Irene Polo
email: ipolo@fis.ucm.es

Albin Ullmann

Pascal Roucou

Bernard Fontaine

Weather Regimes in the Euro-Atlantic and Mediterranean sector and relationship with West African rainfall over the period 1989-2008 from a Self-Organizing Maps Approach

Weather Regimes (WR) have been defined over the Euro-Mediterranean region [60°W-60°E; 15°N-70°N] from May to October season using the daily Sea Level Pressure, 700 hPa geopotential height and specific humidity from ERA-interim over 1989ERA-interim over -2008 period. period. Computations are based on a neural network classification technique referred to as Self Organizing Maps and the WR produced can be use d by the community for comparison with other periods, projection onto model outputs, seasonal prediction or teleconnection studies.

The article particularly examines the relationship between WR and West African (WA) rainfall and our results suggest that changes in particular WR frequencies can account for part of WA interannual rainfall variability. Thus during anomalous wet (dry) years in WA, both higher frequencies of occurrence of WR re la ted to negative (positive) summer NAO-like pattern and less frequent WR re la te d to positive (negative) summer NAO-like pattern are attested in July and August (hereafter SN-and SN+). This is associated with a zonal symmetric pattern, consistent along the middle troposphere, i.e. low pressure anomaly centered over 50°N-20°W and Eurasia (Greenland) and high pressure anomaly centered over Iceland (central Europe) for SN-(SN+). Another striking characteristic of SN-(SN+) is southeastward (southwestward) surface anomalous winds flowing from (to) the Atlantic Ocean at 20°N and therefore able to enhance (weaken) wet convection. That sea surface temperature associated with SN-shows a warming of the Mediterranean in July and the opposite with SN+ in August, suggestin g that te mperature anomalies could be a precursor in the change of frequency of SN-and SN+.

Introduction

The North Atlantic/ Mediterranean region exhibits a great amount of atmospheric variability acting at different spa tio-temporal scales with large nonlinearities in the system [START_REF] Lamb | North Atlantic Oscillation: Concept and application[END_REF][START_REF] Reusch | North Atlantic climate variability from a self-organizing map perspective[END_REF]Cassou et al., 2004a). For example, a sizeable part of the Se a Level Pressure (SLP) variability is closely associated with that of large-scale teleconnection patterns, usually discriminated by tools based on a linear approach [START_REF] Wallace | Teleconnections in the géopotentiel fields during the Northern Hemisphere winter[END_REF]. In particular, the North Atlantic Oscillation (NAO, [START_REF] Hurrell | An overview of the North Atlantic Oscillation[END_REF][START_REF] Folland | The Summer North Altantic Oscillation: Past, Present and Future[END_REF] which represents the largest percentage of SLP variability exhibits marked differences in opposite phases (Cassou et al., 2004a). Besides tropical/extratropical interactions do exist: the tropical convection can be responsible for North Atlantic variability, favoring occurrence and/or persistence of particular structures [START_REF] Cassou | Intraseasonal interaction between the Madden-Julian Oscillation and the North Atlantic Oscillation[END_REF]SanchezGomez et al., 2008), while the extratropical variability can impact on the more equatorial regions through Hadley-type circulation anomalies.

Sa helian rainfall variability, from intraseasonal to mutlidecadal timescale, is largely affected by variations of global and regional Sea Surface Temperature (SST) patterns [START_REF] Folland | Sahel rainfall and worldwide sea temperatures[END_REF][START_REF] Giannini | Oceanic Forcing of Sahel Rainfall on Interannual to Interdecadal Time Scales[END_REF][START_REF] Rowell | The Impact of Mediterranean SSTs on the Sahelian Rainfall Season[END_REF]. These SST variations include anomalies in the tropical Atlantic (Lamb, 1978, Vizy and[START_REF] Vizy | Mechanisms by which Gulf of Guinea and eastern North-Atlantic sea surface temperature anomalies can influ ence African rainfall[END_REF], the central and east Pacific [START_REF] Janicot | Sahel drought and ENSO dynamics[END_REF]Rowell, 2001), the Indian Ocean (Shinoda and Kawamura, 1994) and the Mediterranean Sea [START_REF] Rowell | Variability of summer rainfall over tropical North Africa (1906-92): observations and modelling[END_REF][START_REF] Rowell | The Impact of Mediterranean SSTs on the Sahelian Rainfall Season[END_REF][START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF][START_REF] Gaetani | Influence of the Mediterranean Sea on the West African Monsoon: intraseasonal variability in numerical simulations, accepted on 21[END_REF]. Moreover, the stronger thermal contrast between the Sahara and surrounding favours a penetration of the mean circulation towards the African Sahel from the eastern tropicalequatorial Atlantic and the Mediterranean [START_REF] Fontaine | Atmospheric water cycle and moisture fluxes in the West African monsoon: mean annual cycles and relationship using NCEP/NCAR reanalyses[END_REF]. According to Raic ic h et al. (2003), [START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF] and [START_REF] Gaetani | Influence of the Mediterranean Sea on the West African Monsoon: intraseasonal variability in numerical simulations, accepted on 21[END_REF], warmer conditions over the Mediterranean (in JAS period) favour (i) a moderate weakening of the northeasterly flow east of 10°E and (ii) an enhancement of the northerly moisture flow from the eastern Mediterranean which reaches the eastern and central Sahel and increase the specific humidity.

Changes on rainfall over the Sahel region have been observed, like the contrast between the wet 50's and 60's and the dry 70's and 80's, or more recently the current ra in fall recovery.

The period from the late 70s exhibits also a change in the northern hemisphere teleconnection patterns [START_REF] Johnson | The continuum, of northern Hemisphere Teleconnection Patterns and a Description of the NAO Shift with the Use of Self-Organizing Maps[END_REF], which could be due to a warming trend in the tropical Sea Surface Temperatures (SST) [START_REF] Hurrell | Twentieth Century North Atlantic Climate Change. Part I: Assessing Determinism[END_REF]. [START_REF] Werner | Recent Climate Change in the North Atlantic/European sector[END_REF] identified the decade 1981-1990 as the onset of climate change in the North Atlantic European sector with a trend starting in the beginning of the 70s. Several authors interpreted these changes in the variability patterns as a change in the intensity and frequency of Weather Regimes [START_REF] Werner | Recent Climate Change in the North Atlantic/European sector[END_REF][START_REF] Johnson | The continuum, of northern Hemisphere Teleconnection Patterns and a Description of the NAO Shift with the Use of Self-Organizing Maps[END_REF], which, in turn, can be driven by changes in the Tropical SSTconvection (Cassou et al., 2004b;Hurrell, 2004;[START_REF] Cassou | Intraseasonal interaction between the Madden-Julian Oscillation and the North Atlantic Oscillation[END_REF]SancheGomez et al., 2008).

A Weather Regime (WR) can be defined as a recurrent and persistent atmospheric state [START_REF] Vautard | Multiple weather regimes over the North Atlantic: Analysis of precursors and successors[END_REF][START_REF] Michelangeli | Weather regimes: Recurrence and quasi stationarity[END_REF]. The concept assumes that the atmosphere stays in discrete preferential states which are peaks in the probability density function of the phase space and are usually identified by cluster analysis [START_REF] Vautard | Multiple weather regimes over the North Atlantic: Analysis of precursors and successors[END_REF][START_REF] Michelangeli | Weather regimes: Recurrence and quasi stationarity[END_REF]Plaut and Simonnet, 2001;[START_REF] Ghil | Waves » vs « particles » in the atmosphere's phase space : A pathway to a long-range forecasting ?[END_REF]Cassou et al., 2004a;[START_REF] Cassou | Intraseasonal interaction between the Madden-Julian Oscillation and the North Atlantic Oscillation[END_REF][START_REF] Moron | The impact of El Niño southern oscillation upon weather regimes over Europe and the North Atla ntic during boreal winter[END_REF]SanchezGomez et al., 2008). Moreover, the centroids of the nonlinear clustering methods can be seen as the corresponding "attractors" to the WR [START_REF] Lorenz | Deterministic nonperiodic flow[END_REF]. In this way, low frequency variability can be interpreted as a change in the amplitude of these WR or in the preferred transitions between them [START_REF] Robertson | The influence of Atlantic sea surface temperature anomalies on the North Atlantic Oscillation[END_REF][START_REF] Ghil | Waves » vs « particles » in the atmosphere's phase space : A pathway to a long-range forecasting ?[END_REF]Cassou et al., 2004a).

Can such changes over the northern Atlantic/Mediterranean Sector be important in terms of climate impacts and in particular for the subtropical/tropical convection variability over West Africa (WA)? This question requires to define WRs from daily atmospheric data from May to October season and to project them onto rainfall information. The European Centre for Medium-Range Weather Forecast (ECMWF) reanalysis ERA-Interim has been chosen as the atmospheric information because it provides daily meteorological data at 1.5° horizontal resolution on the period 1989-2008 and represents an improvement on certain key aspects of ERA40 such as the representation of the hydrological cycle (see [START_REF] Berrisford | The ERA-Interim archive[END_REF] for more details). Rainfall estimation is provided by the Global Precipitation Climatology Project (GPCP) for which 6,000 rain gauge stations, satellite geostationary, low-orbit infrared, passive microwave and sounding observations have been merged to estimate 5 days averaged rainfall on a 2.5-degree global grid from 1979 to the present [START_REF] Gruber | Assessment of Global Precipitation Products[END_REF].

The aim of the present manuscript is twofold: (i) to present a collection from May to October WRs over a northern hemispheric region centered on the Mediterranean Sea (MS) from the illustrating and consistent Self Organizing Maps method and, (ii) to evaluate if the changes in frequency of these WRs are related to WA rainfa ll variability.

The Method

Se lf organizing maps (SOM, [START_REF] Kohonen | Se lf -Organizing Maps[END_REF] are based on artificial neural networks which allow the representation of high dimensional data space onto a reduced and ordered space.

Therefore, it provides an alternative to other nonlinear cluster analysis techniques for defining WRs. Since a spatially organized set of patterns of variability of the data is obtained from SOM, it has been used in different fields with notable success. It has been recently used for oceanographic studies [START_REF] Leloup | Detecting decadal changes in ENSO using neural networks[END_REF][START_REF] Tozuka | Tropical Indian Ocean variability revealed by self-organizing maps[END_REF], for climate characterization over the northern hemisphere [START_REF] Hewitson | Self-organizing maps: applications to synoptic climatology[END_REF]Reusch et al., 2007, Johnson and[START_REF] Johnson | The Continuum of North Pacific Sea Level Pressure Patterns: Intraseasonal, Interannual and Interdecadal Variability[END_REF] and also for rainfall prediction in the monsoon systems [START_REF] Cavazos | Intraseasonal Variability Associated with Wet Monsoons in Southeast Arizona[END_REF][START_REF] Chattopadhyay | Objetive Identification of Nionlinear Convectively Coupled Phases of Monsoon Intraseasonal Oscillation: Implications for Prediction[END_REF] as well as for the study of the relationship between synoptic fields and heavy rainfall extreme events [START_REF] Cavazos | Large-scale circulation anomalies conducive to extreme events and simulation of daily rainfall in northeastern Mexico and southeastern Texas[END_REF][START_REF] Nishiyama | Identification of typical synoptic patterns causing heavy rainfall in the rainy season in Japan by a Self Organizing Map[END_REF]. Several advantages of this methodology have been described, starting for being a powerful visualization approach [START_REF] Reusch | North Atlantic climate variability from a self-organizing map perspective[END_REF][START_REF] Leloup | Detecting decadal changes in ENSO using neural networks[END_REF]. The patterns captured from the SOM present continuity in the data space due to the conservation of the topology of the structure in contrast with other clustering techniques, as agglomerative hierarchical algorithm [START_REF] Mangiameli | A comparison of SOM neural network and hierarchical clustering methods[END_REF] or partitioning algorithm of k-means (Cassou et al., 2004a;[START_REF] Fereday | Cluster Analysis of North Atlantic-European Circulation Types and Links with Tropical Pacific Sea Surface Temperatures[END_REF] or partitioning algorithm of dynamic cluster method (Michelangelie et al., 1995). This continuity in the SOM structures provides an efficient way to easily identify the largescale atmospheric slow motions. These smoothly transitioning climate states of SOM array are in deed obtained from nonlinear classification of the continuum of atmospheric conditions. This continuum perspective is feasible with SOM and more realistic than commonly used discrete approach [START_REF] Reusch | North Atlantic climate variability from a self-organizing map perspective[END_REF][START_REF] Johnson | The continuum, of northern Hemisphere Teleconnection Patterns and a Description of the NAO Shift with the Use of Self-Organizing Maps[END_REF]Jonhson and Feldstein, 2009). In fact, Jonhson et al. (2008) and Jonhson and Feldstein (2009) showed that the Northern Hemisphere intraseasonal to interdecadal atmospheric variability can be understood in terms of changes in the frequency distribution of the cluster patterns that comprise this continuum.

Although works using SOM and other clustering methods as k-means for describing the winter North Atlantic variability have shown similar results (Cassou et al., 2004a;[START_REF] Reusch | North Atlantic climate variability from a self-organizing map perspective[END_REF][START_REF] Fereday | Cluster Analysis of North Atlantic-European Circulation Types and Links with Tropical Pacific Sea Surface Temperatures[END_REF], the SOM method reduces subjectivity which make it more attractive from usability point of view [START_REF] Hewitson | Self-organizing maps: applications to synoptic climatology[END_REF][START_REF] Reusch | North Atlantic climate variability from a self-organizing map perspective[END_REF]. Since SOM assumes that the data are continuous, the method allows the representative patterns to span the phase space even when data are not highly clustered [START_REF] Johnson | The continuum, of northern Hemisphere Teleconnection Patterns and a Description of the NAO Shift with the Use of Self-Organizing Maps[END_REF]. [START_REF] Mangiameli | A comparison of SOM neural network and hierarchical clustering methods[END_REF] showed that the SOM is superior to hierarchical clustering methods in accuracy and robustness when applied in a synthetic set of data with different levels of imperfections. Therefore, SOM method strikingly distinguishes itself from other clustering methods through its topological ordering of the representative patterns.

For this study, an area centered over the MS [60°W-60°E; 15°N-70°N] has been chosen after checking that reducing/enlarging the size of this window does not modify the main WR features. 5x5 SOM-array of coupled SLP, geo-potential height and specific humidity at 700 hP a pressure level are used to identify 25 seasonal WRs occurring from May to October.

Using three atmospheric variables gives us an accurate description of the WRs, although results do not substantially change when using the first two variables only (not shown).

Before combining the three variables, each grid-point long-term anomalies have been normalized by their standard deviations to ensure that the three fields account for the same weight in the analysis. In addition, all data have been weighted by the cosine of latitude to take into account the dependence of the grid-point density on latitude. After verifying that results are not sensitive to the topology, we have chosen a hexagonal lattice structure of the array and a Gaussian neighborhood function. We have also tested the result's sensitivity to the size of the array. Ultimately, the choice of 5x5 SOM-array represents a good compromise between economy and the detail's resolution in the representative patterns, since the quantization error and border effects are reduced as the chosen number of SOM increases [START_REF] Johnson | The continuum, of northern Hemisphere Teleconnection Patterns and a Description of the NAO Shift with the Use of Self-Organizing Maps[END_REF].

The algorithm presents the input data to a layer of neurons, which describe the positions on the two-dimensional grid that contains the representative patterns. The neurons have associated a weight or reference vector with the same dimension than the input data. The neurons are orderly disposed in a regular lattice and a global map shape is defined to evaluate how the neurons are connected, thus describing a topological map. Then a linear initialization is performed which implies that the weight vectors are initialized in an orderly fashion along the linear subspace spanned by the two leading eigenvectors of the covariance matrix of the training input data (this initialization have given better results than the random one for our data). Then an initial neighborhood function and initial learning rate are defined: several authors have showed that the choice of the learning rate and the initial neighborhood function radius does not change the results in a range of values [START_REF] Mangiameli | A comparison of SOM neural network and hierarchical clustering methods[END_REF][START_REF] Liu | Performance evaluation of the selforganizing map for feature extraction[END_REF]. We have used the so-called batch training algorithm to train the data in two phases. In this training algorithm two main processes take place: 1) The Best Matching Unit (BMU) is sought in each training step using the minimum Euclidean distance criterion between the neurons and the input data, 2) The weight vectors of the BMU and of its defined neighborhood are updated. The neighborhood function and the learning parameter are also updated and decrease at each time step. The first set of training iterations uses relatively large initial learning ra te and neighborhood radius: this produces a first broad distribution of nodes.

In the second phase, both learning rate and neighborhood radius are small from the beginning in order to refine the mapping [START_REF] Mangiameli | A comparison of SOM neural network and hierarchical clustering methods[END_REF][START_REF] Hewitson | Self-organizing maps: applications to synoptic climatology[END_REF]. The iterative phase is repeated until convergence of the parameters. More deta ils are available in [START_REF] Vesanto | SOM Toolbox for Matlab[END_REF] and at http://www.cis.hut.fi/projects/somtoolbox/.

After determining the representative patterns, we classify each daily field to one of the SOM patterns using the criterion of minimum Euclidean distance; therefore each observation in the input data is associated with one and only one neuron via a weight vector. Later on, following the definition of WR, a criterion of persistence is imposed to keep only patterns that occur at least two consecutive days. The spatial correlation between the centroids (the 25 SOMs) and each corresponding patterns of the composite of daily data always shows a good agreement (from 0.9 spatial correlation scores). In this way, the final composites maps of theses recurrent and persistent structures are considered and hereafter referred to as WR (Figure 1).

Results

a. Pattern Analysis

Figure 1 shows the composite maps of anomalous SLP/Z700 with respect to the long-term May-October mean (Figure 2a), related to the SOM-array performed as described above. The mean months of the occurrence together with the frequency of occurrence over the season are displayed for each WR. Let us recall that four winter North Atlantic weather regimes have been described in the literature [START_REF] Vautard | Multiple weather regimes over the North Atlantic: Analysis of precursors and successors[END_REF][START_REF] Moron | The impact of El Niño southern oscillation upon weather regimes over Europe and the North Atla ntic during boreal winter[END_REF]Cassou et al., 2004a;[START_REF] Ullmann | Weather regimes and sea surge variations over the Gulf of Lions (French Mediterranean coast) during the 20 th century[END_REF]SanchezGomez et al., 2008, among Finally, (iv) the Greenland Anticyclone pattern (negative phase of the NAO) dominated by a strong positive anomaly centered over western Greenland. The NAO pattern is also found in summer season by empirical orthogonal function and cluster analyses. Summer NAO has a near-equivalent barotropic structure on daily and monthly time scales and is characterized by a more northerly location and smaller spatial scale than its winter counterpart [START_REF] Fereday | Cluster Analysis of North Atlantic-European Circulation Types and Links with Tropical Pacific Sea Surface Temperatures[END_REF][START_REF] Folland | The Summer North Altantic Oscillation: Past, Present and Future[END_REF].

The SOM-based composites of daily SLP/Z700 shown in Figure 1 is in accordance with the patterns found in previous studies described above. These results exhibit mainly the seasonal cycle leading the highest variability of the variables (re la tive to the long-term mean in Figure 2a). Although the patterns are phase-locked with the seasonal cycle, it is also confirmed that the WR are the time-averaged signature of the quasi-daily atmospheric pattern variations (Cassou et al., 2004a) Frequency analysis illustrates how the seasonal variability is well captured (Figure 2c). The beginning of the season (from May to June) is characterized by high frequencies of WR 21 to 25 (bottom row in Figure 1). The sequence of WR 16 to 20 (the 4th row) is also associated with the spring season, although less frequent than WR 21 to 25 (Figure 2c). The seasonal peak (July-August) is defined mainly by WRs 20, 15, 10 and 5 (5 th column in Figure 1). In Se ptember-October, the decreasing in the WR number reports a sequence from summer to the beginning of autumn: from 5 to 1, 1 st row in Figure 1, but also albeit less frequent from 9 to 6 and from 14 to 11, 2 nd and 3 rd rows respectively in Figure 1. October is characterized by WR numbers 1, 6, 11 and 16 (1 st column in Figure 1). These above-mentioned sequences mainly represent the seasonal cycle with respect to the mean climatology (Figure 2a); the reinforcement of the Azores High and the Arabian-Sa hara thermal low pressure systems from spring to summer and the enhancement of the Greenland Low pressure system from summer to autumn. However, although varying jointed in the seasonal cycle WR numbers describe also in te rannual variability, as referred later on by analyzing a particular interannual phenomenon.

Finally to complete the description of the seasonal cycle, figure 2d presents the 2 first Principal Components (PC) of an Empirical Orthogonal Function applied on the WR occurrences (WR frequency along the season, therefore the size of the matrix is 25 x 184). It is found that the summer WR (5 th column in Figure 1) are mostly associated with the PC1 (18% of variance), while the PC2 (14% of variance) describes mainly the WR occurring at the beginning (June) and at the end (September) of the season. This discrimination will be used later to examine the relationship between summer WR and WA rainfall.

Figure 3a presents the confidence level of the probability of the transition from WR i to WR j . This has been calculated following [START_REF] Vautard | Sta tistical Significance Test for Transition Matrices of Atmospheric Markov Chains[END_REF], where the transition matrix can be reliable tested with a Monte Carlo test by assuming the representation of the large-scale atmosphere variability as a Markov chain of multiple flow regimes or WRs. A simple interpretation of Figure 3a would be that, along the 5x5 SOM-array in Figure 1, the flow diagram more likely of transitions occurs along the columns. For instance, WRi given i=10, the most likely transition occurs towards WR 5, 15 or 20, which corresponds to the same column in figure 1 or WR family. Also some groups of patterns belo nging to the same row, such as 21-22-23; 3-4-5 and 6-7-8 are linked in the transitions. The largest probability is found in the persistence of each WR in remainin g its own state (dia gonal of the Transition matrix). This could be also measured as the mean persistence in days of each event and each WR. An event is defined as a sequence of days with the same characteristics lasting at least 2 days as in [START_REF] Vautard | Sta tistical Significance Test for Transition Matrices of Atmospheric Markov Chains[END_REF]. Figure 3b displays the mean persistence of each WR together with the number of events counted for each WR. The WRs with the highest persistence are 1, 5, 16, 21, 24 and 25; the number of events is high for WRs 1,5,10,11,15,20,21 and 22. Changes in the persistence of a particular WR could account for the probability of a phenomenon to occur such as more precipitation in a given area. This could be also interpreted as the consequence of an external forcing such as SST forcing (SanchezGomez et al., 2008). In the next section we investigate changes in the WR frequency associated with anomalous rainfall.

b. Relationship with West African rainfall

Are such changes in particular WR associated with the WA rainfall variability? In order to answer, we first classify years of positive and negative rainfall over the Sahel region. To this end, we perform a PC analysis for the anomalous rainfall along the year and the entire season averaged over the box 10°W-40°W; 5°N-20°N. From the leading PC index, positive and negative years of WA rainfall are classified and the differences of the mean frequency for each WR computed between these pools of years. Figure 4a shows the significant differences (at p = 0.10) of the mean frequency for each WR regarding the wet and dry years. The most striking result is the more frequent occurrence at the beginning of the season of WR 17 and WR 21 in wet and dry years respectively and of WR 20 and WR 5 in July-August. The se differences in mean frequency could be explained by changes in persistence of each WR and/or in its number of events. Thus in figure 4b the number of events and persistence of WR 5 increases when years of negative anomalous rainfall occur over WA while WR 20 shows such an in crease for years of positive WA rainfall anomalies. In contrast WR 21 presents a larger number of events but le ss persistence for dry years while WR 17 shows more number of events in wet years but no significant difference in persistence. The significant differences detected in WR 5 and WR 20 occurs at the peak of the WA rainy season and exhibit similar frequency. This is the most intriguing feature for our purpose, since regarding wet and dry years, these WRs have opposite behavior in terms of persistence and number of events. Thus, in July-August, rainfall, SLP, Z700 and surface wind composites regarding WR 5 and WR 20 are first computed. Then the climatology is subtracted for considering only the WRs that do not match with the corresponding WR 5 and 20 for each map (Figure 5): this allows us to enhance the differences and to keep nonlinearities.

Figure 5 presents the differences between WR 5 and WR 20 occurrences. Figure 5a shows that WR 20 (WR 5) is related to more (less) Sahelia n rainfall in July-August, which confirms our previous analyses. A zonally symmetric SLP/z700 pattern is also attested for both WRs (Figure 5b): WR 5 exhibits a summer positive NAO-like structure (hereafter WR 5 will be referred to as SN+) with anomalous high pressures over Europe and anomalous low pressures over Greenland; WR 20 displays a summer negative NAO-like pattern (hereafter WR 20 will be referred to as SN-) with positive anomalies over Island and negative ones over North Atlantic and the continent. The eastward displacement of the southern branch of the positive NAO-like pattern (centered over 20W) with respect to the negative phase (centered over 10E) is consistent with the winter and summer NAO phases (Cassou et al., 2004a;[START_REF] Folland | The Summer North Altantic Oscillation: Past, Present and Future[END_REF]. Surface winds associated with SN-(SN+) display anomalous northwesterlies (northeasterlies) related to lower (higher) pressures, and a subsequent inland (offshore) flow around 20°N. This flow tends to enhance (to weaken) deep wet convection over 10W-0E while in upper levels one notices an intensification (weakening) of the Tropical Easterly Jet and of the barotropic dipole structures in the extratropics (not shown). Additionally, the whole Mediterranean basin suffers a negative (positive) anomalous pressure. SN-and SN+ WR behaviors in the month of maximum occurrence (i.e. July for SN-and August for SN+) are illustrated in figure 5c. Basic SST patterns are discernable: an Atlantic Tripole-like pattern over the North Atlantic and warm SSTs over the MS for SN-; a cooler Subtropical North Atlantic and eastern MS for SN+. Such SST features in Eastern MS have been already discussed by other authors [START_REF] Xoplaki | Mediterranean summer air temperature variability and its connection to the large-scale atmospheric circulation and SSTs[END_REF][START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF] and related to in terannual WA rainfall variability [START_REF] Polo | Part I: Time-Evolving SST Modes Related to West African Rainfall[END_REF][START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF]. Our results demonstrate that differences in SLP/Z700 fields associated with positive and negative anomalous rainfall over WA can be attributed to specific changes in the frequency distribution within the continuum of patterns. Moreover lead/lag analyses between SST fields and WR occurrences (not shown) reveal that MS SSTs could act as a precursor of these WRs and in turn explain part of their relationship with WA rainfall.

Summary and Discussion

This study provides evidence that Self Organizing Maps are a useful tool that provides a reliable and quasi-objective representation of Weather Regimes. This information can then be used in numerous diagnostic studies. The se lected application concerns only their relationship with May-October West African rainfall. The results suggest that interannual rainfall anomalies over the Sahel are associated with changes in the frequency of occurrence of typical WRs. Thus, during abnormally wet (dry) years in WA, both higher frequencies of WR number 20 (5) and less frequent WR 5 (20) have been attested in July-August. This is associated with a zonal symmetric pattern, consistent along the middle troposphere, i.e., a low pressure anomaly centered over 50°N-20°W and Eurasia (Greenland) and a high pressure This study shows that anomalous positive rainfall in the Sahelian region in JAS is associated with the SN-atmospheric pattern, highlighting the importance of the large-scale extratropical atmospheric circulation in the deep wet convection around 10W -0E favouring Sahelian rainfall in JAS. A point of interest is that SN-is also associated with an anomalous warm Mediterranean that could feed the lower troposphere with additional moisture transported southwards northerlies and available for Sahelian rainfall [START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF][START_REF] Gaetani | Influence of the Mediterranean Sea on the West African Monsoon: intraseasonal variability in numerical simulations, accepted on 21[END_REF]. Moreover, lead/lag correlations have shown that an anomalous warm Mediterranean could be a precursor of higher frequency of daily WR 20 during July-September period. In order to confirm this hypothesis, we have performed preliminary analysis with Atmosphere General Circulation Models (AGCM) outputs forced by anomalous positive and negative SST anomalies over the MS and observed climatology elsewhere [START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF]. Although this requires further analysis and is out of scope of this paper, the projection of the WR presented here onto the daily model outputs has shown that SN+ WR has higher (lower) frequency when considering the negative (positive) SST anomaly simulations. To summarize, this paper highlights the Mediterranean as a "key region" for Sahelian rainfall as a source of moisture as well as a possible precursor of the large-scale atmospheric circulation associated with anomalous positive rainfall anomalies in this region.

This study also implies further work in relation to WR. First of all, the utilities of the presented collection of WR include hence the projection onto model outputs. This work extension is being carried out by the authors through the AMMA project sensitivity experiment. Moreover, projection of WR on model outputs could also include AR4 and AR5 simulations in order to analyze changes in WR frequency. Then, still in the AMMA framework, an interesting work could be to analyze changes in frequency of WR (i.e. SN-and SN+) around the onset period and the by end of the rainy season in order to better understand how large-scale atmospheric circulation and Mediterranean SST impact intraseasonal variability. Indeed, the presented WR can be used for a lot of other studies (i.e. heat wave in the Mediterranean). That's why we plan to make the collection of the daily WR available in the AMMA and HYMEX project databases. May-June-July-August-September-Oc to ber (MJJASO) Weather Regimes (composite anomalies with respect to the long-term seasonal mean represented in Figure 2a) of Sea Level Pressure (shaded areas CI=2 hPa) and geopotential height at 700hPa (contour lines CI=20 m, zero line in grey) of 5x5 SOM-array analysis from Sea Level Pressure, geopotential height at 700 hPa and specific humidity at 700hPa over the region and the period 1989-2008. The percentage at the top-right of each map gives the global population of a given WR. In parenthesis is given the months more likely of occurrence for each WR. b) Same as a) but for SLP (shaded areas, in hPa), Z700 (contour lines, CI=20 m, zero line in grey) and surface winds (vectors, in m s -1 ). c) Same as a) but for the SST (in K) and the subtraction of the climatology is done for July and August for SNand SN+ respectively.

List of Figures

  others): (i) The Blocking regime with a strong anomalous high over Scandinavia; (ii) The Zonal regime (positive phase of the NAO) characterized by an enhanced zonal flow crossing the North Atlantic area due to a re inforced Island Low and Azores High; (iii) The Atlantic Ridge re gime exhibiting a positive anomaly over the North Atlantic basin and low pressure over Northern Europe.

  . The negative NAO-like pattern (bottom-left WR 21) and the positive NAO-like pattern (top-right, WR 5) are well captured in spring and summer seasons respectively. A mid-North Atlantic pattern reflecting variability over Iceland low and the thermal Arabian low extends over Europe with opposite sign of these centers (top-left WR 1), and opposed in the SOM-array position and sign of the former WR 1, (bottom-right WR 25), frequent in early autumn and summer respectively. The remaining maps could be interpreted as intermediate patterns reflecting slow motions of the atmosphere state during transitions between the four main patterns, or simply belonging to one of these four ma in families of atmospheric circulation (see also the relative distances of the SOMs which are shown as a distortion su rface according to Sammon (1969) in Figure 2b).

  anomaly centered over Island (central Europe) for WR 20 (5) showing a SN-(SN+). Another striking characteristic of SN-(SN+) is a southeastward (southwestward) surface anomalous winds flowing from (to) the Atlantic Ocean at 20°N and therefore able to enhance (weaken) wet convection.

  Figure 1.

  May-June-July-August-September-October (MJJASO) Weather Regimes (composite anomalies with respect to the long-term seasonal mean represented in Figure2a) of Sea Level Pressure (shaded areas CI=2 hPa) and geopotential height at 700hPa (contour lines CI=20 m, zero line in grey) of 5x5 SOM-array analysis from Sea Level Pressure, geopotential height at 700 hPa and specific humidity at 700hPa over the region and the period 1989-2008. The percentage at the top-right of each map gives the global population of a given WR. In parenthesis is given the months more likely of occurrence for each WR.

Figure 2 .

 2 Figure 2. a) Long-term Mean Sea Level Pressure (shaded areas in hPa) and geopotential height at 700hPa (contour lines in m) for the MJJASO season. b) Sammon mapping of the SOMs computed and referred to Figure 1. It represents relative distance between the SOMs after the training. The reference of the four corners is given for a better visualization of the map's lattice. c) Mean frequency of each WR along the season (days year -1 ). d) Two first Principal Components of the WR occurrence along the season. The PCs describe the seasonal cycle corresponding to a pool of WR. The correlation coefficient scores between the two PCs and WR 5 (SN+) are [0.74; -0.18] and between the two PCs and WR 20 (SN-) are [0.78; -0.29]. These two PCs represent ~30% of variance.

Figure 3 a)

 3 Figure 3a) Confidence level of the probability of transition for each WR (i) with the rest WR (j). The transition matrix has been computed assuming the WRs behave as a Markov Chain and tested with a Monte Carlo test with 10000 permutations of the WR along the timeseries resting the number of events for each WR at p=0.10. b) Mean persistence (in days, blue line) and number of events (green line) of each WR.

Figure 4 a

 4 Figure 4 a) Difference of the mean frequency occurrence for all the WRs between years of anomalous positive and negative rainfall over West Africa. Statistical significance has been tested with a Monte Carlo test with 10000 realizations at p=0.10. b) Difference of persistence (in days, blue line) and number of events (green line) for all the WRs between years of anomalous positive and negative rainfall over West Africa.

Figure 5 a)

 5 Figure 5 a) Composite anomalies of ra infall (in mm day -1 ) of occurrence of WR 20 (SN-, top panel) and WR 5 (SN+, bottom panel) in July and August minus the mean of July and August without considering the WR selected.b) Same as a) but for SLP (shaded areas, in hPa), Z700 (contour lines, CI=20 m, zero line in grey) and surface winds (vectors, in m s -1 ). c) Same as a) but for the SST (in K) and the subtraction of the climatology is done for July and August for SN-and SN+ respectively.

Figure 1 .

 1 Figure1. May-June-July-August-September-Oc to ber (MJJASO) Weather Regimes (composite anomalies with respect to the long-term seasonal mean represented in Figure2a) of Sea Level Pressure (shaded areas CI=2 hPa) and geopotential height at 700hPa (contour lines CI=20 m, zero line in grey) of 5x5 SOM-array analysis from Sea Level Pressure, geopotential height at 700 hPa and specific humidity at 700hPa over the region and the period 1989-2008. The percentage at the top-right of each map gives the global population of a given WR. In parenthesis is given the months more likely of occurrence for each WR.

Figure 2 .

 2 Figure 2. a) Long-term Mean Sea Level Pressure (shaded areas in hPa) and geopotential height at 700hPa (contour lines in m) for the MJJASO season. b) Sammon mapping of the SOMs computed and referred to Figure 1. It represents relative distance between the SOMs after the training. The reference of the four corners is given for a better visualization of the map's lattice. c) Mean frequency of each WR along the season (days year -1 ). d) Two first Principal Components of the WR occurrence along the season. The PCs describe the seasonal cycle corresponding to a pool of WR. The correlation coefficient scores between the two PCs and WR 5 (SN+) are [0.74; -0.18] and between the two PCs and WR 20 (SN-) are [0.78; -0.29]. These two PCs represent ~30% of variance.

Figure 3 .

 3 Figure 3.a) Confidence level of the probability of transition for each WR (i) with the rest WR (j). The transition matrix has been computed assuming the WRs behave as a Markov Chain and tested with a Monte Carlo test with 10000 permutations of the WR along the timeseries resting the number of events for each WR at p=0.10. b) Mean persistence (in days, blue line) and number of events (green line) of each WR.

Figure 4 .

 4 Figure 4. a) Difference of the mean frequency occurrence for all the WRs between years of anomalous positive and negative rainfall over West Africa. Statistical significance has been tested with a Monte Carlo test with 10000 realizations at p=0.10. b) Difference of persistence (in days, blue line) and number of events (green line) for all the WRs between years of anomalous positive and negative rainfall over West Africa.

Figure 5 .

 5 Figure 5. a) Composite anomalies of rainfall (in mm day -1 ) of occurrence of WR 20 (SN-, top pa nel) and WR 5 (SN+, bottom panel) in July and August minus the mean of July and August without considering the WR selected.b) Same as a) but for SLP (shaded areas, in hPa), Z700 (contour lines, CI=20 m, zero line in grey) and surface winds (vectors, in m s -1 ). c) Same as a) but for the SST (in K) and the subtraction of the climatology is done for July and August for SNand SN+ respectively.

Ackno wledgments

The authors would like to thank Thierry Castel for providing the ERA-Interim data. This research was supported by the European AMMA project from the 6 th Framework Research Programm and through a CNRS contract. The study was mainly supported by the Global change and Ecosystems programme EU Integrated project: African Monsoon Multidisciplinary Analysis (AMMA) and the French component of AMMA. Based on a French initiative, AMMA was built by an international scientific group and is currently funded by a large number of agencies, especially from France, UK, US and Africa. It has been the beneficiary of a major financial contribution from the European Community's Sixth Framework Research Programme. Detailed information on scientific coordination and funding is available on the AMMA International web site http://www.amma-in te rnational.org


Kerem Ege
email: kerem.ege@polytechnique.edu

Xavier Boutillon

Vibrational and acoustical characteristics of the piano soundboard

The vibrations of the soundboard of an upright piano in playing condition are investigated. It is first shown that the linear part of the response is at least 50 dB above its nonlinear component at normal levels of vibration. Given this essentially linear response, a modal identification is performed in the mid-frequency domain Hz by means of a novel high resolution modal analysis technique (Ege, Boutillon and David, JSV, 2009). The modal density of the spruce board varies between 0.05 and 0.01 modes/Hz and the mean loss factor is found to be approximately 2%. Below 1.1 kHz, the modal density is very close to that of a homogeneous isotropic plate with clamped boundary conditions. Higher in frequency, the soundboard behaves as a set of waveguides defined by the ribs. A numerical determination of the modal shapes by a finite-element method confirms that the waves are localised between the ribs. The dispersion law in the plate above 1.1 kHz is derived from a simple waveguide model. We present how the acoustical coincidence scheme is modified in comparison with that of thin plates. The consequences in terms of radiation of the soundboard in the treble range of the instrument are also discussed.

INTRODUCTION

The purpose of this study is to describe the vibration regime of the soundboard of an upright piano in playing condition in a large frequency range Hz with only a few parameters. To this end, we have investigated the modal behaviour of the soundboard by means of a recently published high-resolution modal analysis technique [START_REF] Ege | High-resolution modal analysis[END_REF]. Compared to techniques based on the Fourier transform, it avoids the customary frequency-resolution limitation and thus, gives access to a larger frequency-range and to a better precision on damping determinations. In the first section, we study the linearity of the board. Given the essential linear response, we present in the second section the results of two modal identifications of the soundboard from which we derive the modal density and the loss factor up to 2.5-3 kHz. The frequency evolution of the modal density of the piano soundboard reveals two well-separated vibratory regimes of the structure. The low-frequency behaviour (homogeneous isotropic plate) is presented in section 3 and the mid-and high-frequency behaviour (as exhibited by a set of waveguides) in section 4.

LINEARITY

Nonlinear phenomena (such as jump phenomenon, hysteresis or internal resonance) appear when the vibration of a bi-dimensional structure reaches amplitudes in the order of magnitude of its thickness [START_REF] Touzé | Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: Theory[END_REF]. In the case of the piano, the soundboard displacement w measured at the bridge remains in a smaller range, even when played ff and in the lower side of the keyboard. Askenfelt and Jansson report maximum values of displacement at the bridge w max ≈ 6 • 10 -6 m in the frequency range Hz [START_REF] Askenfelt | On vibration sensation and finger touch in stringed instrument playing[END_REF]. This maximum value is less than 10 -3 times the board thickness. We can therefore assume that large displacements are far to be reached and the vibrations of the soundboard can be expected as linear to a high level of approximation.

The technique

In order to quantify experimentally the (non)linearity, we performed measurements on an upright piano soundboard. An exponential sine sweep technique proposed by Farina [START_REF] Farina | Simultaneous measurement of impulse response and distortion with a swept-sine technique[END_REF], mathematically proved by Rébillat et al. [START_REF] Rébillat | Identification of cascade of hammerstein models for the description of non-linearities in vibrating devices[END_REF], is used. It gives access both to the linear part of the impulse response of either system and to the nonlinear part of the response, that is the distortion level in the frequency-domain.

The technique goes as follows: (a) Let's consider first a linear system excited by x(t), a sweptsine of duration T with initial and final angular frequencies ω 1 and ω 2 :

x(t) = sin [φ (t)] with the instantaneous phase φ (t) = ω 1 t + ω 2 -ω 1 T t 2 2
. The impulse response γ imp (t) can be reconstructed by a deconvolution process: the measured signal γ meas (t) (acceleration for example) is convolved with the timereversal of the excitation signal, that is γ imp (t) = γ meas (t) * x(-t). (b) For a system with a weakly non-linear behaviour, Farina proposes to use a sine sweep for which the frequency varies exponentially with time -exponential sine sweep -in order to separate the linear and nonlinear parts of the impulse response:

x(t) = cos [φ (t)] φ (t) = ω 1 T ln (ω 2 /ω 1 ) e t T ln (ω 2 /ω 1 ) -1 -π/2
(1)

This signal verifies the fundamental property [START_REF] Rébillat | Identification of cascade of hammerstein models for the description of non-linearities in vibrating devices[END_REF]:

∀k ∈ N * , cos [kφ (t)] = cos [φ (t + ∆t k )]
where

∆t k = T ln k ln (ω 2 /ω 1 ) (2)
Multiplying the phase of a logarithmic sweep by a factor k shifts it up in time by ∆t k . Rébillat et al. shown moreover that a logarithmic sweep to the power n, x n (t) = cos n [φ (t)], can be written as a linear function of cos[kφ (t)] with k ∈ [1, n] [START_REF] Rébillat | Identification of cascade of hammerstein models for the description of non-linearities in vibrating devices[END_REF]). This property is at the basis of the method for testing nonlinearity. The convolution product of the output signal y(t) with the inverted excitation signal x(-t) yields the linear impulse response preceded in time by the non-linear impulse responses of successive orders. If the excitation time T is long enough, the responses do not overlap and can be separated in time by simple windowing. The experimental problem consists in separating the sources of nonlinearity.

Results

An upright piano of no particular merit has been put in a pseudoanechoic room (anechoic walls and ceiling, ordinary ground). The piano was tuned normally but its strings were muted by strips of foam inserted between them or by woven in two or three places. Two configurations -{loudspeaker, room} and {loudspeaker, piano, room} -have been analysed with the following procedure. The electrical excitation of the loudspeaker was a logarithmic swept-sine [50-4000] Hz with a 40 kHz sampling frequency and a T = 26 s duration. The amplitude of the loudspeaker was set at the beginning of the study in order to obtain displacements of the soundboard corresponding to the ff playing: ≈ 10 -6 m at ≈ 370 Hz in this case.

In the first configuration {loudspeaker, room}, the response of the room is measured with a microphone (prepolarised pressurefield 1/2" -Brüel & Kjaer 4947) placed in front of the loudspeaker (Bose -802 Series II). The spectrogram of the complete pressure response of the room (without the piano) is shown in Fig. 1. Some distortion is clearly visible which may safely be attributed to the loudspeaker rather than to the microphone. The spectra of the linear and nonlinear impulse responses separated with the method exposed above are shown in Fig. 2. By convention, the non-linear response of order n as displayed at frequency f in these spectra, is the response to a sinusoid at f , measured at frequency n f in a Fourier transform of the response. In other words, what is common to points belonging to different curves with the same abscissa is the frequency of the excitation signal. Except below 500 Hz where the distortion of the loudspeaker is large, the nonlinear response level is about 50-60 dB lower than the linear contribution. Linear impulse response (order 1) Nonlinear impulse response (order 2) Nonlinear impulse response (order 3) Nonlinear impulse response (order 4) without the piano) are shown in Fig. 3. The distortion level appears to be approximately the same in both configurations. This shows that the soundboard intrinsic nonlinearity (distrosion rate) is of the order of -60 dB. Linear impulse response -order 1 (without piano) Linear impulse response -order 1 (with piano) Nonlinear impulse response -order 2 (without piano) Nonlinear impulse response -order 2 (with piano) To conclude this preliminary study, it appears that a linear model is sufficient to predict the vibratory behaviour of a piano soundboard in playing situations, within the precision of customary measurements. Given this essentially linear character of the response, modal identifications of the soundboard have been performed. When a loudspeaker was used to excite the piano, the excitation level was comparable to the one used in this linearity study and the linear contribution was extracted. The ordinary impulse excitation does not permit to separate the linear and nonlinear contributions. Nevertheless, in the light of the results presented above, and considering the small amplitudes of displacement caused by the impacts on the tables (typically less than 8 • 10 -6 m, mostly due to a very low-frequency displacement and still less than 1/100 of the board thickness) we consider the linear approximation is also verified.

MODAL IDENTIFICATION BY A HIGH-RESOLUTION METHOD The method

The modal behaviour of the upright soundboard is investigated by means of a recently published high-resolution modal analysis technique [START_REF] Ege | High-resolution modal analysis[END_REF]) which avoids the frequencyresolution limitations of the Fourier transform. This new technique is particularly well suited for structures made of moderately damped materials such as spruce, and at frequencies where the modal overlap is high (more than 30%). Based on the ESPRIT algorithm [START_REF] Roy | ESPRIT -estimation of signal parameters via rotational invariance techniques[END_REF], it assumes that the signal is a sum of complex exponentials and white noise; it projects the signal onto two subspaces: the subspace spanned by the sinusoids (signal subspace) and its supplementary (noise subspace). Rotational invariance property of the signal subspace is used to estimate the modal parameters (frequencies, damping factors and complex amplitudes). The dimensions of both subspaces must be chosen a priori and the quality of the estimation depends on a proper choice for these parameters. The best choice for the dimension of the modal subspace is the number of complex exponentials actually present in the signal. This number, called K, is twice the number of decaying sinusoids. It is therefore advisable to estimate this number prior to the analysis. This is done by means of the ESTER technique [START_REF] Badeau | A new perturbation analysis for signal enumeration in rotational invariance techniques[END_REF], recently developed: it consists in minimising the error on the rotational invariance property of the signal subspace spanned by the sinusoids. The block diagram of the method given in Figure 4 describes the three main steps of the method: (a) reconstruction of the acceleration impulse response (b) signal conditioning (c) order detection and determination of modal parameters.

αk

Data acquisition and normalisation

Timereversal

Band-pass filter

Frequencyshift

Downsampling

Signal conditioning

Impulse excitation

The experimental study presented here aims at estimating the modal parameters (modal frequencies, modal dampings and modal shapes) of the upright piano soundboard in the [0-500] Hz frequency range. The piano was put in a pseudo-anechoic room and excited with an impact hammer (Kistler -type 9722A) at the nodes of a rectangular mesh of 12 × 10 points regularly spaced (Figure 5). The motion of the board is measured at five points with accelerometers (two B&K 2250A-10 and three B&K 4393) located in different zones of the board (Figure 5). For each of the 120 × 5 measures, the impulse response is recon-structed and analysed with the high-resolution modal analysis method. Results are summarised in Figure 6. In order to measure the damping with some precision, it proved necessary to band-filter the impulse responses prior to analysis, as shown by the comparison between (a) and (b) of Fig. 6 and as illustrated by Fig. 7.

Except for the first four low-frequency resonances -for which the rim probably adds non-negligible losses to the distributed ones -modal dampings are very close of the loss factors of the spruce (about 1-3%). The average modal spacing (inverse of the modal density) is about 22 Hz for these 21 lowest modes, in agreement with comparable low-frequency studies (≈ 24.8 Hz for a similar upright piano [START_REF] Dérogis | Analyse des vibrations et du rayonnement de la table d'harmonie d'un piano droit et conception d'un système de reproduction du champ acoustique[END_REF]), ≈ 22.3 Hz for a baby grand one [START_REF] Suzuki | Vibration and sound radiation of a piano soundboard[END_REF])). Above 550 Hz no cloud of points is clearly identifiable in Fig. 6(a) owing to a too-high Signalto-Noise Ratio (≈ 35 dB). Note that the modal overlap µ1 is around 30% at 150 Hz and reaches 70% at 550 Hz.

Acoustical excitation

In order to improve the SNR and thus extend the estimation of the modal parameters towards higher frequencies (higher modal overlap, in fact), we have replaced the impulsive mechanical excitation by a continuous acoustical one (Fig. 8).

The excitation of the loudspeaker is the same as in section 1 (a

Piano Loudspeaker

Accelerometer The modal damping factors as identified up to 3 kHz are reported in Fig. 10, together with available bibliographical results (limited to ≈ 500 Hz). The advantage of the acoustical excitation technique in terms of frequency spanning is obvious. Up to around 1200 Hz, loss factors range from 1% to 3% (mean of η ≈ 2.3% for the 55 estimations, lowest in frequency). Between 1200 and 1500 Hz, damping increases from a mean value of ≈ 80 s -1 below 1200 Hz to ≈ 130 s -1 in this domain. This increase can be attributed to the acoustical radiation of the structure since this frequency-band is the critical domain of the soundboard where modes radiate most efficiently. It is interesting to note that these results (obtained on an upright piano) coincide with the ones obtained by Suzuki [START_REF] Suzuki | Vibration and sound radiation of a piano soundboard[END_REF]) on a small grand of which he estimated the critical domain around 1400 Hz. Above 1.8 kHz the loss factors are of the order of the internal losses of spruce and influence of radiation is no more visible. A reason for this may be the alteration of the acoustical coincidence phenomenon at those high-frequencies, due to the localisation of the waves between the ribs (see last section).

Modal density

The modal density n is a global descriptor of the vibratory behaviour of the structure in the mid-and high-frequency domain. It is given as a function of frequency in Fig. 11 at four points of measurements (see Fig. 5 for the exact locations), as derived from an estimation of the modal spacing (moving average, six successive modes retained for each estimation). The frequency evolution of n reveals two well-separated vibratory regimes of the structure. a. Below 1.1 kHz, the 4 experimental curves are almost similar. n( f ) raises slowly and tends to a constant value of ≈ 0.05 modes Hz -1 independently of the zones of the board where the measure is done. The ribbed board behaves as a homogeneous isotropic plate (see next section). The slow rise confirms moreover that the boundary conditions are constraint. b. For frequencies above 1.1 kHz, n( f ) decreases significantly. Ribs confine the wave propagation: the soundboard behaves as a set of waveguides (see last section).

MODELLING THE LOW-FREQUENCY BEHAVIOUR: A THIN HOMOGENEOUS PLATE IN AN ISOTROPIC EQUIVALENT MATERIAL A simple finite-element model

The modal density of the soundboard below 1.1 kHz suggests that a homogeneous isotropic plate represents a good model in this frequency domain. In order to confirm this numerically, we realised a two-dimensional finite-element model (FEM) of the soundboard by means of the free software CAST3M (triangular elements of 2 cm). The ribbed zone was replaced by an isotropic plate with a dynamical rigidity equal to the one of the orthotropic spruce plate in the direction of the grain:

D H /(ρ H h H ) = D p x /(ρ p h p ) = 156 m 4 s -2 (3)
where the constants of rigidity are generically D = Eh 3 /(12(1ν 2 )), with the Young's modulus E and the Poisson's ratio ν.

The plate density and thickness are ρ and h respectively. The superscript p refers to the unribbed orthotropic spruce plate and H refers to the homogeneous equivalent isotropic plate. The x-axis corresponds to the direction of the grain of the wood, orthogonal to the ribs (see Fig. 5). In the numerical model, the two dead zones in the upper right and bottom left corners of the soundboard are modelled as orthotropic spruce plates with a constant thickness of 8 mm. The two bars delimiting these two zones are made of fir. The mechanical characteristics used in the numerical model for spruce and fir are summarised in Table 1, as derived from measurements made by Berthaut [START_REF] Berthaut | Contribution à l'identification large bande des structures anisotropes[END_REF]) on spruce and fir species selected for piano soundboards.

E L E R G LR ν LR ρ [kg m -3 ] Spruce
11.5 0.47 0.5 0.005 392 Fir 8.86 0.54 1.6 0.005 691

Table 1: Mechanical characteristics of spruce and fir species selected for piano soundboard, after [START_REF] Berthaut | Contribution à l'identification large bande des structures anisotropes[END_REF]). The longitudinal and radial Young's moduli (E L and E R ) and shear modulus (G LR ) are given in GPa.

Results

The numerical solution of the eigenvalue problem is obtained in the conservative case and for clamped boundary conditions. The numerical average modal spacings is presented in Fig. 12 together with the experimental estimations. The theoretical average modal spacing of an isotropic clamped plate is also included. Numerical results match well the experimental estimations in the frequency domain of interest. Moreover the first numerical modal shapes for the isotropic plate -presented in Fig. 13 -are remarkably similar to those obtained experimentally, whereas the one obtains with a numerical model of the ribbed board without bridges are less similar. This confirms that on a typical piano soundboard, the ribs and the bridges compensate globally the anisotropy of spruce. Figure 12: Average modal spacing of the piano soundboard: measured at pointsA 1 (•), A 2 (∆), A 3 (∇), A 5 ( * ), numerically calculated with a finite-element model of a homogeneous isotropic clamped plate ( ) and theoretical for the same plate (-).

To conclude this section, we reproduce on figure 14 the relation of dispersion of flexural waves in the isotropic plate. The coincidence frequency (for which the dispersion curve in the plate intersects the one in air) is ≈ 1500 Hz which is consistent with previous experimental conclusions. However, the half-wavelength λ /2 in the plate become equal or less to the average distance p between two consecutive ribs (≈ 13 cm) at ≈ 1160 Hz so that beyond this frequency, the structure cannot be modelled as a homogeneous plate any more (see next section).

MODELLING THE MID-AND HIGH-FREQUENCY BEHAVIOUR: SET OF WAVEGUIDES AND MOD-IFIED COINCIDENCE PHENOMENON

Numerical observations

For frequencies above 1.1 kHz, n( f ) falls significantly. At 1.1 kHz, the transverse waves in a soundboard without ribs would have a half-wavelength equal to the average distance p between two consecutive ribs. Berthaut et al. [START_REF] Berthaut | Contribution à l'identification large bande des structures anisotropes[END_REF]) made the observation that ribs confine the wave propagation. This is confirmed here by numerical simulations. Modal shapes obtained through the finite-element model of the ribbed board (Fig. 15) exhibit a localisation of the waves for frequencies above 1.1 kHz. The soundboard behaves as a set of waveguides.

The waveguide model

We adopt the simplest possible waveguide model (Fig. 16) in order to derive a dispersion law of this non-homogeneous plate and to calculate its modal density.

The hypotheses are :

• The inter-rib region behaves like an orthotropic plate with clamped boundaries. • Wavenumbers k x m in the direction x normal to the ribs2 According to the first hypothesis, the dispersion law is:

k 4 y + k 2 y D 2 + D 4 D 3 k 2 x m + D 1 D 3 k 4 x m - ρ h ω 2 D 3 = 0 (4)
where the D i are the constants of rigidity of spruce, considered as an orthotropic material (of main axes x and y):

D 1 = E x h 3 /(12(1 -ν xy ν yx )), D 2 = ν yx E x h 3 /(6(1 -ν xy ν yx )), D 3 = E y h 3 /(12(1 -ν xy ν yx )) and D 4 = G xy h 3 /3. With A = D 2 + D 4 D 3 k 2 x m , B = D 1 D 3 k 4 x m and C = ρ h D 3
, the equation 4 becomes a second order equation in k 2 y : k 4 y + A k 2 y + B -C ω 2 = 0. Finally the dispersion law in each waveguide is:

k y = ± √ A 2 + 4Cω 2 -4B -A 2 1/2 (5) The pulsation ω c,p,m = B/C = mπ p 2 D 1
ρ h is a low cutoff pulsation for the transverse modes m in ribs separated by p. Below this pulsation, there is no real solution and waves are evanescent in the y direction.

For a given k x m , the modal density in the wave guide can be calculated as follows. The wavenumbers k y n are approximated by nπ/L y with n ∈ N * . The number of modes of pulsation less than ω is N(ω) = k y (ω)L y /π and the modal density:

n(ω) = dN dk y dk y dω = L y π √ 2Cω √ A 2 + 4Cω 2 -4B √ A 2 + 4Cω 2 -4B -A 1/2 (6)
In the high-frequency limit, the waveguide has the same modal density as that of a beam of length L y , with a ω -1/2 dependency:

n(ω) → ω→+∞ L y π C 1/4 2 √ ω = L y 2π √ ω ρ h D 3 1/4 (7)
The modal density of waveguides with different values of p and L y are reported in Fig. 11.

The modified acoustical coincidence

The acoustical coincidence phenomenon is deeply modified in comparison with the one occurring in a thin plate (see figure 17).

The dispersion curve of a waveguide can present one, two, or no coincidence frequencies depending on the value of p. This creates a nonuniformity in the radiation of the soundboard in the treble range of the instrument compared to the lower range and this may explain the difference in timbre. For example, for the key D 6 having a fundamental frequency around 1245 Hz, the damping factor due to the acoustical radiation of the fundamental may be higher (supersonic waves) than the damping factors of the next two partials (subsonic waves).

CONCLUSION

We have investigated the vibrations of the soundboard of an upright piano in playing condition. At normal levels of vibration, the linear part of the mechanical response to an acoustical response is ≈50-60 dB above the nonlinear component. Given the essentially linear response, a modal identification was performed in the mid-frequency domain Hz by means of a novel high-resolution modal analysis technique: the modal density and the loss factor could be measured up to 2.5-3 kHz. The frequency evolution of the modal density of the piano soundboard reveals two well-separated vibratory regimes of the structure. Below 1.1 kHz, the modal density is very close to that of a homogeneous isotropic plate with clamped boundary conditions. Higher in frequency, the soundboard behaves as a set of waveguides defined by the ribs. A numerical (FEM) determination of the modal shapes confirms that the waves are localised between the ribs. The measured modal density is consistent with an estimation based on the dispersion law of waves in each waveguide. The acoustical coincidence phenomenon is deeply modified in comparison with that occurring in thin plates. The dispersion curve of a waveguide can present one, two, or no coincidence frequencies. This creates a nonuniformity in the radiation of the soundboard in the treble range of the instrument compared to the lower range and this may explain the difference in timbre.

Finally, we would like to notify the reader that the measurements presented in this paper have been used to give a synthetic description of the piano soundboard mechanical mobility (admittance): see the ISMA companion-paper [START_REF] Ege | Synthetic description of the piano soundboard mechanical mobility[END_REF]).

Figure 1 :

 1 Figure 1: Spectrogram of the acoustical response of the loudspeaker, room system to an electrical signal. The distortion appears as successive harmonics of the swept-sine. The spectrogram is calculated with FFT over overlapping windows of 0.1 s with an 50% overlap.

Figure 2 :

 2 Figure 2: Spectra of the linear and the nonlinear responses shown in Fig. 1. Except below 500 Hz, the nonlinear part of the response is ≈ 50-60 dB less than the linear part.

Figure 3 :

 3 Figure 3: Nonlinearities of the two systems {loudspeaker, room} and {loudspeaker, piano, room}. The distorsion rate is comparable in both situations.

Figure 4 :

 4 Figure 4: Block diagram of the high-resolution modal analysis method (after (Ege et al. 2009))

Figure 5 :

 5 Figure 5: Rear view of the upright piano, with the mesh for modal analysis (in red) and the locations of the five accelerometers (in black).

Figure 6 :

 6 Figure 6: High-resolution modal analysis for the [0-600] Hz frequency-band (impulse excitation). Modal frequencies/damping factors map. (a) first rough analysis. (b) After narrow band-pass filtering. (c) After suppression of the (lowprecision) estimations in nodal regions. • : retained modal parameters. × : weighted mean of the modal parameters estimated at four points of the soundboard (acoustical excitation, see following section). -• -: constant loss factors (η = 1 to 3 %).

Figure 7 :

 7 Figure 7: Necessity of the narrow band-pass filtering step. Results for five modes in the [230-330] Hz frequency-band before filtering (top diagrams), after filtering (bottom diagrams).

Figure 8 :

 8 Figure 8: Acoustical excitation of the piano placed in a pseudoanechoic room. The acceleration of the board is measured at four points.

Figure 9 :

 9 Figure 9: Typical bank-filtering analysis of a reconstructed impulse response between 550 et 1150 Hz (acoustical excitation). -: Fourier spectrum of the impulse response at point A 2 . • : modes estimated by ESPRIT (modal amplitudes and frequencies). • • • : amplitude responses of the narrow pass-band filters.

Figure 10

 10 Figure 10: (•) : Modal damping factors estimated at point A 2 up to 3 kHz. Comparison with bibliographical results: • (Suzuki 1986); • (Dérogis 1997); • (Berthaut et al. 2003).

Figure 11 :

 11 Figure11: Modal densities of the board measured at points A 1 (•), A 2 (∆), A 3 (∇), A 5 ( * ) and calculated for the first waveguide mode (1,n) of the corresponding inter-rib space (-, with the corresponding color). Asymptotic values of the modal densities of the inter-rib plates (--, with the corresponding color). Theoretical modal density of the homogeneous isotropic equivalent clamped plate (-• -), see next section.

Figure 13 :

 13 Figure 13: First three modal shapes of the upright piano soundboard: measured (first line), numerical (FEM) after replacement of the ribbed zone by a homogeneous isotropic equivalent plate (middle line), numerical (FEM) for the ribbed board without bridges (bottom line).

Figure 14 :

 14 Figure14: curves in the isotropic plate (-), and in air (--). The critical frequency of radiation of the plate is ≈ 1500 Hz. At ≈ 1160 Hz, the half-wavelength λ /2 is equal to the average distance p between two consecutive ribs.

Figure 16 :

 16 Figure 15: Modal shapes obtained by a FEM of the ribbed board.The positions of the ribs and of the two bars are indicated in black. At 1593 Hz, the antinodes of vibration are localised between the ribs (nodes of vibration). This behaviour is already visible for the modal shapes at 1089 Hz, but not at 776 Hz. In (c), the sign of the phase is represented: red and blue zones denote regions vibrating with opposite phases.

Figure 17 :

 17 Figure 17: Relations of dispersion for flexural waves in the orthotropic plate (• • • along x and • • • along y), in the air (-) and for the two first waveguide modes (-and --) of the waveguide between the second and third ribs. We add the discretes values corresponding to the waveguide modes (1, n) in • and (2, n) in (•), together with the supposed perfectly harmonic partials of the D 6 strings in .

The modal overlap is the ratio between the half-power modal bandwidth and the average modal spacing.

The direction x is parallel to the grain of the spruce board and, by construction, perpendicular to the ribs (see figure5).