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Atmospheric Predictors for Annual Maximum Precipitation in North Africa

BOUCHRA NASRI

Canada Research Chair on the Estimation of Hydrometeorological Variables, Eau Terre Environnement Research Centre,

Institut National de la Recherche Scientiﬁque, Québec, Québec, Canada

YVES TRAMBLAY

UnitéMixte de Recherche Hydrosciences, Institut de Recherche pour le Développement, Montpellier, France

SALAHEDDINE ELADLOUNI

Département de Mathématique et de Statistique, Universitéde Moncton, Moncton, New Brunswick, Canada

ELKE HERTIG

Institute of Geography, University of Augsburg, Augsburg, Germany

TAHA B. M. J. OUARDA

Canada Research Chair on the Estimation of Hydrometeorological Variables, Eau Terre Environnement Research Centre,

Institut National de la Recherche Scientiﬁque, Québec, Québec, Canada, and Masdar Institute of Science

and Technology, Abu Dhabi, United Arab Emirates

(Manuscript received 5 May 2014, in ﬁnal form 21 July 2015)

ABSTRACT

The high precipitation variability over North Africa presents a major challenge for the population and the

infrastructure in the region. The last decades have seen many ﬂood events caused by extreme precipitation in

this area. There is a strong need to identify the most relevant atmospheric predictors to model these extreme

events. In the present work, the effect of 14 different predictors calculated from NCEP–NCAR reanalysis,

with daily to seasonal time steps, on the maximum annual precipitation (MAP) is evaluated at six coastal

stations located in North Africa (Larache, Tangier, Melilla, Algiers, Tunis, and Gabès). The generalized

extreme value (GEV) B-spline model was used to detect this inﬂuence. This model considers all continuous

dependence forms (linear, quadratic, etc.) between the covariates and the variable of interest, thus providing a

very ﬂexible framework to evaluate the covariate effects on the GEV model parameters. Results show that no

single set of covariates is valid for all stations. Overall, a strong dependence between the NCEP–NCAR

predictors and MAP is detected, particularly with predictors describing large-scale circulation (geopotential

height) or moisture (humidity). This study can therefore provide insights for developing extreme precipitation

downscaling models that are tailored for North African conditions.

1. Introduction

Heavy precipitation events are causing extensive

damage to the populations and infrastructure of the

countries located in the southern part of the Mediter-

ranean basin. The last decades saw several deadly ﬂood

events caused by extreme precipitation, including the

2001 ﬂood near Algiers, Algeria, causing more than 700

fatalities (Argence et al. 2008), the 1969 ﬂoods in the

region of Kairouan, Tunisia, with 150–400 fatalities

(Poncet 1970), or the 1995 ﬂood in the Ourika valley,

Morocco, with more than 200 fatalities (Saidi et al.

2003). To better mitigate the impacts of these extreme

events, there is a need to evaluate their predictability on

different time scales. In particular, it is necessary to es-

timate their return periods in a climate change context

Corresponding author address: Bouchra Nasri, INRS-ETE, 490

Rue de la Couronne, Québec, QC G1K 9A9, Canada.

E-mail: bouchra.nasri@ete.inrs.ca

APRIL 2016 N A S R I E T A L . 1063

DOI: 10.1175/JAMC-D-14-0122.1

Ó2016 American Meteorological Society

since several countries experienced an increased vul-

nerability to these events during the last decade (Di

Baldassarre et al. 2010). Several recent studies have

focused on seasonal precipitation and its extremes in the

Mediterranean region, with the objective of identifying

the associated large-scale patterns and the relevant

predictors (Knippertz et al. 2003;Xoplaki et al. 2004;

Martín-Vide and Lopez-Bustins 2006;Toreti et al. 2010;

Tramblay et al. 2011;Kallache et al. 2011;Hertig et al.

2012;Tramblay et al. 2012a,b;Hertig et al. 2013;

Ouachani et al. 2013;Donat et al. 2014). Indeed, to re-

solve the mismatch of scales between general circulation

models and the locations of interest for impact studies,

there is a need to develop downscaling techniques tai-

lored for extreme precipitation (Fowler et al. 2007;

Maraun et al. 2010a). To overcome the limitations of

climate models in reproducing extremes (Sillmann et al.

2013), several studies have used covariates in non-

stationary extreme precipitation frequency analysis

(e.g., Vrac and Naveau 2007;Aissaoui-Fqayeh et al.

2009;Beguería et al. 2010;Friederichs 2010;Kallache

et al. 2011;Tramblay et al. 2011;Maraun et al. 2010b;

Ouachani et al. 2013;El Adlouni and Ouarda 2009;

Cannon 2010;Ouarda and El Adlouni 2011). However,

even if several authors have shown the efﬁciency of at-

mospheric humidity and moisture ﬂux as predictors for

daily rainfall modeling and downscaling (Cavazos and

Hewitson 2005;Bliefernicht and Bárdossy 2007;

Tramblay et al. 2011,2013), the best predictors may

differ from one site to another (Kallache et al. 2011;

Hertig et al. 2013;Chandran et al. 2016). In addition, it

should be noted that the above studies have mostly ap-

plied polynomial dependence (linear or quadratic) be-

tween the covariate and the variable of interest. In the

present work, the nonstationary generalized extreme

value (GEV) model with B-spline dependent function

(Nasri et al. 2013;Chavez-Demoulin and Davison 2005)

is applied. B-spline functions are piecewise polynomial

functions that have certain advantages. A smoothing

B-spline basis is independent of the response variable

and depends only on the following information: (i) the

extent of the explanatory variable, (ii) the number and

position of the knots, and (iii) the degree of the B-spline.

These advantages make it a suitable option for use in the

GEV model with covariates to explain the effect of co-

variates on the response variable. Therefore, the goal of

this study is to identify relevant large-scale predictors

inﬂuencing the annual maximum precipitation at coastal

stations in the southern part of the Mediterranean re-

gion using the GEV B-spline model. This model will

help describe the predictors’ inﬂuence on the pre-

cipitation records within the study period. A number of

studies on the impact of climate variability on extreme

precipitation in the Mediterranean region have em-

ployed atmospheric–oceanic teleconnection indices

such as the North Atlantic Oscillation (NAO; Wanner

et al. 2001), the Mediterranean oscillation (Conte et al.

1989), or the western Mediterranean oscillation (WEMO)

(Knippertz et al. 2003;Vicente-Serrano et al. 2009).

However, in Morocco Tramblay et al. (2012a) observed a

possible dependence of precipitation extremes with these

indices only at 2 stations (Larache and Tangier) out of 10.

In the present study, we tested the effect of different

predictors measured with NCEP–NCAR reanalysis

data to evaluate their inﬂuence on the maximum annual

precipitation (MAP) time series. To our knowledge, no

studies have previously examined these extreme precip-

itation events and their relationship to large-scale atmo-

spheric inﬂuences in this area at the daily time step with

rain gauge data, mainly because of the limited access to

the data. Since reanalysis data are available at a spatial

scale similar to that of global circulation models (GCMs),

this study provides the ﬁrst step toward the development

of extreme precipitation downscaling methods that are

tailored for North African conditions.

2. Datasets

In the present study, we collected long daily pre-

cipitation time series maintained by the governmental

hydrological services of Algeria [Agence Nationale des

Ressources Hydrauliques (ANRH)], Morocco [Di-

rection de la Recherche et de la Planiﬁcation de l’Eau

(DRPE)], and Tunisia [Direction Générale des Re-

ssources en Eau (DGRE)]. The daily data of the Melilla

station located in northern Morocco were obtained

from the European Climate Assessment and Dataset

(ECAD; http://eca.knmi.nl). The data from each station

were carefully scrutinized, in particular to look for shifts,

absurd values, and missing data (Tramblay et al. 2013).

The stations that were subsequently selected had less

than 5% missing days between September and May. The

years with more than 5% missing days during this period

were discarded. Figure 1 illustrates the geographic lo-

cation of all stations, and Table 1 presents a description

of the selected stations with long precipitation records.

Reanalysis data from the National Centers for Envi-

ronmental Prediction (NCEP; Kalnay et al. 1996;Kistler

et al. 2001) are used to compute several large-scale

predictors. Various variables were extracted to be tested

in the model; the selection of covariates is based on the

previous studies of Cavazos and Hewitson (2005),

Kallache et al. (2011),Tramblay et al. (2011), and Hertig

et al. (2013). The NCEP–NCAR reanalysis data have

been selected over more recent products such as

ERA-Interim because the time span of the NCEP–

1064 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 55

NCAR reanalysis is larger and encompasses the whole

period of observations, up to the present. The advan-

tages of recent reanalysis products are manifold, in-

cluding new atmospheric and assimilation systems and

ﬁner grid spacing. However, they cover only the recent

period (from 1979 to present for MERRA, CFS re-

analysis, or ERA-Interim; Hofer et al. 2012). Thus the

advantage of the choice of the NCEP–NCAR reanalysis

is to cover the whole period where observations are

available using a single reanalysis product in order to

identify possible associations with large-scale climate

dynamics. It must be noted that our goal is not to check

the adequacy of the particular NCEP–NCAR reanalysis

product but to evaluate if the distribution of extreme

precipitation can be related to large-scale predictors, if

the same predictors are valid for different stations, and

at which time scales the large-scale forcings are relevant.

For these objectives, the bias of a given reanalysis

product compared to other products is of little rele-

vance, and because of the strong interannual variability

of precipitation in North Africa, it is important to

evaluate the relationships with large-scale dynamics

over long time periods to obtain robust results. The

gridded NCEP–NCAR data (six grid cells covering the

study area) were interpolated by the inverse distance

method to the station locations in order to provide in-

dividual descriptor sets for each station. The selected

variables include the following:

dgeopotential height at 500 and 850 hPa (geopot_500

and geopot_850),

dvertical velocity at 500 and 850 hPa and at surface

(omega_500, omega _850, and omega _surf),

dpotential temperature at surface (ptemp_surf),

dprecipitable water content (pwater_surf),

drelative humidity at surface (rhum_surf),

dspeciﬁc humidity at 500 and 850 hPa (shum_500 and

shum_850),

dmean sea level pressure (slp_surf),

dsurface temperature (temp_surf),

dzonal wind at surface (uwind_surf), and

dmeridional wind at surface (vwind_surf).

The homogeneity of these covariates has been

assessed by following Pettitt (1979) and by using the

modiﬁed version of the standard normal homogeneity

test (SNHT) by Khaliq and Ouarda (2007). Indeed, the

gradual introduction of satellite data into reanalysis

products can introduce an artiﬁcial changepoint leading

FIG. 1. Geographic location of all stations (three selected stations in Morocco, one in Algeria,

and two in Tunisia).

TABLE 1. Description of the selected stations with long records of precipitation.

Station Country Lat Lon Alt (m) Record length (yr) Starting year Ending year

Algiers Algeria 36.7483.068140 47 1951 2005

Larache Morocco 35.18826.1585 51 1942 2011

Tangier Morocco 35.77825.8085 33 1972 2006

Melilla Morocco 35.29822.94847 46 1907 2009

Gabès Tunisia 33.88810.1084 57 1950 2009

Tunis Tunisia 36.83810.23866 58 1950 2009

APRIL 2016 N A S R I E T A L . 1065

to the false detection of trends or homogeneity breaks

(Sterl 2004). The Pettitt and SNHT tests agree only on a

signiﬁcant changepoint at the 5% level in relative hu-

midity, in 1957 for SNHT and in 1963 for the Pettitt test.

Therefore, no changepoints are detected in the beginning

of the 1980s following the introduction of satellite data.

These covariates are considered at different time

steps. In the ﬁrst case, we considered the maximum

observed daily precipitation during the extended winter

season (October–March) and the simultaneous daily

covariate in the reanalysis data associated with this ex-

treme rainfall event. This gives one observation of

maximum daily winter rainfall and its associated co-

variate for each year (hereafter case 1). In case 2, we

considered the maximum winter precipitation and the

average value of each covariate 5 days before the date of

the annual maximum rainfall during the winter. These

two cases can be considered as dealing with the short-

term effect of covariates on MAP. In case 3, we calcu-

lated the 30-day average of the covariate before the date

of maximum winter precipitation. Finally, in case 4, we

considered the maximum daily winter precipitation and

the value of each covariate for the entire season

(October–March average). These last two cases can be

considered as dealing with the long-term effect of co-

variates on MAP.

3. Methods

For modeling extreme rainfall events, we used the

GEV distribution (Coles 2001).TheroleoftheGEV

distribution is to describe a sample that follows a

maximum of distributions introduced by Fisher and

Tippett (1928). The GEV distribution is ﬂexible and

has been the subject of several theoretical studies

and applications for modeling extreme ﬂood, pre-

cipitation, and wind events (El Adlouni et al. 2007;

Hundecha et al. 2008). The development of stationary

GEV distribution models for univariate extreme value

analysis can be found in the literature (Coles 2001;

Olsen et al. 1999). The use of this distribution in the

frequency analysis of extreme events is based on a

number of speciﬁc hypotheses concerning the variable

of interest. Indeed, the observations must be in-

dependent and identically distributed. However, the

stationarity assumption is often not met for observed

hydroclimatic datasets (Khaliq et al. 2006). In this case,

the distribution parameters and the distribution itself

could be changing in time. Therefore, it is essential to

develop the GEV model in the multivariate space,

where extreme events can be associated with other

variables. To model the relationship between the co-

variates and the extreme variable of interest, we can

use the GEV B-spline approach (Nasri et al. 2013). This

approach has been developed to describe the associa-

tion of an external covariate with the variable of

interest. The estimation of the parameters of the GEV

B-Spline model is done in a Bayesian framework to

obtain the posterior distribution by applying Markov

chain Monte Carlo (MCMC) algorithms.

a. The GEV distribution

The GEV distribution is characterized by three

parameters: the location m,scales,andshapejparameters.

Depending on the value of the shape parameter we have

three types of extreme value distributions—namely, the

Gumbel (j50), Fréchet (j.0), and Weibull (j,0).

Considering a sample Y5(y1,⋯,yn), the GEV distri-

bution function is as follows:

F(y,m,s,j)5expn2h11jy2m

si2(1/j)oj6¼ 0

F(y,m,s,j)5exph2exp2y2m

si j50.

(1)

This classical GEV distribution is based on the statio-

narity assumption and does not consider the de-

pendence of extreme events on other variables. In the

following section, the nonstationary GEV approach is

presented to consider the effect of a covariate on

extreme values.

b. The nonstationary GEV B-spline model

In the nonstationary case of a GEV distribution, the

parameters of the GEV distribution are assumed to

change in time or depend on a covariate. In the present

form of the GEV, parameters sand jare assumed to be

constant. Having a random variable Ythat follows the

GEV(mx,s,j) and a vector of pcovariates given by

X5(X1,X2,⋯,Xp), the location parameter of the

GEV is written as follows:

mx5

p

i51

fi(Xi)5f1(X1)1f2(X2)1⋯1fp(Xp), (2)

where fiis a function that represents the relationship

between the parameter and the covariates X

i

. This

function can be described by the following B-spline

function:

fi(xi)5b0,i1

m

j51

bj,iBj,i,d(xi), (3)

where Bj,d(x) is a polynomial function of degree dand m

is the number of control points (Nasri et al. 2013).

Therefore, Eq. (2) can be rewritten as follows:

1066 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 55

mx5

p

i51

fi(xi)5

p

i51"b0,i1

m

j51

bi,jBj,i,d(xi)#. (4)

The predictors’ interaction can be expressed in our

model by using multivariate B-spline functions (de

Boor 2001). These functions allow considering the

correlation between the predictors. In this study 14

predictors are used. Consequently, in order to simplify

the model we did not consider the interaction between

predictors.

c. Parameter estimation

In this study, the estimation of the parameters of

the GEV B-spline model is carried out in a Bayesian

framework. In the Bayesian approach, the unknown

parameters are not constant and are considered as

random variables with a prior distribution p(u).

Bayes’s theorem therefore gives the following

deﬁnition of the posterior distribution of these

parameters:

f(ujy)5f(yju)3p(u)

f(y), (5)

where

u5(mx,s,j)5(b,s,j) and b5b0

b. (6)

According to Nasri et al. (2013), we choose a multivar-

iate normal distribution as a prior for a location pa-

rameter b;N(0, Sb3I), a noninformative prior for

scale parameter 1/s, and a beta distribution as prior for a

shape parameter [b(6, 9)].

The posterior distribution is written as follows:

f(ujy)}1

s8

>

>

>

<

>

>

>

:

11j2

6

6

6

4

y2

j

(1 B)b0

b

s3

7

7

7

5

9

>

>

>

=

>

>

>

;

2(1/j)21

exp0

B

B

@

28

>

>

<

>

>

:

11j2

6

6

4

y2

i

(1 B)b0

b

s3

7

7

59

>

>

=

>

>

;

2(1/j)1

C

C

A

[2pdet(Sb)]2(k/2)exp 2jjbjj2

2s2!1

sjﬃﬃﬃﬃﬃﬃ

2p

pexp"2(j20:1)2

2s2

j#1

s. (7)

The GEV B-spline model, which takes into account

nonstationarity and nonlinearity, offers a great ﬂexibil-

ity and takes into account the heavy-tailed character of

the extreme distribution.

The posterior distribution is estimated by the

Metropolis–Hasting algorithm (see appendix).

To select the number of knots (kt; 1 kt 50.51 m s

21

)

and the degree of B-spline functions used in this

study, we compared several combinations of degrees

and knots using the maximum likelihood method.

The following algorithm explains how these parameters

are chosen:

(i) set d2[1, 10] and m2[1, 10],

(ii) calculate Eq. (7) for all combinations of (d,m), and

(iii) choose values of (d,m) that maximize Eq. (7).

In this case, we apply the B-spline functions with 38

and 3 kt. This choice was found to be optimal for the

majority of the stations’ data.

d. Validity of the model with covariates

To validate the inﬂuence of covariates on the vari-

able of interest, the log likelihood of the GEV B-spline

(M1) model and the stationary GEV (M0) model

(without covariates) are compared using the test of

deviance:

D52[l(M1) 2l(M0)], (8)

where lis the maximum log likelihood function for

model M. Large values of Dindicate that model M1 is

more adequate at representing the data than model M0.

The Dstatistic is distributed according to a chi-square

distribution x2, with ydegrees of freedom, where yis

the difference between the number of parameters of

the M1 and M0 models. For a given aconﬁdence level,

we reject H0 hypothesis (H0: M1 and M0 are similar)

when D $x2

12a. This statistic is often used to compare

two models when one model is a special case of the other

(M0 M1; Coles 2001;El Adlouni and Ouarda 2009).

This test accounts for differences in model complexity to

avoid overﬁtting.

e. Quantile estimation

The MCMC algorithm also produces the conditional

quantile distribution for an observed value x0of the

covariate Xt. Indeed, for each iteration iof the MCMC

algorithm i51, ...,N, the quantiles corresponding to

APRIL 2016 N A S R I E T A L . 1067

the nonexceedance probability 1 2p,x(i)

p,x0, and the pa-

rameter vector [m(i)

x0,s(i)

x0,j(i)

x0] are estimated using the

inverse of the cumulative distribution function of the

GEV distribution:

y(i)

p,x0

5m(i)

x0

2s

j(i)f12[log(1 2p)]2j(i)g, (9)

where m(i)

x0is the position parameter conditional on the

particular value x0of X.

4. Results

a. Tests for independent and identically distributed

random variables

In the ﬁrst step of using the nonstationary GEV model

(in this case GEV B-spline) we checked stationarity,

homogeneity, and independence using the Mann–

Kendall (Mann 1945), Mann–Whitney (Wilcoxon 1945),

and Wald–Wolfowitz tests (Wald and Wolfowitz 1940),

respectively, for MAP series for each station. The results

of these tests showed that all MAP series are non-

stationary, except for the Algiers station where a nega-

tive trend is observed and is signiﬁcant at the 5% level,

as previously observed by Reiser and Kutiel (2011).

However, all the time series of MAP respect the hy-

potheses of homogeneity and randomness. Figure 2

shows the variation of all MAP series versus time, and

Fig. 3 shows the monthly frequency of occurrence of

annual maximum daily precipitation.

b. Predictors from reanalysis data

We selected the 14 NCEP–NCAR covariates

extracted from reanalysis (see section 2) and tested our

models with these covariates considering the four time

scales (cases 1–4). The negative log likelihood and de-

viance between the GEV B-spline model and the sta-

tionary GEV model are analyzed to detect the inﬂuence

of NCEP–NCAR predictors on extreme rainfall for each

of the four cases. To avoid overﬁtting, each covariate is

considered separately in the GEV B-spline model. This

allows evaluating whether each covariate leads to a

better ﬁt than the stationary GEV model. As there are

14 covariates for each case, the results are presented in

Table 2 only for the signiﬁcant covariates on MAP at

each station at the 5% and 10% signiﬁcance levels, ac-

cording to the test of deviance. We note that all 14 co-

variates, depending on the station, are selected into

nonstationary GEV models that better reproduce ex-

treme precipitation than a standard stationary model,

with both short-term and long-term associations.

Overall, a similar number of signiﬁcant covariates is

selected for the four cases tested (i.e., from daily to

seasonal averages of covariates), with 11 covariates

identiﬁed for case 1, 13 for case 2, 16 for case 3, and 13

for case 4. This shows that all considered covariates may

have an impact on extreme daily precipitation at dif-

ferent time steps, from daily values to seasonal averages.

It is observed that the geopotential height (geopot_

500 and geopot_850) generally affects rainfall at all

FIG. 2. Variation of all MAP series vs time for selected stations.

1068 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 55

stations excluding Melilla (station in northern Mo-

rocco). For the stations of Tangier and Larache, the

geopotential heights have a short-term association with

MAP (case 1 and case 2). On the other hand, for the

Algiers station, these variables have an inﬂuence

generally at the seasonal time scale (case 4), and for the

stations of Tunisia (Gabès and Tunis) these variables

inﬂuence MAP in both the short and long term (cases 1,

2, and 4). Humidity predictors (rhum_surf, shum_500,

and shum_850) generally inﬂuence precipitation at all

FIG. 3. Monthly frequencies of occurrence for daily MAP in each selected station.

TABLE 2. The signiﬁcant covariates at 5% and 10% signiﬁcance levels for each station.

Station

Predictors signiﬁcant

at 5% (case)

Predictors signiﬁcant

at 10% (case) Station

Predictors

signiﬁcant at 5% (case)

Predictors signiﬁcant

at 10% (case)

Algiers Slp_surf (case 4) pwater_surf (case 1) Melilla omega_surf (case 1) rhum_surf (case 4)

geopot_500 (case 4) omega_500 (case 2)

omega_850 (case 2)

rhum_surf (case 2)

ptemp_surf (case 3)

slp_surf (case 3)

Gabès geopot_500 (case 1) geopot_850 (case 2) Tangier pwater_surf (cases 1 and 2) geopot_850 (case 1)

geopot_850 (cases 1, 2, and 4) rhum_surf (case 4) shum_850 (case 1) temp_surf (case 4)

pwater_surf (case 3) omega _850 (case 2) uwind_surf (case 4)

slp_surf (cases 1 and 4) uwind_surf (case 2)

shum_500 (case 3) ptemp_surf (case 3)

shum_850 (case 1) shum_850 (case 3)

vwind_surf (case 3)

geopot_500 (case 4)

Larache temp_surf (case 2) pwater_surf (case 4) Tunis pwater_surf (case 2) rhum_surf (case 1)

omega_500 (case 2) rhum-surf (cases 2 and 3) shum_850 (case 1)

shum_850 (case 2) shum_500 (case 2) geopot_850 (case 2)

geopot_500 (case 3) shum_850 (case 2)

ptemp_surf (case 3) omega_500 (case 3)

pwater_surf (case 3) slp_surf (case 3)

rhum_surf (case 3) omega_850 (case 4)

omega_surf (case 4)

ptemp_surf (case 4)

APRIL 2016 N A S R I E T A L . 1069

stations, excluding Algiers. For stations in Morocco,

these predictors appear in almost all cases (cases 1 and

3 for the Tangier station, case 4 for the Melilla station,

and cases 1, 2, and 3 for the Larache station). For sta-

tions in Tunisia, these predictors have both short- and

long-term effects on MAP time series at the stations of

Gabès (cases 1, 3, and 4) and Tunis (cases 1, 2, and 3).

Velocity predictors (omega_500, omega_850, and

omega_surf) have more effects on precipitation in

Morocco. We see a stronger inﬂuence of these pre-

dictors on rainfall in Morocco. Wind predictors

(uwind_surf and vwind_surf) inﬂuence the MAP only

at the Tangier station. Overall, we note the small in-

ﬂuence of wind covariates on precipitation extremes at

all stations. The potential temperature at the surface

(ptemp_surf) inﬂuences MAP at stations located in

Morocco for the long term (cases 3 and 4). The surface

temperature inﬂuences the MAP in Morocco at dif-

ferent time scales (case 3 for Melilla, case 1 for Larache,

and case 4 for Tangier stations). The precipitable water

content has an inﬂuence on MAP at all stations, usually

only for the short-term cases (1 and 2) at all stations.

The mean sea level pressure inﬂuences MAP in only

the Tunis, Gabès, and Algiers stations, generally in the

long-term cases 3 and 4.

c. Principal analysis of components for NCEP–

NCAR predictors

After the analysis of the dependence of MAP with

individual covariates, the possible relationships are also

investigated in a multivariate context. Principal com-

ponent analysis (PCA; Preisendorfer 1988) is used to

that end. The reason for using PCA is to take into con-

sideration the common signals in multivariate datasets.

PCA represents a method for dimensionality reduction.

PCA has been used for this purpose in many other

studies (e.g., Wetterhall et al. 2005;Maraun et al. 2010a).

The objective of this analysis is to summarize as much

information as possible by transforming interrelated

variables into new components (principal components)

that are uncorrelated with each other. In this study, we

ﬁrst applied PCA on the 14 covariates for each station.

Figures 4 and 5show the results of the projections of the

14 covariates on the ﬁrst and second components (F1

and F2, respectively) for the ﬁrst and last case in each

station. A number of criteria, such as the Kaiser crite-

rion (Kaiser 1960), can be used for the selection of the

factorial axis. The Kaiser criterion lies on the factorial

axis choices, where their eigenvalues are greater than 1.

In the present study, we noticed that for all factors that

FIG. 4. Contributions of the 14 NCEP–NCAR reanalysis covariates on the two principal components (F1 and F2) in selected station

(results for case 1). The numbers in the parentheses represent the percentage of explained variance for the represented axes (F1 and F2).

1070 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 55

have an eigenvalue greater than 1, those are generally

factors 1 and 2. This justiﬁes the choice of two factorial

axes. It can be seen that at all stations, there are signif-

icant correlations between the covariates, depending on

the case (1–4) considered for temporal aggregation. To

avoid overﬁtting, each component is considered sepa-

rately in the GEV B-spline model to evaluate if it

provides a better ﬁt than a stationary GEV model. The

results show, ﬁrst, that most variables contribute to the

formation of the components F1 and F2, with some co-

variates having a larger contribution, such as the geo-

potential height (geopot_500 and geopot_850), velocity

(omega_500, omega_850, and omega_surf), and hu-

midity (rhum_surf, shum_500, and shum_850). Pre-

dictors such as uwind_surf and vwind_surf contribute

more in stations close to the Mediterranean coast such

as Tangier, Tunis, and Gabès. We then applied the GEV

B-spline model of the MAP series for each station and each

case using F1 and F2 as covariates. Next, we calculated

the deviance between the results of the GEV B-spline

and GEV0 models to investigate the inﬂuence of these

components on MAP data. Table 3 shows the results of

deviance with a threshold of 5% and 10%. According to

these results one can see that, at all stations, there is at

least one component (F1 or F2) that inﬂuences the MAP

series. For stations in Morocco (Larache, Melilla, and

Tangier), we note that the component that contains

more information about the geopotential height, hu-

midity, velocity, and wind has the largest inﬂuence on

MAP. For the station in Algeria (Algiers), the MAP is

more inﬂuenced by the geopotential height predictors

rather than by others. For stations in Tunisia (Tunis and

Gabès), we can see that the inﬂuence of geopotential

FIG.5.AsinFig. 4, but for case 4.

TABLE 3. The results of the deviance for PCA.

Station Signiﬁcant component Case

Algiers F1 at 5% 4

Larache F1 at 5% 1

Tangier F1 at 5% 1

F2 at 5% 1

Melilla F2 at 5% 2

F1 at 10% 3

F2 at 5% 4

Tunis F1 at 10% 3

F2 at 5% 3

Gabès F1 at 10% 1

F2 at 5% 1

F1 at 5% 2

F2 at 5% 2

F2 at 5% 4

APRIL 2016 N A S R I E T A L . 1071

height, velocity, temperature, and sea surface pressure

on MAP is important.

d. Quantile estimation

We can also see the impact of the covariates on the

estimated quantile level for each of the models. In the

case of the GEV B-spline model, quantile values depend

not only on the nonexceedance probability 1 2pbut

also on the covariate values. This allows computing

quantiles on a seasonal or annual basis, depending on

the values of the covariates. To demonstrate the co-

variates’ impact on quantile values, we show some

quantile estimation examples for each station. Figure 6

displays a nonstationary quantile estimation example

for each station for the 2-yr return period (non-

exceedance probability 50.5), which represents the

median value of MAP.

For each station, we observed different values of the

2-yr quantiles estimated with the GEV B-spline model

since quantile values are dependent on covariates. In

contrast, the GEV0 model provides just one estimate for

the 2-yr quantile (e.g., for the Algiers station, the me-

dian precipitation value corresponding to the 2-yr

quantile is 100 mm for the GEV B-spline and 64 mm

for the GEV0 model, and for the Tunis station, the

stationary quantile is equal to 50 mm and the median of

the nonstationary quantiles is equal to 70 mm). Ac-

cording to this ﬁgure, we notice that the covariate-

dependent quantile values are more ﬂexible and allow

reaching more extreme data values, unlike the station-

ary quantiles that do not take into consideration the

interannual climatic variability. The estimated quantiles

show the advantage of incorporating additional in-

formation into nonstationary models.

5. Conclusions

In this work, the inﬂuences of climatic variables such

as geopotential height, pressure, or temperature on

maximum annual daily precipitation have been studied

at six stations located in North Africa with long pre-

cipitation time series. A total of 14 variables were

computed from NCEP–NCAR reanalysis data. To

study the inﬂuence of these covariates at the different

stations, the GEV B-spline model (Nasri et al. 2013)

was used. The originality of this model, as opposed to

other nonstationary models, is that it takes into con-

sideration the nonstationary and the nonlinear tem-

poral ﬂuctuations of covariates. Nonstationary models,

such as the GEV1 (linear dependence) and the GEV2

(quadratic dependence), deﬁne in advance the form of

dependence between the variable of interest and the

covariates. On the other hand, the GEV B-spline

model takes into consideration all continuous de-

pendence forms between the covariates and the vari-

able of interest.

The results of this study are divided into two parts. In

the ﬁrst part, the possible dependencies between the

FIG. 6. Example of nonstationary and stationary median for each station using the ﬁrst or the second principal component analysis as

covariates.

1072 JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY VOLUME 55

maximum annual precipitation and each of the individ-

ual climatic covariates were considered. The GEV

B-spline model was used to detect these dependencies, and

the deviance likelihood ratio test was used to identify

the nonstationary models with covariates that provide

an improvement in comparison to stationary models in

each station. In the second part, the combined de-

pendencies were analyzed using principal component

analysis of the different atmospheric predictors. From

the results of the principal component analysis, we an-

alyzed the inﬂuence of the combined variables using the

two principal components (F1 and F2) for each station in

the GEV B-spline model. Our results indicate that no

single combination of atmospheric predictors is optimal

for stations. The relevant covariates may vary from one

station to another and also depend on the considered

time scale, from daily to annual averages. These results

are consistent with the fact that extreme precipitation

is a process exhibiting a high spatiotemporal variability

between different locations. Given this variability, it

must be noted that the covariates describing the mois-

ture ﬂux in the atmosphere (relative or speciﬁc humid-

ity) or in atmospheric circulation (pressure and

geopotential heights) are often selected in the different

stations as valid predictors. During winter, when most of

the annual maximum precipitation occurs, geopotential

height might be more important because of the south-

erly position of the extratropical westerlies. In other

seasons thermodynamic predictors like humidity may

gain signiﬁcance because of the convective nature of

precipitation in these seasons.

The present work provides a ﬁrst step prior to the

development of statistical downscaling methods tai-

lored for extreme precipitation in North Africa. The

next step would be to use GCM outputs to ﬁrst validate

the method in the present climate, with the covariates

that are correctly reproduced in historical climate

simulations, and then to make future projections.

However, in this case the use of the nonstationary GEV

model with B-spline functions would probably be less

appropriate because of some limitations: (i) the in-

troduction of several covariates within these types of

models increases the number of hyperparameters,

which increases the number of parameters to estimate

as well as the estimation errors; (ii) the interactions

between the predictors make the model much more

complex since we need to take into consideration

multivariate spline functions (de Boor 2001)oruse

some decisional model, such as an artiﬁcial neural

network as in Cannon (2010); and (iii) this type of

model allows the description of the impact of co-

variates on the variable of interest and is not able to use

them for prediction outside this period of study.

Consequently, an alternative to this type of model is

quantile regression methods (Buchinsky 1998). Unlike

linear regression, which results in the estimation of the

conditional mean for the response variable given cer-

tain values of predictor variables, quantile regression

aims at estimating either the conditional median or

other quantiles of the response variable. Quantile re-

gression was considered by Jagger and James (2006) for

wind speed and by Friederichs and Andreas (2008) for

precipitation, based on several climatic covariates.

Future work can focus on the comparison of extreme

value models and the quantile regression approach to

distinguish the relative beneﬁts of the use of these two

types of models for downscaling purposes.

Acknowledgments. The datasets were provided by

the Agence Nationale des Ressources Hydrauliques

(Algeria), Direction de la Recherche et de la Planiﬁca-

tion de l’Eau (Morocco), Direction Générale des

Ressources en Eau (Tunisia), and European Climate

Assessment and Dataset. Special thanks are given to

H. Ben-Mansour, R. Bouaicha, L. Behlouli, K. Benhattab,

R. Taibi, and K. Yaalaoui for their helpful contribution

to database collection. The authors are indebted to ed-

itor Thomas Mote and to two anonymous reviewers

whose comments helped considerably improve the

quality of the manuscript.

APPENDIX

MCMC Algorithm for GEV B-Spline Model

The basic idea of the MCMC method is, for each

parameter, to construct a Markov chain with the pos-

terior distribution being a stationary and ergodic dis-

tribution. After running the Markov chain of size N

for a given burn-in period N0, one obtains a sample

from the posterior distribution f(ujy). One popular

method for constructing a Markov chain is via the

Metropolis–Hastings (MH) algorithm (Metropolis

et al. 1953;Hastings 1970). We simulated the re-

alizations from the posterior distribution by way of a

single-component MH algorithm (Gilks et al. 1996).

Each parameter was updated using a random-walk MH

algorithm with a Gaussian proposal density centered at

the current state of the chain. Some techniques to as-

sess the convergence of the MCMC methods, such as

the Raftery and Lewis diagnostic (Raftery and Lewis

1992,1995) and subsampling methods (El Adlouni

et al. 2006), make it possible to determine the length of

the chain and the burn-in time. In all cases, the con-

vergence methods indicated that the Markov chains

converged within a few iterations. In this study, we

APRIL 2016 N A S R I E T A L . 1073

considered chains of size N515 000 and a burn-in

period of N058000 runs. In every case, a sample of

N2N057000 values is collected from the posterior of

each of the elements of u.

The principal steps of the MH algorithm can be

summarized as follows:

(i) Initialization: assign initial value u(0)and choose

an arbitrary proposal probability density Q(u*ju).

In this case we propose a multivariate normal

distribution.

(ii) For each iteration t, generate u*, a candidate for the

next sample, by picking from the distribution

Q(u*jut).

(iii) Calculate the acceptance ratio, given by

a(u*, ut)5[p(u*jy)/p(utjy)].

(iv) If a$1, then the candidate is more likely than ut;

automatically accept the candidate by setting

ut115u*. Otherwise, accept the candidate with

probability a; if the candidate is rejected, set

ut115utinstead.

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