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WATER RESOURCES RESEARCH, VOL. 45, W07422, doi:10.1029/2008WR007485, 2009

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Hierarchical Bayesian modeling of multisite daily rainfall occurrence: Rainy season onset, peak, and end C. H. R. Lima1 and U. Lall1 Received 28 September 2008; revised 7 March 2009; accepted 13 May 2009; published 23 July 2009.

[1] A quantitative definition of and ability to predict the onset and duration of the

dominant rainfall season in a region are important for agricultural and natural resources management. In this paper, a methodology based on an analysis of daily rainfall occurrence is proposed and applied to define the onset and end of the rainfall season in northeast Brazil. Multiple rainfall gauges are considered simultaneously, and a hierarchical Bayesian framework is used for parameter estimation. The proposed model can be used to identify the rainy season onset by finding the first day of year in which the estimated probability of rainfall is greater than a specified number, e.g., 0.5. The determination of the end of the rainy season follows a similar procedure. Thus, the rainfall season can be defined as a period during which rainfall is most likely to occur. Monotonic trends are identified for the maximum probability of daily rainfall across the region, suggesting evidence of climate change. However, only the southern stations in the region exhibit a trend in the corresponding date. An examination of the correlations of dates of maximum rainfall probability with leading climate indicators leads to a promising direction for 1–3 month ahead forecasts of onset, which will be very useful for effective planning. Particularly, El Ninõ events in December are found to be associated with delays in the oncoming rainy season onset over a large part of southern northeast Brazil. Negative anomalies in the tropical South Atlantic sea surface temperature between January and March can anticipate the rainy season onset in the northern region and along the eastern coast of northeast Brazil. Citation: Lima, C. H. R., and U. Lall (2009), Hierarchical Bayesian modeling of multisite daily rainfall occurrence: Rainy season onset, peak, and end, Water Resour. Res., 45, W07422, doi:10.1029/2008WR007485.

1. Introduction [2] The understanding of the interannual variability as well as the forecast of the onset and end of the rainy season with some months or weeks in advance is as important as the prediction of rainfall amounts, especially for agriculture and hydroelectricity, and for related areas such as banks and local governments. It is also of scientific interest in understanding factors that influence the regional hydrologic cycle. In regions with rain-fed agriculture, such as northeast Brazil, the timing of the onset and end of the rainy season becomes even more important than the rainfall distribution or amount, as pointed out by Ingram et al. [2002] and Sultan et al. [2005]. For instance, by knowing the best period to seed a specific crop, farmers can mitigate losses by anticipating or delaying the planting of specific varieties or by selecting another more appropriate crop. A formal and reproducible definition of onset and end of the rainy season emerges as the first challenge in such analysis. A variety of methods have been proposed in recent years, but still there is no agreement among researchers of a method that can be effectively applied everywhere. It turns out that the different

1 Department of Earth and Environmental Engineering, Columbia University, New York, New York, USA.

Copyright 2009 by the American Geophysical Union. 0043-1397/09/2008WR007485$09.00

mechanisms that drive the rainy season (e.g., monsoon systems versus cold fronts) have led to the development of different models for each specific region. [3] A simple method to define the start of the rainy season is to take the first day of year in which the total daily rainfall is greater than a threshold value, usually subjectively chosen. For instance, Stern [1982] has used a Markov chain model for the total daily rainfall in order to estimate the probability distribution of the rainy season onset. However, in regions where the wet season rainfall is mainly driven by deep convection, an isolated and unusual event of high rainfall may mask the onset of the rainy season. As an alternative method, the pentad, i.e., 5-day average rainfall, associated with a different threshold is used to define the start of the rainy season. Subjective selections of threshold values lead to a lack of robust conclusions unless sensitivity to these choices is investigated. [4] As an alternative to the rainfall pentad, especially in regions with few rainfall observations, such as the oceans, an outgoing longwave radiation (OLR) pentad [e.g., Wu and Wang, 2000], which is indicative of the presence of clouds and consequently the probability of rainfall, has been used. Since the OLR signal is a proxy, a combination of the rainfall and OLR pentads [Marengo et al., 2001] has also been applied. Alternative methods to define the timing of the rainy season have also been based on the accumulated

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LIMA AND LALL: HIERARCHICAL BAYESIAN MODELING

rainfall [Liebmann and Marengo, 2001; Grantz et al., 2007], cumulative percentage mean rainfall [Odekunle, 2004] and the daily rainfall probability [Garbutt et al., 1981; Odekunle, 2004]. Examples of climate variables used to define the onset of the rainy season include the shift of the ITCZ [Sultan et al., 2005], vertically integrated moisture transport [Fasullo and Webster, 2003], global daily normalized precipitable water [Zeng and Lu, 2004] and the meridional gradient of tropospheric temperature [Goswami and Xavier, 2005]. Of these, some have investigated climate teleconnections associated with delays or advances in the onset of the rainy season by looking at extreme climate events and their effects on the rainy season onset or by doing composite analysis. No one has attempted to develop an operational forecast model for the onset and end of the rainy season. [5] In this paper we propose a model to identify the onset and end of the rainy season based on the seasonally varying rainfall occurrence as done in a similar context (on the basis of the probability of rain instead) by some authors [e.g., Garbutt et al., 1981; Odekunle, 2004], but here we consider data from multiple rainfall sites and we allow the single site model parameters to vary across years; that is, we consider model parameters that are nonstationary in time and space. The parameter estimates are made under a hierarchical Bayesian framework. This allows a formal uncertainty analysis for model structure and parameters, including the interpretation of differences across sites and the relationship of the estimated parameters and exogenous climate variables. Examples of Bayesian models in hydrology and climate include the modeling of monthly maximum temperature [Wikle et al., 1998], rainfall [Gaudard et al., 1999], surface winds [Berliner et al., 1999; Wikle et al., 2001], airsea interaction [Berliner et al., 2003], a variety of hydrologic applications [Parent et al., 1998], rainfall extremes [Coles and Tawn, 1996] and amounts [Sanso and Guenni, 1999] and hydrologic models [Thiemann et al., 2001; Frost et al., 2007]. Wikle et al. [1998] pointed out that Bayesian inference is particularly useful when the model to be constructed involves the estimation of many parameters and the data are nonstationary in space and time, which is the case for most rainfall data. We apply our model to evaluate the daily rainfall occurrence at 504 rainfall sites located in northeast Brazil. Annually varying parameters allow us to assess temporal trends and climate teleconnections associated with delays and advances in the rainy season. This analysis sets up a potential capacity for forecasting the temporal aspects of the rainy season. [6] Background information on northeast Brazil, its associated climate regime and the source of data are presented next. Section 3 describes the hierarchical Bayesian model proposed to identify the onset of the rainy season. The results obtained are provided in sections 4 and 5, followed by a discussion and summary.

2. Northeast Brazil and Rainfall Data [7] The northeast region of Brazil, hereafter referred to as Nordeste, is characterized by high climate variability, rainfed and irrigated agriculture and medium to high levels of poverty, particularly concentrated in the central region often called the dry polygon. Floods and severe droughts alternate across years in the Nordeste. Hastenrath [1994] points out

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that the Nordeste is one of the regions of the world with the highest social and economical impacts due to severe droughts (such as those of 1877, 1915, 1958, 1983, 1993 and 1997). On the other hand, the tropical location of Nordeste and the influence of the Atlantic and Pacific sea surface temperature (SST) on rainfall lead to some of the highest skills for seasonal climate predictions. Motivated by this high rainfall predictability and by the importance of predictions for the economical and social developments of the region, many researchers and institutions have proposed dynamical and statistical forecasting models for the rainfall over Nordeste [e.g., Hastenrath, 1990; Folland et al., 2001; Moura and Hastenrath, 2004; Sun et al., 2006]. [8] The rainfall season of Nordeste shows a spatial variability that is associated with different climate mechanisms. The main rainy season occurs between March and April [Strang, 1972] and is most associated with the displacement of the Intertropical Convergence Zone (ITCZ) [e.g., Moura and Shukla, 1981; Hastenrath, 1994]. Several studies have demonstrated that the variation of the interhemispheric SST gradient has a significant impact on the position and intensity of the ITCZ, which in turn influences the climate of Nordeste and also over Northwest Africa. This tropical Atlantic gradient is usually defined as the difference between North (5°N – 25°N) and South (5°S– 25°S) Atlantic SSTs. The main rainfall period in Nordeste coincides with negative values of this gradient (Tropical South Atlantic (SAI) warmer than Tropical North Atlantic (NAI)), which is associated to a displacement of the ITCZ to its southern most position. The end of the Nordeste main rainfall season is marked by the return of the ITCZ to its location in the North hemisphere [Uvo et al., 1998]. On the other hand, an anomalous positive gradient (NAI warmer than SAI) forces anomalous fields of meridional wind component and an anomalously far northerly ITCZ location, which reduces Nordeste rainfall. Moura and Shukla [1981] suggested also that the interhemispheric SST gradient with warm waters to the North and cold waters to the South corresponds to a Hadley cell circulation in the atmosphere with descending branch over Nordeste. [9] In southern Nordeste the main rainy season is between December and February [Strang, 1972; Chu, 1983; Hastenrath, 1994]. It is modulated by cold front systems from the Southern hemisphere that enter into the region during the year [Kousky, 1979; Hastenrath, 1994]. Along the eastern coast of Nordeste the principal rainfall season is from April to July. It is associated with southeasterly winds that propagates perpendicular to the shoreline [Rao et al., 1993] and the land-sea breeze system [Kousky, 1980]. [10] ENSO events also play a big role on the spatiotemporal variability of Nordeste rainfall, with negative rainfall anomalies in northern Nordeste during extreme El Ninõ events. The drier than normal conditions are in part due to the enhancement of atmospheric subsidence over Nordeste caused by an eastward displacement of the Walker cell during El Ninõ years [Ropelewski and Halpert, 1987; Uvo et al., 1998; Giannini et al., 2001, 2004; Kayano and Andreoli, 2006], with its ascending branch along the east coast of the equatorial Pacific, dominated by anomalous warm SST, and the descending branch over tropical Atlantic. Opposite patterns in the general circulation and SST are

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precision and our model results were not sensitive to modest changes in this value. The rainfall data was obtained through the NOAA NCDC Global Daily Climatology Network and can be obtained from the International Research Institute for Climate and Society at http://iridl.ldeo.columbia. edu/SOURCES/.NOAA/.NCDC/.GDCN/.

3. Methodology

Figure 1. Location of the 504 rainfall stations in northeast Brazil that are used in this work. The three black circles labeled as 1 (northern), 2 (eastern), and 3 (southern) are representative stations of the three regimes of rainfall season in northeast Brazil. Longitude and latitude are shown along the x and y axes, respectively.

associated with positive rainfall anomalies during La Ninã years [Kayano and Andreoli, 2006]. Conversely, the southern Nordeste experiences opposite effects of ENSO. The enhancement of rainfall over southern Nordeste during El Ninõ events often occurs during the rainy season (December – May) of the year subsequent to the El Ninõ year. One of the potential causes of such an increase in rainfall is associated with stronger subtropical westerlies in El Ninõ years, which in turn favors the intensification of mesoscale convection centers in South Brazil [Ropelewski and Halpert, 1987; Diaz et al., 1998; Grimm et al., 1998, 2000; Coelho et al., 2002; Grimm, 2004; Cardoso and Dias, 2006]. Anomalous circulations southeast of Brazil and southwest of South America that favor baroclinic instabilities and changes in rainfall patterns over southern Brazil have also been linked to ENSO events [Grimm et al., 1998, 2000]. Although these studies depict a general pattern and associated climate teleconnections with the Nordeste rainy season, they did not analyze or forecast delays in the onset of the rainy season as well as the interannual variability of the onset on daily and weekly scales. [11] The set of daily rainfall data used in our investigations is composed of 504 stations, as shown in Figure 1, and covers the period from 1940 to 2000. Most stations have a full period of record from 1960 to 1980. The average number of available years for each station is 45, with a few stations having less than 30 years of data. None have less than 20. We consider 1 year as complete if there are data available on at least 200 days. For the seasonality analysis, we first obtain the rainfall occurrence time series of all sites by classifying the rainfall amounts data into wet (y = 1) and dry (y = 0) days. We define a wet state when the rainfall amount exceeds 0.254 mm and a dry state otherwise. This threshold value is commonly used [e.g., Katz, 1977] to censor low rainfall values at the measurement

3.1. Main Model [12] Our primary assumption is that rainfall occurrence can be used to characterize the rainfall season onset, duration and end, in a manner similar to that of Garbutt et al. [1981] and Odekunle [2004]. This assumption is reasonable for Nordeste, where the rainfall season is predominantly in the first half of the year, while there is little or no rainfall occurrence in the second half. We note that the definition we propose can easily address seasonality associated with different target amounts by varying the threshold used to define wet or dry states. If extreme rainfall events were of primary interest, e.g., for defining a flood season, then a higher rainfall threshold could be used to classify the data into wet and dry days. [13] The daily rainfall occurrence time series are binary data (1 = rain, 0 = dry) with a marked seasonal cycle, usually unimodal or bimodal. Let yst(n) be a Bernoulli random variable of rainfall station s, taking the value 1 when there is rainfall on the nth day of year t and 0 otherwise. A natural candidate model for this random variable is the logistic applied to a Fourier series, representing the probability of rainfall as a function of seasonal components. This is similar to the models derived for the daily probability of rainfall [Coe and Stern, 1982; Garbutt et al., 1981]: yst ðnÞ Binð1; pst ðnÞÞ;

ð1Þ

where

pst ðnÞ ¼ logit

1

ast þ

X

! bstk sinðwk nÞ þ cstk cosðwk nÞ

ð2Þ

k

is the probability of rain at station s for the nth day of year t, wk is the angular frequency of the kth Fourier harmonic and the inverse of the logit function is defined as logit1 ð xÞ ¼

expð xÞ : 1 þ expð xÞ

ð3Þ

This guarantees that pst(n) lies between 0 and 1. [14] Coe and Stern [1982] suggested for daily rainfall models that the number of Fourier harmonics to be kept in (1) is usually between two and three [Coe and Stern, 1982]. The power spectrum densities (not shown here) of the rainfall occurrence time series of most rainfall stations in Nordeste show a significant peak at a frequency of 0.0027 d1, which corresponds to the annual cycle. No other frequency appears to be significant. Therefore, for the sake of simplicity and in order to reduce the number

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Figure 2. (a, b, c): Maximum likelihood estimates (MLE) of the Fourier harmonics in (4) for station 3 (see location in Figure 1). The shaded region shows the 95% confidence interval for the estimates. (d) Fitted curve using the MLE for year 1959 (thick solid line) and the time-averaged (across years) MLE (thin solid line) and Bayesian (dashed line) point estimates (based on the expected value of the posterior distribution of parameters) for year 1959. The dots show observations (1, rain; 0, dry).

of parameters in the model, we consider only the first Fourier harmonic to represent the annual cycle: yst ðnÞ Bin 1; logit1 ðast þ bst sinðwnÞ þ cst cosðwnÞÞ ;

ð4Þ

where the angular frequency w of an annual cycle is given 2p = 0.0172 rad d1. Implicitly, we are assuming by w = 365 that, given the Fourier parameters, the random variables yst(n) in (4) are conditionally independent in time (day-today and year-to-year dependence) and space. [15] Rainfall lag correlations across years are negligible and hence are not accounted in (4). A Markov chain model [e.g., Coe and Stern, 1982] could be considered for the daily rainfall process. However, in the work presented here a simpler model is used. 3.2. Parameter Estimates and Hierarchical Bayesian Modeling [16] In general, enough data is available to estimate the time varying Fourier parameters in (4). At least 200 days are available to estimate only three parameters, therefore maximum likelihood estimates (MLE) using standard statistical

software would potentially lead to robust estimates. However, one faces two additional problems: the distribution of the available data along the year and data quality. For instance, Figure 2 displays the maximum likelihood estimates of (4) for station 3 (see location in Figure 1). Clearly the intercept parameter ast when t = 1959 is most likely an outlier, also suggested by a simple c2 outlier test. Figure 2d shows many missing days in the beginning of the hydrological year 1959, which results in a larger than usual confidence interval for the Fourier parameters (Figures 2a – 2c) and model outputs (thick line in Figure 2d) that significantly differ from the output (thin line in Figure 2d) obtained using the time averaged (across years) Fourier parameters, which is indicative of the average probability of rainfall. In addition to this, some stations (not shown here) show very unusual distribution of rainfall occurrence (e.g., a single rainy day in 1 year) along the year, most probably due to data quality, which in turn significantly increases parameter uncertainties and model analysis. One approach to systematically assess and reduce parameter uncertainties in this context is to estimate the parameters under a hierarchical Bayesian framework [Gelman et al., 2004], where more

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reliable model outputs are obtained (e.g., dashed line in Figure 2d). [17] In the first stage of the hierarchical Bayesian model, we assume that, for each station s, the 3 Fourier coefficients ast, bst and cst are sampled from a Gaussian process which has the mean value for these coefficients, As, Bs, Cs, respectively, with a corresponding 3 3 covariance matrix Ss. This way we account for the temporal variability of the coefficients at each station. Mathematically, we can represent this through a prior distribution:

[20] For the covariance matrix of (5) we propose a multivariate Jeffreys prior density [Gelman et al., 2004], since the number of dimensions d = 3 is relatively small: pðSs Þ / jSs jðdþ1Þ=2 :

0

ð5Þ

The large interannual variability of Nordeste rainfall and the observation that rainfall is essentially uncorrelated across years motivated a simpler model where correlation of ast, bst and cst across years is neglected. [18] In the second stage of the hierarchical model we define priors for As, Bs and Cs, which represent temporal averages of the coefficients defined in (4). Note that only years in which there is data available are used to estimate these hyperparameters. By defining prior distributions for each of those parameters, we aggregate more information and constraint the estimates, reducing their variance [Gelman and Hill, 2007]. Our prior information here is the existence of a spatial structure in seasonality and hence among the coefficients. We adopt an approach similar to that of Wikle et al. [1998], where the coefficients are modeled as simple linear spatial trends: As N a0 þ a1 longs þ a2 lats ; s2A ;

Bs N b0 þ b1 longs þ

b 2 lats ; s2B

ð6Þ

Cs N g 0 þ g 1 longs þ g 2 lats ; s2C ;

pðsA ; sB ; sC Þ / 1:

where longs and lats refer to the longitude and latitude of rain site s, respectively. A further justification of the use of linear spatial trends is presented in section 3.2.2. [19] In a more comprehensive approach, one could consider a full spatial model, such as the vector regression model used by Wikle et al. [1998], which would explicitly account for the spatial dependence across rainfall sites. However, it would require the estimation of a variogram and its associated parameters (see Cressie [1993] for some definitions) for each hyperparameter and year in the record. As we discuss in section 3.2.2, the residuals from the linear spatial trend models (6), (7) and (8) show no spatial correlation for distances greater than 500 km, which is the spatial scale of most interest in this work. Therefore, we chose to work with a simplified version of this model, which implicitly accounts for the spatial dependence of the rain probability across sites through the Fourier parameters.

ð11Þ

3.2.1. Parameter Estimation [21] The estimation of the parameters described in equations (4), (5), (6), (7) and (8) is done using simulation methods. From (4) we can write the likelihood function of the binary data as pð yjqÞ ¼

Ts Y S Y 366 Y

Binðyst ðnÞj1;

s¼1 t¼1 n¼1 logit1 ðast þ

bst sinðwnÞ þ cst cosðwnÞÞ;

ð12Þ

where q refers to all parameters being estimated, S is the total number of rainfall stations and Ts is the total number of years of station s. [22] The Bayes’ rule gives the posterior density for q:

ð7Þ

ð8Þ

ð10Þ

For the scale parameters in (6), (7) and (8), since there is enough data available (504 data points correspondent to the Fourier coefficients of the 504 sites) to estimate them, we can just assume independent uniform priors as suggested by Gelman [2005]:

pðqjyÞ ¼ ;

ð9Þ

In the last stage of the hierarchical model, since we do not have any prior information of the regression coefficients in (6), (7) and (8), we specify independent uniform priors: pða0 ; a1 ; a2 ; b 0 ; b 1 ; b 2 ; g 0 ; g 1 ; g 2 Þ / 1:

1 0 1 As ast @ bst A N@ Bs ; Ss A: cst Cs

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pðq; yÞ / pð yjqÞ pðqÞ; pð yÞ

ð13Þ

where p(q) is referred as the prior distribution of the parameters. [23] Substituting the prior of the parameters as defined in (5) to (11), and the likelihood function (12) into (13), yields the joint posterior distribution pðqjyÞ /

YS s¼1 1

YTs Y366 t¼1

n¼1

Binðyst ðnÞj1;

logit ðast þ bst sinðwnÞ þ cst cosðwsÞÞ 0 1 ast As B C N@ bst Bs ; Ss A N As ja0 þ a1 longs þ a2 lats ; s2A cst Cs N Bs jb 0 þ b 1 longs þ b 2 lats ; s2B ð14Þ N Cs jg 0 þ g 1 longs þ g 2 lats ; s2C jSs j2 :

[24] Equation (14) involves the estimation of several parameters with nonconjugate prior distributions. The integral over all parameters cannot be directly solved. In this case, we have to turn to other methods. We adopt here the widely used Markov chain Monte Carlo (MCMC) method to draw values of q from its posterior distribution (14).

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Figure 3. (left) Sample variograms for the (a) As, (b) Bs, and (c) Cs coefficients and (middle) residuals of a linear spatial trend and (right) residuals of a quadratic spatial trend fitted on the As, Bs, and Cs coefficients. In particular, we combine the Gibbs sampler and the Metropolis algorithm for simulating from the posterior distribution [Gelman et al., 2004]. We apply the Gibbs sampler for conditionally conjugate parameters (i.e., when prior and posterior are in the same family of distributions) and when there is a closed form for the conditional posterior distribution, so that it is possible to draw samples directly from it. When the model is not conditionally conjugate and there is no closed form for the conditional posterior, we draw posterior samples using the Metropolis algorithm. Details about the conditional posterior distributions are shown in the appendix. A MCMC simulation was run with five chains to verify the convergence of the results (or the mixing) on the basis of the methodology suggested by Gelman et al. [2004]. 3.2.2. Spatial Variability of Fourier Coefficients [25] An important step in building the hierarchical Bayesian model is to identify the spatial structure among the coefficients defined in (6), (7) and (8) in order to define prior distributions. We use here the concept of sample variogram [Cressie, 1993; Pebesma, 2004] to evaluate the spatial structure and check whether or not the linear trend

suggested in (6), (7) and (8) is reasonable to model the spatial pattern across the Fourier parameters. [26] Figure 3 shows the sample variogram of the time averaged coefficients corresponding to As, Bs and Cs assuming isotropy. A linear trend as well as a quadratic trend are fit on the latitude and longitude coordinates and the variogram of the residuals from this fit is analyzed. For all coefficients, a great amount of the semivariance is reduced with the addition of a linear trend. A quadratic trend does not provide any further reduction in the semivariance. The residuals of the linear trend fit still show a spatial dependence structure related to smaller-scale variations that may require further modeling. A model that addresses this residual structure would lead to a change in the estimates of uncertainty of the parameters defined in (6), (7) and (8). However, the complexity added to the model with one more stage of spatial variance modeling may not justify the increment of uncertainty reduction gained. Moreover, the linear trend residuals displayed in Figure 3 show that most of the residual correlation is limited to less than 500 km and in the current paper we are more interested in analyzing temporal trends and climate teleconnections over

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Figure 4. Daily probability of rainfall for the three selected stations during 1968, 1972, and 1980 obtained from the Bayesian estimates of the time-varying Fourier coefficients. The red line is the fitted model. The black dots indicate occurrence (at 1) or absence (at 0) of rainfall. The horizontal dashed line shows the 0.5 probability of rainfall, while the vertical dashed line presents the day of year of in which the maximum probability of rainfall is achieved. The blue lines show the 95% Bayesian interval of the average probability of rainfall.

relatively larger areas. Therefore, a simple linear trend was used to model the main spatial structure of those parameters.

4. Features of the Rainfall Season Cycle [27] In order to illustrate how the rainfall occurrence model can be used to draw inferences about the rainfall season cycle, we select three benchmark stations (Figure 1) representative of the three different regimes of rainfall seasonality in Nordeste and evaluate the models outputs for those stations using the Bayesian estimates. [28] On the basis of an areal average of 67 stations in North Nordeste over the February – May period [Secretaria de Infraestrutua Hı´drica, 2000], we select 1968, 1972 and 1980 as benchmark years of wet, dry and normal conditions, respectively. As in some regions of Nordeste the rainfall season starts in November, we chose to work with a hydrological year instead of the usual calendar year. The start of the hydrological year is taken to be the 274th day of year (1 October for non – leap years) and the end on the

273rd day of the following calendar year. The average and the 95% interval of the rainfall probability for the selected stations and those years are presented in Figure 4. Shifts in the onset, peak and end of the rainy season can be easily observed in Figure 4. This methodology also allows us to identify an anomalous wet season, which is usually marked by an increase in the number of rainy days and consequently in the maximum probability of rainfall. The converse applies to the identification of dry years. For instance, Figure 4 shows an early peak in the rainy season of the station 1 in 1980 as well as a low rainfall total for this period. The wet season in 1968 is marked by high rainfall probabilities. The late peak during 1972 is also illustrated. Similarly, Figure 4 depicts a dry 1968 and an earlier rainfall peak in 1980 for station 2. [29] The average and the 95% interval of the rainfall mean probability obtained from the Bayesian estimates of As, Bs and Cs for the three selected stations is shown in Figure 5. The proposed model fits well the daily probability on the basis of the observed frequency of rain days. The

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Figure 5. Historical frequency of rainfall days (black line) for the three selected stations and associated fitted model (red line) with Bayesian estimates of Fourier coefficients. The horizontal dashed line shows the 0.5 probability of rainfall, while the vertical dashed line presents the day of year in which the maximum probability of rainfall is achieved. The blue lines shows the 95% Bayesian interval of the average probability of rainfall. The rainy season onset can be defined as occurring when p > p*, where p is the daily probability of rainfall and p* can be any choice that the user makes.

daily probability model for the northern and eastern stations shows values that are under 0.5. The maximum of those stations occurs around March and May, respectively. The southern station presents the rainy season between November and February. All these results agree with the literature, as detailed in section 2. [30] Another more intuitive way to look at the spatial distribution of As, Bs and Cs is to analyze the maximum probability of rainfall Pmax and the day of year J max in which it occurs. We can easily derive the expression for Pmax for each rainfall station s: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Psmax ¼ logit1 As þ B2s þ Cs2 :

ð15Þ

[31] By solving the equation Js ¼ arctan

Bs 365 Bs 1 ¼ arctan ; 2p w Cs Cs

ð16Þ

we find J max s :

Jsmax

J; J þ 365 2 ; ¼ 365 Jþ 2 ; J þ 365;

Bs Cs > 0; Bs > 0 Bs Cs > 0; Bs 0 Bs Cs 0; Bs > 0 Bs Cs 0; Bs 0:

ð17Þ

[32] Note that Pmax and J max are random variables s s obtained for each sample of the posterior distribution of As, Bs and Cs. Figure 6a illustrates the spatial distribution of the expected value of Pmax s , i.e., the Bayesian point estimate

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Figure 6. (a) Spatial distribution of maximum probability of daily rainfall Pmax obtained from Bayesian point s estimates and (b) spatial distribution of day of year J max s in which Pmax is achieved. s

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and in the day of year jmax s,t in which it takes place by fitting a linear curve on those time series. Note again that pmax s,t and jmax s,t are random variables and posterior estimates of them were obtained for each posterior sample of as,t, bs,t and cs,t. and jmax are used as point The expected value of pmax s,t s,t estimation values in a weighted linear fit on the time variable, where the uncertainty is accounted by using weights defined as the inverse of the variance of pmax s,t and . jmax s,t [35] The spatial distribution of the regression slope of (not shown here) suggests a positive trend in the pmax s,t maximum probability of rainfall over northern Nordeste and a decreasing trend in some rainfall stations along the east coast. In order to be field significant at the 95% confidence level, the spatial map should present at least 21 locations (out of 504) with significant slopes, according to the method proposed by Livezey and Chen [1983]. The map of pmax s,t exhibits 202 significant slopes, which is a large number and strongly indicates a field significant trend. The (not spatial distribution of the regression slope of jmax s,t shown) has a spatial pattern in eastern Nordeste, with a tendency to later days of maximum probability of rainfall. The map has 98 stations with statistically significant slopes. [36] Another way to look at temporal trends is by assessing the spatial average of variables that share similar rainfall patterns. We apply the k means technique [Hastie et al., 2001] to define spatial clusters on the basis of the day of (Figure 6). We year of maximum probability of rainfall J max s set k = 3, given the three rainfall regimes of Nordeste reported in the literature. The clusters identified are shown in Figure 7. Note that they correspond to the spatial patterns of rainfall over the northern, eastern and southern regions of Nordeste. Figure 8 shows the spatial average of the maximum probability of rainfall pmax s,t and day of year in each it is achieved ( jmax s,t ) across the rainfall stations within each cluster. Monotonic trends in the maximum probability of rainfall can be observed in all three curves. Evidences of

of Pmax s . The maximum probability of rainfall is very low in the central region of Nordeste. This is consistent with the literature, which refers to this area as the dry polygon. The probability reaches the highest values along the coast and toward southern Nordeste. [33] The day of year of maximum probability of rain depicted in Figure 6b is also in agreement with previous findings. In southern Nordeste the rainfall peak is around January [Strang, 1972], then the displacement of the ITCZ brings maximum rainfall to northern Nordeste around April [Moura and Shukla, 1981; Hastenrath, 1994] and finally the propagation of southeasterly winds perpendicular to the shoreline [Rao et al., 1993] is associated with the rainfall along the Nordeste coast in June.

5. Temporal Trends and Climate Teleconnections [34] We assess trends in the time series of the maximum probability of rainfall pmax s,t , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pmax as;t þ b2s;t þ c2s;t ; s;t ¼ logit

ð18Þ

Figure 7. Clustering based on the Bayesian estimates of the average day of year of maximum probability of rainfall.

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Figure 8. (top) Time series of maximum probability of rainfall for the three clusters depicted in Figure 7. (bottom) Time series of day of year of maximum probability of rainfall for the same clusters. Shown from left to right are northern, eastern, and southern clusters. A linear curve is fit on the data.

monotonic trends in the day of year of maximum probability of rainfall is observed only for the southern stations. [37] Previous work indicated that the Pacific and the Atlantic sea surface temperature (SST) play a big role on the Nordeste Rainfall. We analyze climate teleconnections by examining temporal correlations between the day of maximum probability of rainfall, jmax s,t , and four indices of SST: NINO3, representative of El Ninõ events, the North Atlantic index (NAI), the South Atlantic index (SAI) and

the difference NAI minus SAI, which is indicative of the meridional position of the Atlantic intertropical Convergence Zone (ITCZ). We use the extended SST data [Kaplan et al., 1998; Reynolds and Smith, 1994] provided by The International Research Institute for Climate Prediction (IRI), through http://iridl.ldeo.columbia.edu/SOURCES/.Indices/ .nino/.EXTENDED/.NINO3/. Climate teleconnections associated with the interannual variability of the maximum probability of rainfall are not evaluated given the extensive

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Figure 9. (a) Correlation of climate indexes and day of year of maximum probability of rainfall for the northern cluster of stations, as shown in Figure 7. The vertical solid line indicates the month of the average maximum probability of rainfall, as shown in Figure 5. Correlations above (below) the top (bottom) dashed line are statistically significant at the 95% confidence level. (b) Scatterplot of a potential climate predictor and the day of year of maximum probability of rainfall. The black line shows a linear curve fitted on the data.

literature about remote climate influences on Nordeste rainfall [e.g., Moura and Shukla, 1981; Rao et al., 1993; Uvo et al., 1998; Giannini et al., 2001; Kayano and Andreoli, 2006]. [38] The lagged correlations of northern stations jmax s,t and the four climate indices (Figure 9) suggest that NAI minus SAI has a meaningful influence on jmax s,t in the months that precede the main rainfall season. Such association is expected because of the well known role of NAI minus SAI on Nordeste rainfall. The scatterplot in Figure 9 illustrates a variability of about 2 months in jmax s,t , where earlier onsets of the rainfall season are associated with positive NAI minus SAI anomalies in January. Significant effects of SAI are also observed in March, when the average probability of rainfall in northern Nordeste achieves its maximum value, as showed in Figure 6b. Therefore, forecasts of jmax s,t could be issued in January using NAI minus SAI as predictor and updated in March on the basis of the SAI anomalies. The influence of NINO3 is not well determined. An analysis (not shown here) using weighted correlation coefficients defined as the inverse of the variance from the Bayesian simulations shows similar results. [39] The jmax s,t of the eastern cluster of stations is significantly affected by SAI in March (figure not shown here). The influence of the tropical south Atlantic SST on eastern Nordeste rainfall have already been reported in the literature [Kousky, 1980]. Positive anomalies are associated with later rainfall season onsets. The lag time between the peak in the correlation (March) and the timing of maximum probability of rainfall (June) suggests the possibility of long lead time

forecasts for the onset of the rainfall season in eastern Nordeste. [40] In southern Nordeste the NINO3 index turns out to be the best climate predictor for the onset of the rainfall season (figure not shown here). Positive anomalies of NINO3 in December are associated with advances in the peak of rainfall. La Ninã years yield to earlier rainy season onsets but with higher variability. Correlations with Atlantic indexes are not statistically significant before the timing of the rainy season. [41] Figure 10 further illustrates the spatial pattern of the and the potential climate predictors correlations of jmax s,t discussed above and suggested in the scatterplot of Figure 9. The positive correlation of NINO3 and the rainfall season onset in southern Nordeste appears clearly. The tropical Atlantic SST interhemispheric gradient (NAI minus SAI) has a significant influence on the rainy season onset in northern Nordeste. SAI is linked with the rainfall onset along the east coast and over northern Nordeste.

6. Summary and Discussion [42] In the present paper we analyzed daily rainfall occurrence in northeast Brazil. A hierarchical Bayesian model that uses common information to constrain parameter estimates was developed. The three main regimes of rainfall in Nordeste usually reported in the literature were correctly identified on a daily time scale with an estimate of the associated uncertainties.

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Figure 10. Correlation map of day of year of maximum probability of rainfall and (top) previous December NINO3 (Dec(1)), (middle) current January NAI minus SAI (Jan(0)), and (bottom) current March SAI (Mar(0)). Solid circles are statistically significant at the 95% confidence level.

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[43] The estimates of daily rainfall probability can be used to define attributes of rainfall seasonal timing. For instance, in southern Nordeste, we could define the onset of the rainy season when the probability of rainfall crosses 0.5 and the end when it crosses it again and becomes lower than 0.5. Looking at Figure 5, for example, we could declare the rainfall onset of station 3 on the 326th day of year (22 November for non– leap years) and the end on the 54th day (23 February) of the following calendar year, with a total duration of 95 days. The variability around these estimates would be ±7 days according to the uncertainty of the coefficients. On the other hand, in regions where the historical rainfall probability is usually lower than 0.5, we could outline the onset and end of the rainy season on the basis of a specific rainfall season duration defined a priori. For instance, let us assume that a specific rain-fed crop located close to station 2 needs water for 30 days. On the basis of the day of maximum probability of rainfall, as illustrated in Figure 5, we would say that the best period for planting is between the 142nd day of year (22 May) and the 171st day (20 June). In this case, the uncertainty of this period is only ±1 day. [44] Field significance of trends in the maximum probability of daily rainfall showed a positive trend for northern Nordeste. We note that this trend is likely to be associated with the recent (after the 1980s) positive trend in the SAI (not shown here) and more southward position of the ITCZ as reported by some authors [e.g., Rao et al., 2006]. Suggestive trends of earlier onsets in the rainfall season over eastern and southern Nordeste were observed but the associated correlations are weak. An examination of whether these trends may be related to natural or anthropogenic factors is a subject for climate model– based investigations. [45] Correlation maps identified three basic spatial patterns of climate teleconnections. Positive anomalies in the tropical Atlantic SST gradient (NAI minus SAI) in January prior to the rainfall season tend to anticipate its onset over a large area of northern Nordeste, where the reduced rainfall is due to the more northern ITCZ meridional position during positive anomaly events. Later rainy season onsets over eastern and northern Nordeste are also highly associated with positive anomalies in the tropical South Atlantic SST in March. El Ninõ events lead to later rainy season onsets in a large area of southern Nordeste. Anomalies of rainfall in this region are associated with an intensification of the subtropical westerly jets and consequently the increase of mesoscale activities, as indicated in section 2. [46] The high correlations found suggest that the onset of the rainy season in Nordeste could be potentially forecast one to four months ahead of time with reasonable accuracy. Future studies will focus on the development of a real time forecast model for the onset and end of the Nordeste rainy season with the inclusion of climate indexes as prior distributions for the model parameters. We also plan to expand this work for other regions of Brazil and include other climate variables such as pressure fields and the subtropical Atlantic SST. The model structure will be reviewed to consider a Markov chain or other model for persistence in daily rainfall and to more extensively consider the spatial correlation structure in the parameter estimates. The scale of the region that is best to model

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and/or the selection of relatively homogeneous regions for analysis will be part of these extensions to the current work.

[50] For the covariance matrix Ss we have the following conditional posterior distribution: 0 1 pðSs jÞ /

Appendix A: Bayesian Model [47] From the joint posterior distribution (14), the conditional posterior distribution of the intercept parameter in (4) can be written as pðast jÞ /

Y365 n¼1 1

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Ts Y t¼1

ast As N@ bst Bs ; Ss A jSs j2 ; cst Cs

which can be written as [Gelman et al., 2004] pðSs jÞ ¼ Inv-WishartTs 1 ðSs jSs Þ;

Binðyst ðnÞj1;

logit ðast þ bst sinðwnÞ þ cst cosðwnÞÞ 0 1 ast As B C N@ bst Bs ; Ss A; cst Cs

ðA7Þ

ðA8Þ

where Ss is the sum of squares matrix: ðA1Þ

Ss ¼

Ts X

ð½ast bst cst ½As Bs Cs ÞT ð½ast bst cst ½As Bs Cs Þ:

t¼1

where p(astj) indicates the posterior distribution of ast conditional on the data and remaining parameters. [48] The conditional posterior distribution (A1) is not conjugate and it has no closed form; that is, we cannot sample directly from it. Therefore, we use the Metropolis algorithm inside a Gibbs sampler to estimate the posterior distribution of ast. Details of the algorithm can be found in work by Gelman et al. [2004]. Note that p(bstj) and p(cstj) have similar conditional posterior distributions. The conditional posterior of the first set of hyperparameters (5) can be written as 1 ast As Tn Y B C pðAs ; Bs ; Cs jÞ / N@ bst Bs ; Ss A t¼1 cst Cs 0 1 As a0 þ a1 longs þ a2 lats B C N@ Bs b 0 þ b 1 longs þ b 2 lats ; Ss A; Cs g 0 þ g 1 longs þ g 2 lats 0

ðA9Þ

[51] The joint conditional posterior distribution of a = [a0 a1 a2] is given by pðajÞ /

S Y

N As ja0 þ a1 longs þ a2 lats ; s2A ;

ðA10Þ

s¼1

which yields, after completing the square, the following density: ^ ; Va s2A ; pðajÞ ¼ N a

ðA11Þ

^ = (X TX)1 X Ty, Va = (X TX)1, y = [A1 A2, . . . AS]T where a and 0

1 B2 X ¼B @: S

ðA2Þ

long1 long2 : longS

1 lat1 lat2 C C: : A latS

where Ss is the diagonal matrix: 0

s2A @ Ss ¼ 0 0

0 s2B 0

1 0 0 A: s2C

ðA3Þ

[49] Completing the quadratic of (A2) [Gelman et al., 2004] yields the following joint conditional posterior distribution of As, Bs, Cs: 1 pðAs ; Bs ; Cs jÞ / exp ðPs ms ÞT L1 ð P m Þ s s s 2 ¼ NðPs jms ; Ls Þ;

1 1

ms ¼ S1 s þ T s Ss L1 s

¼

S1 s

þ

0

0

1

as B C m1 ¼ @ bs A: cs

1 XS 2 ðAs a0 a1 longs a2 lats Þ : s¼1 S1 ðA12Þ

The conditional posterior distributions of sB and sC are obtained similarly. ;

Ts S1 s

1 As B C Ps ¼ @ Bs A ; Cs

1 S1 s m0 þ Ts Ss m1

YS p s2A j / N As ja0 þ a1 longs þ a2 lats ; s2A s¼1 ¼ Inv-c2 s2A jS 1;

ðA4Þ

where

[52] Similar posterior distributions are obtained for b = [b 0 b 1 b2] and g = [g 0 g 1 g 2]. [53] The same technique yields also the distribution of the variance parameter:

ðA5Þ 0

1 a0 þ a1 longs þ a2 lats B C m0 ¼ @ b0 þ b 1 longs þ b 2 lats A

[ 54 ] Acknowledgments. The first author thanks CAPES and Fulbright for financial support through grant 1650/041. We thank Andrew Gelman for several suggestions and discussion regarding several aspects of this paper.

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g 0 þ g 1 longs þ g 2 lats ðA6Þ

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U. Lall and C. H. R. Lima, Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA. (ula2@ columbia.edu; [email protected])

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