Pierre Tandeo
email: pierre.tandeo@telecom-bretagne.eu

Emmanuelle Autret

Bertrand Chapron

Ronan Fablet

R Garello

SST spatial anisotropic covariances from METOP-AVHRR data

Keywords: SST, METOP-AVHRR, Spatial Variability, Anisotropy

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Sea Surface Temperature (SST) is a key geophysical variable for many applications (see e.g. [START_REF] Donlon | Toward improved validation of satellite sea surface skin temperature measurements for climate research[END_REF] and reference therein). For instance, global-scale, complete and realistic SST fields are key input data in the Ocean General Circulation Models (OGCMs). One may also cite the use of SST data to track, characterize and reconstruct mesoscale dynamics (cf. [START_REF] Klein | Diagnosis of vertical velocities in the upper ocean from high resolution sea surface height[END_REF]; [START_REF] Isern-Fontanet | Potential use of microwave sea surface temperatures for the estimation of ocean currents[END_REF]; [START_REF] Tandeo | Segmentation of mesoscale ocean surface dynamics using satellite SST and SSH observations[END_REF]). In this respect, satellite sensing data provide invaluable global-scale data with relatively high spatial and temporal resolutions. Such data are however vulnerable to various atmospheric contaminations (e.g., cloud cover, aerosols, heavy rains, etc...) and they may present large missing data rates. Therefore, spatial and temporal interpolation techniques are necessary to reconstruct the hidden information (see e.g. [START_REF] Reynolds | Daily High-Resolution-Blended Analyses for Sea Surface Temperature[END_REF]). Understanding and modeling of mesoscale SST variability appears crucial to provide interpolation priors.

The temporal variability of the SST has been investigated for short time scales from in situ data (cf. [START_REF] Wainer | Time scales of upper ocean temperature variability inferred from the PIRATA data (1997-2000)[END_REF]) and from remote sensing sources (cf. [START_REF] Tandeo | Linear Gaussian state-space model with irregular sampling: application to sea surface temperature[END_REF]) but, its spatial variability, especially spatial anisotropy and non-stationarity, remains poorly explored. Whereas regional studies addressed the isotropic spatial variability of the SST, (e.g. [START_REF] Gohin | Using geostatistics to merge in situ measurements and remotely-sensed observations of sea surface temperature[END_REF] in the bay of Biscay), evidences for local spatial anisotropies have been reported for SST fields in some studies (e.g. [START_REF] Park | Spatial and temporal scale variations of sea surface temperature in the East Sea using NOAA/AVHRR data[END_REF] in the Sea of Japan) as well as for other geophysical variables such as ocean colour (cf. [START_REF] Doney | Mesoscale variability of SeaWiFS satellite ocean color: Global patterns and spatial scales[END_REF]). However, to our knowledge, no study has explored thoroughly at a global-scale these mesoscale spatial SST variabilities. This study relies on geostatistic tools to address these issues from the analysis of the Advanced Very High Resolution Radiometer (AVHRR) instrument on-board the METOP-A satellite (cf. Le [START_REF] Le Borgne | Operational SST Retrieval from MetOp/AVHRR[END_REF]; O&SI SAF ( 2013)). Using compact parametric model of the spatial covariance of the SST fields and the associated model fitting scheme, we produce observation-based atlases of the mesoscale spatial variability of the SST for the global Ocean, in terms of local anisotropy, spatial scales and residual variances. These atlases provide new insights on global and regional mesoscale ocean dynamics and new priors for missing data interpolation. This paper is organized as follows. The METOP-AVHRR SST data and the geostatistical methodology are introduced in Section 2. We discuss the results obtained in the Malvinas current and the global Ocean in Section 3. Section 4 further discusses our key contributions and prospects for future work.

Method

METOP-AVHRR data

METOP-A satellite has a high resolution coverage of the world Ocean. The data are delivered in satellite projection at full resolution and remapped onto a regular 0.05 • ×0.05 • global grid every 12 hours (see O&SI SAF (2013) for more details). METOP-AVHRR is one of the most efficient SST data source due to the high quality measurements and the number of observations (cf. Le Borgne et al. ( 2007)). In this study, we only use night-time data to avoid the vari-ability due to the diurnal cycle, good-quality pixel data corresponding to best proximity confidence levels (flags 4 and 5). We considered the year 2008 and the global Ocean. The resulting METOP SST fields are clearly non-stationary with important seasonal components. The non-stationary components have complex features and we could not find any appropriate parametric model to describe them. To remove these components, we used the daily 0.25 • resolution Optimal Interpolation Version 2 (OIV2) analysis provided by the National Climatic Data Center (NCDC). This analysis is derived from different satellite sources independent of METOP data (cf. [START_REF] Reynolds | Daily High-Resolution-Blended Analyses for Sea Surface Temperature[END_REF]). We assume that they provide a good estimate of the low-variations of the SST conditions. In this paper, our work is focused on the SST anomaly given by the difference between the METOP and the corresponding OIV2 fields. This anomaly characterizes the mesoscale (horizontal scales of 50 km to few hundred kilometers) non-stationary spatial variability of the SST. An example of SST anomaly field in the Malvinas current is given in Fig. 1. The white pixels correspond to flagged data due to the proximity to clouds. The use of the OIV2 fields instead of a spatial filter version of the METOP-AVHRR SSTs is motivated by the following reasons. Firstly, due to the important cloud coverage in certain regions, it is often impossible to generate realistic spatially smoothed SST fields from METOP-AVHRR. Secondly, to be consistent with a previous analysis of the temporal variability of the SST anomalies (cf. [START_REF] Tandeo | Linear Gaussian state-space model with irregular sampling: application to sea surface temperature[END_REF]), we decided to use the same year 2008 and the same reference fields (i.e., OIV2 fields).

Anisotropic semivariogram model

To spatially interpolate a missing SST anomaly at a specific location s noted as Z(s), we need to characterize the local variability of the neighborhood. Thus, we define the semivariogram function γ between two locations s 1 and s 2 (separated by a distance ∆) as

2γ(∆) = V ar (Z(s 1 ) -Z(s 2 )) = V ar (Z(s 1 )) + V ar (Z(s 2 )) -2Cov (Z(s 1 ), Z(s 2 )) (1)
The process {Z} is said to be second-order stationary if the mean is the same at each location and if its spatial covariance only depends on the spatial distance ∆ between s 1 and s 2 such as

Cov (Z(s 1 ), Z(s 2 )) = C (∆) (2)
where C is the spatial covariance function. According to the previous state- ments (1), ( 2) and knowing that V ar (Z(s i )) = C (0) for any spatial location s i , we obtain

γ(∆) = C(0) -C(∆). ( 3
)
To account for spatial anisotropy, i.e. the existence of principal directions of variations, the spatial lag ∆ (in kilometers) is decomposed according to a zonal component ∆ x and a meridional ∆ y one such as

∆ 2 = ∆ 2 x + ∆ 2 y .
A classical empirical estimation of the semivariogram (cf. [START_REF] Cressie | Statistics for Spatial Data[END_REF]) is given by

γ(∆ x , ∆ y ) = (s i ,s j ) (Z(s i ) -Z(s j )) 2 2n (4)
where n denotes all pairs of points (s i , s j ), for which the longitude and latitude spatial distance equal ∆ x and ∆ y .

The semivariogram measures the local spatial variabilities of the process {Z} as a function of the zonal and meridional distances, ∆ x and ∆ y . Then, we fit a positive-definite parametric model (e.g. exponential, Gaussian, spherical, power or Matérn models) to the empirical semivariogram γ. In our case, according to previous analyses (not shown in this paper) and to reduce the number of parameters, we decided to use an anisotropic exponential model given by

γ(∆ x , ∆ y ) = σ 2 0 + σ 2 1 -exp - ∆ L(θ) (5)
where (∆, θ) refers to the polar representation of (∆ x , ∆ y ). This model involves two variance parameters: the nugget σ 2 0 which represents low spatial variations (typically instrumental uncertainties) and the sill σ 2 (such as σ 2 > σ 2 0 ) which accounts for the stationary variance of the process. The graphical representation of the theoretical semivariogram ( 5) is given by an ellipse as shown in Fig. 2. This ellipse is parameterized according to the length of the minor and major axes, respectively denoted by L min and L max and its orientation φ ∈ (0, π) defined as the angle of the major axis with respect to the North. Given these ellipse parameters, distance L(θ) in Eq. ( 5) is defined as

L(θ) = L min L max L 2 max cos 2 (θ -φ) + L 2 min sin 2 (θ -φ) (6)
with L min ≤ L(θ) ≤ L max for any angular value θ. Physically, the orientation φ defines the preferred direction of variation, L min /L max the strength of the anisotropy. Besides L max relates to the correlation range along the preferred direction of variation.

To fit the parameters of the spatial covariance, we minimize the difference between the empirical semivariogram estimates (4) and the theoretical semivariogram (5) using the weighted least squares criterion and a Gauss-Newton algorithm. This iterative procedure is initialized as follows. The minor and major ranges L min and L max are chosen as the Rossby radius. Then, φ, σ 2 0 and σ 2 are respectively initialized as the main orientation, the minimum and maximum values of the empirical semivariogram computed in Eq. ( 4).

Results

The estimated parameters σ 2 0 (nugget), σ 2 (sill), φ (anisotropic angle), L min (minimum range) and L max (maximum range) are discussed in this section. They are computed for a 1 • × 1 • grid over the global Ocean. Locally, we can suppose that the spatial variability is second-order stationary. The computation of the empirical semivariograms proceeds as follows. For a given 1 • × 1 • area, we first compute daily empirical semivariograms: we sort all locations within the considered area and retrieve the combinations of pairs in all the zonal and meridional directions such as described in Eq. ( 4). In a second step, the daily semivariograms are cumulated into a mean semivariogram over the year 2008. The mean is computed as a weighted version of the daily semivariograms where we account for the actual number of pairs of points separated by ∆ x and ∆ y . As stated in Section 2, we fit to each empirical model a parametric exponential and anisotropic model given by Eq. ( 5) using a Gauss-Newton optimization and the weighted least squares criterion.

An example of empirical semivariogram is reported in Fig. 2 for a 1 • × 1 • in the Malvinas current, off the Argentinian coasts. The empirical semivariogram is clearly anisotropic. This region involves a strong current which results in a low spatial SST variability along this direction. The corresponding fitted exponential semivariogram is illustrated as the contour plot. It might be noted that due to a larger number of pairs of points, the estimation of the empirical semivariogram is typically better for small spatial distance. The estimated model parameters in this particular region are given in Table 1.

We map in Fig. 3 the fitted model parameters for the Malvinas current. The estimated anisotropic spatial variability is represented by ellipses (the same representation was adopted in [START_REF] Morrow | Surface eddy momentum flux and velocity variances in the Southern Ocean from Geosat altimetry[END_REF]). Each ellipse is characterized by its minor range L min , major range L max and the angle of its anisotropy direction φ. In addition, we report the stationary variability σ 2 as a color field. The visualization of the nugget effect will be reported only at the global scale in Fig. 4(c). The detailed analysis of the reported results for the Malvinas current clearly stress the spatial non-stationarity of the SST variability, both in terms of spatial correlation lengths and in terms of anisotropy factor. Especially, SST variability is greater near the Mar del Plata mouth and the estimated anisotropic angles φ are highly correlated to the mean annual currents velocities derived from the mean dynamic topography represented by arrows.

In the global Ocean, we plot the spatial distribution of the parameters in Fig. 4. From these maps, we retrieve relevant patterns regarding the spatial variability of SST and the quality of the METOP-AVHRR data in the global Ocean. First, in Fig. 4(a), the estimated stationary variance σ 2 of the SST anomaly highlights regions with strong surface currents depicting strong SST variabilities (from 1 to 1.5) such as the Malvinas, Kuroshio, Gulf Stream or Agulhas currents. In addition, we identify important upwelling systems (for instance, the Peruvian or Benguela systems) with strong winds yielding to important mixing of the upper ocean layer. By contrast, the mean variance in the rest of the global Ocean is about 0.1. Regarding the analysis of the anisotropy factor 0 <= L min /Lmax <= 1 given in Fig. 4(b), the reported results also clearly discriminates different ocean regions. The lower anisotropic factor, the greater the spatial anisotropy of the local SST variability. Conversely, anisotropic factors closer to 1 refers to almost isotropic situations. Highly dynamic regions appear as specific regions discriminated by their anisotropic spatial SST variabilities. As illustrated for the Malvinas region, the preferred direction relates to the mean surface currents. The Inter Tropical Convergence Zone (ITCZ) also depict intermediate mean anisotropy factors (between 0.4 and 0.6). By contrast, in other areas such as in the Northern Atlantic Ocean, The minor and major of the ellipse depicts both the spatial correlation lengths L min and L max . We report the sill σ 2 as a color field and the quivers refer to the mean geostrophic currents issues from the mean dynamic topography. The black dot corresponds to the locations of the anisotropic spatial semivariogram given in Fig. 2.

we observe high levels of isotropy (more than 0.8), which indicate that a preferred direction of variation of the SST does not exist. In Fig. 4(c), we report the estimated map of the spatial nugget σ 2 0 parameters. The nugget parameter corresponds to the variance of the errors of measurement of the METOP-AVHRR sensor. Previous studies evaluated these variance levels from a comparison between the METOP-AVHRR SSTs and in situ drifting-buoy measurements (cf. Le [START_REF] Le Borgne | Operational SST Retrieval from MetOp/AVHRR[END_REF]). Unfortunately, the number of buoys is very limited and estimates over a whole ocean cannot be determined. Here, the proposed approach only relies on remote sensing data, such that we can derive a synoptic view, at the ocean-scale, of the spatial distribution of this measurement uncertainties. The main sources of contamination of the METOP-AVHRR data documented in the literature are the Saharan dust (cf. [START_REF] Foltz | Impact of Saharan dust on tropical North Atlantic SST[END_REF]), the water vapor in tropical regions such as the Indonesia waters and the Pacific Ocean (West of Colombia and Ecuador) as well as ice or cloud contamination at high latitudes. Such atmospheric processes or contaminations can produce regional biases between the METOP SST data and the OIV2 SST analysis. These biases have synoptic scales from 500 to 1500 km and can be considered as constant in 1 • × 1 • boxes. Therefore, they do not affect the estimation of the semivariogram which represent the second order variability of the SST anomaly. However, a very local variability in the METOP-AVHRR SSTs measurements, known as the nugget effect, may be observed. The spatial extent of this nugget effect is clearly mapped in Fig. 4(c). Other regions such as the Gulf Stream and Agulhas currents also depict strong nugget effects highly correlated with the stationary variance σ 2 . These strong nuggets may be due to sub-mesoscale physical structures at the surface of the ocean that cannot be resolved by the spatial resolution of the METOP-AVHRR sensor. These results are also consistent with our previous temporal analysis of the METOP-AVHRR data (cf. [START_REF] Tandeo | Linear Gaussian state-space model with irregular sampling: application to sea surface temperature[END_REF], where the nugget effect was estimated in the Atlantic ocean. The anisotropic parameters are available at an interpolated 1 • × 1 • spatial resolution as supplementary material and online at: ftp://ftp.ifremer.fr/ifremer/cersat/products/ gridded/sst-anisotropic-covariances/.

Conclusion

In this paper, we have presented a geostatistic method to characterize the non-stationarity of the spatial variability of the SST at mesoscale from 2008 METOP-AVHRR data. We chose an exponential model with an anisotropic parameterization to take into account the differences of the SST spatial variability along the directions. Five parameters were estimated in the global Ocean. They are available online as supplementary material. Among them, we find the stationary variance, the anisotropy parameters and the measurement error variance of the METOP-AVHRR SST data.

Among the key issues to be considered in future works, we plan to perform spatiotemporal interpolation of METOP-AVHRR data to provide realistic SST maps. Especially, we plan to use the statistical parameters of the temporal variability computed in [START_REF] Tandeo | Linear Gaussian state-space model with irregular sampling: application to sea surface temperature[END_REF] and the spatial anisotropic parameters given in this paper. In this context, the anisotropic shape of the spatial variances might be an alternative to the classical isotropic assumptions in (i) the coastal and mouth regions, (ii) the front of SSTs and (iii) the strong surface currents.

Future work may also address the use of the full METOP-AVHRR archive to produce robust SST spatial variability parameters and to study their corre- sponding seasonal variations, especially over the major ocean current systems. The estimation of the parameters could be refined by introducing more flexible parametric models for the calculation of the theoretical semivariogram. For instance, one can use the Matérn (using a modified Bessel function, cf. Stein (1999), p. 31) or the powered-exponential functions (see discussion in [START_REF] Diggle | Model-based geostatistics[END_REF], Section 3.4.2). The exponential and the Gaussian semivariograms are particular cases of these two functions. These models may be interpreted in term of stochastic operators, which might offer additional interpretations of the estimated parameters with respect to local advection/diffusion properties.

Fig. 1 .

 1 Fig. 1. Example of a SST anomaly field in the Malvinas current the 1 st of January, 2008. The anomaly is computed from the difference between the high spatial resolution METOP data and the low spatial resolution OIV2 data.

Fig. 2 .

 2 Fig. 2. 2008 cumulated empirical semivariogram (colors) of the SST anomaly and corresponding fitted anisotropic exponential model (contours) given in Eq. (5) in a 1 • × 1 • region around 42 • S and 55 • W in the Malvinas current off Argentina. The exponential anisotropic variability is represented as an ellipsoid with a minor range L min and a major range L max in the direction φ.

Fig. 3 .

 3 Fig. 3. Ellipse-based representation of the parametric semivariogram model given in Eq. (5) of the local 1 • × 1 • variability of the SST anomaly in the Malvinas current.The minor and major of the ellipse depicts both the spatial correlation lengths L min and L max . We report the sill σ 2 as a color field and the quivers refer to the mean geostrophic currents issues from the mean dynamic topography. The black dot corresponds to the locations of the anisotropic spatial semivariogram given in Fig.2.

Fig. 4 .

 4 Fig. 4. Estimated parameters of the SST spatial variability in the global Ocean: (a) the sill parameter or the stationary variance σ 2 , (b) the anisotropic ratio L min/L max and (c) the nugget effect or variance measurement error σ 2 0 . These maps correspond to the fit of the parametric exponential and anisotropic semivariogram model given in Eq. (5) from the 2008 SST METOP-AVHRR data in 1 • × 1 • areas over the global Ocean.

Table 1

 1 Estimated parameters of the anisotropic exponential semivariogram model of the SST anomaly in the 1 • × 1 • region of the Malvinas current considered in Fig. 2, around 42 • S and 55 • W.

	Parameter	Value
	Nugget σ 2 0	0.01 K 2
	Sill σ 2	0.36 K 2
	Minor range L min	20 km
	Major range L max	47 km
	Anisotropic angle φ	152 •

Acknowledgment

The data from the EUMETSAT Satellite Application Facility on Ocean and Sea Ice are accessible through the SAF's homepage: http://www.osi-saf. org/. We are grateful to Dr. Pierre Le Borgne for his expertise and valuable comments on this work.


Pierre Tandeo

Sileye Ba

Ronan Fablet

Bertrand Chapron

Emmanuelle Autret

SPATIO-TEMPORAL SEGMENTATION AND ESTIMATION OF OCEAN SURFACE CURRENTS FROM SATELLITE SEA SURFACE TEMPERATURE FIELDS

Keywords: Observation-driven model, Patch-based model, Latent class regression, Surface dynamical mode segmentation, Sea surface current

The use of satellite Sea Surface Temperature (SST) fields to retrieve zonal and meridional surface currents (U,V) is now a widespread idea. Since the classical approach involves temporal differencing of SST fields, we investigate in this paper the extent to which mesoscale ocean dynamics may be decomposed into a superposition of dynamical modes, characterized by different linear relationships between surface currents and temperature fields. Based on a completely observation-driven approach, we propose a latent class regression model from local satellite surface currents and patches of SST measurements. Applied to the highly dynamical Agulhas region, we demonstrate and discuss the geophysical relevance of the proposed mixture model to achieve a spatio-temporal segmentation and tracking of the ocean surface dynamical modes. Moreover, we show the accuracy of the proposed model to predict mesoscale surface currents from SST single maps.

INTRODUCTION

Since the last two decades, satellite microwave sensors have been providing an important source of spatial and temporal information of Sea Surface Temperature (SST). By contrast, satellite altimeters provide very sparse measurements of zonal and meridional surface currents (U,V) and therefore mesoscale structures with horizontal scales of 50 km to few hundred kilometers. The classical approach to fill in missing data in the altimetrer data relies on a temporal differencing of SST fields as in the maximum cross-correlation technique or alternate strategies (see e.g. [START_REF] Bowen | Extracting multiyear surface currents from sequential thermal imagery using the maximum cross-correlation technique[END_REF], [START_REF] Marcello | Motion estimation techniques to automatically track oceanographic thermal structures in multisensor image sequences[END_REF], [START_REF] Chen | Surface velocity estimation from satellite imagery using displaced frame central difference equation[END_REF]). Alternatively, recent studies (cf. [START_REF] Jh Lacasce | Estimating subsurface horizontal and vertical velocities from sea-surface temperature[END_REF], [START_REF] Lapeyre | Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory[END_REF], [START_REF] Isern-Fontanet | Three-dimensional reconstruction of oceanic mesoscale currents from surface information[END_REF], [START_REF] Klein | Diagnosis of vertical velocities in the upper ocean from high resolution sea surface height[END_REF], [START_REF] Hausmann | The observed signature of mesoscale eddies in sea surface temperature and the associated heat transport[END_REF]) rationalize and demonstrate that fields of SST can become, to some assumptions, an active tracer coupled to the dynamics leading to strong correlations with surface current fields. This is known as the Surface Quasi-Geostrophic (SQG) theory given in Eq. [START_REF] Bowen | Extracting multiyear surface currents from sequential thermal imagery using the maximum cross-correlation technique[END_REF]. Within this theoretical framework, a linear transfer function shall be identified between (U,V) and SST gradient fields. This strongly advocates for observation-driven studies to explore and characterize the local relationships between SST and surface currents from satellite-based routine observations. Yet, as illustrated in Fig. 1, a single linear transfer function cannot be expected to solely govern the whole mesoscale dynamics in a particular ocean region. As revealed, an overall spatial correlation exists, but this relationship clearly varies spatially and temporally. For instance, in the warmer SST frontal zone (top of the image), SST gradients correspond to large surface currents. In the colder frontal area (bottom of the image), large SST gradients do not reflect in large surface currents. We may also point out that within the clearly detected eddy (top-left of the image), significant velocities are associated with weak SST gradients. Fig. 1. Surface currents and associated temperature gradient norms derived from satellite measurements within the Agulhas current the 1 st of January, 2004.

Within an observation-driven framework, the key statement of the proposed approach is to regard the local dynamics as a mixture of different dynamical modes, i.e., local dynam-ics are stated as the superimposition of a finite set of linear transfer functions. In this paper, we propose to investigate such a model to (i) develop a probabilistic learning-based setting for the inference of such mixture models, (ii) achieve a spatio-temporal segmentation of the identified dynamical modes from satellite SST and surface currents observations, (iii) evaluate the extent to which such mixture models are geophysically relevant to characterize the upper ocean dynamics over active ocean regions, and to (iv) analyze their relevance for the estimations of mesoscale surface currents from SST fields.

Hereafter we consider the Agulhas region as our casestudy region and the paper is organized as follows. Section 2 presents the proposed probabilistic learning-based model. In Section 3, the application to satellite observations is evaluated. We further discuss and summarize the key results of our investigations in Section 4.

METHODS

Patch-based approach

As mentionned above, the SQG model has been shown in recent studies to match the observed 3D dynamics of the ocean surface layer (cf. [START_REF] Klein | Diagnosis of vertical velocities in the upper ocean from high resolution sea surface height[END_REF]). In the SQG model, these dynamics are fully governed by the surface dynamics and the temperature becomes an active tracer which is intrinsically linked to the surface current (cf. [START_REF] Isern-Fontanet | Potential use of microwave sea surface temperatures for the estimation of ocean currents[END_REF]) as follows:

(U,V) = γ∇ ⊥ ∆ -1 /2 SST ( 1
)
where γ is a positive scalar, ∇ ⊥ is the orthogonal gradient and ∆ -1 /2 is the fractional Laplacian operator with dimension -1 /2 (cf. [START_REF] Caffarelli | An extension problem related to the fractional laplacian[END_REF]). This equation states that the surface current is a linear transform of spatial derivatives of the SST. The parameter γ relates to a normalization constraint. Though this parameter is generally assumed to be locally constant, it may not be the case for an entire region as illustrated in Fig. 1. These considerations lead us to hypothesize that surface currents may locally relate to SST derivatives according to a finite set of K transfer functions such as

(U,V) = F k (SST) (2)
where F k characterizes the k th mode of local interaction between the SST and the surface currents. In the SQG setting expressed in Eq. ( 1), it may be noticed that, given finitedifference approximations of these derivatives, the transfer function numerically resorts the local convolution of the SST by a linear filter. Therefore we assume that each dynamical mode F k is parameterized by a linear filter. Using a matricial formulation, Eq. ( 2) is rewritten as a patch-based linear regression such as

Y i = H k X i ( 3
)
where Y i encoded as a 2-dimensional vector given by the surface current (U,V) at spatio-temporal location i and X i is the vectorized version of the local SST centered in i (cf. Fig. 2). The p × 2 regression coefficient matrix H k associated with dynamical mode k is corresponding to F k in Eq. ( 2). Here, p = (2h + 1) 2 defines the size of the local SST neighborhood around i and h is set according to the Rossby radius of the study region, which states the mean size of the mesoscale ocean structures like eddies. In our study region, the Agulhas current, it approximately corresponds to 200 km, i.e. h = 4 for the spatial resolution of the data in this study.

X i Y i 2h + 1 X j Y j X i Y i X j Y j
Fig. 2. Sketch of sea surface temperature patches (contour plot), noted as X, and the corresponding zonal and meridional currents (quiver) noted Y at the central position.

Latent class regression model

Our objective is to identify K different hidden surface dynamical modes from a joint set of SST patches (p-dimensional vector X) and zonal and meridional surface currents (2dimensional vector Y). In this paper, we assume that the conditional likelihood of Y given X resorts to a mixture of Normal distributions such as

p (Y|X, θ) = K k=1 λ k N k (Y; Xβ k , Σ k ) (4)
where N k represents a multivariate Gaussian probability density function with mean Xβ k and covariance Σ k , and λ k is the prior probability of mode k. To simplify the notations, we store the overall parameters of model ( 4) in θ = (λ 1 , . . . , λ K , β 1 , . . . , β K , Σ 1 , . . . , Σ K ). In the literature, this model is referred to as a "latent class regression" or "clusterwise regression" (cf. [START_REF] Desarbo | A maximum likelihood methodology for clusterwise linear regression[END_REF]). By construction, it imposes that 0 ≤ λ k ≤ 1, K k=1 λ k = 1 and Σ k is positive defined. To learn model parameters θ, we resort to a classical maximum likelihood criterion and use an iterative EM procedure (cf. [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF]).

Segmentation maps and current predictions

We exploit the inferred model with parameters θ to perform a spatio-temporal segmentation of the underlying dynamical modes. More precisely, for any spatial location s and time t, using the Bayes' theorem, we evaluate the posterior likelihood that the dynamical mode is of type k such as

πk (s, t) = λk N k Y(s, t); X(s, t) βk , Σk K l=1 λl N l Y(s, t); X(s, t) βl , Σl , ∀k. (5)
Then, the pixel at location s and time t is assigned to the most likely dynamical mode. The estimation of the surface current at the spatial location s and time t is given by the fuzzy regression

Ŷ(s, t) = K k=1 πk (s, t)X(s, t) βk . ( 6
)
3. RESULTS

Remote sensing data

As surface current data (U,V), we use the daily delayed time Maps of Absolute Dynamic Topography (MADT) produced by Collecte Localisation Satellites (CLS) available online at http://www.aviso.oceanobs.com/. This information combines the signal of several altimeters onto a 1/3 degree Mercator projection grid. We use the 2004 data since four altimeters were available (Jason-1, Envisat or ERS-2, Topex/Poseidon and GFO). As SST data, we use optimally interpolated microwave SSTs provided by Remote Sensing System (RSS) available online at http://www.ssmi.com/. It combines the signal of three microwave radiometers (TMI, AMSR-E and WindSAT) which are robust to the presence of clouds. The spatial resolution is 1/4 × 1/4 degrees and the temporal resolution is the same as the MADT data, i.e. daily. We bilinearly interpolate the MADT data onto the SST grid. We focus on the Agulhas region between longitudes 5 • E to 65 • E and latitudes 30 • S to 48 • S.

To fit the latent class regression model and learn the model parameters θ according to the EM procedure, we build a set of joint SST patches {X i } i∈{1,...,n} and surface current data {Y i } i∈{1,...,n} , which refer to randomly selected space-time positions in the considered time series of sea surface observations. Overall we typically consider a sample with n = 10 5 observations to balance between the representativeness of the training set and the computational complexity of the EM estimation.

Segmentation of ocean surface dynamics

In this study, as a trade-off between the likelihood and the complexity of the model, we consider K = 4 hidden sur-face dynamical modes, i.e. four different linear transfer functions as in Eq. ( 2). Then, given the estimated model parameters θ, we determine from Eq. ( 5) the posterior likelihood πk of each spatio-temporal location (s, t) to be assigned to any dynamical model k. This leads to dynamical modes which are spatio-temporally well-segmented as illustrated in Fig. 3. The animations of the time series of these daily maps in the Agulhas current over 2004 are available here: http://tandeo.wordpress.com/communications/articles/. Fig. 3. Segmentation of hidden surface dynamical modes given satellite SST and surface current (U,V) data, the 1 st of January, 2004 within the Agulhas current. We use the colors red, green, cyan and blue to respectively distinguish the first, second, third and fourth dynamical modes.

These maps highlight clear spatio-temporal clusters that can be interpreted from a geophysical perspective in terms of different dynamical modes. The first dynamical mode (red) characterizes very strong current magnitude and warm waters. It is primarily associated with the main Agulhas current that flows down the East coast of Africa through the Agulhas ridge. This mode also involves mesoscale eddies, the socalled warm core Agulhas rings. An example is visible near the location 37 • S, 32 • E in Fig. 3. It corresponds to the eddy identified in the upper left corner of Fig. 1 with strong surface currents and low temperature gradients. The second cluster (green) mainly relates to the eastward Agulhas return current that hits a part of the South Atlantic current. It creates a subtropical front varying from 36 • S to 44 • S with strong eastward currents and middle-range SST gradients as observed in the upper part of Fig. 1. This cluster share some agreement with the SQG theory. The third (cyan) and fourth (blue) modes correspond to weaker surface currents. The third mode is characterized by mid-temperatures and westward currents whereas the fourth one involves colder temperatures and eastward currents. Let us stress that the third cluster involves large SST gradients but weak surface currents such as identified in the lower-part of Fig. 1.

Surface current predictions

We further investigate the geophysical consistency of the identified dynamical modes from the analysis of the surface currents predicted by the fitted latent class regression model in Eq. ( 6). In Fig. 4, we report a comparison to the MADT data, considered as a true surface current references. Overall, a good agreement is obtained between the true surface currents and the predictions of the proposed model. The global correlation coefficient is 0.76, and can locally be very large as illustrated in Fig. 4(c), corresponding to the zone depicted in Fig. 1. As found in Fig. 3 this zone involves the four dynamical modes. The mixture model enables us to retrieve both the large warm eddy (upper left) associated with weak SST gradients, the relatively large surface currents along the large warmer SST gradients (upper part), as well as the rather weak currents along the large but colder SST front (lower part of the zone).

By contrast, if we conceal the information of the hidden dynamical modes and learn other classical models with the same n = 10 5 training data, the resulting surface currents are not consistent. For example, when we fit a linear model as in Fig. 4(d), i.e. corresponding to K = 1 in our proposed model, the single linear transfer function results in underestimating the surface currents within the warm eddy (upper left) and overestimating the currents of the colder frontal zone (lower part). We also consider a classical nonlinear approach such as the weighted 10-nearest neighbor regression in Fig. 4(e). This last model is not able to follow the warm eddy (top-left) and the estimated surface currents globally display discontinuities.

CONCLUSION

In this paper, we propose an observation-driven framework to spatially and temporally discriminate hidden dynamical modes at the surface of the Ocean. We rely on a latent class regression model. This probabilistic setting resorts to locally model the distribution of the sea surface currents (U,V) conditionally to the SST as a Gaussian mixture of hidden linear transfer functions. The statistical parameters of the model are estimated using a maximum likelihood approach. We applied the proposed methodology to the 2004 daily 1/4 × 1/4 degrees satellite image series of microwave SSTs and surface currents. The reported results retrieved a relevant spatiotemporal decomposition of ocean surface dynamical modes in the Agulhas region according to four dynamical modes: (i) the main Agulhas current and warm core rings characterized by strong currents and hot temperatures, (ii) the return Agulhas current with lower temperatures and currents but where the SST is dynamically coupled as in the SQG theory, (iii) local fronts regions where strong SST gradients do not seem to affect the current velocities, and (iv) a weaker dynamical mode.

Given a set of identified ocean surface dynamical modes, we show that the latent class regression model is able to make relevant predictions of sea surface currents. In particular, the four-latent class regression overcomes the results given by a linear model or classical nonlinear approaches corresponding to situations where we conceal the dynamical modes. This result confirms the fact that a spatio-temporal tracking of the hidden surface dynamical modes is essential to retrieve accurate surface currents given single SST fields. The latent class model presented in this paper constitutes a serious strategy to fill the gaps of satellite surface currents using abundant microwave SST data.

COPYRIGHT FORMS

Fig. 4 .

 4 Fig.[START_REF] Jh Lacasce | Estimating subsurface horizontal and vertical velocities from sea-surface temperature[END_REF]. Sea surface temperature with the associated true MADT or estimated surface currents (U,V) using a linear regression, a four-latent class regression and a weighted knearest neighbor regression within the Agulhas region the 1 st of January, 2004.


Mathieu Gatumel
email: mathieu.gatumel@iut2.upmf-grenoble.fr

Sabine Lemoyne De Forges

Understanding and Monitoring Reinsurance Counterparty Risk Literature review concerning reinsurance counterparty risk

Keywords: Reinsurance Counterparty Risk, Solvency, Market Discipline, Regulation JEL classification: G22, G28, G32, G33

Since 2008, catastrophic losses and financial turmoil have deeply shaken the insurance and reinsurance industries. Severe difficulties encountered by sector leaders like AIG and Swiss Re have shed light on the potential fragility of the players, and have increased attention on the subject of reinsurance counterparty risk. This corresponds to the exposure of an insurance company to reinsurer failure and is difficult to assess due to a scarcity of reliable measures. It has long been considered as largely auto-regulated by the insurance market. The impact of reinsurance credit on an insurers' balance sheet, market complexity and lack of coordinated responses among states begs questions concerning the role of control and regulation. In this article, we address the current state of reinsurance counterparty risk and existing means by which to measure it. We then discuss the impact of market discipline on this risk and point out the importance of control within the reinsurance industry.

We particularly look at the key role of regulation in providing better risk measurement tools to assist in assessing the importance of reinsurance counterparty risk on insurance levels and the systematic development of risk management tools.

Introduction

Even if insurance and reinsurance companies remained stable after the catastrophic losses, stock market volatility and decline in investment yields in 2010 and 2011, the 2008 financial crisis shed new light on the question of risk measurement and transfer in financial markets. The insurance industry has been deeply shaken, essentially on the asset side of its activity, as a result of events like the fall of AIG, the downgrade of Swiss Re and the resulting impact on means of alternative risk transfer. The question of insurance company exposure to reinsurance counterparty risk remains an important issue in insurance firm risk management.

In this article, we analyze the importance of default risk in light of recent events and discuss ways to mitigate it. Section 1 presents the state of the market and defines risk linked to reinsurer default.

Section 2 explores current means of measuring counterparty risk, and highlights critical underlying issues. Section 3 discusses insurance industry market discipline concerning reinsurance counterparty risk. Finally, section 4 analyses current regulation solutions.

A market with growing exposure to reinsurance default risk

The reinsurance industry has greatly evolved over the last fifteen years. In 2007, some 150 reinsurance companies received a total premium volume of roughly $200 billion (Standard and Poor's, 2011) and had a total capitalization estimated at $129 billion for the last quarter of 2007 (Benfield Group, 2008). Historically, the reinsurance market has been dominated by specialized reinsurance companies, concentrated mostly in Europe and Bermuda and has been very intertwined on a worldwide level. Because transactions are primarily over the counter, information is scarce and difficult to obtain. Since the beginning of the nineties, reinsurance companies have become more intertwined. From 2003 to 2009, the market share of the ten first reinsurers (both Life and non-Life) rose from 56% to 65% (Global Reinsurance Highlights report of Standard and Poor's, 2011). Details of this phenomenon for the top 5 reinsurers are given in Table (1 Simultaneously, the past ten years have seen the downgrading of reinsurance companies as illustrated in Figure (1). This may be explained by a variety of factors: the cost of some particular events for the industry (Andrew, 9/11, Katrina), reserve strengthening for non-life reinsurers, purchase of certain reinsurers at inflated prices or under-performing equity markets. From this perspective, 2011 was particularly turbulent for insurance markets and was characterized by catastrophes that implied losses similar to those the reinsurance market sustained from U.S. hurricanes in 2004 and 2005.

According to Aon -Benfield Group (2012), global property catastrophe losses for 2011 were estimated at $434 billion. In addition, financial markets in 2011 were impacted by volatile stock market conditions, record low investment yields, downgrade of U.S. debt and the European sovereign debt crisis. Fortunately, most carriers avoided direct hits because of strong capital positions held for 3 or 4 years. Nevertheless, for the future, external capital is likely to remain difficult and costly to acquire, and constraints on capital markets will force the industry to reconsider how to transfer it. For example, in 2011, outstanding catastrophe bond issuances were equal to $4.6 billion, below the level in 2006. In 2008 and 2009, among the various reinsurers in the market, Swiss Re was strongly weakened by its financial activities, rendering Berkshire Hathaway's capital injection necessary as well as the purchase of an insurance cover. On the contrary, Munich Re conserved its rating whereas Berkshire Hathaway, following some financial losses due to its banking activity, was downgraded by two rating agencies.

Table (2) details the claim amounts due but not paid to the cedent by the reinsurance company for certain countries. The weight of balance sheet reinsurance recoverables on insurance companies, as well as on reinsurance market structure, point out the importance of studying reinsurance counterparty risk. According to the UK's Financial Services Authority definition, a credit risk is incurred whenever a firm is exposed to loss if a counterparty fails to perform its contractual obligations, including failure to perform in a timely manner. More precisely, a credit event may be classically defined as the failure for an issuer to pay a coupon or to redeem the principal value of an obligation on maturity date; it may also include filing for bankruptcy, insolvency, or compulsory restructuring (Swiss Re, 2003). Reinsurer bankruptcy may spring from three areas: risk subscription and investment policy as classically experienced by all insurance companies, retrocession, and risk credit exposure. Historically, there have been very few reinsurer bankruptcies: between 1980 and 2003, only 24 were enumerated by (Swiss Re, 2003) for a total gross premium corresponding to $820 million (see Figure ( 2)). In fact, many bankruptcies are avoided by the fact that reinsurance companies put their investments into runoff before becoming insolvent.

In case of reinsurer default, an insurance company faces higher financial charges and difficulty reimbursing its insured clients in case of high claims. Few studies have focused on the way insurance companies take into account reinsurer credit risk. The question has been raised when considering optimal reinsurance contracts (Gajek and Zagrodny, 2004) for examining existing or changing regulations (Rossi and Lowe, 2002) and particularly in the case of the United States (Cole and McCullough, 2006;Cole et al., 2007). A liquid market for reinsurance claims does not exist and thus creditors are not allowed to modify their position depending on the credit risk of their reinsurers. This exposure may weaken the creditworthiness of one insurer, as was the case for Reliance Group Holdings: this US-based group collapsed when its exposure to reinsurance recoverables reached 600% of consolidated surplus at a time when no liquid assets were available and sub-investment grade debt was due for repayment (Benfield Group, 2007).

Global exposure of insurance companies to reinsurance credit risk may also weaken the industry as a whole, given its intertwined nature. Systemic risk (De Bandt and Hartmann, 2000) exists at industry or at economy levels. Van Lelyveld et al. (2009) estimate that among the risks introduced by use of reinsurance, credit risk is the single failure most likely to threaten financial stability of the concentrated reinsurance market and could induce large losses. Furthermore, retrocessions may increase the phenomena, as the London XL Spiral proved in the nineties. Burkart (2006) and Wade (2008) highlight that a chain of retrocessions built during the eighties weakened the industry, funneling underlying losses through a chain of Lloyd's syndicates, rendering them opaque and difficult to price. When the catastrophes occurred, losses were on a scale that threatened the whole industry. In 2005, the retrocession rate of reinsurance companies was estimated at 15% of ceded premiums (Group of Thirty, 2006). The limited number of reinsurers and their weaknesses to the same extreme shocks (on assets or liabilities) limited the likelihood that they could be responsible for a global systemic risk, but they were capable of weakening the insurance industry, esteemed the Group of Thirty (2006). The prospective study from Van Lelyveld et al. (2009) on the Dutch reinsurance market provides no evidence of reinsurance systemic risk. Nevertheless, the 2008 financial turmoil began by the failure of major financial firms and was characterized by a strong decrease of transactions in the market, requiring state intervention. This did not create a real systemic risk, but the consequences of financial firm bankruptcy on the situation cannot be ignored.

The difficulty of quantifying and estimating counterparty risk

Before monitoring reinsurance counterparty risk, one must have knowledge of such a risk. The modeling of default risk poses a number of challenges, especially in terms of information access. The reinsurance market is quite opaque, information on reinsurance company financial health being available mainly through rating agency notation. Extracting such information from financial markets remains difficult, as Burkart (2006) showed in his study of co-movements of default risk in the reinsurance industry. Issues related to estimating and modeling are as follows:

• Default probability: The probabilities of default within the market cannot be directly observed but need to be calculated based on either historical credit rating experience, deduced from a process involving certain forms of market prices, determined using a model of the underlying risk sources, or drawn from subjective credit assessment criteria.

• Default correlation: Directly modeling default correlation is challenging. This is due to the non-triviality of attempting to simulate correlated binary variables or to a lack of credible historical data for estimating default correlation directly. Flower et al. ( 2007) point out several sources of correlation to be taken into account when modeling the consequences of reinsurance default risk on a primary insurer's balance sheet. Shocks are the first source of correlation because reinsurers are broadly subject to common risks. Similarly, correlations between underwriting years introduce some temporal effects. Furthermore, the market is subject to wide cycle effects, in the sense that sustained soft cycle periods lead to growing reserve issues which can subsequently emerge as credit issues. This type of correlation is enhanced by the intertwined structure of the market: primary insurers and reinsurers are affected by the same types of events; domino effects may appear as a result of the different levels of retrocession.

Additionally, capital requirement calculations at the extreme loss percentiles involve examination of the tails of asymmetric fat-tailed loss distributions. Is it particularly true for dependency issues that are difficult to capture for fat-tailed insurance risk distributions.

Unfortunately, tools like copulae may appear irrelevant because of the limited available data which induce a high volatility when estimating them.

• Recovery rate: The main issue for its assessment is the difficulty to close an operation. As explained by Standard and Poor's (2008b), "an insurer may cease to write new risk but could spend years, sometimes decades, settling the liabilities that arise from the policies written before it closed its doors." In case of run-off, the entity may have difficulty raising capital. That explains why S&P has developed run-off payment assessment which measures the capacity of an insurance company to pay its policies.

• Time horizon: all of these issues are reinforced by the long-term relationships between insurance and reinsurance companies. Indeed, the duration of insurance liabilities -and thus of reinsurance claims -covers many years. On the contrary, the measure of the underlying risks to the reinsurance credit risk, mainly catastrophic and financial risks, is generally limited to a short horizon.

Methodologies which aim at modeling reinsurance default risk take into account, more or less, the Recent development of several models illustrates the fact that assessment of reinsurance counterparty risk is a difficult task and that work is in progress, upheld and moved ahead by the necessity for monitoring risk.

The market discipline for reinsurance default risk

Considering the recent impacts of the financial crisis, as well as past crises, the question of the need to regulate reinsurance counterparty risk must be addressed. As Harrington (2005) describes, the rationale for government intervention must respond to the following considerations (introduced by Breyer (1982)): the potential for market failure in a competitive market must be assessed and there must be positive benefits to regulation. Thus a preamble to the question of regulation is the study of market discipline on reinsurance counterparty risk. This involves, on one hand, reinsurers' capability to monitor their solvency risk, and on the other hand, the capacity of insurance companies to selfmonitor their exposure to reinsurance counterparty risk.

As regards the reinsurance industry, Epermanis and Harrington (2006) point out several features of the market that plead for a certain level of discipline. Standard agency considerations as well as litigation costs provide incentives for reinsurers to limit their default risk. In addition, as highlighted by

Jean-Baptiste and Santomero (2000), the reinsurance market is characterized by long-term relationships between insurers and reinsurers. Moreover, transactions take place between two wellinformed players. Rossi and Lowe (2002) insist on the differences between the insurance and reinsurance industries, arguing that insurance companies are more capable of evaluating reinsurers' products than are customers when faced with evaluating insurers' products. Thus, in markets based on trust, as in the financial services industry, reinsurer self-discipline is all the more necessary because they exercise a business-to-business activity. For example, several studies have shown the impact of quality on reinsurance treaties regulating pricing and contracting. Zanjani ( 2002) considers a theoretical model where risk neutral reinsurers with limited liability face a demand for insurance conditional on quality that explains their choice of a level of internal capital limiting default risk.

Furthermore, econometric studies have shown that there exists a slight premium for credit risk. Weiss and Chung's ( 2004) results support the risky debt hypothesis that consumers are concerned with the financial strength of reinsurers, as demonstrated by policyholders' surplus, past profit or loss on ordinary activities and new capital all of which are related to reinsurance prices.

As for the exposure of insurance companies to reinsurance counterparty risk, the same standard arguments may apply to reinsurance companies concerning self-regulation: agency consideration, protection of the franchise value, debt holder monitoring risk management, and consumer demand for long-term relationships with insurers, all provide incentives to insurance companies to monitor their own default risk and in doing so, their reinsurance coverage and the inherent counterparty risk. The empirical study of Epermanis and Harrington (2006) documents that rating downgrades of U.S non-life insurance companies are followed by revenue declines in comparison with insurers that did not experience rating downgrades 3 . This highlights the importance of insurer financial strength. The rise of risk management over the last ten years has lead to the development of techniques that may be used to mitigate this counterparty risk. Table (3) synthesizes them. Classical tools include those usually used for the monitoring of reinsurance coverage. Diversification of reinsurance companies as a means used to cover risk allows mitigating counterparty risk. Self-retention management, often used in a cost reduction perspective, may also address the issue. On one hand, higher retention increases the capital requirement due to non-transferred risk, and thus the corresponding cost of capital, but reduces the ceded premiums; on the other hand it increases the capital requirement due to reinsurance counterparty risks and the corresponding cost of capital, but it also increases the ceded premiums.

This arbitrage between expected profit and certainty equivalent raises a central question for firms concerning their risk appetite. Obviously, with growing interest for risk management having been identified by the Benfield Group ( 2008), knowledge of reinsurance counterparty risk is also taken into consideration.

Following Plantin ( 2006), a size-based distinction has to be made between insurance companies.

Reinsurance companies do not merely offer external capital to cover extreme risk, they also provide risk expertise to their customers. In the case of relatively small companies, reinsurers are unavoidable partners for risk management. Therefore, since medium or large companies have begun to develop real risk management expertise, they no longer rely as heavily on reinsurance company knowledge.

New tools have also been tried to mitigate reinsurance default risk.

Development of the catastrophe bonds market represented a way to diversify catastrophe coverage without being subject to reinsurance default risk. However, default risk on catastrophe bonds is not zero: it is subject to the counterparty risk of the swap, as illustrated by the consequences of the Lehman Brothers downfall which caused the default of four catastrophe bonds in 2008. With the development of financial instruments to cover credit risk, direct coverage tools have been available for insurers: insurers or reinsurers have been using Credit Default Swaps for some time (Sawyer, 2006).

Yet not all insurers use them, some arguing that the markets are still too young. More sophisticated products have even been structured to allow for hedging a portfolio of credit risks. In 2007, Hannover

Re issued a CDO of reinsurance recoverables. However, the credit crisis has demonstrated the difficulty in pricing and structuring this kind of product, a difficulty enhanced when considering a tail risk such as reinsurance default risk. Furthermore, due to its complexity, the cost of such coverage would be relatively high compared to the expected loss from reinsurer default.

Furthermore reinsurance counterparty risk remains a rare risk that only large insurance companies seem to have the luxury to study. Section 3 showed that measuring reinsurance counterparty risk is delicate, mostly due to imperfect information on the dependency between actors, between long tail risks, and between insurance and financial risks. This limited knowledge of risk handicaps development of an efficient market discipline. Regulation, however, allows partially solving this issue and is necessary to reveal it.

The modalities and rules of regulation

Reinsurance insurance has always been an opaque market and its culture relies on trust built between cedents and their reinsurers in the framework of a long-term relationship. Since the early 1990s, many changes have occurred in the market. It has become increasingly intertwined due to market concentration. There are a few main actors in a very global field. The underlying risks have also evolved, as market locations have changed with the birth of the Bermudians. Furthermore, management styles have evolved, with the more traditional saxon model giving way to the anglosaxon one, as highlighted by Albert (1991). In this context, the recent financial crisis has pointed out the need to reconsider reinsurance default risk regulation, particularly when reinsurance giants like Swiss Re have been shaken by recent events.

Revealing information on reinsurance counterparty risk seems central to answering regulation questions, as reliable information concerning market discipline is necessary. There is thus a real place for regulators in developing better risk understanding, through increased information sharing within reinsurance markets. Until now, part of this information has been provided by rating agencies.

The Group of Thirty ( 2006) characterizes these agencies as de-facto-regulators. Although their estimation of reinsurance counterparty risk remains very limited, it should not be taken for granted.

Global questioning of their role in the current crisis seems relevant when considering precisely this case. There are three different levels of regulator action that have been taken in different countries.

These actions respond unequally to the question of information disclosure. Table (4) provides a summary of the different observed means of regulation and their benefits and drawbacks.

The first possibility is direct regulation at the reinsurance market level. Such regulation is not always advocated, as noted by Harrington (2005) who estimates that as the reinsurance industry has greater self-discipline than either the insurance or banking industries, solvency and capital requirements should be consequently less restrictive. Evolution of this discussion is noticeable at several levels. In the European Union, Solvability II includes capital requirements for reinsurance counterparty risk.

Furthermore the Reinsurance Directive, adopted in 2005, implemented a common regulatory system which particularly aims at incorporating reinsurance into the EU's supervisory system of the insurance industry (Evans, 2007).

A second possibility is to regulate contracts between insurers and their reinsurers. This is predominantly the case for reinsurance companies originating from countries outside the jurisdiction of the cedant. It stems from a desire to fully protect the cedant, and thus the final consumers, from the reinsurance counterparty risk when not being controlled by the same authority. Discussions are underway in the US on collateralization requirements for unauthorized reinsurance companies4 that have the obligation to fully collateralize gross US liabilities.

Finally, regulation concerning the level of insurance is possible, obliging insurers to immobilize capital corresponding to the risk and to their management possibilities. Regulators may have an interesting catalyst role when creating new measures for risk, as is the case in the European Union, with the SCR calculation. Such measures can augment insurance company interest in risk monitoring. It develops research for new tools and methodologies, and eventually leads to the disclosure of more information by reinsurers concerning their default risk. Even if large insurance companies are already relatively autonomous in their risk management, it can be particularly useful for small insurance companies.

Notwithstanding the different possible means of regulation, some key questions must be addressed when opting for regulation. First, the cost of regulating should be carefully considered. Some (2007) highlight the importance of reinsurance for the financial stability of the insurance industry. They underscore the advantages of regulation at a regional level, at least to ease risk transfers. Going further, Vogelgesang and Kubicek (2007) advocate global supervision of reinsurance companies:

reinsurer eligibility to conduct business should be determined by a single body and monitored frequently . Furthermore, national supervision regimes should be compatible. The role of supervisors to pay attention to signs indicating need for intervention also remains important. In the case of the insurance industry, Bohn and Hall (1999) exploit pre-insolvency data to detect premium patterns before insolvency; they find evidence that a large percentage of firms grow quickly before failure.

An alternative to direct regulation would be increased market discipline (Harrington, 2005), encouraging stakeholders to pay more attention to default risk. Regulation at the European level deals with this issue in two ways, first by obliging insurers to assess the capital charge relative to reinsurer default risk (to this end, Solvency II eases transparency and allows the development of standards).

Second, being characterized by a capital charge, each reinsurer is encouraged to control its financial strength. At the market level, such regulation is more able to improve market discipline than use of collaterals which do not make reinsurers aware of their responsibilities -the collateral amount depends roughly on financial health -and directly limits access to the market.

Conclusion

Trust is the basis for insurance and reinsurance industry interactions. Considering the weight of reinsurance recoverables on insurer balance sheets, the increase of insurance risks, and the turmoil in financial markets -particularly related to risk transfers -reinsurance counterparty risk has to be carefully monitored. Market discipline, existant due to the particularities of the market, is not, however, sufficient to monitor this tail risk.

In this context, a certainty remains. Insurance companies have to develop strong expertise concerning their own risks to be able to decide which risks they are not able to support and thus determine their exposure to reinsurance counterparty risk. Ongoing regulation definitely plays an important role in clarifying this issue through development of proper risk measures to establish capital ratios -if possible on the reinsurance level -and elucidation of reinsurance company responsibilities.

Nosce te ipsum (Know thyself)

Socrates

  ). Guy Carpenter cited 59 M & A transactions in 2008 accounting for an aggregate value of $16.6 billion.

  various issues. One of the first approaches proposed was cited by the "Bad Debt Paper'"(Bulmer et al., 2000) and adopted as a US actuary Advisory Note. It promotes factor-based deterministic provisions. The counterparty risk is captured through charge factors applied to expected recoveries.But, with the development of ICAS, Solvency II and Enterprise Risk Management (ERM), economic capital modeling has adopted a more prospective view. The various solutions for modeling reinsurance counterparty risk are detailed in Figure(3). We distinguish between two types of model. The first relies on a financial model of credit risk, as promoted byFlower et al. (2007) and bySachs (2007), where one is confronted with the difficulty of dealing with reinsurance credit risk portfolios that are much less diversified than typical banking credit portfolios. The second type is based on actuarial foundations and models shocks on the reinsurance market at a high level, as inGrinda and Nguyen (2006), or in a more schematic way in Ter Berg(2008).

  companies like Swiss Re(2003), or researchers like Arnold (2008), argue that regulation may limit capacity: it is explicit for strong alien reinsurers whose capacity could be of use, if the capital requirement increases to provide protection. Moreover, the market does not seem to value solvency protection.Cole et al. (2007) point out the absence of a relation between use of unauthorized reinsurance and prices. In any case, intensively regulating reinsurance insolvency risk implies additional costs(Harrington, 2005), among them an increase of the cost of reinsurance. Strict reinsurance regulation implies higher costs of reinsurance, and capital immobilization at the insurer level to take into account reinsurance counterparty risk is costly as well. All these costs are ultimately passed on to the insured parties. Advocating regulation of this risk implicitly recognizes the role of the reinsurance industry in spreading global risk.The question of reinsurance counterparty risk can indeed be addressed in a more global regulation question. This includes the degree to which insurance companies should be protected from reinsurance counterparty risk, as well as the degree to which reinsurance regulation should allow the disappearance of not sufficiently sound insurers or reinsurers in case of crisis. Global harmony in regulation could be beneficial for the reinsurance industry, due to their business model that rests on the widest possible diversification and distribution of risk.Rossi and Lowe (2002) point out that there is no central regulation of the reinsurance industry: it differs from one country to another, and mutual recognition is uncommon. As analyzed byAlexander et al. (2006), the IAIS has certainly promoted some basic standards for reinsurance supervision by encouraging coordination between national regulators, but they have not been introduced by all countries. The Group ofThirty (2006) and Evans

Figure 1 :

 1 Figure 1: Reinsurer downgrades from 2002 to the _rst semester of 2011. Source: Bloomberg, Brett and Singh (2005).

Figure 2 :

 2 Figure 2: Sources of reinsurer defaults. Source: Swiss Re (2003).

Figure 3 :

 3 Figure 3: Main assessment methodologies of the reinsurer counterparty risk.

Ecole Polytechnique, Economics Department. Palaiseau, France.

Note that there is also a rational for risk-loving behaviors, particularly when companies face high insolvency risk in a system where there is a guaranty fund.Bohn and Hall (1999) provide evidence of high premium growth for more than one third of US P&C insurance companies in the two years before their failures.

According to Guy Carpenter's website, we can define authorized reinsurance as the reinsurance placed with a reinsurer which is licensed or otherwise recognized by a particular state insurance department. We call an insurer or reinsurer domiciled outside the U.S. but conducting an insurance or reinsurance business within the U.S., an Alien Company.

• Albert, M. (1991). Capitalisme contre Capitalisme. Editions du Seuil.

• Alexander, K., Dhumale, R., and Eatwell, J. (2006). Global Governance of Financial Systems:

The International Regulation of Systemic Risk. International Association of Insurance Supervisor (IAIS), p.61-65. Oxford University Press.


Sabine Meunier
email: meunier@lma.cnrs-mrs.fr

Lma

Professor Bertram Scharf

An overview of Bertram Scharf′s research in France on loudness adaptation

until the mid-1990s and collaborated with the University of Marseille (Faculté de Médecine) until his death. One of Bertram Scharf's major contributions to the field of psychoacoustics is in the area of loudness. He first studied spectral loudness summation, when he started working at Harvard University. In France, his work on loudness focused mainly on loudness adaptation. He wrote, "Loudness resembles pain in that it decreases as a function of time only under special stimulus conditions." Bertram Scharf's work with his French colleagues defined aspects of loudness adaptation in its direct (simple loudness adaptation) and indirect (induced loudness adaptation) forms. They studied how the auditory system recovers from loudness adaptation and examined a possible physiological basis for loudness adaptation.

INTRODUCTION

Professor Bertram Scharf traveled a lot as a Visiting Scientist in Germany, Finland and France. France became his second home and, since 1978, he split his time between Boston and Marseille (or Cassis). In Marseille, he worked at the "Laboratoire de Mécanique et d'Acoustique" (Mechanics and Acoustics Laboratory) with Georges Canévet who has created the psychoacoustics team in the laboratory. Bertram Scharf was part of this "adventure". Afterward, he worked at the "Université d'Aix-Marseille, Faculté de Médecine" where he started his work on the role of the olivocochlear bundle in hearing.

A lot of Bertram Scharf's contributions in psychoacoustics concerned loudness: spectral loudness summation (among his first publications), binaural summation of loudness, loudness adaptation, induced loudness reduction. Bertram Scharf also orientated his work on frequency selectivity and the role of attention in frequency selectivity. He had several contributions on binaural hearing and auditory localization, especially when working in Marseille with Georges Canévet and Jens Blauert from Bochum (Germany). In Marseille, he collaborated with physicians on the role of the olivocochlear bundle in hearing. A pioneer study that could be developed because of the cooperation between Bertram Scharf and surgeons in Marseille that used vestibular neurotomies in the surgical management of incapacitating Meniere's disease, during which the olivocochlear bundle is severed.

In this paper, we will focus on Bertram Scharf's contributions to loudness adaptation.

A sensory system adapts when the sensation decreases over time while the physical stimulus is constant. A lot of sensations adapt: brightness, odor intensity, taste. For example, when exiting from a dark room, the brightness of the sun seems very strong while it decreases after a certain time of adaptation. For a long time, "the question of: does loudness adapt?" was asked. Bertram Scharf and his colleagues gave a first answer: "Loudness may adapt, after all" [START_REF] Fishken | Loudness may adapt, after all[END_REF]. But "Loudness resembles pain in that it decreases as a function of time only under special stimulus conditions." [START_REF] Scharf | Loudness adaptation[END_REF].

Different studies showed a decrease in the loudness of a steady stimulus over the first 2 or 3 minutes of exposure. However, in most of the experiments, an interaural matching procedure was used [START_REF] Hood | Studies in auditory fatigue and adaptation[END_REF]Egan, 1955 for example). A continuous sound, with constant level and frequency, was presented to one ear; the task of the listener was to adjust the level of a brief sound, played intermittently to the other ear, in order to make it equal in loudness to the continuous sound. After some minutes of exposure to the continuous sound, the level of the intermittent sound was decreased, showing a decrease in the loudness of the continuous sound. However, when no contralateral signal was used, very little adaptation was found [START_REF] Canévet | Adaptation de sonie[END_REF][START_REF] Scharf | Loudness adaptation[END_REF]; when it appears, it is called simple loudness adaptation, in contrast to induced loudness adaptation when loudness adapation is induced by a second sound, such as in measurements using matching procedures.

SIMPLE LOUDNESS ADAPTATION

In order to measure simple loudness adaptation, procedures are required that do not involve matching between two sounds. [START_REF] Scharf | Loudness adaptation[END_REF] reviewed some of those procedures: absolute judgments, cross-modality matching and tracking. Among the absolute judgment procedures, Bertram Scharf proposed the method of successive magnitude estimation "whereby the observer assigns numbers to express the loudness of a sound at successive time intervals" (see figure 1).

Simple loudness adaptation occurs in very few cases. It has been observed only for pure tones, broadband noise adapts little [START_REF] Scharf | Loudness adaptation[END_REF][START_REF] Scharf | Loudness adaptation, loudness function, and decruitment: In Fechner day 89[END_REF]. Simple loudness adaptation is very strong, sometimes complete, at the lowest sensation levels (< 20 dB SL). Complete adaptation means that the sounds become inaudible. For levels above 20 dB SL, adaptation decreases rapidly, and disappears for levels above 40 dB SL [START_REF] Canévet | Adaptation de sonie[END_REF][START_REF] Scharf | Loudness adaptation[END_REF], figure 2). Adaptation increases with frequency [START_REF] Canévet | Adaptation de sonie[END_REF][START_REF] Scharf | Loudness adaptation[END_REF], see figure 3). High-frequency puretones produce more adaptation than low-frequency pure-tones: Bertram Scharf and his colleagues (Miskiewicz et al., 1993) showed that, at a high frequency, simple loudness adaptation can take place at relatively high sensation levels (40 dB SL, see figure 3). Bertram Scharf and his colleagues [START_REF] Hellman | Loudness adaptation and excitation patterns: effect of frequency and level[END_REF]Miskiewicz et al., 1993) assumed that adaptation takes place when excitation is restricted to a narrow region of the cochlea. This hypothesis is supported by the strong adaptation observed for high frequencies as well as for low-level sounds that both provide a restricted spread of excitation.

Modulated sounds produce less adaptation than continuous sounds [START_REF] Scharf | Loudness adaptation[END_REF] and there are large interindividual differences in the magnitude of adaptation. Most listeners show a decrease in loudness of about 50% after one minute of exposure to a high-frequency pure tone at a low level; but some listeners may report no change in the loudness, whereas for other listeners the sound becomes inaudible [START_REF] Scharf | Loudness adaptation[END_REF]. [START_REF] Scharf | Loudness adaptation[END_REF] did not find intraindividual differences in the magnitude of adaptation among different sessions. It seems then, that the capacity of loudness to adapt is an individual characteristic.

CONTRALATERALLY INDUCED LOUDNESS ADAPTATION

As described previously, the loudness of a steady sound at high levels also decreases over time when played simultaneously with an intermittent sound in the contralateral ear (contralaterally induced loudness adaptation, figure 4). Many papers co-authored by Bertram Scharf and his french colleagues have reported this effect (for example [START_REF] Scharf | Loudness adaptation[END_REF][START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF][START_REF] Canévet | Loudness adaptation, when induced, is real[END_REF][START_REF] Canévet | Simple and induced loudness adaptation[END_REF][START_REF] Canévet | Loudnes adaptation: when and why it occurs[END_REF]. Figure 5 shows an example of contralaterally induced loudness adaptation (dichotic condition) for a 1000-Hz pure tone at 60 dB SPL (from [START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF].

Contralaterally induced loudness adaptation was observed at relatively high levels (40 and 60 dB SL) of the continuous sounds [START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF][START_REF] Botte | Loudness reduction and adaptation induced by a contralateral tone[END_REF][START_REF] Canévet | Simple and induced loudness adaptation[END_REF]. It increases with frequency but not as much as simple adaptation [START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF][START_REF] Botte | Loudness reduction and adaptation induced by a contralateral tone[END_REF]. It is observed even when the inducer and the continuous tones frequencies differ [START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF][START_REF] Botte | Loudness reduction and adaptation induced by a contralateral tone[END_REF]. But, contralaterally induced loudness adaptation is selective in frequency; a stronger effect was found for inducer frequencies close to the continuous tone frequencies [START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF][START_REF] Botte | Loudness reduction and adaptation induced by a contralateral tone[END_REF]. An inducer at the same level or lower than the continuous tone, may induce loudness adaptation [START_REF] Botte | Loudness adaptation induced by an intermittent tone[END_REF][START_REF] Canévet | Simple and induced loudness adaptation[END_REF].

IPSILATERALLY INDUCED LOUDNESS ADAPTATION

Loudness adaptation may also be induced by an intermittent sound presented to the same ear (ipsilaterally induced loudness adaptation, figure 6). Bertram Scharf's contribution to ipsilaterally induced loudness adaptation can be found for example in [START_REF] Scharf | Loudness adaptation[END_REF], [START_REF] Canévet | Loudness adaptation, when induced, is real[END_REF][START_REF] Canévet | Simple and induced loudness adaptation[END_REF][START_REF] Canévet | Loudnes adaptation: when and why it occurs[END_REF], [START_REF] Scharf | Additivity of induced loudness adaptation[END_REF]. In figure 5, the monotic condition corresponds to ipsilaterally induced loudness adaptation. The adaptation quotient is plotted as a function of the continuous tone duration.The adaptation quotient is defined as: (Ei-Ef)/Eg (see figure 3). In the control condition the continuous tone was presented alone. The dichotic condition corresponds to contralaterally induced loudness adaptation where an intermittent tone is played to the left ear at the same time as the continuous tone in the other ear. The monotic condition correponds to ipsilaterally induced loudness adaptation where the intermittent tone is played to the right ear with the continuous tone. The continuous tone was on for 4 min. The intermittent tone started 30 s after the beginning of the continuous tone and lasted 180 s. (From Botte et al., 1982).

Ipsilaterally induced loudness adaptation does not depend on the frequency of the continuous tone [START_REF] Charron | Frequency selectivity in loudness adaptation and auditory fatigue[END_REF]. It is highly selective in frequency [START_REF] Charron | Frequency selectivity in loudness adaptation and auditory fatigue[END_REF], more than contralaterally induced loudness adaptation. Ipsilaterally induced loudness adaptation is maximal when the continuous tone and inducer have the same frequency. Ipsilaterally induced loudness adaptation depends on the level of the inducer. The increment in level produced to the continuous tone by the addition of the ipsillateral inducer must be 5 dB or more [START_REF] Canévet | Loudness adaptation, when induced, is real[END_REF]. The greater the increment, the greater is the loudness adaptation [START_REF] Canévet | Loudness adaptation, when induced, is real[END_REF]. Ipsilaterally loudness adaptation is stronger for pure tones than for noises [START_REF] Scharf | Loudness adaptation, loudness function, and decruitment: In Fechner day 89[END_REF]. Below 40 dB SPL, the lower the base level (the continuous tone), the greater is the adaptation [START_REF] Scharf | Loudness adaptation, loudness function, and decruitment: In Fechner day 89[END_REF].

Ipsilaterally and contralaterally induced loudness adaptation differ in a number of ways. Thus, they seems to be independent mechanisms. [START_REF] Charron | Frequency selectivity in loudness adaptation and auditory fatigue[END_REF] found commun rules in ipsilaterally loudness adaptation and temporary threshold shift (TTS) suggesting a common origin between the two.

RECOVERY FROM LOUDNESS ADAPTATION

Bertram Scharf studied the recovery from loudness adaptation induced by a contralateral tone [START_REF] Scharf | Récupération après adaptation induite de sonie[END_REF]. When the inducer is turned off (after 3 minutes of listening to the continuous tone plus the contralateral inducer) the recovery from loudness adaptation is small. In the presence of the inducer, the loudness of the continuous tone decreased by 60% within three minutes. Five minutes after the end of the inducer, loudness was still 50% below its initial value (at the beginning of the test, before any adaptation). After a long silence (30 s), and without any contralateral inducer, if the continuous tone is turned on, its loudness is still lower than it was initially. This partial recovery has not been found in simple adaptation.

FIGURE 2 .

 2 FIGURE 2.Loudness adaptation for a 4-kHz pure tone at different levels.(From Canévet et al., 1981). The loudness estimate (Evaluation de la sonie) is plotted as a function of the time after the signal onset (Durée du signal). The sound pressure level is indicated on the left (Niveau de pression) and the mean sensation level is indicated on the right (Niveau de sensation moyen) The circles are for monaural conditions and the squares for binaural conditions

FIGURE 3 .

 3 FIGURE 3. Loudness adaptation as a function of frequency for different sensation levels. Loudness adaptation is defined as the percentage of decline: 100(Ei-Ef)/Eg Where Ei = the initial loudness estimate, Ef = the loudness estimate at the end of the experiment (after 1 min 40 to 6 min depending on the study), Eg = the greater of the two

Figures 5 .

 5 Figures 5. Loudness adaptation for a continuous 1000-Hz, 60-dB tone in the right ear.The adaptation quotient is plotted as a function of the continuous tone duration.The adaptation quotient is defined as: (Ei-Ef)/Eg (see figure3). In the control condition the continuous tone was presented alone. The dichotic condition corresponds to contralaterally induced loudness adaptation where an intermittent tone is played to the left ear at the same time as the continuous tone in the other ear. The monotic condition correponds to ipsilaterally induced loudness adaptation where the intermittent tone is played to the right ear with the continuous tone. The continuous tone was on for 4 min. The intermittent tone started 30 s after the beginning of the continuous tone and lasted 180 s.(From Botte et al., 1982).

Figures

  Figures 6. Ipsilaterallyloudness adaptation

  Figures 6. Ipsilaterallyloudness adaptation

ACKNOWLEDGMENTS

The author wishes to thank Mary Florentine and Georges Canévet for their precious help in writing this paper. The author also thanks Jacques Chatron for his help in exporting the figures.