A Multiscaling Model of an
Intensity-Duration-Frequency Relationship for Extreme Precipitation
Hans Van de Vyver
Royal Meteorological Institute of Belgium
September 21, 2017
IDF-curves (Uccle, Belgium)
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 50.0
0.51.02.05.010.020.050.0100.0
d (hour)
Maximum intensity / mm h−1
General IDF relationship
General IDF relationship (Koutsoyiannis, Kozonis &
Manetas, 1998):
◮ T -year return level of rainfall intensity:
i T (d ) = a(T )
(d + θ) η , with θ > 0, 0 < η < 1, with
a(T ) = F Y − 1 (1 − 1/T ),
and F Y (y ) is the CDF of the scaled intensity, I (d ) b(d ).
Generalized extreme value (GEV) distribution
◮ Let X 1 , . . . , X m be a sequence of iid random variables.
◮ Block maxima M m = max{X 1 , . . . , X m }.
◮ CPF: P {M m ≤ z}.
◮ For sufficiently large m, P {M m ≤ z} is approximated by the generalized extreme value (GEV) distribution
G (z ; µ, σ, γ) = exp
"
−
1 + γ z − µ σ
−1/γ # .
◮ In practice: blocks of one year.
Generalized extreme value (GEV) distribution
◮ Location parameter µ specifies center of distribution.
◮ Scale parameter σ determines size of deviations.
◮ Shape parameter γ determines rate of tail decay.
◮ Return period
T = 1
1 − G (z ) .
◮ Return level
z (T ) = µ − σ γ
( 1 −
− log
1 − 1 T
−γ )
.
Simple scaling models
◮ Series of d -hourly intensity: I 1 (d ), I 2 (d ),. . .
◮ Annual maximum intensity:
I (d ) = max{I 1 (d ), I 2 (d ), . . . , I N y (d )}.
◮ Simple scaling.
◮ Strict sense:
I (λ d) = λ −η I (d), with 0 < η < 1.
◮ Wide sense:
E [I q (λ d)] = λ − q η E [I q (d)].
◮ Multiscaling:
E [I q (λ d )] = λ −α q E [I q (d )], with α q = q ϕ q η.
Scaling GEV-models
◮ We assume
I (d ) ∼ GEV[µ(d ), σ(d ), γ].
◮ Simple scaling hypothesis (Menabde et al., 1999; Nguyen et al., 1998):
µ(d ) = d −η µ, σ(d ) = d −η σ, γ (d ) = γ,
with 0 < η < 1. ⇒ Equivalent with the general relationship.
◮ Multiscaling hypothesis:
µ(d ) = d −η 1 µ, σ(d ) = d −η 2 σ, γ (d ) = γ, with 0 < η 1 ≤ η 2 , and η 1 < 1.
⇒ Not covered by the general relationship.
Illustration: Uccle precipitation
1 2 5 10 20 50
12510
Rainfall duration (hour)
µ (mm/h)
Slope = −0.724 MLE for each duration Regression Line
1 2 5 10 20 50
0.20.51.02.05.0
Rainfall duration (hour)
σ (mm/h)
Slope = −0.845
◮ Location parameter: slope = -0.724.
◮ Scale parameter: slope = -0.845.
Bayesian inference
◮ N year of IDF-data:
i =
i 11 . . . i 1M
.. . . .. .. . i N1 . . . i NM
| {z }
M durations .
◮ Density function of IDF-data i is L(i; ψ) = L(i | ψ).
◮ Bayesian rule
π(ψ | i) = L(i | ψ) π(ψ) R
Ψ L(i | ψ) π(ψ)d ψ , with
◮ π(ψ): prior distribution.
◮ π(ψ | i): posterior distribution.
◮ Van de Vyver, H. (2015) Bayesian estimation of
Intensity-Duration-Frequency relationships. J. Hydrol. 529,
1451–1463.
Bayesian inference: scaling exponents
0.7 0.8 0.9 1.0
0 5 10 15 20 25 30
η (−)
Density
η
1η
2Bayesian inference: 20-year return levels
25 30 35
0.000.050.100.150.200.250.30
d = 1 h
Intensity (mm/h)
Density
4.0 4.5 5.0 5.5
0.00.51.01.52.0
d = 12 h
Intensity (mm/h)
Density
2.2 2.4 2.6 2.8 3.0 3.2 3.4
0.00.51.01.52.02.5
d = 24 h
Density
Simple scaling Multiscaling
IDF-curves (Uccle, Belgium)
1 2 5 10 20 50
125102050
Simple scaling
Rainfall duration (hour)
Maximum intensity (mm/h)
1 2 5 10 20 50
125102050
Multiscaling
Rainfall duration (hour)
Maximum intensity (mm/h)
