Nicolas Le Sommer

Romeo Said

Yves Mahéo Valoria

A Proxy-based Model for Service Provision in Opportunistic Networks *

This paper presents a proxy-based model for an enhanced provision of application services in opportunistic networks. This model makes it possible for service providers to specify what services can be delivered by proxies, and what mobile devices are expected to act as proxies. The proxy selection can be performed either in a distributed and autonomic manner or in a centralised manner. This paper also presents some results of the evaluations of this model we have performed using simulation techniques.

Introduction

Nowadays, handheld devices equipped with wireless interfaces supporting ad hoc communications are widespread (e.g., smartphones with a Bluetooth interface, PDA with a Wi-Fi interface). The prospect of using such a communication mode to provide nomadic people with application services appears attractive, especially for the institutes, the companies and the local authorities that cannot (or that do not want to) resort to licenced frequency bands (e.g., UMTS, GPRS) in order to provide end-users with a wide service access area. Opportunistic networking has recently emerged as a solution to support communication in large scale mobile ad hoc networks (MANETs), which are in realistic conditions fragmented in several distinct communication islands due the sparse and irregular distribution of the mobile devices and the fixed infostations that composed them. Opportunistic networking exploits ad hoc communication and device mobility to exchange data, and does not make any assumptions about the existence of a complete path between two nodes wishing to communicate. Furthermore, nodes are not supposed to have or acquire any knowledge about the network topology, which is instead necessary in the traditional MANET routing protocols -which aim at defining and maintaining end-to-end routes between the nodes wishing to communicate. Spatial and temporal paths are defined dynamically, while messages are forwarded to the neighbouring destination(s), and any possible host can opportunistically be used as next relay, provided it is likely to bring the message closer to the final destination(s). An additional delay is usually observed in message delivery, since messages are often buffered in the network waiting for a path towards the destination to be available. Although many application services can tolerate this additional delay, it can be detrimental to the quality of service offered to end-users.

In this paper, we propose a new proxy-based model providing an enhanced multi-hop service discovery and delivery in opportunistic networks. This model allows service providers to specify which services can be delivered by a proxy or a set of proxies, and to select which mobile devices are expected to act as proxies either implicitly using conditional rules or explicitly by specifying the addresses of the mobile hosts. The explicit selection of the proxies enables a centralised management of proxies, whereas the implicit selection of proxies allows a distributed and autonomic management of proxies. This model is based on the "store, carry and forward" principle and on a content-based management of service messages. This model has been implemented in a service-oriented middleware platform supporting opportunistic communications, and has been evaluated using simulation techniques.

The remainder of this paper is organised as follows. Section 1 first introduces the background of our model using a scenario, and then describes this model. Section 2 gives details about the implementation of this model in a serviceoriented middleware platform. Section 3 presents some results we obtained by running our middleware on a network simulator. Section 4 presents related works. Finally, Section 5 provides a summary of our contribution and gives some perspectives.

Opportunistic service provision using proxies 2.1 Background scenario

In opportunistic networks, messages are not simply routed in the network. While travelling from host to host in the network they can also be stored temporarily on certain hosts, and be forwarded later when the circumstances are favourable. Service messages (i.e., service discovery requests, service advertisements, service invocation requests, service responses) are thus stored locally in the caches of mobile devices, and should be returned on demand by these devices if needed. If the mobile hosts were able to process messages according to their content, they could correlate the service discovery requests with the service advertisements, and the service invocation requests with the service responses, and therefore could act as proxies for stateless application services, which are the most important type of services used in ubiquitous and pervasive computing environments (e.g., information services). For the sake of illustration, let us consider a disconnected mobile ad hoc network deployed in a campus. An illustration is shown in Figure 1. This network is composed of several mobile devices used by students and of a service provider SP offering application services to students such as course schedules, campus maps, etc. Moreover in this network, the mobile devices are expected to act as relays.

At time T , only the students who are in the same network island as SP can discover and invoke the services offered by SP (S 1 and S 2 can access SP directly because they are in its vicinity). Let us consider that at time T + ∆t S 2 leaves the network island of SP and joins the network island including S 3 and S 4 , after having discovered and invoked one of the services offered by SP. Since S 2 has stored locally the service messages it has exchanged with SP, S 2 may either spontaneously send an advertisement for the services it has discovered in the network, or reply to a discovery request sent by students S 3 and S 4 for such services. Indeed, since S 2 can process service messages stored locally, it should be able to correlate the service discovery requests sent by S 3 and S 4 with the service discovery request it sent itself previously or with the service advertisement it received previously from SP. As far as the service invocation is concerned, when S 3 and S 4 try to invoke SP with the same requests as the ones S 2 has previously addressed to SP, S 2 is expected to reply with the responses it obtained from SP if these ones are still valid. Thus S 3 and S 4 could access the services offered by SP even if they are not in the same island as SP. In this scenario the mobile host S 2 thus acts as a proxy for the service provided by SP. In contrast to the mobile hosts that only implement the "store, carry and forward" principle, the mobile hosts that act as proxies can process messages according to their content, and can establish a correlation between the requests and the responses sent in the network by the mobile nodes.

In the remainder of this section, we present a taxonomy of services and we show how service providers can select proxies either explicitly or implicitly, how they can manage them, and how the service discovery and delivery are performed. Table 1: Taxonomy of application services for a delivery using proxies.

Overview of the model

A taxonomy of application services Two kinds of application services are likely to be deployed in a pervasive environment and to be provided using opportunistic communications: (1) the stateless application services, whose behaviour is recurrent and independent of the time and that can be cloned by some mobile hosts (i.e., the proxies); (2) and the stateful application services whose behaviour cannot be cloned. Some of stateful applications can have a time-dependent behaviour, which can be reproduced temporarily by proxies. An example of such an application is an application service delivering weather forecast information. Moreover, the responses returned by some application services can depend on the parameters they received at invocation time. Hence, the behaviour of such application services cannot be fully reproduced by proxies. The application service taxonomy given in Table 1 is expected to be used by service providers in order to characterise the services they offer, and to help mobile hosts in deciding how to process the messages they receive for these services. In this taxon-omy, five kinds of application services have been identified: 1) the services whose behaviour can be reproduced without any specific constraints (the proxyable services); 2) the services whose behaviour cannot be reproduced (the nonproxyable services); 3) the services whose behaviour depends on parameters and which therefore can only be partially cloned (the partially proxyable services); 4) the services whose behaviour is time-dependent and which can be reproduced during a given lease (the temporary proxyable services); 5) and the services whose behaviour is timedependent and parameter-dependent (the partially and temporary services).

Proxy selection and management A proxy is expected to clone as faithfully as possible the behaviour of the service provider for which it acts as a proxy both from the viewpoint of the service discovery and invocation during a given lease -such a lease is expected to be defined by a service provider either explicitly or implicitly. In this context, it must be able to publish service advertisements spontaneously or to send service advertisements in response to service discovery requests. Similarly, it must be able to respond to the invocation requests it receives.

In order to decide which mobile devices are likely to act as proxies, the service provider can currently use information such as the frequency and the number of the invocations made by the mobile hosts. In the future, we plan to support other kinds of information such as the location or the mobility degree of the mobile clients. In our model, service providers can select and manage explicitly a collection of proxies for the services they offer. This management mode is well suited for mobile ad hoc networks where some devices are relatively static and reachable at any time by the service providers in order to be invalidated easily. For instance, a relatively static mobile host invoking a service frequently can be considered as a good candidate for playing the role of proxy by the provider that offers this service and that performs a centralised management of proxies. In the centralised proxy management mode, the service provider selects and invalidates proxies by sending control messages in the network explicitly. Such messages notably contain the name of the service, and also the period after which the proxy is expected to renew its lease with the provider. If the proxy does not renew its lease by sending a request, the provider considers that the proxy is not reachable and thus removes it from its list of proxies. On its side, the mobile host stops to act as a proxy. The provider can also invalidate this proxy by sending a control message. Figure 2 presents a grammar describing how the providers can define in their control messages the characteristics of the service, the proxy management rules and the service management policy in terms of service discovery and invocation. For instance, a provider can send to a particular mobile host a control message specifying that this host must act as a proxy for a proxyable service for a given lease period. Moreover it can also specify, that the mobile host is expected to support a proactive service discovery by sending periodically an advertisement, and that the responses it returns do not depend on the time. In addition to a centralised selection and management of proxies, it could be convenient to have an autonomic and distributed selection and management of proxies. Indeed, in opportunistic networks composed only of mobile devices that appear and disappear dynamically, the centralised selection and management of proxies cannot be extremely difficult. The autonomic and distributed management scheme of proxies is not achieved explicitly by the service provider itself, but implicitly by specifying rules in the service advertisement messages These rules will be used by mobile hosts in order to decide if they can act as proxies or not. These rules pertain to the number of different invocations and on a method coverage rate (i.e. the number of the methods invoked by the clients divided by the number of methods offered by the service). Such rules are presented in Figure 2, and an example of the utilisation of these rules is given in the next section. In the future, such rules should also refer to other contextual properties such as geographical properties and mobile host characteristics (e.g. CPU, Memory). Indeed, some services can be relevant only in a given geographic area, or can be provided by mobile hosts having specific hardware characteristics.

As far as the distributed management of time-dependent application services by proxies is concerned, no additional properties are required. Indeed, in opportunistic communication, messages generally include spatial and temporal properties so as to control their propagation in the network, and to avoid that messages disseminate eternally in the network. Typically these messages include a number of hops -which is equivalent to the TTL of IP-and a lifetime. Such information can be used by a proxy in order to decide if a message can be considered as still valid or not, and thus whether it can return this message in response to a request sent by a client. Proxies can also exploit this information in order to decide how many times they can act as providers for a given service. The capability of a mobile host to act as a proxy for a time-dependent application service thus decreases naturally in time. In the distributed mode, each mobile node acting as a proxy is expected to autonomously evaluate if it must stop to act as a proxy.

3 Implementation overview

Middleware architecture

In order to validate our model, we have developed a middleware in Java over an OSGi platform. This middleware is mainly structured in two layers, namely an opportunistic message switching layer and a service management layer (see Figure 3). Actually, both layers are implemented as OSGi services, which can be used by the local application services in order to discover and invoke the application services offered by remote devices opportunistically. The first layer implements a Publish/Subscribe API and is composed of four main elements: a publisher, a message receiver, a cache of messages and a message switching orchestrator that can be configured using strategies. For example, strategies using periodic message emission, or strategies implementing gossip mechanisms can be considered.

The service management layer is composed of several elements collaborating in the service discovery and the service invocation process. Service discovery is mainly achieved by the service tracker, the service publisher and the service register. The service tracker is designed so as to be a subscriber to service advertisement messages and to register the discovered services in the service register. The service publisher is responsible for publishing the local services in the network using the communication layer.

The service invocation is mainly performed by the service invoker and the service response handler which are implemented so as to act respectively as a publisher of service invocation messages and as a subscriber to service response messages. These elements must be used by local services in order to invoke remote services. A further detailed description of these elements can be found in [START_REF] Sommer | A Framework for Service Provision in Intermittently Connected Mobile Ad hoc Networks[END_REF]. In our middleware platform, the proxy-based model we propose is implemented by a specific OSGi service behaving as a client of the communication layer in order to be notified about the service discovery requests and the service invocation requests for the services for which it must act as a proxy. It can also invoke the communication layer in order to look for the responses for the requests it received. If the responses are not available in the cache, or if they are not valid, the proxy forwards the request to the service provider in order to obtain a response. Each service provider can also run a proxy manager that is responsible for the centralised management of the proxies. The distributed and autonomous management of the proxies is, as for it, performed by the mobile hosts themselves. The conditional rules used for the proxy selection are specified by the local services when they register themselves in the service register. The local services considered are OSGi services. OSGi services are characterised by both the Java interface they implement and their non-functional properties. In the current implementation, the messages sent by these application services (or by the middleware platform) are serialised as XML documents structured in two main parts, namely the headers and the content of the messages. The headers provide information that can be used by both the communication layer and the service management layer. A header can contain for example the destination, the origin, the lifetime, the type of messages and a content description. In the current implementation of our platform the content-based management of messages is performed using a MD5 sum of the content. In the messages returned in response to a request, the MD5 sum of the content of the request is also specified in order to make it possible for proxies to quickly establish a correlation between a request and a response. In Figure 4, we present an example of a partial service advertisement returned in response to a service discovery request. The service advertisement messages also include additional information to support the distributed selection of proxies and the provision of services by these proxies. An example of the specification of such information is given in Figure 4. In this example, the service is considered as fully-proxyable. The mobile hosts that have a service method coverage rate higher than 0.9 and that have least 5 different invocation requests and responses are expected to act as a proxy, and to support a proactive and reactive service discovery with a period of service advertising of 30 seconds.

<message id="fb0097820f0b371" type="service-advertisement"> <headers> <header name="origin" value="00:0F:1F:C5:2F:F5"/> <header name="destination" value="*"/> <header name="number-of-hops" value="5"/> <header name="date" value="Nov 29 16:09:47 CET 2006"/> <header name="lifetime" value="12:00:00"/> <header name="md5sum-content" value="186c322f04579a179f1cce23b78e7a555"/> <header name="discovery-request-id" value="db0192810f0b21"/> <header name="md5sum-discovery-request-content" value="186c322f04579a179f1cce23b78e7a555"/> <header name="proxy-selection-rules" value="5,0.9"/> <header name="proxy-service-status" value="fully-proxyable,5,0.9"/> <header name="proxy-service-discovery" value="both,30"/> <header name="proxy-service-invocation" value="time-independent"/> ....

Simulation results

In order to evaluate our model, we have made a series of simulations using the Madhoc simulator 1 , a metropolitan ad hoc network simulator that features the components required for both realistic and large-scale simulations, as well as the tools essential to an effective monitoring of the simulated applications. This simulator, which is written in Java, allow us to run our middleware platform on it.

In this section, we focus on a particular experiment whose objective was to measure the ability to satisfy the client service discovery and invocation efficiently using proxies. For that, we compared our proxy-based service provision model with one implementing a 1-hop discovery and invocation model, and with one implementing a purely epidemic discovery and invocation model. In the 1-hop model, clients must be in the vicinity of a provider offering the service they require in order to discover and to invoke this service, while in the epidemic model, their service messages are forwarded by the other mobile hosts opportunistically (each mobile host forwards all the messages it receives in addition of its own messages).

The simulation environment we consider is an open area about 1km 2 populated with 100 mobile devices equipped 1 http://agamemnon.uni.lu/~lhogie/madhoc/ with Wi-Fi interfaces that evolve following a random way point mobility model. Each mobile host moves at a average speed between 0.5 and 2 m/s. Among these 100 hosts, 8 act as service providers, 60 are potential clients of the services offered by these providers, 32 are neither clients nor providers of services. In our simulation scenario, a provider offers only one service, and the same service is provided by two different providers. The clients are interested in only one service, and they are expected to initially send a service discovery request in the network in order to discover the providers of the service they look for (the clients do not send their service discovery request at the same time). When they have discovered a provider, the clients invoke this provider periodically (every 5 minutes) with a different request. The clients of the same service are designed so as to send the same invocation requests. Each host sends its messages with a periodicity of 20 seconds. Each message has a lifetime of 10 minutes. The services considered in this scenario are stateless application services. Each mobile host having relayed (and put in cache) at least a discovery request, an advertisement, and five invocation requests and responses is expected to start acting as a proxy for this service. The proxy lease is automatically managed by the mobile hosts themselves according to the lifetime of the messages they handle. In Figure 5, we present the network load for the simulations we have made, and in Table 2 we summarize the simulation results we have obtained. In contrast to the epidemic and the proxy-based provision models, the 1-hop service provision model has the advantage of offering a low network load, but to the detriment of the service provision quality since only 0.15% of the service invocation requests have been satisfied during the simulation (i.e., 1 hour). As shown in Figure 5 and Table 2, the proxy-based model has approximately the same invocation success ratio as the epidemic model, but with a better response delay and a lower network load.

Related work

Opportunistic networking has recently appeared as a promising approach to allow applications to communicate when synchronous communications based on end-to-end connectivity are not possible. Many ongoing works aim at defining communication supports for opportunistic and/or disruption/delay-tolerant networking [START_REF] Fall | A Delay-Tolerant Network Architecture for Challenged Internets[END_REF][START_REF] Li | Sending Messages to Mobile Users in Disconnected Ad-hoc Wireless Networks[END_REF][START_REF] Musolesi | EMMA: Epidemic Messaging Middleware for Ad hoc networks[END_REF][START_REF] Pelusi | Opportunistic Networking: Data Forwarding in Disconnected Mobile Ad Hoc Networks[END_REF][START_REF] Shah | Delivering Messages in Disconnected Mobile Ad-Hoc Networks[END_REF]. The epidemic routing protocol [START_REF] Vahdat | Epidemic Routing for Partially Connected Ad Hoc Networks[END_REF], the disconnected transitive communication protocol [START_REF] Chen | Enabling disconnected transitive communication in mobile ad hoc networks[END_REF], the asynchronous probabilistic protocol [START_REF] Lindgren | Probabilistic Routing in Intermittently Connected Networks[END_REF], the context-aware adaptive protocol [START_REF] Musolesi | Adapting Asynchronous Messaging Middleware to Ad Hoc Networking[END_REF][START_REF] Musolesi | Adaptive Routing for Intermittently Connected Mobile Ad Hoc Networks[END_REF], the history-based routing protocol [START_REF] Boldrini | Hibop: a history based routing protocol for opportunistic networks[END_REF], the Opportunistic Spatio-Temporal Dissemination System [START_REF] Leontiadis | GeOpps: Geographical Opportunistic Routing for Vehicular Networks[END_REF] and the Time-Aware Content-based dissemination system [START_REF] Sollazzo | TACO-DTN: A Time-Aware COntent-based dissemination system for Delay Tolerant Networks[END_REF] are some examples of protocols for opportunistic and/or disruption/delay-tolerant networking. Nevertheless, in contrast to our work such protocols do not offer content-based message management functionalities. Such functionalities are necessary in order to implement a proxy-based model for the service provision. Service discovery protocols designed specifically for such mobile ad hoc networks can be divided into two categories: network layer-based and application layer-based service discovery protocols. In the service discovery protocols coupled with routing layer, service query and response messages are often piggybacked on to ad hoc routing messages. In this way, a host requesting a service in addition to discovering the service will also be informed of the route to the service provider at the same time. Examples of such protocols are [START_REF] Zhu | PrudentExposure: A Private and User-centric Service Discovery Protocol[END_REF], [START_REF] Ververidis | Routing Layer Support for Service Discovery in Mobile Ad Hoc Networks[END_REF] and [START_REF] Kozat | Network Layer Support for Service Discovery in Mobile Ad Hoc Networks[END_REF]. Protocols that implement service discovery at routing layer generally reduce the communication and energy consumption overheads. However, the proposals made along this line rely on the assumption that communication between two devices in a network is possible only if these devices are both simultaneously active, and if a transmission route can be established between them whenever needed using reactive or proactive techniques.

In service discovery protocols based on the application layer, the service discovery functionality is supported above the routing layer and usually do not make any assumption regarding this one, and thus seems to be well suited for context-aware and opportunistic networking. Most of these service discovery protocols are implemented in service-oriented middleware platforms. Konark [START_REF] Helal | Konark : Service Discovery and Delivery Protocol for Ad-hoc Networks[END_REF], and DEAPspace [START_REF] Hermann | DEAPSpace: Transient Ad-Hoc Networking of Pervasive Devices[END_REF] and PDS [START_REF] Meier | Proximity-Based Service Discovery in Mobile Ad Hoc Networks[END_REF][START_REF] Nedos | Service*: Distributed Service Advertisement for Multi-Service, Multi-Hop MANET Environments[END_REF] have investigated the service discovery and delivery issues in ad hoc networks with middleware platforms. DEAPspace provides a support for the discovery and the delivery of services in wireless singlehop ad hoc networks. DEAPspace implements a push based service discovery paradigm. The Konark middleware has objectives similar to those of DEAPspace, but considers multi-hop wireless ad hoc networks. Like in DEAPspace, each device in Konark maintains in a directory a vision of services available in the network, and acts both as a client and a provider of services. It supports both push and pull discovery style, and it is limited to one-hop. Moreover, Konark makes some assumptions regarding the network by considering that a transmission route can be established between a client and a service provider whenever needed. Thus it does not support the service provision in intermittently connected mobile ad hoc networks.

PDS, hides the complexity of the underlying communication system by providing an asynchronous advertise/discover programming model to the application layer that is also independent of the structure of service representation. PDS can therefore work with multiple service representations. Moreover, it proposes a discovery approach of ad hoc services that exploits the fact that the relevance of such services is often limited to a specific geographical scope. Service providers are thus expected to define the areas (so-called proximities) in which their services are available. Clients register their interest in specific services and are subsequently informed whenever they come into a "proximity" within which these services are available [START_REF] Nedos | Service*: Distributed Service Advertisement for Multi-Service, Multi-Hop MANET Environments[END_REF].

Conclusion

In this paper, we have presented a proxy-based service provision model for opportunistic networking. This model relies on a taxonomy of application services and a contentbased management of services messages, and allow service providers to specify which mobile hosts should acts as proxies and which services can be proxified. Moreover it supports a centralised control or alternatively a distributed and autonomic management of proxies. This model was evaluated using simulations that show that it is a relevant model for the provision of services. In the future, we plan to improve this model by taking into account geographical properties. In this way service providers could specify in which area the services they offer are considered as relevant, as well as to control the spatial distribution of the proxies. We also would like to improve this model by supporting a proxy collaboration in the service provision through an overlay of proxies.

Figure 1 :

 1 Figure 1: Example of a disconnected mobile ad hoc network.

  :=non-proxyable | proxyable | partially-proxyable | temporary-proxyable-r | temp-part-proxyable-r temporary-proxyable-r:= temporary-proxyable lease-period temp-part-proxyable-r:= temp-part-proxyable lease-period proxy-management:=centralised | distributed centralised := address(, address)* lease-period address:= byte* lease-period:= int distributed := naming-rules naming-rules:=number_of_requests method_coverage_rate number_of_requests := int method_coverage_rate := float discovery-management:= proactive_discovery | reactive | both_discovery proactive_discovery:= proactive period both_discovery:= both period period := int

Figure 2 :

 2 Figure 2: Grammar rules for implicit and explicit proxy naming and management.

Figure 3 :

 3 Figure 3: Middleware architecture

Figure 4 :

 4 Figure 4: An XML-formatted message exchanged by application-level services.

Figure 5 :

 5 Figure 5: Network load.

Table 2 :

 2 Simulation results for service discovery and invocation.

* This work is done in the project SARAH. This project is supported by the French ANR (Agence Nationale de la Recherche) in the framework of the 2005 ARA SSIA program.

http://www-valoria.univ-ubs.fr/SARAH


Nicolas Pinel
email: nicolas.pinel@gmail.com

Christophe Bourlier
email: christophe.bourlier@univ-nantes.fr

LOCAL UNIFIED MODELS OF BACKSCATTERING FROM OCEAN-LIKE SURFACES AT MODERATE INCIDENCE ANGLES

In the context of electromagnetic wave backscattering from ocean-like surfaces, by using the SSA-1 model, Bourlier et al. proposed a technique to reduce the number of numerical integrations to two for easier numerical implementation. To be consistent with microwave measurements, closed-form expressions of the Fourier coefficients of the backscattering RCS are obtained. For Gaussian statistics, previous work is extended to kernels of unified models expanded up to the order two, like the SSA2 and LCA2. Electromagnetic scattering by rough surfaces, Random media, Sea surface, Water pollution, Multistatic scattering

I. INTRODUCTION

From the 1960s, the derivation of the microwave backscattering normalized radar cross section (BNRCS) from ocean surfaces is a topic of investigation which makes progress and remains a challenging task. The first developed model is the TSM [START_REF] Kur'yanov | The scattering of sound at a rough surface with two types of irregularity[END_REF][START_REF] Wright | A new model for sea clutter[END_REF]. Recently, this approach was improved [START_REF] Soriano | A cutoff invariant two-scale model in electromagnetic scattering from sea surfaces[END_REF][START_REF] Sajjad | An improved two-scale model for the ocean surface bistatic scattering[END_REF][START_REF] Boukabara | Development of a modified two-scale electromagnetic model simulating both active and passive microwave measurements: Comparison to data remotely sensed over the ocean[END_REF]. In the last two decades, another group of scattering models was proposed, namely the local unified models. The term local means that the multiple scattering phenomenon is neglected, and the term unified means that the model satisfies the highand low-frequency limits, given by the geometric optics approximation (GOA) and the small perturbation method (SPM), respectively. For more details, see the thorough review of Elfouhaily and Guérin [START_REF] Elfouhaily | A critical survey of approximate scattering wave theories from random rough surfaces[END_REF]. One of the most popular is the SSA2 published by Voronovich [START_REF] Voronovich | Small slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces[END_REF][START_REF] Voronovich | Wave scattering from rough surfaces[END_REF][START_REF] Voronovich | The effect of the modulation of Bragg scattering in small-slope approximation[END_REF]; more recently, models based on the same decomposition of the scattering matrix as SSA2, like the LCA2, were published by Elfouhaily et al. [START_REF] Elfouhaily | Local and non-local curvature approximation: a new asymptotic theory for wave scattering[END_REF][START_REF] Elfouhaily | Extension of the local curvature approximation to third order and full tilt invariance[END_REF].

It is well known that such backscattering models are extremely difficult to implement in the full three-dimensional case, because of the four-fold integral that is involved (two space variables and two frequency variables) and because of the strongly oscillating behavior of the integrand. That is why, recently, Bourlier and Pinel presented [START_REF] Bourlier | Azimuthal harmonic coefficients of the microwave backscattering from a non-Gaussian ocean surface with the first-order SSA model[END_REF][START_REF] Bourlier | Numerical implementation of local unified models for backscattering from random rough sea surfaces[END_REF] an original technique to reduce this computation to a two-fold integral (one space variable and one frequency variable) by resorting to azimuthal harmonic expansion of the BNRCS and by using Bessel functions. This is done for two-harmonic spectra such as the Elfouhaily et al. spectrum [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF].

In this paper, this model is presented and tested for microwave frequencies and different wind speeds. The paper is organized as follows. In section 2, the technique developed by Bourlier and Pinel is briefly summarized, and in section 3, the BNRCS is compared with its first order, easier to compute than the second order. The last section gives concluding remarks.

II. BNRCS OF UNIFIED SCATTERING MODELS

In the literature, from microwave (C and K u bands for instance) experimental data [START_REF] Wentz | A model function for ocean radar cross section at 14.6 GHz[END_REF][START_REF] Bentamy | Ocean surface wind fields estimated from satellite active and passive microwave instruments[END_REF][START_REF] Mouche | Dualpolarization measurements at C band over the ocean: Results from airborne radar observations and comparison with ENVISAT ASAR data[END_REF], it was established that the BNRCS σ pq can be expressed for pq={VV,HH} copolarizations in the form where φ is the observation azimuthal angle with respect to the wind direction, θ the observation elevation angle, and u the wind speed. The isotropic backscattering term σ 0 pq mainly describes the wind speed, σ 1 pq the surface asymmetry between the up (φ=0) and down (φ=180°) wind directions, and σ 2 pq the surface asymmetry between the up (φ=0) and cross (φ=90°) wind directions. For Gaussian statistics, σ 1 pq =0. Non-Gaussian statistics is addressed in [START_REF] Bourlier | Azimuthal harmonic coefficients of the microwave backscattering from a non-Gaussian ocean surface with the first-order SSA model[END_REF] by considering only the SSA-1 kernel.

By considering the first two orders of the kernels of unified models, , which depend on the chosen model, the NRBCS is then equal to the sum of two terms, σ pq = σ 11 pq + σ 12 pq . The subscript "11" results from the correlation of the first-order scattered field, whereas the subscript "12" results from the cross-correlation between the first-and second-order scattered fields. The vectors k 0 and k (boldface denotes a vector) are the horizontal components of the incident and the scattered waves (whereas q 0 and q k are the vertical ones, see figure 1). 2 ), Q z = 2Kcosθ, k B = 2Ksinθ, K=2π/λ, in which λ is the radar wavelength. W 0 (r) and W 2 (r) stand for the isotropic and anisotropic parts of the height autocorrelation function, respectively. The hat over W, as a function of ξ, corresponds to the associated spectrum. In polar coordinates (ξ,φ ξ ), the sea spectrum is assumed to be , which is consistent with the Elfouhaily et al. sea spectrum [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF]. J m and I m are the Bessel functions of the first and second kinds, respectively, and of order m. The above equations show that the BNRCS is obtained from a single numerical integration over the radial distance r, instead of two, with one over r and one over an angular variable. Thus, the numerical evaluation of the order "11" is rather simple. For the SSA model, Ν 1 pq is given by the SPM model, whereas for the LCA model, it is given by the Kirchhoff approximation model. Their expression can be found in [START_REF] Bourlier | Numerical implementation of local unified models for backscattering from random rough sea surfaces[END_REF].

For the LCA2 model, we have (ν=0)

(1)
where

(2) (3)
and δ n,m is the Kronnecker Symbol, such that δ n,m =1 if n=m ; 0 otherwise. Equation ( 2) is the Fourier series of the kernel expressed from the first-and secondorder kernels. In the backscattering direction, their expression can be found in [START_REF] Bourlier | Numerical implementation of local unified models for backscattering from random rough sea surfaces[END_REF]. To simplify the derivation of the BNRCS, the PPT is often applied, which implies a new definition of the kernel . Under the PPT, in Eq. ( 3), the term is derived from the BNRCS (for instance, see Eqs. ( 21) and ( 22) of [START_REF] Bourlier | Numerical implementation of local unified models for backscattering from random rough sea surfaces[END_REF]) by using the series expansion e x ≈ 1+x. By comparison, the scattered field (for instance, see Eq. ( 10) of [START_REF] Bourlier | Numerical implementation of local unified models for backscattering from random rough sea surfaces[END_REF]) derived under the PPT uses the reverse way, that is to say the approximation 1+x ≈ e x .

For the SSA2 model, we have (ν=1)

(4)
where and with real. Unlike the LCA2 model, the SSA2 model requires the computation of a sum over s. For s=0, corresponding to the LCA2 model, Eq. ( 1) is found.

The SSA2 requires the computation of a sum because its second-order sub-kernel depends on the angle φ ξ , whereas the LCA2 second-order sub-kernel is isotropic, since it is independent of φ-φ ξ . This fundamental difference implies that the derivations led for the SSA2 model are more complicated than that of the LCA2 model. It is also important to note that Eq. ( 4) is valid for any kernel obeying the same properties as SSA2 or LCA2.

III. NUMERICAL RESULTS

A. Numerical implementation

For incidence angles of interest for remote sensing applications, θ∈[0;60]°, and for the VV and HH copolarizations, this section presents numerical results of the incoherent BNRCS by assuming an anisotropic height spectrum given by the Elfouhaily et al. [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF] model. Two radar frequencies are studied, f=5.3 GHz (C-band, sea relative permittivity ε r =67+j35 [START_REF] Ellison | New permittivity measurements of seawater[END_REF]) and f=14 GHz (Ku-band, ε r =47+j38 [START_REF] Ellison | New permittivity measurements of seawater[END_REF]).

The numerical evaluations of the BNRCSs are rather simple. First, the isotropic W 0 (r) and anisotropic W 2 (r) parts of the correlation function are computed over r ∈[0;r max ]. For frequencies f={5.3,14} GHz, the maximum radial distance r max ≈{5,1} meters. It decreases when the incidence angle decreases, the wind speed increases and the frequency increases. It is important to note that since the sea correlation function is independent of {θ,f}, {W 0 (r),W 2 (r)} were computed and stored in a data file. In addition, the sampling over the radial distance r is done in a logarithmic scale with 200 samples.

The numerical evaluations of the BNRCSs are more complicated, because they require an additional integration over the sea wavenumber ξ∈[ξ min ;ξ max ]. We choose ξ min =0.25k p , which corresponds to the value for which the sea height isotropic spectrum falls down to 10 -5 times its maximum, which occurs at k p . The value ξ max equals 4K. Thus, the double integration over r ∈[0;r max ] and over ξ∈[ξ min ;ξ max ], is done in a logarithmic scale with 200x200 samples. Finally, the Fourier series coefficients defined by Eq. ( 2) are calculated by using a sampling step over φ-νφ ξ of 3 degrees. The BNRCSs require the computation of a sum over s from 0 to s max . With the LCA model, s max =0 whereas for the SSA model, s max must be determined. This aspect is presented in details in [START_REF] Bourlier | Numerical implementation of local unified models for backscattering from random rough sea surfaces[END_REF]. It is shown that the integer s max increases slightly with the angle θ, is few sensitive to the polarization and the order n of the harmonic. Typically, for θ ≤ 30°, s max ≤ 3 whereas for θ∈]30;60]°, s max reaches the maximum value 7 in average. Thus, the sum over s converges rapidly.

With these parameters, for a given incidence angle θ, on a PC with 4GB of RAM and a processor of 3 GHz, the computing time is of the order of 0.9 second. The scope of the paper is not to compare the different backscattering models with measurements. This was thoroughly already done in previous works [START_REF] Voronovich | The effect of the modulation of Bragg scattering in small-slope approximation[END_REF][START_REF] Bourlier | Azimuthal harmonic coefficients of the microwave backscattering from a non-Gaussian ocean surface with the first-order SSA model[END_REF][START_REF] Mouche | A simplified asymptotic theory for ocean surface electromagnetic wave scattering[END_REF]. For near-nadir incidence angles, figures 2 and 3 reveal that the LCA11 and the SSA11 models are similar, which means that SSA11 reproduces the KA (Kirchhoff Approximation) reduced to the SPA (stationary Phase Approximation).

B. Comparison of SSA2, LCA2 and SPM1

Theoretically, SSA11 does not reproduce the KA, but since the sea surface is highly conducting and the backscattering angle vanishes, the SPM polarization matrix is close to the Kirchhoff one.

It can be noted that by construction, the LCA11 model is the same as the KA+SPA model. For incidence angles ranging from 0 to approximately 20 degrees, only the gravity waves contribute to the scattering and therefore the KA+SPA can be applied. However, a smooth transition for scattering angles θ∈ [20;40]° is observed, for which the KA+SPA model becomes invalid and the Bragg scattering regime (given by SPM) contributed increasingly. In this region, SSA11 tends to SPM, and the higher orders of SSA2 and LCA2 contribute. For the LCA2 model, this contribution is positive for the VV polarization, whereas it is negative for the HH polarization. For the SSA2 model, this contribution is negative for the VV polarization and it is weak, whereas it is positive for the HH polarization and it is much smaller than that of the LCA model. Thus, the behavior of the kernel of each model is very different.

It is important to note that the LCA2 results plotted in figures 2 and 3 use the PPT, which consists in adding the term in Eq. ( 3). If the PPT is not applied, then for larger scattering angles, simulations not reported in this paper, show a non-physical behavior of the BNRCS for the HH polarization. With the SSA model, the results obtained with the PPT are the same as the ones plotted in figures 1 and 2. For a onedimensional sea surface, as explained in details in the paper of Bourlier et al. [START_REF] Bourlier | Comparison of asymptotic backscattering models (SSA, WCA, and LCA) from one-dimensional gaussian ocean-like surfaces[END_REF], although the LCA kernel reaches the SPM and the KA+SPA limits and is tilt invariant, the BNRCS does not converge toward the Bragg regime. A theoretical explanation in given in the conclusion of [START_REF] Bourlier | Comparison of asymptotic backscattering models (SSA, WCA, and LCA) from one-dimensional gaussian ocean-like surfaces[END_REF]. Thus, the use of the PPT in the LCA model allows us to remove this drawback. This is why, in the paper of Mouche et al. [START_REF] Mouche | A simplified asymptotic theory for ocean surface electromagnetic wave scattering[END_REF], the LCA model with the PPT gives satisfactory results on the BNRCS.

C. Optimization of the SSA2 and LCA2 computation

Eqs. ( 1) and ( 4) require the computation of two-fold numerical integrations over the radial distance r and the sea wavenumber ξ. First, integrating over r, the resulting integrand depends then only on ξ. Plotting this integrand versus ξ, we observe that two wavenumbers {ξ l , ξ h } mainly contribute to the scattering process. The first one is defined as ξ l =1.66k p (low frequency), in which k p corresponds to the wavenumber for which the sea height isotropic spectrum is maximum. In fact, 1.66k p corresponds to the maximum of the slope isotropic spectrum (heigh spectrum multiplied by ξ 2 ). The second one is defined as ξ h =k B (high frequency). Thus, the second-order BNRCS results from the inteference of two waves of wavenumbers {ξ l , ξ h }. More precisely, for ξ= ξ h , one can observe that the adjacent wavenumbers also contribute to the scattering process, and we show that

ξ h ⇒ ξ h ∈[k B -Δ ξ ;k B +Δ ξ ] with Δ ξ =0.5k B .
It is equivalent to multiply the sea spectrum by a pulse function centered around k B and of width 2Δ ξ .

Figure 4 plots the ratio of the SSA2 model versus the incidence angle θ, with different choices of the integration over ξ. The labels in the legend mean that for MP1, ξ=ξ l =1.66k p for the integration, for MP2, ξ=ξ h for the integration with a sampling step of 0.1 k B (n ξ =11), for MP3, ξ={ξ l , ξ h } for the integration, else ξ∈[0.25 k p ;4K] (full spectrum) with n ξ =200 (number of samples).

As one can see, for low incidence angles, only the sea wavenumber ξ=ξ l contributes to the scattering process, whereas for moderate incidence angles (Bragg regime), both the low-(ξ l ) and high-frequencies (ξ h ) contribute. Moreover, the results are very close to that obtained from the full spectrum. Thus, with this new integration, the computing time is reduced to 0.07 second instead of 0.90 second, when the full spectrum is used.

Same simulations done with the LCA2 model and not reported here led to different conlusions. Indeed, the results computed from the LCA2+MP2 model are the same as the ones obtained from the LCA2 (full spectrum) model. This shows that only the high-frequency components of the sea spectrum contribute to the scattering process, unlike the SSA2 model, for which both the low-and high-frequency components contribute.

IV. CONCLUSION

In this paper, closed-form expressions of the Fourier coefficients of the BNRCS were expressed for a Gaussian process. Then, the "11" order, resulting from the correlation of the first-order scattered field, requires two independent numerical integrations over the wavenumber ξ (for the calculation of the height correlation function), and over the radial distance r. The "12" order, resulting from the crosscorrelation between the first-and second-order scattered fields, requires two-fold numerical integrations over the radial distance r and over the wavenumber ξ, and one numerical angular integration for the computation of the Fourier series coefficients of the second-order kernel. The SSA2 and LCA2 kernels were tested for microwaves frequencies and different wind speeds. The numerical results showed that the SSA2 and LCA2 have different behaviors, and the correction from the "12" order is larger for the LCA2 model than for the SSA2 model. In addition, an optimization for the numerical integration over ξ was proposed, leading to a computing time of the "12" order less than 0.1 second on a standard office computer for a given wind speed, a given frequency and a given incidence angle.

Figure 1 .

 1 Figure 1. Geometry of the problem By assuming Gaussian statistics, Bourlier and Pinel [13] recently showed that the BNRCS harmonics related to "11" can be expressed as

Figures 2

 2 Figures 2 and 3 plot the harmonics of the BNRCS versus the incidence angle θ for f={14, 5.3} GHz, u 10 =10 m/s and for VV and HH polarizations. The labels in the legend mean SSA11 corresponds to computed from the SSA-1 model with n={0,2} and pq={VV,HH}, SSA11+12 corresponds to computed from the SSA2 model with n={0,2} and pq={VV,HH}, LCA11 corresponds to computed from the LCA-1 model with $ n={0,2} and pq={VV,HH}, LCA11+12+PPT corresponds to computed from the LCA2 model combined with the PPT with n={0,2} and pq={VV,HH}, SPM-1 corresponds to computed with SPM and given by

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. Harmonics of the BNRCS versus the incidence angle θ for f=14 GHz, u10=10 m/s, VV (left) and HH (right) polarizations and for n={0 (top), 2 (bottom)}.


Nicolas Pinel
email: nicolas.pinel@gmail.com

Christophe Bourlier

MODELING OF RADAR SCATTERING FROM OIL FILMS

Oil films on the sea surface damp the capillary waves of the surface height spectrum. This hydrodynamic damping influences the normalized radar cross section (NRCS) of contaminated seas, comparatively to clean seas. First, a simple appropriate damping model is presented with parameters that match experimental results. Second, the twodimensional bistatic NRCS of the contaminated sea surface is presented by comparison with that of a clean sea.

Electromagnetic scattering by rough surfaces, Random media, Sea surface, Water pollution, Multistatic scattering

I. INTRODUCTION

In order to model the power scattered by either clean or contaminated seas (i.e., oil films on the sea surface), a good hydrodynamic modeling of the surfaces is necessary. That is to say, the surface height probability density function (PDF) and the surface height spectrum (equal to the Fourier transform of the surface height autocorrelation function) must be determined. In this paper, the case of insoluble homogeneous oil films on the sea surface is considered, which is valid for wind speeds u 10 inferior to approximately 8 or 10 m/s. Moreover, the two air/oil and oil/sea interfaces are assumed to be identical and parallel, which can be considered for most cases in oil spill detection.

Next, an appropriate electromagnetic modeling for both clean and contaminated seas can be made. This step can be resolved by using "exact" numerical methods, even though this way demands extensive computing time and memory space for the case of contaminated seas. Such methods can then be used as reference methods in order to validate asymptotic models which will preferably be used for the sake of fastness in computing time. For the case of sea-like surfaces, first works on this subject used a two-scale model [START_REF] Ayari | The influence of ripple damping on electromagnetic bistatic scattering by sea surface[END_REF][START_REF] Fuks | Polarization dependence of radar contrast for sea surface oil slicks[END_REF]. More sophisticated asymptotic models can be used, typically, the small slope approximation or the weighted curvature approximation [START_REF] Elfouhaily | A critical survey of approximate scattering wave theories from random rough surfaces[END_REF].

In section II, the influence of an oil film over a sea surface on the surface height spectrum is studied: the surface height spectrum of a contaminated sea (for both air/oil and oil/sea interfaces), S cont , is presented and compared with the one of a clean sea, S clean , given by the Elfouhaily et al. model [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF]. For a contaminated sea surface, the Lombardini et al. model [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF] is used for both air/oil and oil/sea interfaces (which are considered to be strictly identical and parallel in the following), which does not explicitly depend on the oil film thickness. Last, with the knowledge of the relative permittivities of the sea and the oil media, the twodimensional normalized radar cross section (with onedimensional surfaces) of a contaminated sea is compared in section III to that of a clean sea.

II. HYDRODYNAMIC MODELING: SURFACE SPECTRUM OF CLEAN AND CONTAMINATED SEAS

In this section, the hydrodynamic modeling of clean and contaminated seas is led, by considering that the two (air/oil and oil/sea) interfaces of the contaminated case obey the same statistics, i.e. have the same surface height PDF and spectrum. For both clean and contaminated seas, the surface height PDF is assumed to be Gaussian. For a clean sea, the surface spectrum is assumed to obey the Elfouhaily et al. model [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF]. In what follows, the surface spectrum of a contaminated sea is studied more thoroughly.

Lombardini et al. [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF] demonstrated that ripples on a water surface covered by an oil film exhibit a damping effect, which is characterized by a maximum located in the gravity-capillary region of the spectrum, around the frequency f = 10 Hz. This damping effect is expressed by an attenuation coefficient y [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF], which is usually called the Marangoni viscous damping coefficient [START_REF] Alpers | The damping of ocean waves by surface films: a new look at an old problem[END_REF][START_REF] Gade | On the reduction of the radar backscatter by oceanic surface films: Scatterometer measurements and their theoretical interpretation[END_REF]. In the Lombardini et al. model, the damping coefficient y depends on two parameters, namely, the elasticity modulus E 0 , and the characteristic pulsation ω D . Here, a sea covered by an insoluble oil film is considered (for soluble films, see [START_REF] Ayari | The influence of ripple damping on electromagnetic bistatic scattering by sea surface[END_REF][START_REF] Ayari | Marine oil-spills electromagnetic scattering model using two-scale model[END_REF] for instance). Then, ω D depends on the structural relaxation between intermolecular forces. It must be noted that contrary to simulations led by the authors in [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF] which consider organic films, we are here in the case of oil films. This implies that the simulation parameters (E 0 and ω D ) to be used can take a priori different values than the typical ones in [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF]. This was confirmed recently from experiments [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF][START_REF] Sergievskaya | On wave damping due to oil films[END_REF], in which it was concluded that the elasticity values E 0 are smaller for oil films than for organic films, and smaller than hal-00426393, version 2 -26 Oct 2009 Author manuscript, published in "International Radar Conference, Bordeaux : France (2009)" 10 mN/m [START_REF] Sergievskaya | On wave damping due to oil films[END_REF]. Thus, previous works [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF][START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF], which used parameters more typical of organic films, are not very appropriate parameters of the damping model of oil films. Experience gained from experimental results [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF][START_REF] Sergievskaya | On wave damping due to oil films[END_REF] (more precisely, Fig. 4 of [START_REF] Sergievskaya | On wave damping due to oil films[END_REF] and Fig. 3 of [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF]), together with the Cox and Munk experimental results on the surface root mean square (RMS) slopes [START_REF] Cox | Measurement of the roughness of the sea surface from photographs of the sun's glitter[END_REF] allowed us to estimate the range of E 0 between approximately 1 and 4 mN/m. Thus, in what follows, three different cases are presented.

In Fig. 1, the normalized slope spectrum (isotropic part) of a contaminated sea surface, k²S cont = k²M(k)/y, is plotted versus the wave number k for {ω D = 16 rad/s, E 0 = 1 mN/m}, {ω D = 10 rad/s, E 0 = 2 mN/m}, and {ω D = 1 rad/s, E 0 = 4 mN/m}. The wind speed is u 10 = 6 m/s, and the normalization is done by dividing the slope spectrum by the square of RMS slope of the clean sea water σ s clean = 0.186. For comparison, the normalized slope spectrum (isotropic part) of a clean sea surface given by the Elfouhaily et al. model [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF], k²S clean = k²M(k), is plotted versus the wave number k. As expected, one can observe in Fig. 1 that the oil film strongly damps the high frequencies corresponding to the capillary waves. Comparatively to higher values of E 0 , which typically correspond to organic films, the damping in the high frequencies is in general weaker for oil films than for organic films. Moreover, the damping is stronger for {ω D = 1 rad/s, E 0 = 4 mN/m} than for {ω D = 10 rad/s, E 0 = 2 mN/m}, which is stronger than for {ω D = 16 rad/s, E 0 = 1 mN/m}. In what follows, the impact of the oil damping on the surface height auto-correlation function and on the surface RMS slope is studied. Here, we are interested in the height autocorrelation function W 0 , which is equal to the inverse Fourier transform of the surface height spectrum. For a 1-D surface, using the same parameters as for the height spectrum, the normalized height autocorrelation function is plotted in Fig. 2 for a wind speed u 10 = 6 m/s. That is to say, the three autocorrelation functions are divided by the square of the clean sea water RMS height σ h clean = 0.232 m. It can be seen that because the capillary waves are strongly damped by the oil film, the amplitudes of the height autocorrelation function for the surfaces of the oil film are smaller than those for a clean sea surface for small distances. Moreover, it is smaller for smoother films (i.e., the case {ω D = 1 rad/s, E 0 = 4 mN/m} is smaller than {ω D = 10 rad/s, E 0 = 2 mN/m}, which is smaller than {ω D = 16 rad/s, E 0 = 1 mN/m}). As explained in [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF], with the knowledge of the surface slope spectrum k²S of a clean and contaminated 1-D seas, the surface RMS slope can be obtained. Fig. 3 represents the surface RMS slope of 1-D clean and contaminated sea surfaces, versus the wind speed u 10 , in the range [2; 12] m/s. It can be seen that the RMS slope of the contaminated (air/oil and oil/sea) interfaces is significantly lower than that of the clean air/sea interface. Moreover, comparatively to the Cox and Munk experimental model, the Lombardini et al. model is slightly higher, for the higher wind speeds u 10 . This difference can be explained physically. First, the chosen parameters of the Lombardini et al. model were taken to best match experimental results for oil films in Fig. 4 of [START_REF] Sergievskaya | On wave damping due to oil films[END_REF] and Fig. 3 of [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF]. Second, the Cox and Munk experimental results were led for a film which cannot strictly be considered as oil. Indeed, it was a mixture consisting of 40 percent used crankcase oil, 40 percent Diesel oil, and 20 percent fish oil. As a consequence, the comparison with the Cox and Munk experimental results is mainly qualitative. It highlights good qualitative agreement with the simulation results of the Lombardini et al. model, the two last configurations giving the best quantitative agreement. As a consequence, the second configuration, ω D = 10 rad/s and E 0 = 2 mN/m, should be a rather typical configuration of oil films and will be retained for numerical simulations of the electromagnetic modeling to follow. Indeed, it is a good compromise between results in Fig. 4 of [START_REF] Sergievskaya | On wave damping due to oil films[END_REF] and Fig. 3 of [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF] and the Cox and Munk experimental model.

III. HEIGHT AUTOCORRELATION FUNCTION AND RMS SLOPE OF CLEAN AND CONTAMINATED SEAS

IV. ELECTROMAGNETIC MODELING: IMPACT ON THE NORMALIZED RADAR CROSS SECTION

Let us have a look at the modification of the normalized radar cross section (NRCS) from the contaminated sea, in comparison with the NRCS of a clean sea surface. From a hydrodynamic point of view, as explained in section II, the clean sea surface (i.e. the air/sea interface) is modeled by Elfouhaily et al. spectrum model [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF], and the contaminated sea (i.e. the air/oil and oil/sea interfaces, where the two interfaces are assumed to be strictly identical and parallel) by the Elfouhaily et al. spectrum model together with Lombardini et al. damping model [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF]. To resolve the electromagnetic problem, the benchmark numerical method used here is the Propagation-Inside-Layer Expansion method (PILE) [START_REF] Déchamps | Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagation-inside-layer expansion method[END_REF], combined to the Forward-Backward method (FB) [START_REF] Iodice | Forward-Backward method for scattering from dielectric rough surfaces[END_REF] with Spectral Acceleration (SA) [START_REF] Chou | A novel acceleration algorithm for the computation of scattering from rough surfaces with the Forward-Backward method[END_REF], and denoted PILE+FB+SA [START_REF] Déchamps | Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to the forward-backward novel spectral acceleration[END_REF]. This numerical method makes it possible to validate the following semi-empirical approach.

For the case of homogeneous oil slicks covering sea surfaces, the two surfaces are considered as locally smooth (flat) (see Fig. 5 of [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF]). Then, from the knowledge of the scattering coefficient σ oil of the air-oil interface of the contaminated sea, the scattering coefficient σ cont of the contaminated sea is obtained by multiplying σ oil with the square modulus of the ratio of the equivalent Fresnel reflection coefficient, r eq , of the global system air-oil-sea to the Fresnel reflection coefficient of the air-oil interface, r 12 :

oil i i eq cont r r σ χ χ σ × = 2 12 ) ( ) ( (1)
with χ i the local incidence angle. This approximation is consistent with the use of the Kirchhoff approximation (KA), as under this approximation, the considered surfaces are locally smooth (flat). Then, the local incidence angle χ i ≡ χ i 0 is given by the relation 2

) ( i r i θ θ χ - - = (2)
with θ i the incidence angle, and θ r the observation angle. The positive sense is clockwise, then, θ i being directed counterclockwise, it is negative; θ r is directed clockwise in the specular direction. This semi-empirical approach, applied to the KA and reduced to the geometric optics approximation (KA+GOA) in the calculation of the NRCS, gives results in good agreement with the reference method around the specular direction (see Fig. 7 of [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF] which allowed us to validate this approach when used with the KA+GOA). This result is also in agreement with more qualitative previous works [START_REF] Franceschetti | SAR raw signal simulation of oil slicks in ocean environments[END_REF][START_REF] Migliaccio | A physical approach for the observation of oil spills in SAR images[END_REF]. Afterwards, the approach was extended to an asymptotic method which is more adapted to sea surfaces. Fig. 4 represents the semi-empirical approach (denoted as TL for Thin Layer) applied to the first-order of the small slope approximation (SSA-1), denoted as SSATL-1. The wind speed is u 10 = 5 m/s, the radar frequency f = 3 GHz, and the incidence angle θ i = 0 degrees. The parameters of the oil film are {ω D = 10 rad/s, E 0 = 2 mN/m}, with a film thickness H = 10 mm. One can observe a very good agreement of the SSATL-1 with the reference method, at least for observation angles ranging [-40; +60] degrees. Thus, this semi-empirical approach, applied to an appropriate asymptotic model, gives good results for low incidence angles and moderate scattering angles. Moreover, it allows one to obtain the scattering coefficient fast.

In what follows, further investigations are led, in order to test the validity of this semi-empirical approach. For a more precise comparison, the asymptotic models are computed numerically, for a Monte-Carlo process. This has the advantage of comparing the same surfaces. Asymptotic results are obtained from the Kirchhoff Approximation and by using the method of stationary phase (KA+MSP), the first-order small- slope approximation (SSA1), and the Weighted Curvature Approximation (WCA) [START_REF] Elfouhaily | A critical survey of approximate scattering wave theories from random rough surfaces[END_REF].

V. NUMERICAL RESULTS: FURTHER INVESTIGATIONS Figure 5. Bistatic NRCS of a clean sea and of the air/oil interface of a contaminated sea for H = 10 mm: comparison between the reference numerical method and the KA+MSP, the SSA-1, and the WCA for H polarization.

Fig. 5 presents the numerical results of a clean sea, and of the air/oil interface of the contaminated sea with an oil thickness H = 10 mm, for H polarization. The wind speed at 10 m above the sea surface is u 10 = 5 m/s, the radar frequency f = 3 GHz, and the incidence angle θ i = 20 deg. A comparison is made between the reference PILE+FB+SA method, and three models computed numerically. Namely, the KA+MSP without shadowing effect (plotted in full green line), the SSA1 (plotted with cyan circles), and the WCA (plotted with magenta plus signs). The reference PILE+FB+SA method is plotted in blue dashed line for the clean sea and in black dotted line for the air/oil interface.

For V polarization, the numerical results (not presented here) highlight a very good agreement of the asymptotic with the reference method. Only low differences appear for the SSA1, which underestimates the normalized radar cross section (NRCS) for the clean sea. As well, low differences appear for the KA+MSP for very grazing observation angles θ r , owing to the fact that it does not take the effect into account. For H polarization, the same general observations can be done, except from the KA+MSP. Indeed, it overestimates the NRCS for both the clean sea and the air/oil interface, so that it is valid only for θ r ranging [-40; +70] deg, where good agreement is found though. Other simulations led to the same conclusions. Thus, for the single interface case, for V polarization the three models are in good agreement with the reference method, the best one being the WCA. For H polarization, the same conclusion can be drawn, except from the KA+MSP which is valid only for moderate observation angles θ r . Figure 6. Bistatic NRCS of a contaminated sea for H = 10 mm: comparison between the reference numerical method and the KA+MSP, the SSA-1, and the WCA for both V and H polarizations, with and incidence angle θi = 0 deg. Fig. 6 presents the numerical results of a contaminated sea for an oil thickness H = 10 mm, for both V and H polarizations. The simulation parameters are the same as in Fig. 5, except from the incidence angle θ i = 0 deg. The reference PILE+FB+SA method is plotted in red dash-dot line. The other models are computed from the calculation of the NRCS of the air/oil interface σ oil , by using equation ( 1) describing the semiempirical approach. This figure can be compared with Figs. 9 and 10 of [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF], where the semi-empirical approach was applied to the SSA1 computed analytically. Results from Fig. 6, where asymptotic models using the semi-empirical approach are computed numerically, confirm that there is a very good agreement with the reference numerical method. The low differences that appear are very similar to the ones for the single interface case, which confirms the validity of this semiempirical approach. Other simulations for thinner layers led to the same conclusion, which is not surprising: semi-empirical approach is valid for thin layers, and is getting more precise when the layer thickness decreases. Similarly, numerical simulations for other incidence angles and wind speed lead to the same conclusion and allow to confirm the validation of the thin-layer approach.

Thus, this semi-empirical approach, applied to appropriate asymptotic models like the KA+MSP, the SSA, and the WCA, makes it possible to quantify the scattering from an oil film over the sea surface. This new approach gives results in general very good agreement with the reference method, and makes it possible to obtain numerical results very fast.

Figure 1 .

 1 Figure 1. Normalized slope spectrum (isotropic part) of clean and contaminated sea surfaces versus the wave number k. The wind speed is u10 = 6 m/s.

Figure 2 .

 2 Figure 2. Normalized height autocorrelation function W0 of clean and contaminated seas versus the distance. The wind speed is u10 = 6 m/s.

Figure 3 .

 3 Figure 3. Surface RMS slope of 1-D clean and contaminated sea surfaces versus the wind speed u10. A comparison is also made with the Cox and Munk experimental model.

  ω D = 1 rad/s, E 0 = 4 mN/m

Figure 4 .

 4 Figure 4. Bistatic scattering coefficients of clean and seas for V = 10 mm: comparison between the reference numerical method and the SSA-1. The wind speed is u10 = 5 m/s, the radar frequency f = 3 GHz, and the incidence angle θi = 0 degrees.

  u 10 =5m/s, f=3GHz, θ i =0deg. -H=0.01m)

  u 10 =5m/s, f=3GHz, θ i =20deg. -H=0.01m)


Nicolas Pinel

Christophe Bourlier

UNPOLARIZED INFRARED EMISSIVITY OF OIL FILMS ON SEA SURFACES

Keywords: Sea surface, Water pollution, Infrared detectors, Optical propagation, Geometrical optics

The unpolarized infrared emissivity of oil films on sea surfaces is derived under the geometric optics approximation. The multiple reflections at each of interface are ignored, but the multiple reflections between the upper and lower interfaces of the oil film are taken into account. The oil film is assumed to be thin, so that the two interfaces can be considered as parallel and identical. Thus, under the geometric optics approximation, the oil film can locally be seen as a Fabry-Pérot interferometer. Thus, the emissivity of the contaminated sea can easily be obtained from the emissivity of the air/oil interface only. Emissivity comparisons between clean and contaminated seas are then presented to study the oil film detectability.

INTRODUCTION

Remote sensing, by either radar or optical imagery, can be used to detect and monitor possible oil slicks on sea surfaces [START_REF] Wismann | Radar signature of marine mineral oil spills measured by an airborne multi-radar[END_REF][START_REF] Espedal | Detection of oil spills near offshore installations using synthetic aperture radar (SAR)[END_REF]. For optical applications, in order to act quickly when marine oil pollutions occur, it is then essential to dispose of a means that can predict the emissivity, which quantifies the intrinsic radiation of the surface. Then, by calculating the contrast between a sea covered by an oil film (called contaminated sea) and a clean sea, it is possible to study the detectability of marine pollutions.

For optical or infrared (IR) applications, at moderate angles, since the electromagnetic wavelength λ is much smaller than the sea surface mean curvature radius R c , the tangent plane approximation (usually called Kirchhoff approximation, KA) can be applied. Moreover, the electromagnetic wavelength λ being also much smaller than the surface root mean square (RMS) height σ h for both clean and contaminated seas, the geometric optics approximation (GOA) can be applied. Indeed, at optical or IR frequencies, the capillary waves of sea surfaces also have a large mean curvature radius and a large RMS height comparatively to the wavelength λ.

From an analytical approach based on previous work [START_REF] Bourlier | Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics[END_REF], the unpolarized emissivity of a thin oil film over a 2D anisotropic rough sea surface (i.e., for a general 3D problem) is derived under the GOA. Attention is focused here on the case of homogeneous insoluble oil films, which restricts the validity domain of the study to low to moderate wind speeds at 10 meters above the sea surface u 10 less than 8 -10 m/s [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF][START_REF] Lamkaouchi | Water: a dielectric standard. Permittivity of water-petrol mixtures at microwave frequencies[END_REF]. The multiple reflections at each interface are ignored (i.e., for both the upper air/oil interface and the lower oil/sea interface for the contaminated case). By contrast, the multiple reflections between the upper and the lower interfaces of the oil film are taken into account by assuming a locally flat and planar thin oil film. In other words, the two surfaces of the oil film are assumed to be strictly identical and parallel, so that the film can be locally seen as a Fabry-Pérot interferometer. This implies that the whole oil film problem can be treated from the single air/oil surface problem, by substituting the Fresnel reflection coefficient of air/oil single interface for the equivalent Fresnel reflection coefficient of the air/oil/sea film. The calculations are led for IR wavelengths, inside the two windows regions 3 -5 µm and 8 -13 µm at the wavelengths 3.4 µm and 10 µm, for which the refractive indexes of the sea water (taken from the refractive index of pure water by Hale and Querry [START_REF] Hale | Optical constants of water in the 200-nm to 200-µm wavelength region[END_REF] with the salinity adjustment from Friedman [START_REF] Friedman | Infrared characteristics of ocean water (1.5-15 µ)[END_REF]) and the oil [START_REF] Al'perovich | Optical constants of petroleum samples in the 0.25-25 µ range[END_REF][START_REF] Osadchy | The airborne identification of oil films at the caspian sea surface using CO 2 lidar[END_REF] are given in Table 1. Since under the GOA, the emissivity depends on the slope PDF (Probability Density Function), the RMS slope along the wind direction of a contaminated sea surface must be calculated. In this paper, one model of surface slope damping due to oil films is studied: the Lombardini et al. damping model [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF], is a rather simple model which is independent of the oil film thickness.

In what follows, in section 2, the hydrodynamic modeling of the surfaces of clean and contaminated seas is described. In section 3, the emissivity contrast between a clean sea and a contaminated sea is calculated, and the oil film detectability is studied.

HYDRODYNAMIC MODELING OF THE SURFACES OF CLEAN AND CONTAMINATED SEAS

The hydrodynamic modeling of the surfaces of clean and contaminated seas is based on the Elfouhaily et al. model [START_REF] Elfouhaily | A unified directional spectrum for long and short wind-driven waves[END_REF] for the case of a clean sea. By comparison, for the case of a contaminated sea, a damping effect occurs, which has a maximum located in the gravity-capillarity region of the surface wave spectrum, around the surface wave frequency ν s = 10 Hz. This damping effect is expressed by an attenuation coefficient y [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF], which is usually called the Marangoni viscous damping coefficient. For more details, one can refer to [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF].

To describe this damping effect, we refer to two distinct hydrodynamic models of the literature. The first one, namely the Lombardini et al. damping model [START_REF] Lombardini | Modulation of the spectra of short gravity waves by sea surface films: slick detection and characterization with a microwave probe[END_REF], is rather simple, as it depends on only 2 hydrodynamic parameters, but in return is independent of the film thickness H. The second one, namely the Jenkins and Jacobs damping model [START_REF] Jenkins | Wave damping by a thin layer of viscous fluid[END_REF], is more sophisticated, as it depends on 9 hydrodynamic parameters and depends on the film thickness H, but is valid only for thin films. Here, we concentrate on the first model, and represent in Fig. 1 the surface slope spectrum of clean and contaminated seas for a wind speed at 10 meters above the sea surface u 10 = 7 m/s, with different values of the two parameters ω D and E 0 of the Lombardini et al. model (ω D being the oil characteristic pulsation, and E 0 the oil elasticity modulus). These two parameters are chosen to match experimental results of both the surrface wind wave spectrum damping [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF][START_REF] Sergievskaya | On wave damping due to oil films[END_REF] and the RMS slope [START_REF] Cox | Measurement of the roughness of the sea surface from photographs of the sun's glitter[END_REF]. As expected, one can observe in Fig. 1 that the oil film strongly damps the high frequencies corresponding to the capillary waves. Comparatively to higher values of E 0 , which typically correspond to organic films, the damping in the high frequencies is in general weaker for oil films than for organic films. Moreover, the damping is stronger for ω D = 1 rad/s, E 0 = 4 mN/m than for ω D = 10 rad/s, E 0 = 2 mN/m, which is stronger than for ω D = 16 rad/s, E 0 = 1 mN/m.

Then, this hydrodynamic modeling is applied in section 3 to the infrared emissivity of clean and contaminated seas, in order to study the detectability of oil films. To do so, it must be noted that both the upper (air-oil) and lower (oil-sea) interfaces of the oil film obey the same hydrodynamic modeling, with identical parameters. Moreover, as dealing with thin oil films, it is assumed that the two interfaces of the oil film are strictly identical and parallel (see Fig. 5 of [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF]). In what follows, the surface slope spectrum is used to calculate the surface RMS slopes, in the up-wind direction σ sx and in the cross-wind direction σ sy , for both clean and contaminated seas. Indeed, σ sx and σ sy are parameters of the slope probability density function which appears in the expression of the infrared emissivity under the GOA [START_REF] Bourlier | Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics[END_REF].

INFRARED EMISSIVITY OF CLEAN AND CONTAMINATED SEAS

Under the GOA, which is based on the KA, the two interfaces of the oil film can be considered as locally flat and parallel. Consequently, for moderate emission angles and for thin films, the oil film can be considered as locally flat, and appears as a local Pérot-Fabry interferometer. Then, the multiple reflections between the upper and the lower interfaces of the oil film can be modeled in a simple way. Indeed, starting from the single (air/oil) interface case, under the GOA, the Fresnel reflection coefficient of a single interface can be substituted for the equivalent Fresnel reflection coefficient of the air/oil/sea film, calculated by considering an infinite number of reflections inside the film. Its expression is given by equation ( 9) of [START_REF] Pinel | Modeling of the bistatic electromagnetic scattering from sea surfaces covered in oil for microwave applications[END_REF]. In this paper, from an analytical approach based on the work of Bourlier [START_REF] Bourlier | Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics[END_REF], the unpolarized emissivity from a thin oil film over a two-dimensional anisotropic sea surface is derived by using the GOA.

Then, the infrared emissivities of clean and contaminated seas can be computed. Fig. 2 represents the unpolarized infrared emissivities of clean and contaminated seas with respect to the emission angle θ, for a wavelength λ = 3.4 µm, a wind speed u 10 = 7 m/s, and a wind direction φ = 0. The contaminated sea is a heavy oil film of thicknesses H = {0; 10; 50; 100} µm, described by the Lombardini et al. damping model with parameters ω D = 11 rad/s and E 0 = 1 mN/m. The case H = 0 is plotted to highlight the influence of the surface wave damping due to the presence of oil. The result of the contaminated sea for zero thickness H = 0 is very close to the one of the clean sea. Indeed, differences appear only for relatively high emission angles θ, owing to the damping of the capillarity waves in the surface wave slope spectrum due to the presence of oil. Then, for these typical wind speeds, the damping due to the presence of oil has an effect on the unpolarized emissivity only for high emission angles θ. Similarly, the results with various thicknesses do not highlight significant differences with the clean sea case, except for high θ. In fact, the various thicknesses represented here highlight a small and seemingly constant (negative) difference with zero thickness H = 0. Thus, the oil film detection is possible here only for high emission angles θ. Moreover, the differences between the various thicknesses being negligible, the oil film thickness estimation is impossible at this wavelength.

Fig. 3 presents the same simulations as in Fig. 2, but for a wavelength λ = 10 µm. For zero thickness H = 0, the same observations can be done: differences with the clean sea case are observable only for high θ. This highlights the influence of the capillary wave surface damping onto the emissivity. Indeed, it is well-known that the capillary waves play an increasing role as the observation angle θ increases. Contrary to Fig. 2 where λ = 3.4 µm, here for λ = 10 µm the results with various thicknesses highlight differences with the case H = 0 and, most important, with the clean sea case. The differences being significant, this makes the oil film detectable. Moreover, depending on the emission angle θ, it can be seen that the results with various thicknesses highlight significant differences between theses thicknesses, and a general different behavior with respect to θ. Then, from the knowledge of the emissivity of an oil film for a few values of θ, this makes it possible to estimate the oil film thickness.

Fig. 4 represents the unpolarized infrared emissivities of a contaminated sea for two wavelengths λ = 3.4 µm and for either a heavy or a light oil film of thickness H = 50 µm, the other parameters being the same as previously. For the lower wavelength λ = 3.4 µm, the differences between a heavy and a light oil is not significant, making the oil characterization impossible. By contrast, for λ = 10 µm, differences between the two oil types appear for either low or high observation an- [START_REF] Hale | Optical constants of water in the 200-nm to 200-µm wavelength region[END_REF] with the salinity adjustment from Friedman [START_REF] Friedman | Infrared characteristics of ocean water (1.5-15 µ)[END_REF]), and the oil [START_REF] Al'perovich | Optical constants of petroleum samples in the 0.25-25 µ range[END_REF][START_REF] Osadchy | The airborne identification of oil films at the caspian sea surface using CO 2 lidar[END_REF]

CONCLUSION AND FUTURE WORK

In conclusion, the emissivity of clean and contaminated seas was computed by using the classical geometric optics approximation for one interface. The extension to the contaminated sea was done by using the Lombardini et al. damping model with parameters chosen to get good agreement with experimental results [START_REF] Ermakov | Possibilities of identification of oil films using radar probing of the sea surface[END_REF][START_REF] Sergievskaya | On wave damping due to oil films[END_REF][START_REF] Cox | Measurement of the roughness of the sea surface from photographs of the sun's glitter[END_REF], and by considering an oil film with two identical and parallel interfaces. Numerical results are then presented in the middle of the infrared window regions, at λ = 3.4 µm and at λ = 10 µm. Thus, at the lower wavelength λ = 3.4 µm, the oil film detection is possible mainly for high observation angles θ, and the oil film characterization and thickness estimation are impossible. By contrast, at the higher wavelength λ = 10 µm, the oil film is much easier and it is possible in general at any observation angle θ. Moreover, the oil film characterization is possible for either low or high θ, and the thickness estimation is hoped-for if measurements are made at several observation angles θ.

Fig. 1 .

 1 Fig. 1. Isotropic part of the surface slope spectrum of clean and contaminated seas (using the Lombardini et al. damping model) versus the surface wavenumber k s . The wind speed is u 10 = 7 m/s.

Fig. 2 .

 2 Fig. 2. Unpolarized emissivities of clean and contaminated seas versus the emission angle θ, for λ = 3.4 µm, u 10 = 7 m/s, and φ = 0. The contaminated sea is a heavy oil film of thicknesses H = {0; 10; 50; 100} µm, described by the Lombardini et al. damping model with parameters ω D = 11 rad/s and E 0 = 1 mN/m.

Fig. 3 .

 3 Fig. 3. Same simulations as in Fig. 2, but for a wavelength λ = 10 µm

Fig. 4 .

 4 Fig. 4. Unpolarized emissivities of a contaminated sea versus the emission angle θ, for λ = {3.4, 10} µm, u 10 = 7 m/s, and φ = 0. The contaminated sea is either a heavy or a light oil film of thickness H = 50 µm, described by the Lombardini et al. damping model with parameters ω D = 11 rad/s and E 0 = 1 mN/m.

Table 1 .

 1 Refractive indexes of the sea water (taken from the refractive index of pure water by Hale and Querry

  Thus, this highlights the possibility of characterizing different oil types at this typical wavelength, which can be useful to determine the origin of an oil pollution.

			for
	λ = {3.4; 10} µm, respectively.
	Sea water	Heavy petroleum Light petroleum
	1.426 + j0.019	1.41 + j0.160	1.45 + j0.080
	1.227 + j0.050	1.52 + j0.002	1.53 + j0.001
	gles θ.