T
EMPORAL EVOLUTION AND EXTREME VALUE ANALYSIS
OF PRECIPITATIONS IN
B
URKINA
F
ASO
Liliane Bel
a
, Jérôme Weiss
a
, Jean-Jacques Boreux
b
, Dominique Tapsoba
c
a AgroParisTech/INRA, UMR 518 Math. Info. Appli., F-75005 Paris, France; b The University of Liège, Arlon, Belgium;
c Hydro-Québec research institute (IREQ), Varennes, Québec, CANADA
The study presented here makes use of 41 daily rain gauges of Burkina Faso retrieved from the West Africa daily rainfall database Tapsoba (1997) from 1950 to 1990. These daily rain gauges have been chosen to maximize spatial uniformity of coverage and tem-poral continuity. The data were examined for continuity and miss-ing records. A spatio temporal study shows that the decline is uniform on the region.
41--Stations sans lacunes de 1950-90
N
Introduction
Extreme value theory
The GEV family of distribution function
G(z) = exp n − h1 + ξ z−µ σ −1/ξ + o if ξ 6= 0 exp n − exp h − z−µ σ io if ξ → 0
are limiting approximations to the distribution of Mn =
max(X1, . . . , Xn) linearly rescaled as n → ∞, where the
Xi are i.i.d. random variables.
zp such that G(zp) = 1 − p is the return level associated with the return period −1/ log(1 − p) ' 1/p for small p in unit of years if the GEV corresponds to the annual maximum.
Ha
Ouagadougou series
Total monthly precipitations OUAGADOUGOU VILLEyear mm 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 0 100 200 300 400 1950 1960 1970 1980 1990 40 60 80 100 120 140
annual maxima OUAGADOUGOU VILLE
year
mm
Ha
Parameter estimation
loc µ scale σ shape ξ
Estimates 56.77594 16.11875 0.01380 Standard Errors 2.9253 2.1866 0.1430 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Probability Plot Empirical Model 40 60 80 100 120 50 100 150 Quantile Plot Model Empir ical 40 60 80 100 120 0.000 0.010 0.020 Density Plot Quantile Density 0.2 0.5 2.0 5.0 20.0 100.0 50 100 150
Return Level Plot
Return Period
Retur
n Le
v
el
The return level zp = 260mm corresponds to a return period of 100 000 years!
Generalized Extreme Values
Nevertheless in 2009 a daily rainfall event of 263mm has been recorded, causing dra-matic damages in the Burkina capital city Oua-gadougou. Such an event is more than twice the magnitude of previously observed maximum. A standard extreme value analysis performed on our dataset indicates that this event should be very unlikely. We investigate possible causes of such a gap.
Ouagadougou severe event
Spatial max-stable process
A stochastic process Z(·) with continuous pathes is called max-stable if
max i Zi(·) − an(·) bn(·) d = Z(·)
Zi(·) independent replications of Z(·), an(·), bn(·) continuous functions.
Extremal coefficient
For Z max-stable process, Fréchet margins, the extremal coefficient θN is
P (Z(s1) < z, . . . , Z(sN ) < z) = exp
− θN
z
θN = N, if and only if margins are independent.
The extremal coefficient function for a stationary max stable process
exp θ(h) = P (Z(s + h) < z, Z(s) < z) characterizes the asymptotic
spatial dependence of the process.
Ha
Spatial max stable models
Brown Resnick process (Kabluchko et al.)
Z(s) = max
i ξi exp Wi(s) − σ
2
(s) s ∈ R2
ξi Poisson point process of intensity ξ−2dξ on R+, Wi independent copies of a zero mean Gaussian process with stationary increment and Var(W (s)) = σ2(s).
Extremal coefficient function
θ(h) = 2Φ s γ(h) 2
Φ is the standard normal cumulative distribution function, γ is the W process semi-variogram function.
Modeling the Burkina data
Linear model on the coordinates for the GEV parameters Variogram γ(h) =
khk
λ
α
Maximum composite likelihood estimator.
Ha ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● 0 200 400 600 1.0 1.2 1.4 1.6 1.8 2.0 2.2 h θ ( h ) ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ●●● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●
Extremal coefficient function
0 200 400 600 800 0 100 200 300 400 500 600 700
100−year return level
x y 100 120 140 160 180
The return level zp = 260mm corresponds to a return period of 29 000 years!
Spatial extremes
The National Climatic data Center provides Global Surface summary Of Day data (GSOD) over 9000 Worldwide Stations. Daily precipitations are available for 10 stations from Burkina, including Ouagadagou. OUAGADOUGOU 1990 months precipitations (mm) 1 2 3 4 5 6 7 8 9 10 11 12 0 10 20 30 40 50 60 50−90 data GSOD data OUAGADOUGOU years precipitations (mm) 1990 1993 1996 1999 2002 2005 2008 0 100 200 300 400 500
The GSOD data coincides with our 1950-1990 database for year 1990. This data show no flood event in September 2009 in Ouagadougou, neither in the 9 other stations. But there is one severe event in 2004.
A GEV study on the maxima series gathered from the two database give this time a return period corresponding to the return level 260mm equal to 72 years.
GSOD data
The ECMWF (European Centre for Medium-Range Weather Forecast) provides 10-daily and monthly outputs of global circulation model. They are reanalysis time series.
station OUAGADOUGOU VILLE
decades 10−da ys r ainf all (mm) 1980 1990 2000 2010 0 100 200 300 400 50−90 data ECWMF data −4 −2 0 2 10 11 12 13 14 15
ECWMF precipitations (mm) first decade 09/09
longitude latitude 0 100 200 300 400 ● Ougadougou
These data are consistent with our 1950-1990 database. They show no severe event in Ouagadagou for that period but a spot 250km away from Ouagadagou.
ECWMF data
This study shows that assessing future levels of extreme rainfall from collected data is a very difficult task. An important issue is the reliability of past data: a severe flood as the september 2009 in Ouagadougou one, does not figure in databases. Models of extreme have to take into account sufficiently large periods and all information available as spatial asymptotic dependence.
Concluding remarks
[1] Coles S. and Pericchi L. (2003) Anticipating catastrophes through extreme value modelling. Appl. Statist. 52,Part4, pp405-416.
[2] Kabluchko, Z., Schlather, M., and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab. Volume 37, Number 5 , pp 2042-2065.
[3] Ribatet, M. SpatialExtremes, R-package. [4] Stephenson, A. evd, R-package.
[5] Tapsoba, D. (1997) Caractérisation évènementielle des régimes pluviométriques Ouest Africains et de leur récent changement. Thèse de l’Université ParisXI, 145 pp.
