Large scale climate and rainfall seasonality in a Mediterranean...
Transcript of Large scale climate and rainfall seasonality in a Mediterranean...
Received: 9 July 2015 Accepted: 15 October 2016
DO
I 10.1002/hyp.11061R E S E A R CH AR T I C L E
Large scale climate and rainfall seasonality in a MediterraneanArea: Insights from a non‐homogeneous Markov model appliedto the Agro‐Pontino plain
Francesco Cioffi1 | Federico Conticello1 | Upmanu Lall2 | Lucia Marotta3 | Vito Telesca2
1Dipartimento di Ingegneria Civile Edile
Ambientale, Università di Roma ‘La Sapienza’,Rome, Italy
2Department of Earth & Environmental Eng,
Columbia University, New York, New York,
USA
3Scuola di Ingegneria, Università della
Basilicata, Potenza, Italy
Correspondence
Marotta Lucia, Scuola di Ingegneria, Università
della Basilicata, Potenza, Italia.
Email: [email protected]
Hydrological Processes 2016; 1–19
AbstractIn the context of climate change and variability, there is considerable interest in how large scale
climate indicators influence regional precipitation occurrence and its seasonality. Seasonal and
longer climate projections from coupled ocean–atmosphere models need to be downscaled to
regional levels for hydrologic applications, and the identification of appropriate state variables
from such models that can best inform this process is also of direct interest. Here, a Non‐
Homogeneous Hidden Markov Model (NHMM) for downscaling daily rainfall is developed for
the Agro‐Pontino Plain, a coastal reclamation region very vulnerable to changes of hydrological
cycle. The NHMM, through a set of atmospheric predictors, provides the link between large scale
meteorological features and local rainfall patterns. Atmospheric data from the NCEP/NCAR
archive and 56‐years record (1951–2004) of daily rainfall measurements from 7 stations in
Agro‐Pontino Plain are analyzed. A number of validation tests are carried out, in order to: 1) iden-
tify the best set of atmospheric predictors to model local rainfall; 2) evaluate the model perfor-
mance to capture realistically relevant rainfall attributes as the inter‐annual and seasonal
variability, as well as average and extreme rainfall patterns.
Validation tests show that the best set of atmospheric predictors are the following: mean sea
level pressure, temperature at 1000 hPa, meridional and zonal wind at 850 hPa and precipitable
water, from 20°N to 80°N of latitude and from 80°W to 60°E of longitude. Furthermore, the
validation tests show that the rainfall attributes are simulated realistically and accurately. The
capability of the NHMM to be used as a forecasting tool to quantify changes of rainfall patterns
forced by alteration of atmospheric circulation under climate change and variability scenarios is
discussed.
KEYWORDS
Climate Change, Hidden Markov Model (HMM), Nonhomogeneous Hidden Markov Model
(NHMM), Statistical Downscaling
1 | INTRODUCTION
The Agro‐Pontino Plain is a coastal reclamation region of Central Italy,
whose hydro‐geological features make it particularly vulnerable to even-
tual future changes of hydrological cycle such as those induced by climate
change. It is of naturalistic and economic importance in the
Mediterranean region. The Mediterranean coastal regions have been
noted as onesof themost vulnerable “hot‐spots” to future climate change
(Giorgi, 2006; IPCC, 2013; Cudennec, Leduc, & Koutsoyiannis, 2007).
wileyonlinelibrary.com/journa
Several researchers (Ulbrich et al., 2006; Giorgi & Lionello, 2008;
Sheffield & Wood, 2008), by using General Circulation Models (GCM)
or Regional Circulation Models (RCM), predicted significant impacts
globally on entire Mediterranean region, due to changes in annual pre-
cipitation and their interannual and seasonal variability.
However, rainfall simulations by GCMs and RCMs tend to have
large uncertainties and biases, particularly at the local scale, that is
the most relevant to assess vulnerability, resilience, and adaptation
measurements of communities to climate change.
Copyright © 2016 John Wiley & Sons, Ltd.l/hyp 1
2 CIOFFI ET AL.
The larger goal of our research is to construct a tool able to fore-
cast future hydrological cycle alterations on Agro‐Pontino Plain. A first
step, pursued here, consists of modeling the link between large scale
atmospheric circulation and local rainfall pattern by using a statistical
downscaling model (SDM).
Due to the coarse parameterization of multi‐scale hydrologic
processes, and the limited ability to resolve significant sub‐grid scale
features such as topography, clouds and land use (Grotch & Mac
Cracken, 1991), GCMs fail to reproduce several statistics of the
regional or local rainfall series, including the frequency and intensity
of daily precipitation (Bates, Charles, & Hughes, 1998; Charles, Bates,
& Hughes, 1999; Busuioc, Bates, Whetton, & Hughes, 1999a; Dibike,
Gachon, St‐Hilaire, Ouarda, & Nguyen, 2008; Baguis, Roulin, Willems,
& Ntegeka, 2009; Willems & Vrac, 2011). Large scale temporal and
spatial features of atmospheric circulation are significantly better
simulated by GCMs than precipitation, therefore, methods for down-
scaling local rainfall from the GCMs projections, using links between
the large scale atmospheric circulation and local rainfall patterns have
evolved.
During the last 20 years, a wide range of statistical downscaling
methods (SDMs) have been developed (Hewitson & Crane, 1996;
Zorita & von Storch, 1997; Wilby et al., 2004a, 2004b; Fowler,
Blenkinsop, & Tebaldi, 2007; Maraun et al., 2010). A detailed bibliogra-
phy analysis of working principles, strengths and weaknesses of differ-
ent SDMs is proposed in a number of papers, as for instance, (Fowler
et al., 2007; Hashmi, Shamseldin, & Melville, 2009; Wilby et al.,
1998; Wilby, Hayc, & Leavesley, 1999; Wilby et al., 2004a, 2004b;
Xu, 1999). Here, from the different statistical approaches proposed
in literature, the application of the Hidden Markov Model (HMM)
and Non‐homogeneous Hidden Markov Model (NHMM) is chosen.
HMM and NHMM have found widespread application in meteo-
rology and hydrology, for studies of climate variability or climate
change, and for statistical downscaling of daily precipitation from
observed and numerical climate model simulations (Zucchini &
Guttorp, 1991; Hughes & Guttorp, 1994; Hughes, Guttorp, & Charles,
1999; Bellone, Hughes, & Guttorp, 2000; Robertson, Kirshner, &
Smyth, 2004; Betrò, Bodini, & Cossu, 2008; Charles et al., 1999). In
the recent past, NHMMs have been successfully used to downscale
precipitation in different regions of the world (Khalil, Kwon, Lall, &
Keheil, 2010; Robertson et al., 2004; Kwon, Lall, Moon, Khalil, &
Ahn, 2006; Kwon, Brown, Xu, & Lall, 2009, Cioffi, Conticello, & Lall,
2015). An important characteristic of HMMs and NHMMs is that rain-
fall patterns at the local scale, as recorded by a number of rain‐gauges,
may be associated with synoptic or large‐scale atmospheric patterns.
Thus, these downscaling methods may also be used as diagnostic tools
to investigate the atmospheric features generating regional rainfall.
The HMM, represents a doubly stochastic process, involving an under-
lying hidden, or not observable, stochastic process, interpreted in the
present case as a hidden weather state, that is translated into another
stochastic process that yields the sequence of observations (rainfall
occurrence and amount at the different rain gauges) (Rabiner & Juang,
1986). The observed process (e.g., precipitation occurrence or/and
amount at a network of sites) is conditional on the hidden states which
evolve according to a first order Markov chain. Transitions from one
state to the next have fixed probabilities that depend only on the
current state. The NHMM is obtained as a generalization of a HMM
model, by allowing the transition probabilities between the hidden
states to be time‐varying, being themselves conditioned by atmo-
spheric predictors varying in time. Thus, NHMMs generalize the class
of mixtures of multivariate regression models with concomitant
variables (Wang et al., 1998) to allow for temporal dependence.
In the present paper, several potential predictors derived from the
NCEP‐NCAR re‐analysis (Kalnay et al., 1996) for a 56‐years record
(1950–2005) and the corresponding daily rainfall measurements from
7 stations in the Agro‐Pontino‐plain are analyzed.
Data on temperature at 1000 hPa (T1000), mean sea level pres-
sure (MSL), meridional winds (MW850) and zonal winds at 850 hPa
(ZW850), precipitable water (P), from 20°N to 80°N of latitude and
from 80°W to 60°E of longitude, are used for the identification of
the main meteorological features that influence daily rainfall patterns
in Agro‐Pontino Plain. As in (Cioffi et al., 2015), the atmospheric circu-
lation fields are used as the determinants of changes in the seasonality
of precipitation, rather than a pre‐specification of the seasonality and
its change.
By performing NHMM validation tests, we were able to test the
capability of the model to realistically simulate: a) seasonality of local
rainfall pattern in Agro‐Pontino plain; b) extreme daily rainfall
frequency and amount; c) trends for the entire examined period
(1951–2004) of annual and seasonal rainfall amounts, as well as, of
daily extremes.
In section 2, the data and method are described; the applications
of HMM and NHMM to the study case are detailed in section 3, while
in section 4 the results of the study are summarized and future per-
spectives are outlined.
2 | DATA AND METHODS
2.1 | Climate context and Data
The study is focused on a coastal area of Central Italy, the
Agro‐Pontino plain which is typical of the hydro‐geological features
of Mediterranean coastal environments. It is densely populated and
is the site of important agricultural and industrial activities. Potential
climate changes may translate into hydrologic hazards and adversely
affect the future socio‐economic development of the area and the
biodiversity of the National Park ‘Circeo’.
Geographically, the Agro‐Pontino plain covers the coastline
between the Tirrenean Sea and the Apennine dorsal called
“Lepino‐Ausona”. The length of this territory is about 50 kilometers,
its width 20 kilometers and it is extended along the NW‐SE direction.
The coordinates of the center of the area are 41°27′N, 12°53′E.
The hydrological basin receives water from Lepini, Ausoni and
Colli Albani mountains, and from Karst soft water springs that outcrop
along the all edge piedmont; this system also includes coastal lakes like
Paola, Monaci, Fogliano and Caprolace.
The area has a Mediterranean climate it is mild and wet during the
winter and hot and dry during the summer (Giorgi & Lionello, 2008).
Winter climate is mostly dominated by the eastward movement of
storms originating over the Atlantic and impinging upon the western
CIOFFI ET AL. 3
European coasts. The winter Mediterranean climate, and most impor-
tantly precipitation, is thus affected by the North Atlantic Oscillation
(NAO) over its western areas (Hurrell, 1995), the East Atlantic and
other patterns over its northern and eastern areas as East Atlantic
West Russia (EAWR) and or Scandinavia (Trigo et al., 2006). The El
Nino Southern Oscillation (ENSO) has also been suggested to signifi-
cantly affect winter rainfall variability over the Eastern Mediterranean
(Mariotti et al., 2002). In addition to Atlantic storms, Mediterranean
storms can be produced internally to the region associated with cyclo-
genesis in areas such as the lee of the Alps, the Gulf of Lyon and the
Gulf of Genoa (Lionello et al., 2006b). In the summer, high pressure
and descending motions dominate over the region, leading to dry con-
ditions particularly over the southern Mediterranean. Summer
Mediterranean climate variability has been found to be connected with
both the Asian and African monsoons (Alpert, Ilani, Krichak, Price, &
Rodò, 2006) and with strong geopotential blocking anomalies over
central Europe (Xoplaki, González‐Rouco, Luterbacher, & Wanner,
2004; Trigo et al., 2006).
A 56‐years record (1950–2005) of daily rainfall collected at 32 sta-
tions in Agro‐Pontino plain (“Istituto Idrografico e Mareografico di
Roma” and “Aeronautica Militare”) is assembled. However most of
the stations had time series that had severe gaps. On the basis of com-
pleteness of the time series and tests of homogeneity of data
(Wijngaard et al., 2003), 7 stations (Table 1) are selected for use in
the downscaling models (HMM‐NHMM). The location of these
stations is shown in Figure 1. The selected stations cover most of the
considered area.
As discussed in the current section, the climate of the Mediterra-
nean region is strongly forced by planetary scale patterns. Due to the
regional orography and the presence of the Mediterranean Sea, the
temporal and spatial behavior of the regional features associated with
such large‐scale forcing is complex. For using NHMM, it is important to
identify a number of candidate atmospheric predictors on a region suf-
ficiently wide to account also of possible remote influences, such as
NAO, EAWR, Scandinavia Oscillation and ENSO.
The literature about the climatology of Mediterranean region
(Lionello et al., 2006a; Alpert et al., 2006; Ulbrich et al., 2006; Giorgi
& Lionello, 2008; Sheffield & Wood, 2008; Langousis & Kaleris,
2014), suggests that atmospheric fields from a region bounded by lat-
itude: 20 N to 80 N and longitude: 80 W to 60E, should be wide
enough for capturing the influence of large scale atmospheric circula-
tion on local rainfall in the region. Most of the climate indices cited
above refers to dipolar atmospheric features within this region. As
shown in (Cioffi, Lall, Rusc, & Krishnamurthy, 2015), the influence of
TABLE 1 Rainfall gauging stations
N. LAT LON NAME
1 41.35 13.05 CESARELLA DI SABAUDIA
2 41.59 12.82 CISTERNA DI LATINA
3 41.47 12.99 FORO APPIO
4 41.55 12.91 LATINA 1
5 41.47 12.9 LATINA 2
6 41.51 13.41 OSTERIA DI CASTRO
7 41.46 13.1 PONTE FERRAIOLI
ENSO on precipitation patterns in Europe is not direct; rather, it is
via its influence on the NAO.
A global climate data set referred to as “the NCEP/NCAR 40‐
YEAR Reanalysis Project” (Kalnay et al., 1996) was acquired. This
NCEP/NCAR reanalysis data set (1950–2005) is continually updated
to represent the state of the Earth’s atmosphere, incorporating obser-
vations and numerical weather prediction (NWP) model output from
1950 to 2005. It was a joint product from the National Centers for
Environmental Prediction (NCEP) and the National Center for Atmo-
spheric Research (NCAR).
From the reanalysis archive, the following atmospheric fields were
identified as candidates for NHMM application: geo‐potential height,
air pressure at mean sea level, temperature, meridional & zonal wind,
precipitable water, cloud area fraction, sensible heat flux, vertical wind
(“NCEP/NCAR”).
2.2 | Model description
The Hidden Markov and Non‐homogeneous Hidden Markov models ‐
as presented by Kirshner, 2005a, 2005b, Khalil et al., 2010, Robertson
et al., 2004 ‐ are adapted for the applications pursued in this paper.
The main ideas are summarized below for completeness and the reader
is referred to these papers for details.
A HMM is a doubly stochastic model where multivariate time
series are generated conditionally on few discrete underlying hidden
states, via some distribution. The hidden states undergo Markovian
transitions. In hydrological applications, a hidden state is thought as
an unobserved weather state affecting rainfall occurrence and amount
in a number of locations simultaneously (Hay, McCabe, Wolock, &
Ayers, 1991). In the present application of the model rainfall occur-
rence is modelled as in Kirshner (2005a, 2005b), Robertson et al.
(2004), Hughes and Guttorp (1994) and Khalil et al. (2010), while rain-
fall amounts are incorporated directly into the formulation of the
HMM, similarly to the approach of Bellone et al. (2000).
Let Rt be a M‐dimensional vector of observations at time t,
representing the daily rainfall amount at M different rain gauges. Let
St be a discrete variable (St = [1, S]) representing one of the S possible
hidden states at the same time t. Let R1:T = (R1 , . . . , RT ) and S1:
T = (S1 , . . . , ST) be, respectively, the corresponding sequences of
rainfall amount at the M locations and of the hidden states.
The log‐likelihood of the data of the model can be written as
l ¼ log P Rð Þ ¼ log ∑s
P S1ð Þ ∏T
t¼2P St St−1jð Þ
� �∏T
t¼1P Rt Stjð Þ
� �(1)
In Equation 1, P(St| St − 1) is the transition probability between two
temporal subsequent hidden states, and P(Rt| St) indicates the emission
probabilities, i.e. the conditional distributions of the observed variable
Rt from the specific state St.
For a first‐order homogeneous HMM, stationary transition matrix
Γ with entries γij = P(St = i| St‐1 = j) is used to characterize P(St| St − 1).
Introducing additional observed variables X = (X1, X2 , …XT),
−where Xt ¼ X1t ;……::;Xp
t
� �represents a sequence of p exogenous
atmospheric variables at time t ‐ and making St dependent on both
St − 1 and Xt, a nonhomogeneous HMM (NHMM) is obtained. In the
FIGURE 1 Location of stations in Agro‐Pontino Plain
4 CIOFFI ET AL.
NHMM the probability of hidden state transitions is allowed to vary
with time, as a function of the exogenous variables X.
The log‐likelihood of the data for NHMM becomes:
l ¼ logP RjXð Þ ¼ log∑S
P S1jX1ð Þ ∏T
t¼2P StjSt−1;Xtð Þ
� �∏T
t¼1P RtjStð Þ
� �(2)
FIGURE 2 Log‐likelihood vs number of hidden states
In Equation 2, the hidden state transitions are modeled by multi-
nomial logistic regression depending on Xt, (Hughes et al., 1999):
P St ¼ jjSt−1 ¼ i;Xt ¼ xð Þ ¼exp
�σji þ ρjxt� �
:
∑Kk¼1 exp σjk þ ρkxt
� � (3)
FIGURE 3 BIC vs number of hidden states
CIOFFI ET AL. 5
where σjk and ρk are parameters for the K‐order multinomial
regression.
Conditional‐Chow‐Liu (CL), Independent delta‐gamma (CI‐gamma)
or Independent delta‐exponential (CI‐exponential) can be used to
model the multivariate, rainfall probability distribution P(Rt| St). The last
two models are a mixture of a Dirac delta functions and a number
of gamma or exponential distributions. Details of the Conditional
Chow‐Liu approach, which considers a factorization of the spatial
dependence of rainfall as a tree structure, are given in Kirshner, Smyth,
and Robertson (2004).
For CI‐gamma, assuming Rt ¼ rt ¼ r1t ;…:rMt� �
; the probability dis-
tribution P(Rt| St) can be written as
P rtjSt ¼ ið Þ ¼ ∏M
m¼1P rmt jSt ¼ i� � ¼ ∏
M
m¼1aim where
FIGURE 4 Occurrence and mean amount of daily rainfall for each Hidden
aim ¼pim0 rmt
¼ 0
∑num2−1
c¼1pim1
βαimcimc rmt
� �αimc−1 exp −βimc rmt
� �Γ αimcð Þ rmt >0:
8>><>>:
(4)
In Equation 4, pim0 is the probability of no precipitation for
state St = i for station m, pim1 is the complementary probability of rain-
fall, and βimc, and αimc are parameters of the Gamma distribution for
each component c in the mixture model.
For CI‐exponential the following relationship holds
aim ¼pim0 rmt ¼ 0
∑num2−1
c¼1pim1 δimc exp −δimcrmt
� �rmt >0
8><>: (5)
State (5HSs)
FIGURE 5 Ten‐day moving average seasonality of HS mean dailyfrequency (period 1951–2004)
FIGURE 6 a Air pressure at mean sea level (MSL), b Zonal & meridional winwater (P) from 10 to 1000 hPa
6 CIOFFI ET AL.
Where δimc is the parameter of exponential distribution for each
component c in the mixture model.
The maximum likelihood estimate of the set of parameters of
Equations 3–5 is computed using the expectation maximization (EM)
algorithm (Baum, Petrie, Soules, & Weiss, 1970, Dempster, Laird, &
Rubin, 1977). Full details of the specific EM procedure used for param-
eter estimation can be found in (Kirshner et al., 2004) and (Robertson
et al., 2004).
Finally, the Viterbi algorithm is used to identify the most prob-
able sequence of hidden states associated to the sequence of
observations (Viterbi, 1967). The Viterbi algorithm seeks to assign
a state to each day, such that the model likelihood is maximized.
The details of the dynamic programming algorithm used for the
purpose are provided in (Bellone et al., 2000) and (Kirshner,
2005a, 2005b).
d at 850 hPa (UA‐VA), c Air temperature at 1000 hPa (T), d Precipitable
CIOFFI ET AL. 7
3 | APPLICATION TO AGRO‐PONTINOPLAIN
The steps in model construction are now briefly described. First, the
HMM is applied to daily rainfall for the entire period of observation.
The HMM is run assuming different model configurations (e.g., target
probability density functions (PDFs) or structure of spatial dependence
between the differently located rain‐gauges), as well as different num-
ber of hidden states.
The optimal HMM is identified by comparing two different metrics
quantifying the accuracy of the HMM: log‐likelihood and Bayesian
Information Criteria (BIC). The BIC introduces a penalty term for the
number of parameters in the model to avoid overfitting (Schwarz,
1978). Generally the model with highest log‐likelihood and lowest
BIC are preferred. These metrics are calculated in the learning or train-
ing phase. In such phase, the probabilities of daily rainfall occurrence
and the parameters of PDF of daily rainfall amount, for each state
and each rain‐gauge, are also calculated.
FIGURE 6 Continued
The temporal sequence of the hidden states of HMM is then calcu-
lated by the Viterbi algorithm. Given this assignment, the probability of
a hidden state to occur on a particular data, as well as the probability of
transition to another state are computed, as a function of calendar date.
As discussed in section 2.1, a large number of potential atmo-
spheric variables and their domains of influence on local rainfall, can
be potential predictors in NHMM. A parsimonious model, i.e., one that
uses an appropriately small number of predictors, is constructed. A
heuristic procedure is used for a preliminary identification of candidate
NHMM predictors.
The procedure consists of calculating the composite anomaly
field of each potential atmospheric predictor, i.e. the average anom-
aly field of the temporal sequence of the variable associated with a
given hidden state, as it appears in the Viterbi sequence of HMM.
Then, on the basis of physical, meteorological or thermodynamic
considerations, evaluate whether this anomaly is consistent with the
rainfall statistics that are expressed corresponding to that hidden
state. The final rigorous and quantitative verification of the
8 CIOFFI ET AL.
suitability of the selected set of predictors in modelling the local
rainfall pattern is performed by using the BIC for each candidate
NHMM model.
Following the criteria described above, an initial exploration of
potential predictors is performed, from which the following set is
retained: mean sea level pressure (MSL), zonal & meridional wind at
850 hPa (UA‐VA), air temperature at 1000 hPa (T), precipitable water
(P) integrated from 10 to 1000 hPa over a domain ranging of latitude
from 20°N to 80°N and of longitude from 80°W to 60°E.
As in (Cioffi et al., 2015), the possibility to construct an NHMM
which simulates the rainfall features during the entire year is explored.
The rainfall seasonality is thus determined using the atmospheric circu-
lation fields as the determinants of changes in the seasonality of pre-
cipitation, rather than a pre‐specification of the seasonality and its
change. Such an approach is necessary when we are interested to
downscaling rainfall from GCMs to explore also the possible changes
in rainfall seasonality induced by global warming. This is in contrast
to most of the applications of NHMM where prescribed seasons are
used to estimate NHMM parameters.
FIGURE 6 Continued
The NHMM is fit with different sets of candidate predictors
whose spatial and temporal fields were reduced to a smaller number
of predictors using Principal Component Analysis. The BIC as well as
the computation of the likelihood of the model on data reserved as a
validation set are used to choose the model predictor set and the asso-
ciated parameters that, with best accuracy, simulate seasonality,
extremes and trends of significant indices of rainfall.
3.1 | Identification of Hidden states (HS) and spatialrainfall dependence
To model the spatial distribution of rainfall on any given day when it
rains, we considered: a) Conditional Independence model (HMM‐CI),
in which rain at each gauge is assumed to be conditionally independent
given a hidden state assigned to all the stations for that day; and b) the
Chow Liu model (HMM‐CL) (Kirshner et al., 2004) which considers a
parsimonious factorization of the multivariate spatial dependence
structure.
FIGURE 6 Continued
FIGURE 7 Correlation matrix of the PCs. PCs refer to the followingpredictors: MSL from 1 to 5; T from 6 to 7; UA from 8 to 12; VAfrom 13 to 17; P from 18 to 22
CIOFFI ET AL. 9
In each case, candidate PDFs for rainfall amount were considered
as described in Section 2.2. For each combination of proposed HMM
type and PDFs, a learning or training phase is performed by varying
the number of the hidden states from two to ten. The values of the
metrics ‐ Likelihood, Bayesian Information Criteria (BIC) ‐ of these pre-
liminary HMM runs are shown in Figures 2 and 3.
TABLE 2 Different combinations of model and predictors
ID MODEL PREDICTORS
1 HMM ‐
2 NHMM MSL
3 NHMM T
4 NHMM UA ‐ VA
5 NHMM P
6 NHMM MSL ‐ T
7 NHMM MSL ‐ T ‐ UA ‐ VA
8 NHMM MSL ‐ T ‐ UA – VA ‐ P
TABLE 3 In the first column are the models, in column two and threerespectively the posterior log‐likelihood (P. L‐L) and the BIC (BayesInformation Criteria) associated to the models
ID (MODEL) P. L‐L BIC
1 ‐1.4900e + 05 2.99e + 05
2 ‐1.2059e + 05 2.42e + 05
3 ‐1.20696e + 05 2.42e + 05
4 ‐1.19732e + 05 2.41e + 05
5 ‐1.20877e + 05 2.43e + 05
6 ‐1.2038e + 05 2.42e + 05
7 ‐5.5148e + 04 1.12e + 05
8 ‐4.3532e + 04 8.91e + 04
10 CIOFFI ET AL.
From these figures it appears that for both HMM‐CI and
HMM‐CL, the models with Gamma PDF for rainfall amount perform
better than those with the Exponential one. Furthermore, improve-
ments in model accuracy, beyond 5 hidden states are negligible.
Thus, 5 hidden states are selected. For 5 hidden states, HMM‐CL
and HMM‐CI have a very similar performance, with just a slight
superiority of HMM‐CL. However, given that the HMM‐CL has
FIGURE 8 Comparison between monthly median of observed (black) and
more parameters, and requires a much higher computation time
we chose the HMM‐CI.
For the selected model (HMM‐CI), with Gamma PDF for rainfall
amount, the rainfall occurrence probability and the average daily
rainfall for each of the 5 states is shown in Figure 4. Figure 5 repre-
sents the frequency of hidden state occurrence, during the calendar
year, of each of the five states, calculated for the entire period
(1951–2004).
From these figures we can describe these states as follows:
1. represents a dry condition that is nearly homogeneous for all the
stations and is present mainly in the late autumn and winter (from
October to March). Its rainfall occurrence probabilities are low but
rainfall amounts are significant.
2. is a very dry homogeneous condition for all the stations; the state
is dominant from May to August, i.e, in the late spring and sum-
mer. In this period, this state dominates (probability occurrence
about equal to 90%);
3. represents a wet but a non‐homogeneous condition; it is present
mainly in autumn and winter; it has average rainfall amounts that
simulated (red) rainfall amount in the period 1995–2004
FIGURE 9 Comparison between monthly median of observed (black) and simulated (red) number of wet days in the period 1995–2004
TABLE 4 CVRMSE of monthly rainfall amount for each month
ID J F M A M J J A S O N D
1 0,152 0,165 0,110 0,129 0,171 0,731 0,962 0,275 0,213 0,203 0,210 0,187
2 0,141 0,148 0,133 0,157 0,206 0,692 0,905 0,258 0,216 0,149 0,124 0,134
6 0,141 0,173 0,183 0,180 0,184 0,543 0,635 0,208 0,214 0,158 0,126 0,127
7 0,108 0,140 0,127 0,113 0,110 0,343 0,348 0,135 0,126 0,124 0,109 0,087
8 0,097 0,107 0,090 0,085 0,068 0,092 0,090 0,082 0,104 0,075 0,084 0,069
TABLE 5 CVRMSE of monthly wet days for each month
ID J F M A M J J A S O N D
1 0,106 0,094 0,134 0,138 0,130 0,329 0,498 0,300 0,169 0,145 0,172 0,143
2 0,096 0,094 0,104 0,092 0,118 0,328 0,494 0,304 0,160 0,091 0,101 0,100
6 0,079 0,096 0,132 0,089 0,094 0,243 0,318 0,183 0,169 0,098 0,104 0,082
7 0,081 0,089 0,111 0,074 0,063 0,200 0,199 0,126 0,131 0,075 0,093 0,053
8 0,088 0,075 0,093 0,072 0,067 0,090 0,091 0,100 0,098 0,048 0,096 0,051
CIOFFI ET AL. 11
12 CIOFFI ET AL.
are lower for the stations closer to the coast;
4. is a wet homogeneous situation present mainly in autumn and
winter. In this case there is higher rainfall for all the stations;
5. can be defined as a very wet homogeneous condition dominant in
autumn and winter. It disappears from April to August. In this case
both rainfall occurrence and amount are high and homogeneous
for all the stations.
3.2 | Atmospheric patterns of the potentialpredictors associated with the hidden states
The composite anomaly field of each potential atmospheric predictor,
associated with a given hidden state, are calculated and the consis-
tence of such anomaly field with the rainfall statistics corresponding
to that hidden state is evaluated. For the atmospheric variables
selected as potential predictors, these patterns are shown in Figure 6
a, b, c and d. Specifically, in the figures, referred to each variable, in
the plot at the upper left, the annual mean composite field is shown,
while the other plots show the anomalies of composite fields associ-
ated with each hidden state.
From Figure 6a, the most evident difference is that between the
state 2 “very dry homogeneous” which is typical of summer and the
winter state 5 “very wet homogeneous”. Locally wetter conditions in
state 5 (but also for the remaining state 1, 3, 4 of Figure 4) correspond
to lower pressure in the Mediterranean Region and Azores but high
pressure in the North Atlantic. Instead, for state 2, dry conditions are
associated with more uniform high pressure on Mediterranean Region
and low pressure in the North Atlantic. Anomaly fields associated with
wetter configurations differ based on the position and intensity of low
pressure in the Mediterranean region.
Zonal & Meridional wind at 850 hPa (UA,VA) anomaly fields
(Figure 6b), are consistent with the pressure features at MSL: in fact,
states 1–3 – 4 – 5, for which low pressure is present in the Mediterra-
nean, are characterized by counter‐clockwise winds (from South‐East),
while for state 2 clockwise winds (from North‐East) are characteristic
FIGURE 10 Comparison between Frequencyand total precipitation indices simulated(median) and observed for the period1995–2004 (Points refer to the differentstations). Extreme values refer to 90thpercentile
CIOFFI ET AL. 13
of the same area. Also in this case differences can be seen in the anom-
aly fields associated with the different hidden states in term of inten-
sity and position of cyclonic region.
The temperature anomaly fields (Figure 6c) are similar for the
states 1–3, that are typical in the autumn, and for the states 4–5
expected in the winter; state 2 differs from the others in intensity
because it is more typical of summer conditions, as the positive values
of the anomaly evidences.
The composite fields of the precipitable water (P) anomaly
(Figure 6d) show different patterns associated with the different hid-
den states. Precipitable water P has a strong seasonal dependence that
is evident when the summer dry condition, represented by state 2, is
compared with the very wet homogeneous state 5.
From the analysis of the figures discussed above, one can infer
how each atmospheric variable among those discussed above might
be selected as a potential predictor of NHMM. In fact, the composite
fields of air pressure at mean sea level (MSL) and Air temperature at
1000 hPa (T) are determinants of the rainfall patterns which typically
occur in summer and winter in Agro‐Pontino Plain. Zonal & meridional
FIGURE 11 Comparison between Frequencyand total precipitation simulated (median) andobserved for the period 1995–2004 (Pointsrefer to the different stations). Extreme valuesrefer to 95th percentile
wind at 850 hPa (UA‐VA) represents the atmospheric circulation con-
figurations responsible of moisture transport and in particular the
westward movement of winter storms originating over the Atlantic
and impinging upon the western European coasts (Garaboa‐Paz,
Eiras‐Barca, Huhn, & Pérez‐Muñuzuri, 2015). Precipitable water (P)
from 10 to 1000 hPa, which is the atmosphere water vapor content
in the atmospheric column, has spatial patterns highly coherent with
the rainfall occurrence and amount associated to the different hidden
states.
3.3 | NHMM calibration and validation
The first step in building the NHMM is to reduce the dimension of the
potential predictors. For each atmospheric circulation field of the
potential predictors, a Principal Component Analysis (PCA) is per-
formed. The leading PCs that explain 80% or more of the variance of
the field are retained. This led to 2 PCs for temperature, and 5 each
for the other fields. In order to avoid model overfitting and multi‐
collinearity (Khalil et al., 2010), due to the possible existence of
FIGURE 12 Trend of 20 year moving average of a) Annual rainfallamount b) Winter rainfall amount; c) Summer rainfall amount
14 CIOFFI ET AL.
significant correlation between the PCs of individual predictors, a cor-
relation analysis across the PCs from each field is performed. From
Figure 7, it is clear that the leading PCs of the predictors are mutually
correlated. Thus, a further reduction of data by PCA of these predic-
tors is performed to obtain fewer uncorrelated predictor PCs, retaining
a number of new PCs that explain the 99% of the variance of the orig-
inal PCs series. For the model that considers the entire set of predic-
tors (model 8 of Table 2), this procedure reduces from 22 original
PCs to 17 uncorrelated PCs.
Using NHMMwith 5 hidden states, calibration and validation tests
are carried out with different combinations of these atmospheric pre-
dictors. Parameters of the NHMM are identified in the learning or
training phase, by fitting the NHMM for different combinations of can-
didate predictors on the 1951–1994 historical data. Validation is per-
formed by checking the performance of each model with respect to
seasonality and other attributes, calculated by simulations and obser-
vations, for the period 1995–2004. To validate the model, one hun-
dred simulations for each model configuration are carried out for the
decade 1995–2004.
In Table 2 the models are listed together with their corresponding
combination of predictors.
In Table 3, the posterior log‐likelihood and the BIC of each model
are shown. It shows that the best model appears to be model 8, the
one in which all the predictors are taken into account, whose
posterior likelihood increases an order of magnitude with respect to
the others.
In the following sections a validation of the NHMM is carried out
in order to verify if it is able to capture: a) seasonality; b) extremes; c)
trends and interannual variability in selected rainfall attributes.
3.3.1 | Seasonality
The ability of the models to reproduce the seasonality of rainfall is
illustrated by the boxplots in Figures 8 and 9, where for sake of sim-
plicity only models 1,2,6,7,8 are shown.
These figures compare the monthly distribution of observed and
simulated (by NHMM) rainfall amount and the number of wet days
for the period of validation 1995–2004 (Figure 8 and 9). The compar-
ison refers to the monthly wet days and amount averaged for the
ensemble of stations. From the figures it is clear that the HMM is
not able at all to reproduce the seasonality observed in monthly wet
days and rainfall amount, while the seasonality is captured to different
degrees depending on the set of atmospheric predictors used by the
NHMM. This suggests that future changes in seasonality contingent
on the use of these predictors with the NHMMmay be quite effective.
From Figure 8 and 9 it is possible to evaluate how the different
models fit the observed seasonal rainfall features on Agro‐Pontino plain.
To provide an estimate of the difference in performance of
models, the coefficient of variation of root mean squared error
(CVRMSE) of each model, for each month, is reported in Tables 4
and 5. That error is defined as:
CVRMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1Na
∑Na
i¼1xmsi−x
moi
� �2s
1Na
∑Na
i¼1xmoi� � (6)
Where xsim and xoim are respectively ‐ for the m‐month ‐ the
median of monthly rainfall amounts from the simulations and the
monthly rainfall amounts from observations, Na is the number
of years.
The improvement of NHMM ‐ Model 8 in reproducing the typical
Mediterranean seasonal rainfall trend of monthly amount and number
of wet days is remarkable. This improvement is particularly significant
in summer months where the introduction of the Precipitable water
CIOFFI ET AL. 15
as a predictor dramatically reduces the error. The CVRMSE ranges,
depending on the months, from 5% to 10%.
3.3.2 | Extreme rainfall
Different criteria may be used to define extreme precipitation.
(Du et al., 2013a; Du, Wu, Zong, Meng, & Wang, 2013b): a) absolute
or arbitrary or fixed threshold method (Jones et al., 1999; Klein Tank
& Können, 2003; Bell, Sloan, & Snyder, 2004); b) standard deviation
method (Henderson & Muller, 1997; Gong & Ho, 2004), which con-
siders events that exceed k‐standard deviations from the long‐term
mean; c) percentile‐based method (Xu, Xu, Gao, & Luo, 2009; Huang,
Qian, & Zhu, 2010; Kothawale, Revadekar, & Rupa Kumar, 2010; Li,
Zheng, Liu, & Flanagan, 2010; Xu, Du, Tang, & Wang, 2011), where
the exceedance of a specified percentile of the empirical marginal dis-
tribution is used to define the event.
In this study approach c), using non‐parametric percentiles, is used
(Cioffi et al., 2015).
For each rain gauge, a threshold is defined in terms of a fixed per-
centile of the daily rainfall amount series, considering only days with
non‐zero rainfall; then, the number of non‐zero precipitation events
during each year is identified and a pre‐fixed percentile (90th and
95th) of this series estimated for each year. The median of these per-
centiles across all years is chosen as the threshold. Thus, two indices
are defined: frequency and total precipitation. The first represents the
number of events per year whose daily rainfall amount exceeds the
threshold; the second the sum of daily rainfall amount of such events.
A time series of frequency at each location is computed as:
fjt ¼ ∑Nd
i¼1I Pijt>P�j� �
(7)
TABLE 7 CVRMSE of Total Precipitation ‐ 95th percentile
Year trend\Station 1st 2nd 3rd
20 0.1708 0.1502 0.2112
15 0.1735 0.1512 0.2095
10 0.1860 0.1647 0.2224
5 0.2134 0.2163 0.2702
TABLE 8 CVRMSE of Frequency ‐ 95th percentile
Year trend\Station 1st 2nd 3rd
20 0.1661 0.1330 0.1812
15 0.1688 0.1380 0.1795
10 0.1838 0.1557 0.1899
5 0.2100 0.1990 0.2354
TABLE 6 CVRMSE of simulated trends for different periods of mov-ing average
Year trend\CVRMSE Annual Winter Summer
20 0.0201 0.0224 0.0384
15 0.0265 0.0351 0.0391
10 0.0539 0.0633 0.0750
5 0.0773 0.0849 0.1127
where t is the year, j the station, Pijt the rain on day i in year t at the
location j and P�j is the rainfall threshold for station j; I(.) is an indicator
function that takes the value 1 if the argument is true and 0 otherwise;
Nd is the number of days of the year (365).
The total precipitation time series is derived as:
rjt ¼ ∑Nd
i¼1I Pijt>P�j� �
*Pijt (8)
A representation for frequency index and total precipitation index
is shown in the following Figure 10 and 11 for all stations averaged in
1995–2004.
From Figure 10 and 11 it is evident for both the percentiles, most
of the values are within the range of +/− 10%; only two stations are
out of this range for the 90th percentile frequency.
3.3.3 | Trends and interannual variability of annual andseasonal rainfall amount and rainfall extremes
In order to verify the capability of the model to capture inter‐annual
variability, 100 simulations are carried out for the entire period of
observation (1951–2004).
In Figure 12, a 20 year moving average for respectively, annual (a),
winter (Oct‐Mar) (b) and summer (Apr‐Sep) (c) simulated and observed
rainfall amount (averaged on all the stations) is shown. The comparison
between observed and simulated rainfall amounts indicates how the
model is able to capture the actual trends, for the period 1951–
1994. In fact, the median of simulations fits very realistically the
observed quantities.
To make a more complete analysis, this comparison has been car-
ried out also for different moving average periods. In Table 6 the coef-
ficient of variation of root mean square error (CVRMSE) is shown for
the different periods considered and the different rainfall amounts.
As expected, the smaller the period of moving average the greater
the error. However, errors are small in most of cases, denoting the pos-
sibility to capture the influence of interannual oscillations which are
typical of mid‐latitude climate on rainfall amount. An analogue analysis
has been carried out for rainfall extreme indices defined in paragraph
3.3.2. No evident trends are present in the indices of the extremes cal-
culated by observations. For sake of brevity, these trends are not
4th 5th 6th 7th
0.1576 0.1415 0.2551 0.1447
0.1610 0.1483 0.2632 0.1495
0.1651 0.1958 0.2836 0.1676
0.2284 0.2747 0.3151 0.2118
4th 5th 6th 7th
0.1210 0.1248 0.2431 0.1181
0.1259 0.1330 0.2501 0.1249
0.1333 0.1852 0.2666 0.1429
0.1994 0.2674 0.2955 0.1887
16 CIOFFI ET AL.
shown. In Tables 7 and 8 the CVRMSE for different moving average
periods and respectively for the two selected indices calculated for
each station are shown. They range from about 14% to 30% depending
on the station and the period considered. This result seems to show
that if the model is used for future projections, changes in extreme
rainfall have to be carefully evaluated, since the uncertainty associated
with the extreme rainfall indices is not negligible.
4 | DISCUSSION AND CONCLUSION
In this paper, the potential of the NHMM, to be used as predictive tool
of local rainfall patterns in future global warming scenarios has been
explored.
An optimum set of atmospheric variables, assumed as predictors
of NHMM, has been identified. This optimum set consists of the atmo-
spheric fields, from 20°N to 80°N of latitude and from 80°W to 60°E
of longitude, of mean sea level pressure (MSL), temperature at
1000 hPa (T), zonal & meridional wind at 850 hPa (UA‐VA), precipitable
water (P) from 10 to 1000 hPa. By performing NHMM validation tests,
we were able to verify and quantify, the capability of the model to
realistically represent: a) seasonality of local rainfall pattern in
Agro‐Pontino plain; b) extreme rainfall frequency and amount of the
events having, respectively, daily amount over the 90th and 95th
percentile threshold; c) the trend for the entire examined period
(1950–2005) of annual and seasonal rainfall amounts and extremes,
filtered by different moving average periods.
The results show NHMM captures realistically most of the previ-
ously described patterns of local rainfall. Specifically, the model simu-
lates seasonal variability, as well as, interannual rainfall variability of
annual, seasonal and monthly rainfall amounts, within an error, as
quantified by CVRMSE, minor of 10%. These indices are important
extensive quantities to assess the impact of change in hydrologic
regime in a region as Agro‐Pontino plain, where, agriculture is largely
practiced, with summer irrigation mainly relying on groundwater with-
drawal. Appreciable reduction of winter (which is in the wet season in
Mediterranean region) rainfall amount can limit the aquifer recharge;
as a consequence, the aquifers of the coastal region, under the impact
of summer overexploitation for irrigation, could be affected by salt
intrusion. This situation further worsens if summer is drier, as it
appears to be the case of Mediterranean regions.
Trend and interannual variations of extreme rainfall are simulated
within larger errors than the above discussed rainfall attributes. The
model underestimates extremes of single stations, and fits, with a
larger range of error, interannual variability of extremes This makes
the estimate of future trends rather uncertain. Predicting the fre-
quency and intensity of extreme precipitation is crucial for risk man-
agement. In recent years, a number of papers have discussed the
possibility of an increase in the frequency of occurrence of extreme
rainfall in response to the increase of global temperature (Yonetani &
Gordon, 2001; Palmer & Räisänen, 2002; Kharin, Zwiers, Zhang, &
Hegerl, 2007; Bengtsson, Hodges, & Keenlyside, 2009; Kundzewicz
et al., 2014; Arnone, Pumo, Viola, Noto, & La Loggia, 2013; Alpert
et al., 2002, Brunetti, Buffoni, Maugeri, & Nanni, 2000; Brunetti,
Colacino, Maugeri, & Nanni, 2001). Experiments with coupled
ocean–atmosphere climate models have shown an increase in the
occurrence of extreme precipitation events in mid‐latitudes (Hennessy,
Gregory, & Mitchell, 1997; Cubasch et al., 2001).
However, it should be noted that no significant trends were evi-
dent in the observed extremes in the period investigated. The absence
of clear trends probably contributes to make the model less effective.
In each case, the modelling of extremes should be improved, both in
the framework of NHMM and/or by approaching and verifying the
effectiveness of other models such as the Dynamic Bayesian networks,
of which NHMM is just one of the possible models.
It should be underlined that NHMM has been constructed without
any “a priori” demarcation of the seasons; in fact, the rainfall variability
was just thought as a function (through the PCs) of temporal variations
of atmospheric predictors.
The role of atmospheric circulation, as driven by pole‐equator and
ocean‐land contrast temperature gradients, on rainfall pattern has
been recently investigated by (Karamperidou, Cioffi, & Lall, 2012) and
(Byrne & O’Gorman, 2015). These authors identify different mecha-
nisms, induced by changes of global temperature gradients, which
affect atmospheric circulation and then, as consequence, moisture
flows, that explain observed and simulated (by GCMs) changes in spa-
tial and temporal rainfall patterns.
It is reasonable to hypothesize that such changes in temperature
gradients could also affect local rainfall seasonal variability, provoking,
for instance, seasonal shifts, that, otherwise, could not captured
through an “a priori” demarcation of seasons.
Finally, some further questions, concerning the use of GCMs, have
to be investigated before to perform future projections under global
warming scenarios. Since we use as predictors the fields of a number
of atmospheric variables, a first question concerns how accurately
GCM simulated fields fit the reanalysis ones, during a common histor-
ical period. Untill now, there are about 70 Global Climate Models
(GCMs); in order to make feasible the analysis we need to formulate
criteria to choose properly outputs from a limited number of GCM
simulations.
There are several approaches proposed by literature, e.g.: extreme
(max/min) approach; ensemble approach; and validation approach. The
extreme (max/min) approach suggests taking into account the extreme
values of a selected variable of interest, coming from the full range of
the values proposed by all the GCMs available. The ensemble approach
suggests taking into account mean or median values from all the GCM
outputs. The validation approach suggests to compare GCM outputs
with reanalysis model in our area of study and to retain four or five
best‐agreement models (Fenech, 2012). The latter approach is applied
by (Cioffi et al., 2015); these authors, in order to project future local
rainfall in a tropical region, such as east Africa, are forced to apply a
variance corrections to the GCM’s PCs. This simple correction is suffi-
cient to correctly reproduce the climatology of the investigated region,
but the extension of this procedure to other regions of the world as
Agro‐Pontino Plain may be not so straightfoward.
ACKNOWLEDGMENTS
The research project has been funded by University of Rome ‘La
Sapienza’ (n. C26A12HEJT, 2012). The authors are grateful to the
CIOFFI ET AL. 17
“Istituto Idrografico e Mareografico di Roma”, “Areonautica Militare”
and “NCEP/NCAR” for providing daily rainfall datasets and re‐analysis
data respectively.
REFERENCES
Alpert, P., Ilani, R., Krichak, S., Price, C., & Rodò, X. (2006). Relationsbetween climate variability in the Mediterranean region and the Tro-pics: ENSO, South Asian and African monsoons, hurricanes andSaharan dust. In P. Lionello, P. Malanotte‐Rizzoli, & R. Boscolo (Eds.),Mediterranean Climate Variability (pp. 149–177). Amsterdam: Elsevier.
Alpert, P., Ben‐Gai, T., Baharad, A., Benjamini, Y., Yeku‐tieli, D., Colacino,M., … Manes, A. (2002). The paradoxical increase of Mediterraneanextreme daily rainfall in spite of decrease in total values. GeophysicalResearch Letters, 29, 1–31.
Arnone, E., Pumo, D., Viola, F., Noto, L. V., & La Loggia, G. (2013). Rainfallstatistics changes in Sicily. Hydrology and Earth System Sciences, 17,2449–2458. doi:10.5194/hess-17-2449-2013
Baguis, P., Roulin, E., Willems, P., & Ntegeka, V. (2009). Climate change sce-narios for precipitation and potential evapotranspiration over centralBelgium. Theoretical and Applied Climatology, 99(3–4), 273–286.doi:10.1007/s00704-009-0146-5
Bates, B. C., Charles, S. P., & Hughes, J. P. (1998). Stochastic downscaling ofnumerical climate model simulations. Environmental Modelling &Software, 13(3–4), 325–331.
Baum, L. E., Petrie, T., Soules, G., & Weiss, N. (1970). A maximization tech-nique occurring in statistical analysis of probabilistic functions ofMarkov chains. Annals of Mathematical Statistics, 41(1), 164 {171,February 1970.
Betrò, B., Bodini, A., & Cossu, Q. A. (2008). Using a hidden Markov model toanalyse extreme rainfall events in Central‐East Sardinia. Environmetrics,19(7), 702–713.
Bell, J. L., Sloan, L. C., & Snyder, M. (2004). Regional changes inextremeclimatic events: a future climate scenario. Journal of Climate,17(1), 81–87. doi:10.1175/1520-0442(2004)017
Bellone, E., Hughes, J. P., & Guttorp, P. (2000). A hidden Markov model fordownscaling synoptic atmospheric patterns to precipitation amounts.Climate Research, 15(1), 1–12.
Bellone E. 2000. Non homogeneous hidden Markov models for downscal-ing synoptic atmospheric patterns to precipitation amount. PhD Thesis,University of Washington, 2000.
Bengtsson, L., Hodges, K. I., & Keenlyside, N. (2009). Will ExtratropicalStorms Intensify in a Warmer Climate? Journal of Climate, 22,2276–2301. doi:10.1175/2008JCLI2678.1
Brunetti, M., Buffoni, L., Maugeri, M., & Nanni, T. (2000). Precipitationintensity trends in northern Italy. International Journal of Climatology,20, 1017–1031.
Brunetti, M., Colacino, M., Maugeri, M., & Nanni, T. (2001). Trends in thedaily intensity of precipitation in Italy from 1951 to 1996. InternationalJournal of Climatology, 21, 299–316.
Busuioc, A., Giorgi, F., Bi, X., & Ionita, M. (2006). Comparison of regional cli-mate model and statistical downscaling simulations of different winterprecipitation change scenarios over Romania. Theoretical and AppliedClimatology, 86(1–4), 101–123.
Busuloc, S. P., Bates, B. C., Whetton, P. H., & Hughes, J. P. (1999a). Valida-tion of downscaling models for changed climate conditions: Case studyof southwestern Australia. Climate Research, 12, 1–14.
Byrne, M. P., & O’Gorman, P. A. (2015). The response of precipitation minusevapotranspiration to climate warming: Why the “wet‐get‐wetter,dry‐get‐drier” scaling does not hold over land. Journal of Climate, 28,8078–8092.
Charles, S. P., Bates, B. C., & Hughes, J. P. (1999). A spatio‐temporal modelfor downscaling precipitation occurrence and amounts. Journal ofGeophysical Research, 104, 31657–31669.
Cioffi, F., Conticello, F., & Lall, U. (2015). Projecting Changes in TanzaniaRainfall for the 21st century. International Journal of Climatology.doi:10.1002/joc.4632
Cioffi, F., Lall, U., Rusc, E., & Krishnamurthy, C. K. B. (2015). Space‐timestructure of extreme precipitation in Europe over the last century. Inter-national Journal of Climatology, 35(8), 1749–1760.
Cubasch, U., Meehl, G. A., Boer, G. J., Stouffer, R. J., Dix, M., Noda, A., …Yap, K. S. (2001). Projections of future climate change. In J. T.Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. Van der Linden, X.Dai, et al. (Eds.), Climate Change 2001: The Scientific Basis, Contributionof Working Group I to the Third Assessment Report of the Intergovernmen-tal Panel on Climate Change (pp. 525–585). New York, NY: CambridgeUniversity Press.
Cudennec, C., Leduc, C., & Koutsoyiannis, D. (2007). Dryland hydrology inMediterranean regions ‐‐ a review. Hydrological Sciences Journal, 52(6),1077–1087.
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihoodfrom incomplete data via EM algorithm. Journal of the Royal StatisticalSociety: Series B: Methodological, 39(1), 1 {38, 1977.
Dibike, Y. B., Gachon, P., St‐Hilaire, A., Ouarda, T. B. M. J., & Nguyen, V.T.‐V. (2008). Uncertainty analysis of statistically downscaled tempera-ture and precipitation regimes in Northern Canada. Theoretical andApplied Climatology, 91(1–4), 149–170.
Du, H., Wu, Z., Li, M., Jin, Y., Zong, S., & Meng, X. (2013a). Characteristics ofextreme daily minimum and maximum temperature over NortheastChina: 1961–2009. Journal Theoretical and Applied Climatology,111(1–2), 161–171. doi:10.1007/s00704-012-0649-3
Du, H., Wu, Z., Zong, S., Meng, X., & Wang, L. (2013b). Assessing the char-acteristics of extreme precipitation over northeast China using themultifractal detrended fluctuation analysis. Journal of GeophysicalResearch‐Atmospheres, 118, 6165–6174. doi:10.1002/jgrd.50487
Fenech, A., 2012. Presentation to the Canadian Association ofGeographers – Ontario Division A Validation Against Observations of24 Global Climate Models over Canada: Which GCMs Model Best,Where? UPEI Climate Lab.
Fowler, H. J., Blenkinsop, S., & Tebaldi, C. (2007). Linking climate changemodelling to impacts studies: Recent advances in downscaling tech-niques for hydrological modelling. International Journal of Climatology,27, 1547–1578. doi:10.1002/joc.1556
Garaboa‐Paz, D., Eiras‐Barca, J., Huhn, F., & Pérez‐Muñuzuri, V. (2015).Lagrangian coherent structures along atmospheric rivers. Chaos, 25,063105. doi:10.1063/1.4919768
Giorgi, F. (2006). Climate change hot‐spots. Geophysical Research Letters,33, L08707. doi:10.1029/2006GL025734
Giorgi, F., & Lionello, P. (2008). Climate change projections for theMediterranean region. Science Direct, 63, 90–104.
Gong, D. Y., & Ho, C. H. (2004). Intra‐seasonal variability of wintertimetem‐perature over East Asia. International Journal of Climatology, 24(2),131–144. doi:10.1002/joc.1006
Grotch, S. L., & Mac Cracken, M. C. (1991). The use of general circulationmodel to predict regional climate change. Journal of Climatology, 4,286–303.
Hay, L. E., McCabe, G., Wolock, D. M., & Ayers, M. A. (1991). Simulation ofprecipitation by weather type analysis. Water Resources Research, 27,493 {501, 1991.
Hashmi, M. Z., Shamseldin, A. Y., & Melville, B. W. (2009). Statistical down-scaling of precipitation: state‐of‐the‐art and application of bayesianmulti‐model approach for uncertainty assessment. Hydrology and EarthSystem Sciences Discussions, 6, 6535–6579.
Henderson, K. G., & Muller, R. A. (1997). Extreme temperature days intheSouth‐Central United States. Climate Research, 8(2), 151–162.doi:10.3354/cr0008151
Hennessy, K. J., Gregory, J. M., & Mitchell, J. F. B. (1997). Changes in dailypre‐cipitation under enhanced greenhouse conditions. Climate Dynam-ics, 13, 667–680.
18 CIOFFI ET AL.
Hewitson, B. C., & Crane, R. G. (1996). Climate downscaling: Techniquesand application. Climate Research, 7, 85–95.
Huang, D. Q., Qian, Y. F., & Zhu, J. (2010). Trends of temperature extremesinChina and their relationship with global temperature anomalies.Advances in Atmospheric Sciences, 27(4), 937–946. doi:10.1007/s00376-009-9085-4
Hughes, J. P., & Guttorp, P. (1994). A class of stochastic models for relatingsynoptic atmospheric patterns to regional hydrologic phenomena.Water Resources Research, 30(5), 1535–1546.
Hughes, J. P., Guttorp, P., & Charles, S. P. (1999). A non‐homogeneoushidden Markov model for precipitation occurrence. Journal of the RoyalStatistical Society: Series C: Applied Statistics, 48(1), 15–30.
Hurrell, J. W. (1995). Decadal trends in the North Atlantic Oscillation:regional temperature and precipitation. Science, 269, 676–679.
IPCC, 2013. Climate. Change 2013. The Physical Science Basis. WorkingGroup I Contribution to the fifth Assessment Report of the Intergov-ernmental Panel on Climate Change. Cambridge University Press.
Jones, P. D., Horton, E. B., Folland, C. K., Hulme, M., Parker, D. E., &Basnett, T. A. (1999). The use of indices to identify changes in climaticextremes. Climate Change, 42(1), 131–149. doi:10.1023/a:1005468316392
Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., …Joseph, D. (1996). The NCEP/NCAR 40‐Year Reanalysis Project.Bulletin of the American Meteorological Society, 77, 437–471.
Karamperidou, C., Cioffi, F., & Lall, U. (2012). Surface TemperatureGradients as Diagnostic Indicators of Midlatitude Circulation Dynamics.Journal of Climate, 25(12), 4154–4171. doi:10.1175/JCLI-D-11-00067.1
Khalil, A. F., Kwon, H.‐H., Lall, U., & Keheil, Y. H. (2010). Predictive down-scaling based on non‐homogeneous hidden Markov model.Hydrological Sciences Journal, 55(3), 333–350.
Kharin, V. V., Zwiers, F. W., Zhang, X., & Hegerl, G. C. (2007). Changes inTemperature and Precipitation Extremes in the IPCC Ensemble ofGlobal Coupled Model Simulations. Journal of Climate, 20,1419–1444. doi:10.1175/JCLI4066.1
Klein Tank, A. M. G., & Können, G. P. (2003). Trends in indices of daily tem‐perature and precipitation extremes in europe, 1946–99. Journal of Cli-mate, 16, 3665–3680. doi:10.1175/1520-0442(2003)016<3665:TIIODT>2.0.CO;2
Kothawale, D. R., Revadekar, J. V., & Rupa Kumar, K. (2010). Recent trendsinpre‐monsoon daily temperature extremes over India. Journal of EarthSystem Science, 119(1), 51–65. doi:10.1007/s12040-010-0008-7
Kundzewicz, Z. W., Kanae, S., Seneviratne, S. I., Handmer, J., Nicholls, N.,Peduzzi, P., … Sherstyukov, B. (2014). Flood risk and climate change:global and regional perspectives. Hydrological Sciences Journal, 59(1),1–28. doi:10.1080/02626667.2013.857411
Kwon, H. H., Lall, U., Moon, Y.‐I., Khalil, A. F., & Ahn, H. (2006). Episodicinterannual climate oscillations and their influence on seasonal rainfallin the Everglades National Park. Water Resources Research, 42,W11404. doi:10.1029/2006WR005017
Kwon, H.‐H., Brown, C., Xu, K., & Lall, U. (2009). Seasonal and annual max-imum streamflow fore‐casting using climate information: application tothe Tree Gorgers Dam in the Yangtze basin, China. Hydrological SciencesJournal, 54(3), 582–595.
Kirshner, S., Smyth, P., & Robertson, A. W. (2004). Conditional Chow‐Liutree structures for modeling discrete‐valued vector time series. In M.Chickering, & J. Halpern (Eds.), Proceedings of the Twentieth Conferenceon Uncertainty in Artificial Intelligence (UAI‐04) (pp. 317 {324).AUAIPress.
Kirshner, S. (2005a). Quick Start Manual for the MVN‐HMM Toolbox. Irvine:Donald Bren School of Information and Computer Science University ofCalifornia.
Kirshner S., 2005b. Modeling of Multivariate Time Series Using HiddenMarkov Models. PhD thesis, University of California, Irvine, March2005.
Langousis, A., & Kaleris, V. (2014). Statistical framework to simulate dailyrainfall series conditional on upper‐air predictor variables. WaterResources Research, 50(5), 3907–3932. doi:10.1002/2013WR014936
Li, Z., Zheng, F. L., Liu, W. Z., & Flanagan, D. C. (2010). Spatial distributionandtemporal trends of extreme temperature and precipitation eventson the Loess Plateau of China during 1961–2007. Quaternary Interna-tional, 226(1–2), 92–100. doi:10.1016/j.quaint.2010.03.003
Lionello, P., Malanotte‐Rizzoli, P., Boscolo, R., Alpert, P., Artale, V., Li, L., …Xoplaki, E. (2006a). The Mediterranean climate: an overview of themain characteristics and issues. In P. Lionello, P. Malanotte‐Rizzoli, &R. Boscolo (Eds.), Mediterranean Climate Variability (pp. 1–26).Amsterdam: Elsevier (NETHERLANDS).
Lionello, P., Bhend, J., Buzzi, A., Della‐Marta, P. M., Krichak, S., Jansà, A., …Trigo, R. (2006b). Cyclones in the Mediterranean region: climatologyand effects on the environment. In P. Lionello, P. Malanotte‐Rizzoli, &R. Boscolo (Eds.), Mediterranean Climate Variability (pp. 325–372).Amsterdam: Elsevier (NETHERLANDS).
Maraun, D., Wetterhall, F., Ireson, A. M., Chandler, R. E., Kendon, E. J.,Widmann, M., … Thiele‐Eich, I. (2010). Precipitation downscaling underclimate change: Recent developments to bridge the gap betweendynamical models and the end user. Reviews of Geophysics, 48,RG3003. doi:10.1029/2009RG000314
Mariotti, A., Struglia, M. V., Zeng, N., & Lau, K. M. (2002). The hydrologicalcycle in the Mediterranean region and implications for the water budgetof the Mediterranean Sea. Journal of Climate, 15(13), 1674–1690.
Palmer, T. N., & Räisänen, J. (2002). Quantifying the risk of extremeseasonal precipiation in a changing climate. Nature, 415, 512–514.doi:10.1038/415512a
Rabiner L. R., Juang B. H. 1986. An introduction to hidden Markov models.IEEE ASSP MAGAZINE. 0740‐7467/86/0100‐0004$01.
Robertson, A. W., Kirshner, S., & Smyth, P. (2004). Downscaling of dailyrainfall occurrence over northeast Brazil using a hidden Markov model.Journal of Climate, 17(22), 4407–4424.
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Sta-tistics, 6(2), 461–464.
Sheffield, J., & Wood, E. F. (2008). Global Trends and Variability in SoilMoisture and Drought Characteristics, 1950–2000, from Observation‐Driven Simulations of the Terrestrial Hydrologic Cycle. Journal of Cli-mate, 21(3), 432–458. doi:10.1175/2007JCLI1822.1
Trigo, R., Zorits, E., Luterbacher, J., Krichack, S., Price, C., Jacobeit, J., …Mariotti, A. (2006). Relations between variability in the Mediterraneanregion and mid‐latitude variability. In P. Lionello, P. Malanotte‐Rizzoli,& R. Boscolo (Eds.), Mediterranean Climate Variability (pp. 179–226).Amsterdam: Elsevier.
Ulbrich, U., May, W., Li, L., Lionello, P., Pinto, J. G., & Somot, S. (2006). TheMediterranean climate change under global warming. Developments inEarth and Environmental Sciences, 4(2006), 399–415.
Viterbi, A. J. (1967). Error bounds for convolutional codes and an asymptot-ically optimum decoding algorithm. IEEE Transactions on InformationTheory, 13(2), 260 {269, 1967.
Wijngaard, J. B., Klein Tank, A. M. G., & Können, G. P. (2003). Homogeneityof 20th century European daily temperature and precipitation series.International Journal of Climatology, 23(6), 679–692.
Wilby, T. M., Wigley, L., Conway, D., Jones, P. D., Hewitson, B. C., Main, J.,& Wilks, D. S. (1998). Statistical downscaling of General CirculationModel output: A comparison of methods. Water Resources Research,34, 2995–3008.
Wilby, R. L., Hayc, L. E., & Leavesley, G. H. (1999). A comparison of down-scaled and raw GCM output: a comparison of methods.Water ResourcesResearch, 34, 2995–3008.
Wilby R.L., Charles S.P., Zorita E., Timbal B., Whetton P., Mearns L.O.2004a. Guidelines Use of Climate Scenarios Developed from StatisticalDownscaling methods. Task Group and Scenario Support for impactsand climate analysis (TGICA).
CIOFFI ET AL. 19
Willems, P., & Vrac, M. (2011). Statistical precipitation downscaling forsmall‐scale hydrological impact investigations of climate change. Journalof Hydrology, 402(3–4), 193–205. doi:10.1016/j.jhydrol.2011.02.030
Wilby, R. L., Charles, S. P., Zorita, E., Timbal, B., Whetton, P., & Mearns, L.O. (2004b). Guidelines for Use of Climate Scenarios Developed FromStatistical Downscaling Methods, Supporting material of the Intergov-ernmental Panel on Climate Change (pp. 27). Cambridge, U. K.,Singapore: [Available from the DDC of IPCC TGCIA.] Cambridge Univ.Press.
Xoplaki, E., González‐Rouco, J. F., Luterbacher, J., & Wanner, H. (2004).Wet season Mediterranean precipitation variability: influence oflarge‐scale dynamics and trends. Climate Dynamics, 23, 63–78.
Xu, C. (1999). From GCMs to river flow: a review of downscaling methodsand hydrologic modelling approaches. Progress in Physical Geography,1999.
Xu, Y., Xu, C. H., Gao, X. J., & Luo, Y. (2009). Projected changes in tem‐perature and precipitation extremes over the Yangtze River BasinofChina in the 21st century. Quaternary International, 208(1–2), 44–52.doi:10.1016/j.quaint.2008.12.020
Xu, X., Du, Y. G., Tang, J. P., & Wang, Y. (2011). Variations of temperatureandprecipitation extremes in recent two decades over China.
Atmospheric Research, 101(1–2), 143–154. doi:10.1016/j.atmosres.2011.02.003
Yonetani, T., & Gordon, H. B. (2001). Simulated Changes in the Frequencyof Extremes and Regional Features of Seasonal/Annual Temperatureand Precipitation when Atmospheric CO2 Is Doubled. Journal of Cli-mate, 14, 1765–1779. doi:10.1175/1520-0442(2001)014<1765:SCITFO>2.0.CO;2
Zorita, E., von Storch H., 1997. A survey of statistical downscalingtechniques, GKSS Rep. 97/E/20, GKSS Research Center, Geesthacht,Germany.
Zucchini, W., & Guttorp, P. (1991). A Hidden Markov Model for Space–Time Precipitation. Water Resources Research, 27(8), 1917–1923.
How to cite this article: Cioffi, F., Conticello, F., Lall, U.,
Marotta, L., and Telesca, V. (2016), Large scale climate and rain-
fall seasonality in a Mediterranean Area: Insights from a non‐
homogeneous Markov model applied to the Agro‐Pontino
plain, Hydrological Processes, doi: 10.1002/hyp.11061