Large scale climate and rainfall seasonality in a Mediterranean …water.columbia.edu/files/2017/01/hyp11061.pdf · 2017-01-06 · regional orography and the presence of the Mediterranean

Received: 9 July 2015 Accepted: 15 October 2016

DO

R 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.

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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