Towards Probabilistic Quantitative Precipitation Estimation:

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