Estimating IDF-Relations consistentlyusing a duration-dependent GEV with spatial covariatesJana Ulrich, Madlen Peter, Oscar E. Jurado and Henning W. Rust

TopicIntensity-Duration-Frequency (IDF) curves are a popular tool in Hydrology for estimating the properties of extremeprecipitation events. They describe the relationship between rainfall intensity and duration for a given non-exceedanceprobability (or frequency).

MethodWe use a duration dependent GEV with spatial covariates, to model the distribution of annual precipitation maximasimultaneously for a range of durations and locations.

Why?This way, we can obtain return level maps for various durations, as well as IDF curves at all locations. Furtheradvantages are parameter reduction and more efficient use of the available data.

Does it work?We use the Quantile Skill Score to investigate under which conditions this method leads to an improved estimatecompared to using the GEV separately for each duration at every station.

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Precipitation Data

Case study in Wupper-Catchment:→ 92 gauge stations in 75 locations (see figure 1)→ different measuring periods (see figure 2)→ varying length of time series

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Alti

tude

[m]

50.8°N

51.0°N

51.2°N

51.4°N

6.8°E 7.2°E 7.6°E

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20 km

Figure 1: Study area: dashed line shows catchment area.Altitude data: [1], River data: [2]

Provided by the German Weather Service (DWD) and theWupperverband (WV)Annual precipitation maxima

→ for durations 1, 4, 8, 16, 32 minutes, 1, 2, 3, 8, 16 hours and1, 2, 3, 4, 5 days

1900 1920 1940 1960 1980 20000

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20

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60

Year

Num

ber

of s

tatio

ns

●

Observations

dailyhourlyminutely

Figure 2: Number of active gauge stations in the study area per year for the differentmeasuring periods.

Provider Number of stations Measuring period Length of time seriesDWD 69 1 day 9-121 yearsDWD 17 1 min 5-14 yearsWV 6 1 hour 38 years

jana.ulrich@met.fu-berlin.de Estimating IDF-Relations consistently EGU2020: Sharing Geoscience Online 2 / 6

Model

Modeling precipitation block maxima z with the GEV [3]: G(z;µ, σ, ξ) = exp{

−[

1 + ξ

( z − µ

σ

)]−1/ξ}

Simultaneously model maxima over different durations and locations using

Duration dependent GEV (d-GEV)Assumptions for the dependency of the GEV-parameters onduration d following [4]:

scale: σ(d) = σ0 · (d + θ)−η

location: µ(d) = µ̃ · σ(d)

shape: ξ(d) = ξ = const.

Resulting in d-GEV:

G(µ(d), σ(d), ξ) = G(µ̃, σ0, ξ, θ, η)↑ ↑ ↑ ↑ ↑5 parameters

Intensity

Dur

atio

n

Prob. D

ensity

Figure 3:Probability density surface of

the d-GEV

with spatial covariatesAllow d-GEV parameters to vary in space:

G( µ̃(~r), σ0(~r), ξ(~r), θ(~r), η(~r) )Using orthogonal polynomials of longitude and latitude [5]for all parameters Φ ∈ {µ̃, σ0, ξ, θ, η}, with maximum orders J = K = 6:

φ = φ0 +∑J

j=1βφj Pj (lon) +

∑Kk=1

γφk Pk (lat)

+∑J

j=1

∑Kk=1

δφj,k Pj (lon)Pk (lat)

Model selection: avoid overfitting, by allowing certain coefficients to remainzero

Figure 4:Model selection result: addedcovariates colored according toorder of their selection. Whitemeans parameter remains zero.

Par

amet

ers

P1(

lon)

P2(

lon)

P3(

lon)

P6(

lon)

P2(

lat)

P3(

lat)

P6(

lat)

P1(

lon)

P1(

lat)

P2(

lon)

P1(

lat)

P3(

lon)

P1(

lat)

P1(

lon)

P2(

lat)

P2(

lon)

P6(

lat)

θη

σ0

µ~ξ

Selected covariates

Ord

er

first

last

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Quantile estimationUsing the d-GEV with spatial covariates, we can obtainreturn level maps for various durations (figure 5) and IDF-curves at all locations (figure 6).

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d= 5 min

p= 0

.99

1

2

54

54.5 55

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5

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d= 30 min

1

2

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37

d= 1 h

1

2

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p= 0

.95

1

2

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45.5

1

2

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26.8

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27.6 27.8

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28.2

28.

4 28.6

1

2

25 38 50 66 76 87 100 115 132 152 174 200

Intensity [mm/h]

Figure 5: 100-year (upper panel) and 20-year (lower panel) return level maps fordurations: 5 minutes, 30 minutes and 1 hour. Diamond and Triangle mark the

locations for the IDF-curves in figure 6

0.51.02.0

5.010.020.0

50.0100.0200.0

Inte

nsity

[mm

/h]

p−quantile

0.990.750.5

Solingen−Hohenscheid1

0.51.02.0

5.010.020.0

50.0100.0200.0

0.05 0.2 1 5 24 120

Duration [h]

Inte

nsity

[mm

/h]

p−quantile

0.990.750.5

ungauged site2

Figure 6: IDF-curves for a station with minutely precipitationmeasurements and an ungauged location. 95% confidence Intervals

were obtained by the bootstrap percentile method.

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VerificationHow does the d-GEV perform in the study area?

1. When using an individual model at each station2. When using one spatial model

Investigation of model performance using the Quantile Score QS [6]corresponding to weighted mean difference between observations os,d at certain station s,for certain duration d and modeled p-quantile qs,d (p)

Mean Quantile Score for probability p and duration d , averaged over stations:

QSd (p) =1

Sd

Sd∑s=1

1Ns,d

Ns,d∑n=1

ρp(os,d,n − qs,d (p)), ρp(u) ={

pu , u ≥ 0(p − 1)u , u < 0.

Compare model score with reference using the Quantile Skill Score QSS

Reference = individual GEV model for each duration at every station

QSS(p)d = 1−QS d-GEV

d (p)

QS GEVd (p)

Results (see figure 7):→ Both approaches: improved modeling of rare events→ d ≥ 24 h (right of dashed line): more data availability, pooling information over durations is less

important→ Improvement for stations with short observation time series

→ Short durations d ≤ 0.5 h: loss in skill for both approaches (dashed circle)→ d-GEV might not be flexible enough to model durations d ≤ 0.5 h properly

0.50.80.9

0.950.980.99

0.995

Pro

babi

lity

p

0.50.80.9

0.950.980.99

0.995

0.02

0.07

0.13

0.27

0.53 1 2 4 8 16 24 48 72 96 12

0

Duration [h]

Pro

babi

lity

QSS−Inf −0.4 −0.2 −0.1 0 0.1 0.2 0.3 1

Figure 7: Mean Quantile Skill ScoreQSSd (p) for station-wise d-GEV (upperpanel) and spatial d-GEV (lower panel).

Positive values (red) indicate improvementcompared to the reference.

jana.ulrich@met.fu-berlin.de Estimating IDF-Relations consistently EGU2020: Sharing Geoscience Online 5 / 6

References

[1] http://www.diva-gis.org/gdata.

[2] https://www.openstreetmap.org.

[3] Stuart Coles, Joanna Bawa, Lesley Trenner, and Pat Dorazio.An introduction to statistical modeling of extreme values, volume 208.Springer, 2001.

[4] Demetris Koutsoyiannis, Demosthenes Kozonis, and Alexandros Manetas.A mathematical framework for studying rainfall intensity-duration-frequency relationships.Journal of Hydrology, 206(1-2):118–135, 1998.

[5] M Fischer, HW Rust, and U Ulbrich.A spatial and seasonal climatology of extreme precipitation return-levels: A case study.Spatial Statistics, 34:100275, 2019.

[6] Sabrina Bentzien and Petra Friederichs.Decomposition and graphical portrayal of the quantile score.Quarterly Journal of the Royal Meteorological Society, 140(683):1924–1934, 2014.

We would like to acknowledge the Wupperverband as well as the Climate Data Center of the DWD,for providing the precipitation time series.

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