A STATISTICAL DOWNSCALING METHOD FOR MONTHLY TOTAL PRECIPITATION OVER TURKEY

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 24: (24) Published online in Wiley InterScience ( DOI: 1.12/joc.997 A STATISTICAL DOWNSCALING METHOD FOR MONTHLY TOTAL PRECIPITATION OVER TURKEY HASAN TATLI, a, *H.NÜZHET DALFES b and ŞSIBELMENTEŞ a a Meteorology Department, Istanbul Technical University, Maslak, Istanbul, Turkey b Eurasia Institute of Earth Sciences, Istanbul Technical University, Maslak, Istanbul, Turkey Received 1 August 23 Revised 12 November 23 Accepted 12 November 23 ABSTRACT Researchers are aware of certain types of problems that arise when modelling interconnections between general circulation and regional processes, such as prediction of regional, local-scale climate variables from large-scale processes, e.g. by means of general circulation model (GCM) outputs. The problem solution is called downscaling. In this paper, a statistical downscaling approach to monthly total precipitation over Turkey, which is an integral part of system identification for analysis of local-scale climate variables, is investigated. Based on perfect prognosis, a new computationally effective working method is introduced by the proper predictors selected from the National Centers for Environmental Prediction National Center for Atmospheric Research reanalysis data sets, which are simulated as perfectly as possible by GCMs during the period of The Sampson correlation ratio is used to determine the relationships between the monthly total precipitation series and the set of large-scale processes (namely 5 hpa geopotential heights, 7 hpa geopotential heights, sea-level pressures, 5 hpa vertical pressure velocities and 5 1 hpa geopotential thicknesses). In the study, statistical preprocessing is implemented by independent component analysis rather than principal component analysis or principal factor analysis. The proposed downscaling method originates from a recurrent neural network model of Jordan that uses not only large-scale predictors, but also the previous states of the relevant local-scale variables. Finally, some possible improvements and suggestions for further study are mentioned. Copyright 24 Royal Meteorological Society. KEY WORDS: independent component analysis; precipitation; recurrent neural networks; sampson correlation; statistical downscaling; Turkey 1. INTRODUCTION The introduction of the concept of downscaling has opened a wide spectrum of applications in many fields (e.g. Klein, 1982; Kim et al., 1984; Wilks, 1989; Karl et al., 199; Wigley et al., 199; Giorgi and Mearns, 1991; Zorita et al., 1992; von Storch et al., 1993; Bardossy, 1994; Matyasovsky et al., 1994; Noguer, 1994; von Storch, 1995; Cubasch et al., 1996; Hewitson and Crane, 1996; Schubert and Henderson-Sellers, 1997; Conway and Jones, 1998; Heimann and Sept, 1998; Kidson and Thomson, 1998; Murphy, 1999, 2; Sailor and Li, 1999; Smith, 1999; von Storch and Zwiers, 1999; Fuentes and Heimann, 2; Wilby and Wigley, 2; Stein et al., 21; Watson, 22; Geerts, 23). The logic behind downscaling and representation of regional climate in nested models (known as dynamical downscaling) may be found in Giorgi and Mearns (1991) and McGregor et al. (1993; McGregor, 1997). In those studies, the methods, large-scale predictors and local-scale predictands were various. For example, in the work of Kim et al. (1984) the predictors were large-scale monthly mean temperature and precipitation. On other hand, monthly surface temperature and precipitation were selected as local-scale predictors. The Correspondence to: Hasan Tatlı, Aeronautics and Astronautics Faculty, Meteorology Department, Istanbul Technical University, Maslak, Istanbul, Turkey; tatli@itu.edu.tr Copyright 24 Royal Meteorological Society

2 162 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ methods they applied were principal component analysis (PCA) and linear regression. For downscaling Iberian winter precipitation, Zorita et al. (1992) and von Storch et al. (1993) selected gridded North Atlantic sealevel pressure (SLP) as a large-scale predictor, and the downscaling method was simple regression based on canonical correlation analysis (CCA). The other very interesting application is the work of Heimann and Sept (1998), who employed both statistical and dynamical downscaling techniques in order to analyse the climatic change of summer temperature and precipitation in the Alpine region. Statistical downscaling methods may be sorted into three groups: model output statistics (predictors from global circulation models (GCMs)); perfect prognosis (predictors from large-scale free atmospheric observations or free atmospheric reanalysis data sets); and, downscaling with surface variables (predictors from large-scale surface observations). Perfect prognosis, based on multivariate regression, is gaining considerable acceptance in downscaling of GCMs in view of its inherent simplicity and flexibility. This technique provides an external description of the system under study and leads to a parsimonious representation of the process. The accurate determination of the predictors is a necessary first step in downscaling, and this continues to be a subject of extensive contemporary research. A literature review reveals that many statistical methods are available for finding the relations between large-scale predictors and local-scale predictands (von Storch and Zwiers, 1999). The selection of a reliable and efficient criterion has been elusive, since most criteria are sensitive to the statistical properties of the processes. Validation of most of the available criteria has generally been through simulated GCMs. More precisely, in the traditional methods, downscaling of surface variables from large-scale processes is thought of as a static map (regression equation). However, the unsolved problem is that the downscaling process has to satisfy constraints. These constraints can be seen as mathematical objects used to make explicit the logic behind of the downscaling problem. In this study, the constraints of the downscaling process are summarized thus: 1. The key assumption is that there is growing evidence of local-scale patterns being driven by large-scale climatic fluctuations. 2. Classical, linear methods may be inappropriate for downscaling local-scale processes from large-scale processes. Here, we consider the local-scale observations as the outcome of a finite-dimensional, nonlinear dynamical system and use recurrent neural networks to recover the nonlinear, causal and naïve skeleton of the underlying static relations and dynamics from the large-scale processes and the previous states of the system. Based on these constraints, the model is not thought of as a curve-fitting problem. Rather, the statistical model must satisfy the probability assumptions, e.g. homogeneity of dependence is the first requirement in distinguishing a model from the curve-fitting transformation. For example, predictions of the local variables from large-scale processes by well-known linear time series techniques (Bras and Rodriguez-Iturbe, 1993; Box et al., 1994; Hipel and Mcleod 1994; von Storch and Zwiers, 1999), such as autoregressive (AR) and autoregressive moving average (ARMA) techniques, are possible ways in modelling to satisfy locality features. On the other hand, to predict the same local processes from large-scale processes by means of regression equations is another possible approach (based on static relations). However, this paper suggests a method of combining these two approaches. Since the suggested method offers a statistical model, the predictors should be independent in the sense of the statistical framework. Details are given in the following sections, but instead of the PCA (also known as empirical orthogonal functions in meteorology literature), which is based on second-order statistics (Diaz and Fulbright, 1981; Preisendorfer, 1988), a new method known as independent component analysis (ICA) based on high-order statistics is employed (Jutten and Herault, 1991; Comon, 1994; Everson and Roberts, 1999; Hyvarinen et al., 21). A number of studies regarding the spatial and temporal properties of precipitation or rainfall in Turkey have been published (Kutiel et al., 1996, 21; Türkeş 1996, 1998; Kadıošlu, 2; Türkeş et al., 22; Touchan et al., 23). However, this paper is the first study regarding statistical downscaling of local variables in Turkey. In this paper, the monthly total precipitation data of 31 stations have been selected as an application Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

3 STATISTICAL DOWNSCALING OF TURKISH RAINFALL 163 of the proposed approach. It is demonstrated that the proposed method can suppress well the increase of the downscaling error due to nonlinearities, improving greatly the accuracy of the downscaling compared with the conventional regression-based techniques. In Section 2, the features of the precipitation climate of Turkey are summarized. In Section 3, the proposed method is described in detail: a new idea in statistical downscaling techniques is discussed along with an algorithm; it represents the main ideas behind the proposed method. In the same section, we also briefly discuss the concept of ICA, which is then used in Section 4 to build a downscaling model for monthly total precipitation series over Turkey. Some comments and conclusions are provided in Section A GENERAL LOOK AT THE PRECIPITATION CLIMATE OF TURKEY In general terms, Turkey is associated with a subtropical climatic regime that is referred to as Mediterranean. Because of its geographic location, the air masses that shape the weather regimes affecting Turkey have both polar and tropical origins, which are dominant in the winter and summer respectively. During the autumns and winters, polar fronts have a determining effect on the mid-latitudes. Large-scale atmospheric motions are translated into local weather conditions due to different land characteristics. These local conditions have both thermal and dynamical features. On the other hand, the different local conditions have sparse local-climate features due to the land sea distribution, elevation and such other local physical geographical properties that make those macro-climate features highly variable. Turkey (despite being surrounded by the sea on three sides), because of its coastal regions characterized by high topography, includes a rather large inland area (central Anatolia) that has a quite continental climate character (Taha et al., 1981; Erinç, 1984; Kadıoǧlu, 2). The topography causes strong variations in the climate. The coastal sides of the mountains receive heavy precipitation, but there are also thermal effects due to topographic height differences coupled with other factors. Summer. Since the maritime-polar (MP) and continental-polar (CP) air masses move towards northern regions, the tropical air masses have a dominant effect on mid-latitudes in this season. The southern and southeastern parts of the country are under the influence of a continental tropical air mass (due to the Arabic low) which is quite dry. The marine air mass that travels from the Atlantic towards Turkey (a northwesterly motion) gets heated and loses most of its relative humidity during its passage across the mainland and presents a stable structure. The Azores high-pressure centre locates over Europe during its northward and eastward displacement, and this situation affects the characteristic motions, especially for the western parts of Turkey. The interaction between the marine and continental systems causes strong winds, especially during the June August period. Both these air masses have stable structures characterized by hot and dry conditions, and cloudless summers are quite typical for Turkey. Exceptions do occur, as sometimes the western, and especially northwestern, parts of the country get heavy precipitation. The precipitation is the result of frontal systems forming as the Atlantic polar marine air mass interacts with the tropical marine air mass. The northern part of the country, having high topography along the Black Sea coast, also experiences some heavy precipitation of orographic origin. In the west, the mountains are not parallel to the coast; as a result, the orographic precipitation has a more local character. The mountains also divert the air flow in various ways, causing central Turkey to possess a peculiar wind regime. Winter. The Mediterranean basin, which includes all the European countries having a coast on the Mediterranean, becomes an active frontogenesis region at the beginning of autumn. The Azores high shifts to the south and the Siberian high-pressure system starts to affect northern and eastern Europe (due to thermal reasons); the Mediterranean belt becomes a convergence zone. The frontal systems that travel along various trajectories towards the east affect Turkey. These cyclonic depressions have two main paths across Turkey. The first affects the western and southern parts of the country and then continues towards the southeastern parts. The second has a trajectory towards the northeast and is the cause of most of the precipitation in the northwestern, northern and central parts of the country. The other trajectory of the polar front that affects Turkey is felt when the Azores high becomes strong enough to affect western Europe. When this comes onto the scene the depressions travel from the Thrace Marmara Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

4 164 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ and western part of the Black Sea region towards southeastern Turkey by northerly and northwesterly flows; most of Turkey is affected by those motions. The effect is more pronounced when the mechanisms related to the Azores high create northerly and northwesterly winds over the Anatolian Plateau. The principal effects of the Mediterranean cyclones on Turkey are southwesterly winds on the Turkish Mediterranean coast. Topography creates important local effects in the western and Black Sea coasts of Turkey: the seaward sides of the mountains are washed by heavy precipitation and the precipitation decreases towards the inner parts of Asia Minor. There exists a very sharp negative orographic precipitation gradient from the coastal regions towards central Anatolia. In central Anatolia, there is a plateau effect rather than a sharp topography effect. During the winter, a convergence zone also forms in the eastern part of the Black Sea. This is a consequence of the interaction between the northerly currents created by the Asian high (Siberian high) and the local southerly currents created by the small-scale high-pressure centres due to thermal effects in eastern Anatolia. Foehn winds and rain shadows are, therefore, common in this region during cold, winter periods. In the winter the passage of the fronts and related cyclones causes precipitation in the coastal regions, but central and eastern Anatolia remain largely continental in terms of climatic features and become a divergence field. During the spring, depressions gradually decrease over Turkey, due to the heating of the land surface and the depression frequencies reaching a minimum during the summer (Tatlı et al., 23). The depressions that are seen during the summer mainly affect the northern part of the Marmara region and over the Black Sea coast. Spring and summer are seasons that are dominated mainly by local effects. The mountains that run parallel to the Black Sea coast of Turkey prevent the northerly currents from penetrating inland. However, the situation in the west is different, as the mountains run perpendicular to the coast, so they cannot block the westerly currents very effectively, and Mediterranean depressions penetrate into the mid-western parts of the country through the gaps between these mountains. 3. METHODS A key assumption that constitutes the foundation of downscaling is that local climate is controlled by largescale processes. For simplicity, consider first the case of two different multivariate random variables: X originates from large-scale physical processes (such as a large-scale SLP series) and Y represents local variables (could be surface precipitation series). Mathematically, the statistical downscaling may be defined in terms a multivariate regression equation. The multivariate random variables X and Y aresaid to be dependent if and only if Y (t) = F [X (t)] + e(t) (1) exists, where F is an operator to be determined and e(t) is the multivariate error term. In other words, the joint probability density P(X, Y ) cannot be factorized into a product of their marginal densities P(X ) and P(Y ). In practice, the probability densities are generally unknown. Therefore, particularly to distinguish between predictor and predictand variables, some generalization of the multiple correlation coefficients may be of use. One such possibility, according to Sampson (1984), is R S = Tr(C YX C 1 XX C XY ) Tr(C YY ) where C XY, C XX and C YY indicate multivariate cross-covariance and covariance matrices and Tr denotes the trace of the related matrices respectively. Another important special case of this approach concerns the correlation between the multivariate X and Y, called CCA. CCA is a multivariate linear technique for portraying the relationship between multiple sets of multivariate data (Glahn, 1968; Nicholls, 1987; Chen et al., 1994; Chen and Chang, 1994), and it is based on symmetrical Pearson correlation. That is, CCA transforms the predictors and predictands into new variables (canonical correlation variates: CCVs) by maximizing the Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24) (2)

5 STATISTICAL DOWNSCALING OF TURKISH RAINFALL 165 correlation between two CCVs. In words, CCA does not recognize predictor and predictand variables as in the case of Sampson correlation (Jackson, 1991), because the correlation between predictors and predictands may not be symmetrical, i.e. correlation (X, Y )? = correlation(y, X). If we put a single grid time series of each large-scale variable (X is now a vector in Equation (2)) and multiple stations time series set (Y is now a matrix in Equation (2)) in Equation (2), then the map of computed Sampson correlations is termed the R S correlation pattern in this study. The interpretation of the R S patterns is similar to the canonical correlation patterns with respect to their statistical properties. Although there is only one R S correlation pattern, CCA extracts a number up to the rank of {X, Y } correlation patterns (rank(x, Y) = min[rank(x), rank(y)]). Before model building, the application of statistical preprocessing of data sets is required (Rummukainen, 1997). In the study, we employ ICA based on PCA, and followed by CCA. First, the data sets are decomposed into their significant principal components (PCs) according to the maximum-likelihood criterion of Jöreskop and Sörbom (1989); second, these uncorrelated PCs are transformed into independent components (ICs) by the FastICA technique, which is an ICA approach of Hyvarinen and Oja (1997). PCA and the closely related principal factor analysis techniques are widely applied in meteorology or climatology literature. Without going into details, assume the first- and second-order statistics of a multivariate X is known or can be estimated from the sample. In PCA transform, the variables of X are centred by subtracting their means, and then the covariance matrix of X is defined as the expected value of the minor products of its variables. In this paper, all matrices are organized such that rows represent simultaneous observations and columns are observed variables at different sites. S XX = E(X T X ) (3) One has to keep in mind that PCs are not invariant under scaling. Without loss of generality, an orthogonal decomposition of S XX is given as follows. S XX = E X D X E T X (4) where E X is a matrix of the orthonormal eigenvectors of S XX and D = diag(λ 1,...,λ k ) is the diagonal matrix of the eigenvalues of S XX in a decreasing magnitude order (Preisendorfer, 1988). The PCs are then computed by V X = XE X (5) where V X illustrates the PCs and the reconstruction of X follows X = V X E T X (6) The PCA model was discussed above as a distribution-free method with no underlying statistical model, but in the case of principal factor analysis (FA) the multivariate X is reconstructed as in the following: X = F X A X + G X (7) where F X and G X are called common factors (CFs; or latent variables) and white noise components (or specific components), respectively. A X is called a factor loading, defined as S XX = A X A T X = E X D 1/2 D 1/2 E T X (8) There are no statistical constraints on PCs, but CFs must be Gaussian in an FA model. Furthermore, PCs are easily calculated in a unique way, whereas there is no unique method for CFs; however, a possible way according to Reyment and Jöreskog (1993) is F X = XS 1 XX AT X (9) Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

6 166 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ The CFs may be rotated for easier interpretation of the patterns (Richman, 1985), e.g. by the varimax method (Kaiser, 1959). On the other hand, ICA may be seen as a special kind of rotation of CFs (or PCs), but using high-order statistics rather the second-order statistics. If the data series have Gaussian density then FA and ICA are identical, because the high-order statistics of Gaussian variables are all zero (assume the variables have zero mean and unit variance). ICA has been widely used in data analysis and decomposition in recent years (Jutten and Herault, 1991; Comon, 1994; Everson and Roberts, 1999; Hyvarinen et al., 21). ICA typically aims to solve the blind source separation problem in which a set of unknown sources is mixed in some way to form the data. An ICA model assumes that the multivariate data set X is a mixing of the ICs as X = SA (1) or in terms of PCA as S = XW = V X E T X W (11) where A = W 1 is called mixing matrix, and S (like PCs or CFs) represents ICs, respectively. The starting point for ICA is the very simple assumption that all the ICs are statistically independent based on a non- Gaussian distribution. However, the basic model of ICA does not assume this distribution is known. After determining the matrix A, its inverse W is then computed to obtain the ICs by Equation (11) from PCs. The restriction in a PCA is that all PCs are mutually uncorrelated, but the restriction in an ICA is that all the ICs are statistically independent. Uncorrelatedness is a weaker form of independence. That is, two random variables are uncorrelated if their covariance is zero. On the other hand, if the variables are independent then they are uncorrelated, from which it follows that uncorrelatedness does not imply independence (Comon, 1994). Let us define independency according to Hyvarinen et al. (21). Two random variables y 1 and y 2 are said to be mutually independent if, given two arbitrary functions h 1 (y 1 ) and h 2 (y 2 ), they satisfy the following condition: E{h 1 (y 1 )h 2 (y 2 )}=E{h 1 (y 1 )}E{h 2 (y 2 )} (12) where E represents the expected value. According to the definition of statistical independency, ICA has a stronger definition than the uncorrelatedness concerning PCA. ICA uses high-order statistics and can separate the data into true sources, unlike PCA or FA, and so guarantees true sources in data. In this study, the ICs are computed via a fixed-point algorithm called FastICA, introduced by Hyvarinen and Oja (1997). The FastICA algorithm is based on maximizing the absolute value of kurtosis of the variable by using an information measure or negative of entropy: J(y) = 1 12 E{y3 } kurt(y)2 (13) where J(y) [, 1] indicates information measure, and the kurtosis of y is illustrated by kurt(y) definedas: kurt(y) = E{y 4 } 3(E{y 2 }) 2 (14) Since the information measure of Gaussian variables among equivalent variances is zero, the information can be used as an indicator of the measure of the degree of normality in the data series. There are some basic ambiguities that hold in an ICA model: 1. The variances of ICs cannot be determined, whereas the variances are explained by eigenvalues of PCs (orcfs)inapca(orfa)approach. 2. The order of ICs cannot be determined. Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

7 STATISTICAL DOWNSCALING OF TURKISH RAINFALL 167 A detailed introduction to ICA for practitioners may be found in Hyvarinen et al. (21). A downscaling model may be divided into a naïve (e.g. AR, ARMA) and causal (e.g. regression) components. One must distinguish between causal models and naïve models. A naïve model approach involves using only historical data. Thus, it assumes that the trend of the dependent variables, which in this study are the monthly total precipitations, remains constant over time, and the value of the dependent variables Y(t + k), where t is the present month and k is the number of the month to predict, can be extrapolated from historical data. On the other hand, a causal model approach assumes that there are some (causal) variables (e.g. GCM outputs) that are responsible for changes in the dependent variables. In the causal models, previous information is available from GCMs; therefore, this information should be considered in the process of the model building for downscaling purposes (i.e. a Bayesian approach). Assuming now that t illustrates the prediction time step and G is an unknown transformation (such as AR coefficients in an AR model), then prediction of the multivariate Y from its own finite countable previous states can be illustrated as in the following: [ ] Y (t + t) = G Y (t k t) + e(t) k =, 1...,n (15) Let Y 1 (t) and Y 2 (t) denote the outputs of the causal model and the naive model respectively, and in order to distinguish and separate the models. Assume that the static transformation (causal) is linear and the dynamic prediction model (naive) is first order (Richardson, 1981); then these two separated prediction models can be written as Y 1 (t + t) = A[X (t + t)] (causal model) (16) Y 2 (t + t) = B[Y (t)](naïve model) (17) where X(t) represents causal variables and, for simplification, we omit the error term e(t). The main problem arises in selecting the best outputs from these models. However, it is not clear which of the model outputs is important for local-scale processes, but one may propose a multivariate difference equation as in the following (Bras and Rodriguez-Iturbe, 1993): Y (t) = M 1 [Y (t t)] + M 2 [X (t)] + M 3 [e(t)] (18) where M 1, M 2 and M 3 are linear operators and the vector e(t) consists of uncorrelated white noise. The vector covariance of e(t) is a diagonal matrix: D e = E(e(t) T e(t)) (19) where E illustrates the expected value operator, which in practice is generally taken as the sample average. The definition of Equation (18) forms a possible way for accommodating the naive and the causal models into a new model with linear structure. Meteorological or climate processes are generally expected to behave nonlinearly, whereas the process represented by Equation (18) has a linear static part and a linear dynamic part; one can claim that it represents a linearization approximation. Therefore, Equation (18) may not be enough to satisfy the nonlinearity constraint. Hence, we develop here a more sophisticated approximate approach for solving this problem. If the two downscaled sets from Equations (16) and (17) are combined as Z (t) = [Y 1 (t), Y 2 (t)], then the suggested model can be written as Y (t) = M [H [Z (t)]] + e(t) (2) where e(t) is the multivariate Gaussian noise. M is a linear operator in the case of Equation (1) and it predicts local variables from the {H [Z (t)]} basis functions, and H is a nonlinear scalar function given by Equation (21) that can generate the basis functions. So we now have a new version of the downscaling method Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

8 168 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ that constitutes both the knowledge from the static model of the causal large-scale variables in the case in Equation (16) and the knowledge from the first-order local linear dynamic model in the case of Equation (17). However, the regression equation in Equation (16) may be applied as long as one assumes homogeneity in the sets of predictors and predictands for which the regression is evaluated. Obviously, linear regression of Equation (16) may not directly be applied due to heterogeneity, since heterogeneity may be introduced in the dependence. In this study, the homogeneity of dependence is an assumption that describes the invariance of the relation (deterministic rule) between predictors and predictands. Hence, a method is proposed to solve this problem. First, using CCA, which may be seen as a classification of predictors and predictands, the significant CCVs are determined. Second, the so-called causal model is evaluated between these CCVs. In this study, CCA is employed after ICA where it has been briefly described in the paper. Now we can determine the structure of the scalar function H. It should be nonlinear in its structure to satisfy the constraint of nonlinearity. So, we transform Z(t) element by element to generate the basis variables of the system in Equation (2) with one of a nonlinear H function (Equation (21)); thereafter, the degrees of problem may be reduced. Two such possible nonlinear functions from neural network applications are (Connor et al., 1994; Haykin, 1999, 21): } H 1 (y) = tan h(αy) H 2 (y) = 1 (21) 1 + exp( αy) where α is a scalar constant. The proposed method algorithm stages are summarized in Table I, and a flow chart diagram that describes the model components is also shown in Figure 1. The proposed approach is, indeed, an originating form of a recurrent neural network (RNN) model (Jordan, 1986; Elman, 199; Robinson and Fallside, 1991; Puskorius and Feldkamp, 1994; Connor et al., 1994; Pearlmutter, 1995; Haykin, 1999, 21; Hochreiter et al., 21). In the conventional RNN models, the training process is based on minimizing the variance of the error between estimated and observed values. Hence, the structure of an RNN is sensitive to the dynamics of the process. The RNN model of Jordan (1986) can be represented as } Z (t + t) = H [M 1 Y (t) + M 2 X (t + t)] (22) Y (t + t) = M 3 Z (t + t) where H is a nonlinear scalar function defined in Equation (21), Z illustrates the states of the system (outputs of hidden neurons), and M 1, M 2 and M 3 are linear operators. In the proposed RNN, the initial weights (as given by M 1 and M 2 in Equation (22)) between inputs and hidden neurons are selected as A given in Equation (16) and as B given in Equation (17). If the initial weights are selected randomly, then the conventional RNN may Table I. The proposed method algorithm 1. Perform a statistical large-scale analysis by ICA based on PCA, and employ CCA between ICs and monthly total precipitation series for reducing the proper predictors and predictands (see text) 2. Divide all the data sets into two parts. Select one part for model identification and leave the second part for validation 3. Normalize all data sets in use to make their means zero and variances one 4. Construct a naïve and a causal model separately 5. Transform the outputs of the linear static (causal) and dynamic (naïve) models by one of the functions given in Equation (21) 6. Construct a multivariate linear regression model between the responses of the fifth step and the predictand variables 7. Apply a training through-time algorithm (e.g. Puskorius and Feldkamp, 1994; Pearlmutter, 1995; Haykin, 21; Hochreiter et al., 21) a case in the RNN models via part (causal) by part (naïve) 8. Test the model to see whether it is appropriate or not (validation) Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

9 STATISTICAL DOWNSCALING OF TURKISH RAINFALL 169 Large-scale X (t) Y (t 1) Local-scale CCA PCA PCs ICA ICs Causal Model CCVs (Eq.16) Naive Model (Eq.17) Y2 (t) Feed Back Basis Functions Z (t) Linear Model Y1 (t) Y (t) Validation X (t) Statistical Preprocessing CCVs Y (t 1) RNN Y (t) Figure 1. The flow chart of the components in the proposed downscaling approach capture only the dynamical features of the local-scale process. To solve this problem, the training process is employed part by part. The elements of A (naïve) are kept constant while updating the elements of B (causal); similarly, the elements of A are updated while keeping constant the elements of recently updated B. Consequently, the proposed RNN can satisfy not only the nonlinearity, such as in the case of a conventional RNN; it also satisfies and distinguishes the naïve and the causal relationships (linear or nonlinear). In this paper, the well-known training process algorithms of RNN are not described, but one may, find the effective algorithms in Puskorius and Feldkamp (1994), Pearlmutter (1995), Haykin (21) and Hochreiter et al. (21), for example. 4. RESULTS The basic data consists of monthly total precipitation of 31 stations over Turkey (Table II) with a record length of 38 years, during the period The weather stations selected have no missing data in the prescribed period, and represent the characteristics of precipitation of Turkey. In order to perform a successful statistical large-scale analysis (Rummukainen, 1997), National Centers for Environmental Prediction National Center for Atmospheric (NCEP NCAR) reanalysis data sets (Kalnay et al., 1996) are used between 1 5 E and 3 6 N, since this area is large enough to represent the largescale climate or synoptic features that affect the monthly total precipitation series over Turkey. The following large-scale data sets from NCEP NCAR reanalysis data sets were used as predictors in this study: 1. The interactions of long (Rossby) and short waves are identified at the 5 hpa level. At this level, the long waves generally show a barotropic character, whereas the short waves are baroclinic. Therefore, the short waves at this level have determining effects on the development of surface cyclonic and anticyclonic systems. The cyclonic and anticyclonic vorticity at all high levels is a result of the interactions of short waves of high-level currents and surface systems. As a consequence of these arguments, 5 hpa geopotential heights are considered as predictors. Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

10 17 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ Table II. The stations of Turkey used in the application Station name ( N) ( E) International station number Zonguldak Sinop Samsun Trabzon Rize Edirne Göztepe (Istanbul) Bolu Kastamonu Sivas Erzurum Kars Iǧdir Çanakkale Bursa Ankara Van Afyon Kayseri Malatya Izmir Isparta Konya Gaziantep Şanliurfa Mardin Diyarbakir Muǧla Antalya Mersin Adana hpa geopotential heights have also been considered as predictors. Cloud and precipitation occur by vertical motions in regions with short waves. In particular, local precipitation is significantly related to the short waves at the 7 and 5 hpa levels. 3. Owing to the thermal effects of the troposphere, 5 1 hpa geopotential thicknesses are considered. 4. In order to represent surface weather systems features, SLP (a prognostic variable in GCMs) is considered. SLP is the most important causal variable of the proposed model, since Kutiel et al. (21) have recently studied the relationships of SLP patterns associated with dry or wet monthly rainfall conditions in Turkey. 5. The diagnostic predictor is the 5 hpa pressure vertical-velocity (omega), since this is considered to represent the vertical dynamics of the troposphere (Holton, 1992; Bluestein, 1993), where vertical dynamics properties are associated with the occurrence of the precipitation process. In order to represent jet streams in the troposphere, one may take the 2 and 3 hpa wind speeds (e.g. Xoplaki et al., 23); but, as seen below, while preprocessing large-scale variables, 5 hpa vertical pressure velocities can satisfactorily represent the vertical dynamics characteristics. The significant CCVs are selected such that the CCV can explain the maximum variance of the monthly total precipitation series over Turkey. The significant PCs and CCVs are shown in Tables III and IV; and Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

11 STATISTICAL DOWNSCALING OF TURKISH RAINFALL 171 Figure 2 displays the contrast between the PCs and the ICs. In fact, a true statistical downscaling may be done within the data sets with similar power spectra. Indeed, the CCVs of the large-scale processes and the CCVs of the local-scale processes show similar spectral densities. As a consequence, the downscaling process is said to be in conformance with the constraint of homogeneity of dependence. As a result, the low-frequency large-scale predictors induce instabilities in the prediction of the high-frequency local-scale predictands, and vice versa. The first three spectral densities of the CCVs of the large-scale and the local-scale data sets are computed using the Climlab2 Software Package (Tourre, 2), and the results are shown in Figure. 3. In order to analyse the relationships between the large-scale data predictors and the monthly total precipitation series, Sampson correlation (R S ) patterns are computed, using Equation (2), between the individual large-scale variables (namely 5 hpa geopotential heights, 7 hpa geopotential heights, SLPs, 5 hpa vertical pressure velocities and 5 1 hpa geopotential thicknesses) and the precipitation series for four seasons; however, in this paper, only the results for winter and summer are shown in Figures 4 8. In these figures, the significance of the R S correlations (based on α-level statistics) is shown in grey scale from light to dark to represent the decreasing order of significance. Figure 4 shows the Sampson correlation coefficients between the seasonal sea-level atmospheric pressures (wet and dry seasons being characterized by the winter and the summer respectively) and the precipitation series of Turkey. The most important correlations are observed in the region that encompasses Italy, the entire Balkans region, Turkey and the eastern Mediterranean. This explains the effect of the low-pressure anomalies on the Turkish precipitation regimes during the winter. These correlations are in agreement with the work of Kutiel et al., (21). The strongest correlations are observed in the central and southern Aegean. For the summer, weaker correlations exist in the eastern Mediterranean. As seen in Figure 5, there is an important correlation between the values of the 5 hpa geopotential heights and the winter precipitation series. The correlation pattern in the area that includes the western part Table III. The significant PC of the large-scale variables according to the criterion of Jöreskop and Sörbom (1989) Large-scale predictors Sum of explained variance (%) Number of PCs 5 hpa geopotential heights hpa geopotential heights SLP hpa thicknesses hpa vertical pressure velocities Table IV. The CCVs of the large-scale variables (seventh CCV is more significant than the sixth CCV) CCVs of the large-scale variables Explained variance (%) of the monthly total precipitation series CCV CCV CCV CCV CCV CCV Sum Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

12 172 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ 4 PCs of SLP 5 ICs of SLP Figure 2. The plots of the first three PC and ICs of SLP of Turkey, the Adriatic Sea, the Balkans and the eastern part of Italy is at an appreciable level. In winter, the geopotential low centre at 5 hpa affects Turkey from the northwest. Such conditions are observed at high levels (5 hpa) when low-pressure centres and related frontal systems are observed at the surface. Significant correlations for the summer occur in the eastern Mediterranean (namely southern Turkey and Cyprus). During the summers, Turkey is under the influence of warm air masses (systems) of tropical origin, which is visible in terms of positive anomalies for the 5 hpa geopotential heights. Spring and autumn form transitions between the above mentioned regimes. In Figure 6, the 7 hpa geopotential height data set gives a similar result to that of 5 hpa in terms of its correlations with the precipitation regimes during the winter. In this figure, the significant correlations are seen over western parts. However, no such significant correlation pattern exists for the summer (according to 95% statistical α level). The anomalies related to the geopotential thicknesses of 5 1 hpa (in a similar way to the 5 hpa geopotential heights themselves) in central Europe have a correlation with the Turkish precipitation regimes, as can be seen in Figure 7. It can be postulated that, especially during the winter, the negative thickness anomalies are related to cold advection, because the thickness itself is an indicator of thermal advection. As can be seen from the contours, a similar correlation is valid for the Caucasus and the Caspian region as well (this correlation is related to the positive thickness anomalies that indicate the lack of precipitation during winter, i.e. winter with dry conditions). As discussed for Figure 4, Turkey is under the influence of Iceland and Mediterranean lows in winter; therefore, those regions are characterized by convergent fields. That is, upward motions are effective. The pronounced nature of such cold advection at high levels (from 1 hpa to 5 hpa) has a determining effect on the occurrence of precipitation. In winter, the eastern part of Turkey (especially the eastern Black Sea part of Turkey) is an important convergent field (see Section 2). The existing cold advection over these upward motions has important effects on the formation of clouds and Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

13 STATISTICAL DOWNSCALING OF TURKISH RAINFALL (a) (b) Figure 3. The spectral densities of the first three CCVs. The horizontal and vertical axes represent frequency (month 1 ) and spectral density respectively: (a) the large-scale variables (on left), (b) the local-scale variables (on right) 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (a) 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (b) Figure 4. The R S pattern between the SLP and the monthly total precipitation series: (a) winter months; (b) summer months precipitation. In summer, the effect of the air system originating from the Sahara (dry and warm) is clearly seen in Figure 7(b). This period indicates the dry period for Turkey. Correlation patterns between the 5 hpa vertical pressure velocity (omega) series and the precipitation series can be observed in Figure 8. For both seasons, there is a strong correlation pattern for the area that includes the east of Greece, the entire Black Sea and the entire eastern Mediterranean. There is also a strong correlation pattern for the southwest corner of Turkey and Cyprus. During the winter, the area of omega Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

14 174 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (a) 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (b) Figure 5. The R S pattern between the 5 hpa geopotential heights and the monthly total precipitation series: (a) winter months; (b) summer months 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (a) 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (b) Figure 6. The R S pattern between the 7 hpa geopotential heights and the monthly total precipitation series: (a) winter months; (b) summer months anomalies is on the trajectory of Mediterranean low-pressure systems (Alpert et al., 199). In summer, the strong correlation patterns have a more patchy nature, having maxima in the Balkans, northern Africa (Sahara), the central Mediterranean, the Arabian Peninsula and the Persian Gulf. These centres have potential to be related to droughts, as they are mainly characterized by the Azores high connected to sinking motions and the northwestern part of the monsoon low (the Arabic low), which has a dry characteristic during summers. According to Figures 4 8, the summer R S patterns are especially very different from each other for each large-scale variable. This is expected, because the polar and tropical air systems have dominant influences on mid-latitude precipitation during winter, whereas these patterns are not so active in summer. Hence, the summer patterns in Figures 4 8 represent and describe the influences of the local physical and geographic conditions (e.g. the correlations near surface levels are relatively weaker than the high levels during summer months) Downscaling results In order to show the proposed model spatial performance, the pseudo PC patterns of both the observed and the downscaled precipitation series are shown in Figures 9 11 (as the amount of data is insufficient to extract the true PC patterns; these are termed pseudo patterns). Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

15 STATISTICAL DOWNSCALING OF TURKISH RAINFALL N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (a) 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (b) Figure 7. The R S pattern between the 5 1 hpa thicknesses and the monthly total precipitation series: (a) winter months; (b) summer months 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (a) 6.N 57.5N 55.N 52.5N 5.N 47.5N 45.N 42.5N 4.N 37.5N 35.N 32.5N 3.N 1.E 15.E 2.E 25.E 3.E 35.E 4.E 45.E 5.E (b) Figure 8. The R S pattern between the 5 hpa vertical pressure velocities and the monthly total precipitation series: (a) winter months; (b) summer months The scatter plots of the downscaled versus the observed monthly total precipitation time series for Göztepe (İstanbul), Ankara, Diyarbakır, İzmir, Rize, Adana and Erzurum are selected from rainfall regions according to Türkeş (1996, 1998) and are shown in Figure 12. In order to distinguish the performance for validation, the correlation coefficients (r 1, r 2 ) between the observed and the downscaled data sets are computed. In this figure, r 1 and r 2 represent the correlation coefficients for the training part of the data (the data part for identifying the model) and the non-training part of the data (validation part) respectively. The performance of the proposed model increases from the continental regions of Turkey to the coastal regions of Turkey except in the eastern part of the Black Sea (Rize) and in southern Anatolia (Diyarbakır) region. This result is expected, because the precipitation conditions over the western and the southern coastal regions (including mainly the Marmara Transition and the Mediterranean rainfall regions of Turkey) are dominantly controlled by large-scale systems, whereas in the continental parts the added precipitation is due to local conditions. In the Black Sea region, the added precipitation is due to the topographical and rain-shadow characteristics that the large-scale processes may not capture. The other unexpected result is for Diyarbakır. The performance Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24)

16 176 H.TATLI,H.NÜZHET DALFES AND Ş. SIBEL MENTEŞ var=39.6 % var= 49.6 % (a) Figure 9. The first PC pattern of the monthly total precipitation series (the map indicates the correlation coefficients between the monthly total precipitation time series and the first PC): (a) observed; (b) downscaled (b) var=12.9 % (a) var=16.5% (b) Figure 1. The second PC pattern of the monthly total precipitation series: (a) observed; (b) downscaled var = 8.7 % var = 11% (a) Figure 11. The third PC pattern of the monthly total precipitation series: (a) observed; (b) downscaled of downscaling at this station is superior to the other non-coastal stations. Diyarbakır s precipitation is affected by both Mediterranean frontogenetical systems and the large-scale drought climate conditions (especially in summers). Therefore, Diyarbakır is not a typical continental station; this result is in agreement with the work of Türkeş (1996, 1998), where he named this region the Transition Mediterranean). Copyright 24 Royal Meteorological Society Int. J. Climatol. 24: (24) (b)