the region: calibration;• Analysis of Kansas and IA networks data: transferability of Analysis of Kansas and IA networks data: transferability of
results;results;• Modeling shorter and smaller scales: FFG;Modeling shorter and smaller scales: FFG;• Implementing and testing a generator of ensembles: Implementing and testing a generator of ensembles:
uncertainty propagation;uncertainty propagation;• Investigating event-type conditioning: removing seasonal Investigating event-type conditioning: removing seasonal
dependence; dependence; • Improving our understanding of the mechanism (physical Improving our understanding of the mechanism (physical
and statistical) causing the observed error behavior!and statistical) causing the observed error behavior!
Thank You! Thank You!
The EndThe End