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Regionalizing StochasticRainfall Generators

Dongkyun Kim and Francisco Olivera

Zachry Department of Civil Engineering

Texas A&M University

American Society Civil EngineersEnvironmental and Water Resources Institute

World Environmental & Water Resources Congress 2010Providence, Rhode Island – May 17, 2010

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Why stochastic rainfall generation?

Synthetic rainfall “data” can be used as input to hydrologic models whenever rainfall data are not available:Basins with rain gages but with missing dataBasins that need thousands of years of rainfall input to

assess the risks associated with hydrologic phenomena (e.g. floods, draughts, water availability, water contamination)

Basins with no rain gages

3Image Source: http://www.meteoswiss.admin.ch/web/en/research/projects/rain.html

Storm components

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– Storm arrival: Poisson process

– Rain cell arrival: Poisson process

– Storm duration: Exponential distribution

m– Rain cell intensity: Exponential distribution

– Rain cell duration: Exponential distribution , - Gamma distribution

MBLRP model parameters

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MBLRP model parameters

For the convenience, the parameters are normalized as = / and = /.

Therefore, the following six parameters are typically used: , , , , and .

The model calibration consists of minimizing the discrepancy between the statistics of observed and simulated precipitation.

λ (1/T): expected number of storms per unit time.

/ (T): expected rain cell duration.

: uniformity of the rain cell durations.

m (L/T): expected rain cell intensity.

: ratio of the expected rain cell duration to the

expected duration of storm activity.

: product of the expected number of rain cells per

unit time times the expected rain cell duration.

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Mean_1Var_1AC_1Prob0_1

Mean_3Var_3AC_3Prob0_3

Mean_12Var_12AC_12Prob0_12

Mean_24Var_24AC_24Prob0_24

Rainfall statistics

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Rainfall statistics

MeanVariance

Prob0 Lag-1 autocorrelation

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Regionalization Estimate the MBLRP parameters at 3,444 NCDC gages across the

contiguous US. Interpolate the parameters using the Ordinary Kriging technique. Cross-validate the parameter maps at all 3,444 gages.

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Regionalization - Interpolation

Ordinary Kriging was used to interpolate the estimated parameters

zi = a1*w1 + a2*w2 + a3*w3 + … + an * wn

The weights wi are determined based on a empirically driven function called “variogram.”

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Expected Results

Expected number of storms per hour in September: (1/hr)

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Regionalization - Multimodality

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2.8

2.3 6

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Number of rain cells

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Regionalization - Multimodality

Set 1

Set 2

Set 3

Set 4

Set 1

Set 2

Set 3

Set 1

Set 2

Set 3

Set 4

Set 1

Set 2

Set 1

Set 2

Set 3

Set 2

Set 3

Set 1

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Results

72 maps = 6 parameters 12 months

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l(1/hr) (hr)

a (mm/hr)

May

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Rainfall CharacteristicsRainfall characteristics according to the

MBLRP model:

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May

Average number of rain cells per storm

Average storm duration (hr)

Average rain cell arrival rate (1/hr)

Average rainfall depth per storm (mm)

Average rain cell duration (hr).

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State Texas Iowa Washingto

n

Florida

Longitude -95.592° -93.591° -123.975° -80.913°

Latitude 30.349° 41.966° 47.862° 27.050°

Rainfall depth per storm

(mm)

22.2 13.5 14.2 19.2

Storm duration (hr) 7.9 8.7 12.1 11.7

Number of rain cells per

storm

4.7 6.0 16.4 4.0

Rain cell arrival rate (1/hr) 0.57 0.61 1.30 0.38

Rain cell Duration (hr) 0.24 0.28 0.21 0.24

Average rainfall characteristics for the month of May for selected locations with mean monthly rainfall depth of 141 mm

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Validation

Cross-validated parameters were used to simulate the accuracy of interpolated points.

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Validation

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Validation

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Summary and Conclusions72 MBLRP parameter maps were developed for the

contiguous US (i.e., 6 parameters 12 months).

Overall, the parameters showed a regional and seasonal variability:Strong : λ , μ Discernible : φ, κ, α Weak: ν

Parameter values from the maps were cross-validation and showed that the rainfall statistics could be reproduced reasonably well except for the lag-1 autocorrelation coefficient.

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Questions?