THE SACRED IN THE LORD OF THE RINGS

Marc Chémali, Université Paris Nanterre I have often wondered at the fact that The Lord of the Rings, with its gruesome battles, acts of heroism and rather rigid moral frame, should have become such a cult book among the hippy community who were, if my fond memories do not deceive me, more bent on promoting love, not war, and whose moral system could be summed up in the caption "if it feels good, do it."

Ethereal, long-haired Elves and the return to nature would not suffice to explain the phenomenon, nor would mere escapism or what Tolkien himself called "arresting strangeness".

A likelier reason for this unexpected success lies in the fact that, for Tolkien, a fervent Catholic who described himself as an anarchist, power corrupts and absolute power corrupts absolutely.

The only ones worthy of holding power are precisely those who are prepared to renounce it. And the only being that can hold absolute power without being corrupted by it is not a creature but the Creator (in "On Fairy Stories", Tolkien talks about "the right to power as distinct from its possession" [START_REF] Tolkien | The Homecoming of Beorhtnoth[END_REF] ). Thus, in addition to awakening a feeling of wonderment within the reader through this "arresting strangeness" 2 , the narrative appears as a profane representation of transcendence and appeals to the sense of the sacred (as distinct from religious belief). This, in addition to his healthy wariness of power, may also explain the unforgivable success of his epic. Daniel Grotta-Kurska describes Middle-earth as "God's world for any Christian travelling through it." [START_REF] Grotta-Kurska D | Architect of Middle-earth[END_REF] But even rabid atheists like me, for instance, cannot wholly forgo their sense of the sacred, and Tolkien does not cater only for Christian believers: what The Lord of the Rings offers all its readers is, as it were, guilt-free natural religion experienced vicariously through the characters.

I shall therefore start with a few remarks on the links between literature and the sacred. If all literature is, as Kathryn Hume contends, "a meaning-giving experience" [START_REF] Hume | Fantasy and Mimesis[END_REF] , it already appears akin to the sacred, which can be described as the meaning-giving experience. This applies to any type of literature but I wholly agree with Jean-Jacques Meslin's claim that fantasy literature is the privileged fictional medium of the sacred, the one best suited to represent it. On the one hand, the laws of the world of fantasy differ from those of the reader's world, and on the other -and this is true of The Lord of the Rings -the omniscient narrator vouches for the supernatural dimension of certain events and characters. In "On Fairy Stories", Tolkien insists on his wariness regarding both the terms "supernatural" and "fantasy". With regard to the latter, in Fantasy: the Literature of Subversion, Rosemary Jackson, for lack of a better word in English borrows Todorov's classification in Initiation à la littérature fantastique and refers to "the marvellous" to designate the genre in which the supernatural occurrences are established beyond doubt. 5 It so happens that the irruption of the marvellous in the narration perfectly corresponds to the irruption of the sacred in profane reality. Following his identification with certain characters, the reader of fantasy or marvellous literature is vicariously led to experience what Rudolf Otto calls "the feeling of the state of creature". Indeed, Otto states that there exist analogical correspondences between "numinous feelings" and profane emotions. He also indicates that the emotions triggered by a narrative, i.e. by a work of fiction, stand halfway between the two. It follows that fantasy literature buffs, more than mainstream literature readers, are thrill seekers who yearn for feelings more acute than profane reality -or fiction -can awaken. 6 This, among other reasons, is why Tolkien appears as an exceptional representative of the genre, as an author whose individual work transcends it while contributing to its definition. In

The Lord of the Rings, he is not content with pulling the strings of the genre in order to fulfil the expectations of readers seeking intense emotions, even though he most convincingly demonstrates that this expectation on the readers' part is perfectly legitimate and modestly describes his approach to writing in the following terms:

The prime motive was the desire of a tale-teller to try his hand at a really long story that would hold the attention of readers, amuse them, delight them, and at times maybe excite them or deeply move them. [He further adds] As for any inner meaning or message, it has in the intention of the author none. 7 What differentiates Tolkien from the plethora of authors who were inspired by his work or, at least, his approach, is, of course, the mastery he displays in his use of the marvellous elements. A mastery which leads to a luxuriance of supernatural manifestations without, in my opinion, his ever falling into the trap of "arresting strangeness" for its own sake. It would be interesting to proceed taxonomically and bring out a structure from this symbolic profusion. For a while, I was tempted by a reading inspired from Gilbert Durand's theory of symbols that could result in quite a rigorous classification of the staggering proliferation of images in the work 8 . But in Tolkien's world, the Ring that brings invisibility to its wearer -a theme as old as the ring of Gyges -the enchanted swords, the gift of healing, and so on and so forth, in addition to manifesting the structures of imagination which Durand classifies in Les Structures anthropologiques de l'imaginaire, are carefully integrated into a complex system aiming at harmonising two diametrically opposed world visions. As I could not satisfactorily deal with both aspects in so short a time, I have chosen to follow the latter rather than apply a reading grid which, although interesting, would yield a rather stark analysis if left to stand on its own. The coexistence of these two visions generates a tension that underlies the narration and determines all its aspects. It originates, on the one hand, in Tolkien's resolutely Christian position. Indeed, according to his own repeated declarations, he had put his pen at the service of his faith and was not wholly undeserving of his "Oxford Christian" title. On the other hand, his writing reflects his fascination for ancient Anglo-Saxon, Germanic and Scandinavian texts which he was among the first scholars to approach from a literary standpoint rather than as mere linguistic material enabling the study of dead languages. I have borrowed from Benvéniste the opposition between the sacred and the holy that underlies Tolkien's story telling. 9 Benvéniste opposes sacer and sanctus, the former referring to the bipolarity of the sacred, the latter to the exclusive manifestation of the divine. The representation of transcendence in The Lord of the Rings operates through both categories, which leads to a redoubling of the opposition that informs the narration.

The multiplication and entanglement of oppositions makes the analysis slightly arduous but highly enthralling. Indeed, the Judeo-Christian vision appears as a backdrop to the pagan representation of the sacred. The Judeo-Christian moral values -love and respect for one's neighbour, selflessness, truthfulness and, above all, renunciation of power -are universalised in the framework of a cosmogony traced on the biblical model. Like the God of the Scriptures, 8 DURAND G. Les Structures anthropologiques de l'imaginaire, Paris, Bordas, 1969. 9 BENVÉNISTE É. Le Vocabulaire des institutions indo-européennes, Paris, Minuit, 1969.

Iluvatar creates Valar, Elves and Men in his own image and, in so doing, provides them with a model of conscience. Obviously, as he develops his own Book of Genesis, Tolkien noticeably veers away from the letter of his biblical model. He manages, however, to inscribe his narration in the biblical perspective by presenting the reader with a postlapsarian world -since the inhabitants of Middle-earth, wise or otherwise, know the difference between good and evil -but still a world that precedes the Revelation, so that this knowledge of good and evil appears to be spontaneous and, as I have just stated, universal. This is what is conveyed by Aragorn's answer to Éomer during their first meeting: 'It is hard to be sure of anything among so many marvels. The world is all grown strange. Elf and Dwarf in company walk in our daily fields; and folk speak with the Lady of the Wood and yet live; and the Sword comes back to war that was broken in long ages ere the fathers of our fathers rode into the Mark! How shall a man judge what to do in such times? 'As he ever has judged,' said Aragorn. 'Good and ill have not changed since yesteryear; nor are they one thing among Elves and Dwarves and another among Men. It is a man's part to discern them, as much in the Golden Wood as in his own house. 10 Likewise, the universalisation of Judeo-Christian values invokes Plato's equation of Beauty and Goodness. Very few examples contradict this position, in fact only two: Sauron's ringultimately beautiful and ultimately evil -and Saruman's voice. To Milton's Satan's "all good to me is lost; evil, be thou my good", Tolkien adds Morgoth's and Sauron's "all beauty to me is lost; ugliness, be thou my beauty." Such a system would seem to leave no room for the very principle of hierophany that characterises the sacred. The cosmos is holy because it manifests through its very existence that of the divine. This principle is clearly stated by Gandalf in one of his confrontations with Denethor: But I will say this: the rule of no realm is mine, neither of Gondor nor any other great or small. But all worthy things that are in peril as the world now stands, those are in my care. And for my part, I shall not wholly fail of my task, though Gondor should perish, if anything passes through this night that can still grow fair or bear fruit and flower again in days to come. For I am also a steward. Did you not know? 11 This resolutely Judeo-Christian vision is contradicted by the mise en abyme of the notion of sub-creation that Tolkien coins to describe his activity as a writer. Indeed, if the Creation is Iluvatar's doing, the actual completion of this great work is entrusted to the Valar. The latter therefore appear as reproductive and tutelary powers and provide the ancestors of Elves and Men with models of behaviour in addition to the model of conscience originating in the Creator.

The Third Age, that ends with the fall of Sauron, can thus be defined as a hybrid period during which the last details of the foundation of the Cosmos -the sacred action par excellence -are left to elvish and human heroes and no longer to beings of divine essence.

Thus is the sacred reintroduced in a Cosmos initially defined as holy. Evil is brought into the world by Morgoth, a fallen Vala, who will establish himself as anti-model in order to thwart the work of the other Valar who then represent the positive polarity of the sacred. In both cases, model and anti-model manifest themselves in the world in the form of hierophanies. In such a context, it was inevitable that profane and holy should occasionally merge as they both come into opposition with the sacred. The latter concept itself embraces an irreducible opposition between the two poles that constitute it, namely hieros and hagios, fas and nefas. By contrast, the irruption of the sacred evacuates the holy dimension to underline the undifferentiated and chaotic nature of the profane universe. The sacred is therefore opposed to the profane, which alternately appears to the reader either as holy or as a pale reflection of a transcendental model.

It is the complexity of this cosmogonic and, let's not deny it, theological system that counterbalances the seemingly Manichean dimension of the conflict opposing those who try to serve the divine design to those who claim to serve a new design, independent from the former; in other words, the conflict between the goodies and the baddies. These tension-generating oppositions culminate in The Lord of the Rings, which tells the story of the sacrificial crisis marking the passing from one Age to another. A Girardian reading of this absolute conflict, mythic in its cyclical repetition, reveals the staging of a highly pagan violence leading to the establishment of a Cosmos deserted by sub-, super-or para-human creatures, inhabited solely by men and subjected to divine laws that, from the point of view of the Christian reader, will be the object of a further revelation; in other words, a Cosmos in which the cyclical dimension of mythic time gives way to the linearity of History, more specifically, Holy History. This leads me to the narrative categories of the sacred, namely time, space and language. My analysis of the first two draws heavily on the work of Mircea Eliade, who actually coined the term hierophany. As regards the representation of time, I would say that the "History" of Middle-earth repeats itself a little too often for the historian not to become a mythologist. If we take The Silmarillion into consideration, it appears that the successive conflicts that have ravaged Middle-earth and that foreshadow the War of the Rings are described by Gandalf and other "loremasters" as the same fight, eternally recurring, between the two poles of the sacred. Gandalf puts it in a nutshell at the outset of the narrative telling Frodo: "Always after a defeat and a respite, the Shadow takes another shape and grows again" 12 , uncannily echoing Paul Ricoeur who declares:

The sacred universe is the universe that emerges from chaos and that may return to it at any moment. Heaven is only order and life is only a blessing because the underlying chaos must ceaselessly be vanquished. The sacred is dramatic. [my translation] [START_REF] Ricoeur | Manifestation et proclamation[END_REF] However much these conflicts may order themselves within a historical frame and feature different protagonists every time, they still remain reactualisations of the primordial conflict between the Valar ab initio. Moreover, the protagonists themselves, despite their historical dimension and their individuated nature, clearly appear as avatars of founding ancestors that have preceded them. This dual status is nowhere more obvious than in the case of the couple constituted by Aragorn and Arwen, for example. They appear both as avatars and historical descendants of a mythical founding couple uniting Elves and Men, namely Beren, a man, and Luthien Tinuviel, an elvish princess. The description of their encounter is, indeed, highly revealing in this regard:

For Aragorn had been singing a part of the Lay of Luthien which tells of the meeting of Luthien and Beren in the forest of Nildoreth. And behold! There Luthien walked before his eyes in Rivendell.

For a moment Aragorn gazed in silence, but fearing that she would pass away and never be seen again, he called to her crying, Tinuviel!, Tinuviel! Even as Beren had done in the Elder days long ago. 14

[p. 1058]

Once the mistaken identity problem has been solved, it appears hardly to have been a mistake.

When Aragorn tells her that she is the perfect likeness of her ancestor Arwen replies: "So many have said […]. Yet her name is not mine. Though maybe my doom will not be unlike hers." 15 Needless to say, her doom turns out to be identical to her ancestor's. Lastly, the time framework of The Lord of the Rings is itself most ambiguous in this regard: it is presented as the outcome of a historical development but, at the same time, it is the founding moment of a new Age and as such, pertains to myth.

To this double time prism there corresponds a representation of space that is heterogeneous in more than one sense. In this semi-mythical context, the Ptolemaic and Copernican conceptions of the universe are superimposed: setting off for Valinor, the Elves, inheritors of the old order, sail upon a flat sea and arrive at their destination by following what Tolkien calls the "Straight Road". When Men travel westwards, on the contrary, they can only come back full circle to where they started from, for as far as they are concerned the earth is round. But even in Middle-earth where the action takes place, the various types of breaks in space lead to a highly complex representation of the latter. Indeed, space is organised in networks where we find an opposition between infernal and Edenic spaces, both these types of space being sacred, that is to say separate, forbidden: Mordor and Mirkwood, on the one hand, and Rivendell and Lothlorien on the other. The characters who do not belong to them and enter them are respectively sullied by them or, on the contrary sully them. In all cases, they die for it or escape death very narrowly. These spaces are distinct from the consecrated places inhabited by Men that are licit and are yet regarded as special, if not altogether separate: in other words, as holy. It is the case of Hobbiton, for example, the celestial Jerusalem, in a way, protected by a hedge but totally vulnerable, and of Minas Tirith, the earthly Jerusalem, protected by battlements and yet hardly less vulnerable, after all. Finally, these spaces, whether sacred or holy are opposed to chaotic, unfounded, perfectly profane spaces systematically designated by the term "wilderness" and systematically described as dangerous and hostile: very close, in fact, to those sacred spaces belonging to the negative polarity of the sacred.

In addition to these breaks in space and time that constitute the foundation of Tolkien's world, we find linguistic breaks that contribute to the process of marking the text with the seal of the sacred and manifesting the principle of spirit/matter permeability. Both the narrator and the protagonists almost systematically underline the irruption of the sacred in the world by adopting archaic and, sometimes, unfortunately slightly bombastic language (but then Tolkien was quite aware of that and did convincingly justify his frequent use of the "high style"). Likewise, in certain cases where a character is confronted with a hierophanic manifestation: speech gives way to song or silence, prose gives way to verse: 'Fair Lady Golberry!' [Frodo] said again. 'Now the joy that was hidden in the songs we heard is made plain to me.

O slender as a willow-wand! O clearer than clear water! O reed by the living pool! Fair River daughter! O spring time, and spring again after! O wind on the waterfall, and the leaves' laughter!

Suddenly he stopped and stammered, overcome with surprise to hear himself saying such things. 16 This leads to the hierophanic dimension of language itself. The invention of Elvish languages was, according to the author, the real starting point of his literary adventure. These languages, notably Quenya, appear as liturgical, a paradoxical status in a context where no reference is ever made to any cult whatsoever, barring the sort of grace Faramir performs in the presence of Sam and Frodo. But as I have just stated, Quenya presents a magico-religious dimension that makes it not only the privileged medium for the expression of the sacred, but also a hierophany in its own right. On several occasions, characters who do not know this language experience a sort of Pentecostal revelation and, uttering incantations, or simply names, in the aforementioned language, gain the upper hand over supernatural forces that are far beyond them. This, for example, happens twice to Sam on the edge of Mordor, and is all the more striking as the character is of humble extraction and quite unlearned.

But the hierophanic dimension of language is not limited to Elvish languages. The common language itself may at any moment become infused with power and act upon the world.

Thus the name of Sauron is often replaced by periphrases for fear that its very utterance should bring about disaster or at least sully its user: "He whom we do not name", "the nameless one", "the Enemy", etc. Likewise there are things it is not permitted to mention. This is a theme Ernst Cassirer develops in Language and Myth, where he declares: That the name and the being should be intrinsically and necessarily related, that the name should not only designate the being, but that it should itself be the being and that the force of the being should be within it, such are a few of the fundamental presuppositions of the mythical intuition itself. [My translation] 17 The most spectacular manifestation of this sacred power of language can be found in the confrontation between Gandalf and Saruman during which the former breaks the latter's staff simply by declaring: "Saruman, your staff is broken." 18 A situation where reality radically changes to fit speech. As Caillois contends, power is first and foremost the power of speech. 19 What these multiple fractures convey is that everything in the universe of homo religiosus is susceptible of acquiring the status of hierophany. Nevertheless, if the reorganisation of the elements of the referential universe that characterises fantasy literature makes the latter more apt to express the sacred than mainstream literature, some of these elements lend themselves more than others to hierophanisation. This leaves us with the representation of the sacred-oriented mentality and the status of power to examine. Once more, we see the principle of coincidentia oppositorum at work. Insofar as the sacred is clearly an experience of limits, it is hardly surprising to note that the sacred nature of an object or a being is characterised by its excessive dimension. In

The Lord of the Rings, the representation of the sacred invokes the categories of the extreme, regrouped in antithetical pairs: the huge and the tiny, the ancient and the newly-appeared, the hideous and the wonderful, etc. If the sacred nature of some of the elements in Tolkien's world seems obvious to the characters who are confronted with them, and through those characters, to the readers, the necessity of interpreting such signs entails a division of the characters into two groups: lore masters and uninitiated characters. Insofar as knowledge of good and evil is presented as an immediate, spontaneous given of conscience, most of the teaching dispensed by the lore masters consists in a rehabilitation of myth and/or legend. At this point we may talk 'Halflings!' laughed the Rider that stood beside Éomer. 'Halflings! But they are only a little people in old songs and children's tales out of the North. Do we walk in legends or on the green earth in the daylight?' 'A man may do both,' said Aragorn. 'For not we but those who come after will make the legends of our time. The green earth, say you? That is a mighty matter of legend, though you tread it under the light of day!' 20 Through this mise en abyme, Tolkien rehabilitates fantasy literature as a whole. For those characters who have the reader's trust tirelessly reaffirm the truth of these myths within the myth and, of course, the events themselves continually prove them right. This is where we find ourselves faced with the theme of power to which I have alluded.

For in Middle-earth as elsewhere, knowledge and power are indissolubly linked. Saruman's Faustian fate and the participation of the Elves in the making of the Rings are obvious examples of this. Admittedly, Tolkien somehow manages to establish a distinction between two sorts of power, one whose aim is to dominate, the other whose aim is to protect: what he respectively calls in his "Essay on Fairy Stories" goetia and magia. Nevertheless, what The Lord of the Rings brilliantly demonstrates is that the teleological justification of power is necessarily limited owing to the very nature of the latter. Indeed the very logic of power itself inevitably leads to the appropriation and negation of the other. In most emphatic tones do both Gandalf and Galadriel refuse to take the Ring that Frodo willingly offers them. Thus it is that the true aim of the Quest is the renunciation of absolute power, the power that has its source in the spirit/matter permeability: that power can only be legitimately exercised by the Creator. The reader therefore witnesses a sort of magico-religious crescendo that culminates with the destruction of the Ring.

In turn, this destruction of the ultimate form of power in Middle-earth entails the disappearance

  17 CASSIRER E. Langage et mythe, Paris, Minuit, 1973.18 TOLKIEN J.R.R. The Lord of the Rings, op. cit., p. 583.19 CAILLOIS R. L'Homme et le sacré, Paris, Gallimard, 1938. about metamyth. Indeed, the sacrificial crisis partly results from the oblivion into which the founding stories have fallen. Historical -or Mythical -truth only survives in degraded, truncated form. The sceptics of Middle-earth call such surviving narrations myths in the pejorative sense of superstitious beliefs held only by the gullible. The role of the wise of Middleearth is therefore to give back to those myths their original meaning of essential truth. Witness Aragorn's reaction to the incredulity manifested by a Rohirrim regarding the existence of Hobbits:

JACKSON R., Fantasy: the Literature of Subversion, London, Routledge, 1988.

OTTO R. Le Sacré, Paris, Payot, 1929.

TOLKIEN J.R.R. The Lord of the Rings, London, Harper Collins, 2005, p. xxiii.

TOLKIEN J.R.R. The Lord of the Rings, op.cit., p. 438.

Ibid., p. 789.

TOLKIEN J.R.R. The Lord of the Rings, op. cit., p. 1058.

Ibid., p. 1058.

Ibid., p. 123/124.

TOLKIEN J.R.R. The Lord of the Rings, op. cit., p. 434.

of this spirit/matter permeability from a world forsaken by all non-human creatures endowed with reason. Once more, at the end of the narration, Myth caves in upon itself to make way for History.

The Lord of the Rings is a hybrid narration, a work founded on a paradox, where the narrative codes of romance harmoniously cohabit with modern novel techniques, for example.

But the supreme paradox lies in Tolkien's project itself. This project consists in trying to regenerate reality and resemanticise it by dipping it, as it were, in fantasy, that is, in the genre furthest removed from it. As far as Tolkien, who, as I said at the beginning, had not usurped his "Oxford Christian" title, was concerned, the very essence of reality was to be found in the Catholic creed. I have tried to show how he managed to express what he considered to be the Truth through a story involving creatures and natural laws that contradict both reality and the text which, he believed, contained its essence, namely the Scriptures. C.S. Lewis had written a letter to him in which he described mythopoetic activity as "breathing lies through silver".

Tolkien's answer in writing The Lord of the Rings could be summed up as "breathing the Truth through silver lies".


Uygar Karabas

Youssef Diouane

Rémi Douvenot

A Variational Adjoint Approach on Wide-Angle Parabolic Equation for Refractivity Inversion

Keywords: Refractivity inversion, inverse problem, wideangle parabolic equation, adjoint model

Radar systems performance under anomalous propagation conditions can be predicted if the atmosphere is properly known. This paper introduces a tomographic approach to estimate refractivity in the troposphere with an adjoint-based inversion method. A new adjoint model is developed for the two-dimensional wide-angle parabolic equation using variational adjoint approach, to invert refractivity from phaseless data measured with an array of radio receivers in open-sea environment. The obtained adjoint model is validated considering propagation through range-independent medium over flat perfectly-electricconducting surface at horizontal polarization. The ill-posedness of the regarded inverse problem is shown with the optimization landscapes. The parametric study indicates the potential use of this method as a refractivity gradient retrieval system under certain circumstances.

I. INTRODUCTION

N ON-STANDARD refractive index variations cause per- formance anomalies for maritime communication and surveillance platforms [START_REF] Hitney | Integrated refraction effects prediction system (IREPS)[END_REF]. Prediction of tropospheric refractivity in abnormal propagation conditions is crucial for survey radars because these conditions create blind zones, increase clutter level, alter detection range and cause range-altitude errors [START_REF] Skolnik | Introduction to Radar Systems[END_REF]. Although accurate radar coverage estimation methods exist taking refractivity as input [START_REF] Baumgartner | Duct propagation modelling for the integrated-refractive-effects prediction system (IREPS)[END_REF]- [START_REF]TERPEM User Guide[END_REF], abnormal conditions are poorly quantified in practical sense [START_REF] Douvenot | A duct mapping method using least squares support vector machines[END_REF]. It is necessary to quantify the refractivity over spatial extent of the platform for the entire operation time dynamically with robust, cost-effective methods [START_REF] Hitney | Tropospheric radio propagation assessment[END_REF]. Refractivity-from-clutter (RFC) is proposed to address these requirements in near-real time [START_REF] Krolik | Tropospheric refractivity estimation using radar clutter from the sea surface[END_REF]- [START_REF] Rogers | Estimating evaporation duct heights from radar sea echo[END_REF]. It is a self-contained remote sensing method for refractivity inversion which potentially outperforms other refractivity estimation techniques [START_REF] Karimian | Refractivity estimation from sea clutter: An invited review[END_REF], [START_REF] Yardim | Statistical estimation and tracking of refractivity from radar clutter[END_REF].

The current objective of RFC is to invert refractivity from sea-surface reflected radar clutter typically under 10 minutes with high reliability in realistic high-dimensional scenarios [START_REF] Rogers | Effects of the variability of atmospheric refractivity on propagation estimates[END_REF], [START_REF] Douvenot | Refractivity from sea clutter applied on vampira and wallops' 98 data[END_REF]. Commonly, the inversion problem is formulated as a simulation-driven nonlinear optimization problem [START_REF] Karimian | Refractivity estimation from sea clutter: An invited review[END_REF]. Computationally-efficient simulation methods exist solving parabolic equations with split-step techniques [START_REF] Dockery | An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation[END_REF], [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF]. The Manuscript received Month Day, Year. This work is supported by ENAC/ISAE-SUPAERO/ONERA Research Federation.

Uygar Karabas ¸, ENAC, Department TELECOM-EMA, 31400 Toulouse, France -ISAE-SUPAERO, Université de Toulouse, 31400 Toulouse, France (e-mail: uygar.karabas@enac.fr).

Youssef Diouane, ISAE-SUPAERO, Université de Toulouse, 31400 Toulouse, France (e-mail: youssef.diouane@isae-supaero.fr).

Rémi Douvenot, ENAC, Department TELECOM-EMA, 31400 Toulouse, France (e-mail: remi.douvenot@enac.fr). inversion technique [START_REF] Karimian | Refractivity estimation from sea clutter: An invited review[END_REF] and the clutter model [START_REF] Rosenberg | Modeling the statistics of microwave radar sea clutter[END_REF] are scenario-dependent open questions, however. Inversion of refractivity from sea-reflected radar clutter or array antenna measurements, as illustrated in Figure 1, is a complex nonlinear, ill-posed problem [START_REF] Yardim | Tracking refractivity from clutter using kalman and particle filters[END_REF]. Historically, the inversion complexity has been reduced thanks to coarse parametrization with functional representations in altitude [START_REF] Dougherty | Recent progress in duct propagation predictions[END_REF], [START_REF] Gerstoft | Inversion for refractivity parameters from radar sea clutter[END_REF]. In parallel, range-independent medium is assumed [START_REF] Goldhirsh | Propagation factor errors due to the assumption of lateral homogeneity[END_REF]. Until the end of 2000s, several RFC techniques have been proposed with varying degrees of simplification on the spatial variability of refractive index (e.g., [START_REF] Tabrikian | Estimating tropospheric refractivity parameters from radar sea clutter[END_REF]- [START_REF] Douvenot | Real time refractivity from clutter using a best fit approach improved with physical information[END_REF]). However, these techniques are not applicable for high-dimensional problems since the computational cost of inversion increases with the number of parameters to invert. Therefore, they are not efficient for problems at high frequencies when the propagation becomes increasingly sensitive to the spatial details of the medium inhomogeneities [START_REF] Karimian | Refractivity estimation from sea clutter: An invited review[END_REF]. That is why some recent works have investigated the inversion of refractivity profiles defined using high-dimensional nonparametric functional models.

The first RFC technique whose computational cost does not depend on the number of parameter to invert has been developed in [START_REF] Zhao | Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation[END_REF] in the framework of variational adjoint approach in 2010. The authors have derived the adjoint model (AM) for two-dimensional narrow-angle parabolic equation (2-D NAPE) which is solved using split-step Fourier (SSF) technique. Following studies have confirmed the severity of ill-posedness in high-dimensional scenarios despite regularization [START_REF] Zhao | Refractivity from clutter by variational adjoint approach[END_REF]- [START_REF]Atmospheric duct estimation using radar sea clutter returns by the adjoint method with regularization technique[END_REF]. Unfortunately, today's adjoint-based RFC techniques still require unrealistically good initial guess with additional apriori information or need to incorporate dimensionality reduction techniques [START_REF] Zhao | Estimating rangedependent evaporation duct height[END_REF], [START_REF] Zhao | refractivity-from-clutter" based on local empirical refractivity model[END_REF]. Thus, reliable inversions which are purely driven by real-time data have not been realized yet in high-dimensional realistic configurations in 2-D. Other recent efforts concentrate on improving RFC in low-dimensional scenarios [START_REF] Fountoulakis | Duct heights inferred from radar sea clutter using proper orthogonal bases[END_REF]- [START_REF] Gilles | A subspace pursuit method to infer refractivity in the marine atmospheric boundary layer[END_REF].

Despite the prominence of forward model (FM) accuracy in inverse problems, no previous study has attempted to fuse the adjoint-based approach with a wide-angle propagation model in RFC community. In this work, we extend the adjoint approach for 2-D wide angle parabolic equation (WAPE) by identifying the adjoint of square-root propagation operator. The proposed solution is validated numerically. Our main motivation is to enrich the pool of available adjoint models in radio and acoustic wave propagation communities with the most accurate WAPE on which split-step technique is applicable. We simulate the adjoint model with split-step wavelet (SSW) technique [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF], which is faster than SSF, to tend towards a real-time accurate system. We aim at capturing any refractive condition at the spatial resolution of FM simulations by adopting point-by-point inversion approach in altitude [START_REF] Zhao | Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation[END_REF].

Unlike previous work, the obtained gradient of the cost function is computed and validated successfully using a finite differences approximation. We also include the optimization landscapes in our analysis in order to visualize the inversion complexity and to explain how much the inversion is sensitive to initial guess for a given setup [START_REF] Barucq | A priori estimates of attraction basins for nonlinear least squares, with application to helmholtz seismic inverse problem[END_REF]. The method is presented for tomographic approach as an intermediate step towards a full effective RFC system. Although regularization techniques are kept out of scope of this work at this stage, our parametric study still shows the potential of our proposed approach.

The outline of the paper is as follows. In Section II, the formulations of our problem are presented. In Section III, the numerical methods are explained. Numerical test results are detailed in Section IV.

II. MODELING

In this section, we first give an overview of adjoint method in generic sense. After defining the forward problem and the inverse problem, the adjoint model is derived. The derivation is done in continuous adjoint form, so the parametrization and data are kept in functional forms. The recipe for the discrete adjoint formulation and finite dimensional setting of data and parametrization are explained in other appropriate sections.

A. Framework overview

In this section, the inverse problem framework is presented following [START_REF] Kern | Numerical Methods for Inverse Problems[END_REF] to put forward the motivation. The inverse problem is seen as the minimization of a square error functional over control parameter m in the bounded admissible space M ad given by:

min m∈Mad J(m) = 1 2 d sim m -d obs 2 D . (1)
The cost function J(m) depends on the parameter m through the simulated measurement d sim m . The observation state d obs is an experimental measurement function. We seek for a solution using an efficient gradient-based optimization method. For that reason, we choose to work with the square error functional (1) as this norm is differentiable. We present the standard formalism to obtain the gradient of the cost function below.

Consider the inverse problem [START_REF] Hitney | Integrated refraction effects prediction system (IREPS)[END_REF] where D = L 2 (Γ R ) and Γ R forms a partition of the domain Ω. The map of misfit quantification is given by:

K : D → R, d → K(d) = 1 2 d -d obs 2 D .6 (2)
The simulated measurements d sim m are taken via an observation operator P that applies to the the state u m ∈ U = L 2 (Ω), i.e., u m → d sim m = P (u m ). The state u is uniquely defined for a given parameter m with the state equation F (u, m) = 0 at u = u m . This latter is calculated from m using the forward map S:

S : M ad → U, 6m → u m = S(m) such that F (u m , m) = 0. (3)
Note that the cost function J is a composite function given by J(m) = K • P • S(m). The gradient of J with respect to m is given by [START_REF] Kern | Numerical Methods for Inverse Problems[END_REF]:

∇ m J = φ (m) * ∇ um J, (4)
where φ(m) = P • S(m) and φ (m) is the differential of the mapping φ(m). The asterisk * denotes its adjoint. The function ∇ um J is the forcing function of the adjoint model [START_REF] Meyer | Optimal nonlocal boundary control of the wide-angle parabolic equation for inversion of a waveguide acoustic field[END_REF]. Since φ is implicitly defined with the solution of the state equation, the implementation of ( 4) is not straightforward in our case. Therefore, we cast (4) using the variational form in order to obtain an explicit adjoint equation which allows the gradient to be identified and to be calculated more simply. In the following sections, the elements given in this section are specified for our particular problem.

B. The forward problem

This section discusses the physical modeling part in the adjoint formalism given by (3). In the context of propagation at open-sea environment with azimuthal symmetry, modeling of electromagnetic wave propagation considers solely forward propagating part of the Helmholtz equation in cylindrical coordinates:

∂ r u -jk 0 (1 -Q) u = 0, (5)
which is also called the one-way equation. The square-root operator Q is given by Q = m 2 (z) + k -2 0 ∂ 2 z with k 0 the wave number in vacuum and m(z) ∈ R + the modified refractive index that accounts for earth roundness. Equation ( 5) is exact for range-independent medium as long as the far field approximation holds, and takes into consideration the entire forward propagating wave field propagating at frequency f = k 0 c/2π. The reduced field u(r, z) ∈ C is related to the power at the receiver antenna by taking into account the distance, antenna gain and medium impedance [START_REF] Levy | Parabolic equation methods for electromagnetic wave propagation[END_REF].

There are numerous choices to model the propagation in the lower troposphere depending on how accurate Q is approximated in [START_REF] Dockery | An overview of recent advances for the TEMPER radar propagation model[END_REF] [START_REF] Levy | Parabolic equation methods for electromagnetic wave propagation[END_REF], [START_REF] Jensen | Computational ocean acoustics[END_REF]. The standard parabolic approximation (NAPE) is the first order approximation of Q [START_REF] Tappert | The parabolic approximation method[END_REF], it has limited accuracy [START_REF] Kuttler | Differences between the narrow-angle and wide-angle propagators in the split-step Fourier solution of the parabolic wave equation[END_REF] and it is the only model used in adjointbased RFC systems so far [START_REF] Zhao | Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation[END_REF]- [START_REF] Zhao | refractivity-from-clutter" based on local empirical refractivity model[END_REF]. In this paper, we use the wider angle approximation (WAPE) as forward model, which is introduced by Thomson and Chapman [START_REF] Thomson | A wide-angle split-step algorithm for the parabolic equation[END_REF] for underwater acoustics according to the splitting of Q as proposed by Feit and Fleck [START_REF] Feit | Light propagation in graded-index optical fibers[END_REF]:

∂ r u + j k 0 (m(z) -1) + k 2 0 + ∂ 2 z -k 0 u = 0, (6a)
u(0, z) = φ(z), (6b)

u(r, 0) = 0. ( 6c
)
This model considers the boundary condition of a flat perfectly-electric-conducting surface at z = 0 with source term φ(z) modeling the initial field of emitted radio waves at horizontal polarization. Note that the state equation (6a) could also take into account a dielectric ground using a variable change proposed by [START_REF] Dockery | An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation[END_REF]. However, reflection coefficient c R = -1 is a good approximation for both polarizations at very grazing angles.

To sum up, the forward problem is to determine the state u m for a given parameter model m by solving [START_REF] Hurtaud | Une préfiguration des futures aides tactiques: filecode PREDEM: Observation des côtes et des océans: Senseurs et Systèmes (OCOSS)[END_REF].

C. The inverse problem

In our inverse problem, we target to determine the parameter model m that minimizes the following cost function:

min m∈Mad J(m) = 1 2 Z 0 d sim m -d obs 2 dz. (7)
The problem ( 7) is designed with some idealizations which help the validation of the inversion process. Namely, for an objective refractivity model m OBJ , the experimental measurements will be synthetically generated using the forward model [START_REF] Hurtaud | Une préfiguration des futures aides tactiques: filecode PREDEM: Observation des côtes et des océans: Senseurs et Systèmes (OCOSS)[END_REF]. According to the tomographic approach given in Figure 1, the measurements are taken at the range r = R such that P :

u(r, z) → |u(R, z)| 2 and data space D is such that Γ R = {R} × [0, Z]. Therefore, the observed data obey d obs = {|u(R, z)| 2 : F (u, m OBJ ) = 0}.
In our study, the parameter model m is function of altitude z only. Note that d obs ∈ Im(P • S) so the existence of global minimum is guaranteed for [START_REF]TERPEM User Guide[END_REF]. But some other solutions become indistinguishable from m OBJ since phase is immeasurable. In a realistic scenario, simulated measurement with (6) would not perfectly produce the experimental measurement. The ground is modeled better as dielectric ground. The refractivity varies with range. In addition, measurements are timeaveraged, contain noise and cannot be pointwised in practice. Modelling and measurement errors will add to the complexity of the idealized inverse problem. Therefore, it is of interest to avoid real-world data during validation of the inversion routine.

We note also that noise needs to be added to synthetic measurements in order to assess the robustness of the inversion algorithm against instabilities. Here, the noise modeling does not aim to be realistic and the inversion results from noisy data are neither an indicator of performance in real operation nor conclusive. Instead, the aim is to deduce the sensitivity of solver of (7) to perturbations. In order to be able to draw conclusion from inversion of noise-free synthetic data, it is necessary to show a certain level of robustness to noise. In that context, additive Gaussian noise can be added to the simulated measurements, i.e.,

d obs noisy = d obs + ζ (8)
where ζ follows a centered normal Gaussian distribution with a standard deviation of τ × rms √ d obs . The function rms(x) is root-mean-square of function x ∈ R and τ is termed as the level of noise in this work.

D. The tangent linear model

Let us denote F (u m , m) = F m for brevity. Assume a linear perturbation on the parameter model in the direction m such that F m+α m = 0. We seek for the operator F which fulfills the following relation:

lim α→0 F m+α m -F m -F (û m , m; m)α m α m = 0. (9)
Here, the operator F is referred to as the Fréchet derivative of operator F . The function ûm ∈ L 2 (Ω) satisfies the tangent linear model (TLM) F (û m , m; m) = 0 and it is referred to as the Gâteaux differential of u m with respect to m:

lim α→0 u m+α m -u m α = ûm . (10)
Substituting the operator F with ( 6) in ( 9), one deduces that the TLM satisfies the following property:

∂ r ûm + j k 0 (m -1) + k 2 0 + ∂ 2 z -k 0 ûm + jk 0 mu m = 0 on Ω, (11a) ûm (0, z) = 0, (11b) ûm (r, 0) = 0. (11c)
In what comes next, for two given functions f and h, we will make use of the following inner products:

f, h = Z 0 R 0 f (r, z) • h(r, z) drdz, f, h Z = Z 0 f (r, z) • h(r, z) dz, f, h R = R 0 f (r, z) • h(r, z) dr, (12)
where h is the complex conjugate of h. We also note that

f, h = R 0 f, h Z (r) dr = Z 0 f, h R (z) dz. (13)

E. Gradient of the cost function

The gradient ∇ m J of the cost function J, at the parameter model m, can be obtained using the fact that:

J (m; m) = ∇ m J, m Z , (14)
where J (m; m) is the Gâteaux differential of J(m) with respect to m, i.e.,

J (m; m) = lim α→0 J(m + α m) -J(m) α . ( 15
)
Then, the gradient of the cost function ( 7) is given by the following:

∇mJ, m Z = J (m; m) = 2 Z 0 |um(R, z)| 2 -d obs um(R, z) • ûm(R, z) dz = 2 |um(R, z)| 2 -d obs um(R, z), ûm(R, z) Z , (16)
where {x} denotes the real part of x ∈ C.

F. The adjoint model

The gradient ∇ m J is defined by deriving the adjoint state equation F * from the variational form of F . A unique adjoint state w m (r, z) ∈ C exists such that w m , F (û m , m; m) = 0 which ( 16) is subject to. Let us drop index m from the function w m , u m and ûm for brevity. The variational form is:

w, ∂ r û + w, jk 0 (m -1)û + w, j((k 2 0 + ∂ 2 z ) 1/2 -k 0 )û + w, jk 0 mu = 0. (17)
Applying the rule of partial integration considering (11b) and (11c) leads to the following variational problem:

-∂ r w, û + w, û Z | r=R -jk 0 (m -1)w, û -j((k 2 0 + ∂ 2 z ) 1/2 -k 0 ) * w, û -jk 0 wu, m = 0. (18)
Let A = k 2 0 +∂ 2 z , then according to [52, Theorem 1], for any positive self-adjoint operator A ∈ L(H D , H R ) where H D , H R are two Hilbert spaces, there exists a self-adjoint square root operator

√ A = √ A *
. In our context, self-adjointness A = A * always holds but positiveness Af, f ≥ 0 is reserved for when ∂ 2 z f, f ≤ k 2 0 f, f . The latter indeed holds for the entire propagating modes of (6a) [START_REF] Levy | Parabolic equation methods for electromagnetic wave propagation[END_REF]. Thus,

√

A * = √ A holds as far as the scope of resolution is restricted on the propagating part of the field. Since we only measure the propagating part of the field for refractivity inversion, selfadjointness of the square-root operator can be asserted in the context of RFC. Numerically this corresponds to ∆z ≥ λ/2 where ∆z is the step size in the axial direction of Ω and λ is the wavelength. Consequently, we have:

-∂ r w, û + w, û Z | r=R -jk 0 (m -1)w, û -j((k 2 0 + ∂ 2 z ) 1/2 -k 0 )w, û -jk 0 wu, m = 0. (19)
Observing the self-adjointness of the operators in the variational form, the aim is to derive an adjoint state equation which is the same as (6a) so that the FM solver can be inherited for AM. Combine ( 16) and [START_REF] Rosenberg | Modeling the statistics of microwave radar sea clutter[END_REF]. Let w m be given by:

∂ r w m + j k 0 (m -1) + k 2 0 + ∂ 2 z -k 0 w m = 0, (20a) w m (R, z) = 2 |u m (R, z)| 2 -d obs u m (R, z), (20b) w m (r, 0) = 0, (20c)
such that [START_REF] Dougherty | Recent progress in duct propagation predictions[END_REF] holds

∇ m J = jk 0 R 0 w m (r, z) • u m (r, z) dr . (21)
The estimation of such gradient will now enable us to use gradient-based optimization methods (e.g., Quasi-Newton methods) which are suitable for large-scale inversion problems. In the next section, we give the implementation details for [START_REF] Yardim | Tracking refractivity from clutter using kalman and particle filters[END_REF] and [START_REF] Dougherty | Recent progress in duct propagation predictions[END_REF] and present an approach to validate both the derivation and the implementation.

III. NUMERICAL METHODS

A. Inversion strategy

The inversion process starts from an initial guess m IG which gives simulated measurement data d sim m using propagation model [START_REF] Hurtaud | Une préfiguration des futures aides tactiques: filecode PREDEM: Observation des côtes et des océans: Senseurs et Systèmes (OCOSS)[END_REF]. Next, the adjoint equation (20a) is solved in order to estimate the gradient using [START_REF] Dougherty | Recent progress in duct propagation predictions[END_REF]. The iteration parameters are estimated with a gradient-based minimization algorithm, which may include inner cost function estimations using the propagation model. The same process continues until a convergence criteria is met. In our study, we do not have access to measurements. Therefore, we generate the measurements d obs synthetically using the forward model given by ( 6) for a known objective refractivity profile m OBJ . The workflow given in Figure 2 summarizes the inversion strategy and the used notations. Next section presents the numerical methods which we use. Eq. ( 6) Eq. ( 20) Eq. ( 21)

Objective Refractivity Model

Eq. ( 6)

EXPERIMENTAL MEASUREMENT END Fig. 2. Schematic of inversion workflow. The algorithm corresponding to the red boxes will be validated.

B. Numerical Methods

The forward problem (6) is modeled using SSW technique. This forward solver implementation has been validated [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF]. The adjoint model solver is derived from the forward model solver where the adjoint initial condition is achieved by replacing (6b) with (20b) in the FM solver. Adjoint backpropagation is achieved by changing sign of range step from ∆r to -∆r and 180 • rotation of computational domain around axial direction in the FM solver. This change of sign occurs in the free-space propagator, boundary condition and phase screen operators in [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF]. The cost function [START_REF]TERPEM User Guide[END_REF] and its gradient [START_REF] Dougherty | Recent progress in duct propagation predictions[END_REF] are computed using Simpson's rule of integration. The optimization algorithm is the well-known BFGS quasi-Newton method [START_REF] Nocedal | Numerical Optimization[END_REF].

C. Validation of the gradient estimation

The problem ( 7) is difficult to solve accurately. The inversion results cannot be used for validating the quality of the work except for oversimplified cases (e.g., [START_REF] Karabas | On the use of adjoint methods for refractivity estimation in the troposphere[END_REF]). Therefore, it is necessary to perform a validation process starting from checking the gradient computation. This process can be seen as an intermediate step towards understanding why RFC may fail. In fact, ill-posedness in RFC can be incriminated only after running the validation process of the gradient.

The part of the inversion algorithm lacking validation is shown in the workflow in Figure 2 in red. Typically, for inverse problems, the adjoint model can be validated with dotproduct test or by comparison of ∇ m J AM (i.e., ∇ m J estimated using ( 21)) with a reference estimation of the gradient. In the context of inverse problems, the validation scope of the latter is slightly wider than that of dot-product test. If the adjoint state w m is formulated and simulated properly, ∇ m J AM would match the gradient computed with a reference method. The reference estimation of the gradient can be obtained using a finite differences (FD) scheme. Typically, for a number n of inversion parameters, the i th component of the FD approximation of the gradient is given by

[∇ m J FD ] i = J(m + e e e i ) -J(m) , (22)
where e e e i is the n-sized column vector whose ith component is one while all the others are zeros, and is chosen as 10 -6 . Note that the estimation of the gradient with a finite differences scheme requires n + 1 evaluations of the cost function, while ∇ m J AM is computed at the cost of 2 FM runs [START_REF] Kern | Numerical Methods for Inverse Problems[END_REF].

To sum up, the gradients ∇ m J FD and ∇ m J AM should follow each other if the adjoint field is formulated properly in (20a) and simulated properly and the numerical integration of ( 21) is done properly. The baseline computational setup of the validation process is given in Section IV-A which also contains some notes on the validation methodology. Numerical validation is given in Section IV-C.

D. Landscapes of the cost function

There is no general technique to tackle a nonlinear inverse problem [START_REF] Kern | Numerical Methods for Inverse Problems[END_REF], [START_REF] Wolpert | No free lunch theorems for optimization[END_REF]. Solving such a problem requires the development of a special strategy tailored for the specificity of the regarded problem.

We are asked to find the global minimum of an error functional. Some of the difficulties in adjoint-based RFC are listed as sensitivity to initial guess [START_REF]Atmospheric duct estimation using radar sea clutter returns by the adjoint method with regularization technique[END_REF], loss of signal at long ranges [START_REF] Zhao | Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation[END_REF], failure of some regularization techniques [START_REF] Zhao | Refractivity from clutter by variational adjoint approach[END_REF], insufficiency of measurement data [START_REF] Zhao | Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation[END_REF], high-dimensionality [START_REF] Zhao | Estimating rangedependent evaporation duct height[END_REF]. The difficulties are commonly seen as a consequence of nonlinearity and ill-posedness. In practice, the two attributes reflect on the problem [START_REF]TERPEM User Guide[END_REF] as nonconvexity and nonsmoothness of the optimization landscape. Therefore, from our perspective, looking at the cost function landscape [START_REF] Barucq | A priori estimates of attraction basins for nonlinear least squares, with application to helmholtz seismic inverse problem[END_REF] can guide the community to build the right strategy. RFC is much more complex than [START_REF]TERPEM User Guide[END_REF], so optimization landscape analysis of ( 7) is valuable as an intermediate step for understanding why RFC is still an open problem.

In this work, we limit our analysis to two different onedimensional optimization landscapes of the high-dimensional problem. They allow to visualize the cost function around some given solutions with a one-dimensional variation. Firstly, we define:

α → J(m 1 (α)), α ∈ [-2, 2], with m 1 (α) = m OBJ + α(m 0 -m OBJ ). (23)
By varying the parameter model using m 1 , we construct the cost function landscape around the objective refractivity profile. Secondly, we define:

α → J(m 2 (α)), α ∈ [-1, 1], with m 2 (α) = m 0 + α m max -m 0 ∞ ∇ m0 J ∇ m0 J ∞ , (24)
where m max is the maximum bound in the admissible space.

The landscape using the parameter model m 2 is constructed around the initial guess in the direction of the gradient.

The two landscapes will be plotted and commented in Section IV-D.

IV. NUMERICAL ANALYSIS

In this section, firstly, the baseline computational setup is presented. Next, a parametric study is performed. After validation of the method, the inversion difficulty is explained with the directional cost function landscapes. The analysis ends with displaying the impact of initial guess on inversion and perturbation robustness tests.

A. Computational setup

In the baseline computational setup, the geometry of the computational domain Ω is a two dimensional plane constructed on axial and radial directions as given in Figure 1. The geometry is truncated such that Ω = [0, R] × [0, Z]. Altitude of interest Z is limited to 150 m. Range of propagation R is considered at different values ∈ {1, 5, 10, 30, 60} km.

The mesh of discretized domain Ω Nr,Nz is a uniform grid. The parameter N r and N z are the number of grid points along the directions r and ẑ, respectively. The cell size in altitude is set to ∆z = 1 m for all cases. The cell size in range ∆r is determined such that N r = 101 is satisfied when ∆r < 100 m, otherwise the cell size is fixed at ∆r = 100 m for the control of numerical error.

The initial condition φ(z) is obtained from a complex-point source positioned at (r s , z s ) = (-100 km, 25 m). It has width of 5 m and emission frequency f s = 2 GHz at horizontal polarization. This setting yields a φ(z) profile with full shape, which is useful for validation [START_REF] Karabas | On the use of adjoint methods for refractivity estimation in the troposphere[END_REF]. The ground is accounted with the local image method [START_REF] Zhou | Modeling the long-range wave propagation by a split-step wavelet method[END_REF] and a Hanning absorbing window is applied on the top of the domain.

The refractivity is given with the modified parameter M (z) = (m(z) -1) × 10 6 . The parameter M which corresponds to d obs is called the objective of the inversion and denoted by M OBJ . Similarly, the initial guess and the inversion result parameters are denoted by M 0 and M INV , respectively. There is no variation of M with range. The simulations take into account the discretized parameter vector with dimension M ∈ R Nz using linear projection of M at grid nodes. To be clear, the aim is to invert N z = 151 refractivity parameters with 151 simulated data sample from 151 experimental data sample. In this scenario, there are 151 receivers on the receiver array extending from ground level to the altitude of interest Z = 150 m, with the receiver spacing of ∆z.

The objective refractivity M OBJ (z) has a trilinear structure of surface based ducts [START_REF] Gerstoft | Inversion for refractivity parameters from radar sea clutter[END_REF]. The duct is given by the following altitude-refractivity tuple: (z, M OBJ (z)) = {(0, 330.0), (50, 350.0), (100, 330.0), (150, 335.90)}. The initial guess is constant at M 0 = 330.0 in the baseline setup.

In the next section, we give the inversion results of the workflow given in Figure 2 at different ranges for the baseline computational setup. The objective and initial guess fields are given in Figure 3 with their difference.

B. Results: inversion

The inversion routine given in Figure 2 is tested for its capability to retrieve M OBJ from the corresponding measurement.

Measurement range R has an influence on inversion. The dependence of M INV (the inverted refractivity model) on varying range R is given in Figure 4. One can see that it may not be possible to retrieve M OBJ but M INV captures the synoptic structure of M OBJ at R = 1, 5, and 10 km with varying degrees of success. The refractivity gradient is retrieved thanks to the fact that the solution of ( 7) is not unique but with some dispersion in M INV due to ill-posedness of [START_REF]TERPEM User Guide[END_REF]. For R = 30 and 60 km, M INV stays closer to the initial guess M IG for increasing R. As failure of the inversion and stocking of M INV around M IG are progressive with the increase of R, we conclude that (7) gets more nonlinear with R.

The mechanisms of nonlinearity are the propagation model, ground reflections and dimensionality of the problem. Firstly, [START_REF]TERPEM User Guide[END_REF] is driven by the simulations of (6a) in which the relation between m and u m (R, z) is nonlinear. Secondly, range R matters due to multipath interference of ducted and reflected field which increasingly complicates the shape of u m (R, z) at long distances. Thirdly, nonlinearity of the problem increases with the number of parameters to invert. Thus, the number of local minima on the optimization landscape increases. The problem [START_REF]TERPEM User Guide[END_REF] becomes such that a local minima can be found in the neighbourhood of initial guess for R = 60 km with 151 optimization parameters.

Even so, we need to verify that the gradient is computed successfully and this complexity phenomenon is driven by the cost function landscape. To achieve this, firstly, we perform the validation with finite differences at different ranges as described in Section III.

C. Results: validation of the gradient

Since the inversion could not retrieve M OBJ , we first put into question the validity of adjoint model. In this section, the adjoint model [START_REF] Yardim | Tracking refractivity from clutter using kalman and particle filters[END_REF] presented in Section II-F is validated with the comparison of ∇ m J FD and ∇ m J AM .

We validate over the gradient of the first descent iteration. AM estimates the gradient that would otherwise have to be computed using FD at higher cost. In Figure 5, the gradient calculated with the AM and with FD are plotted with respect to altitude for the different values of R. Both gradients perfectly match, except that at R = 1 km there exists some differences due to the very low value of the gradient. Thus the gradient estimation is more robust for long range applications.

Numerical issues become prominent in the computation of the gradient when the gradient is low as in Figure 5a. For AM, this occurs when w approaches to null and gradient is estimated near a stationary point according to (20b) and [START_REF] Dougherty | Recent progress in duct propagation predictions[END_REF]. For FD this corresponds to the case where field bending effect of different m is not significant on |u|, according to [START_REF] Gerstoft | Inversion for refractivity parameters from radar sea clutter[END_REF], which is expected at short ranges. Thus, stationarity check is recommended for RFC applications at the end of inversion.

Consequently, the adjoint model is validated with the comparison of the gradient with FD. Failure to invert is not due to the adjoint model. Problem ( 7) is ill-posed within its nature even without measurement noise, modeling and theoretical errors. Potentially, the ill-posedness could be combined with convergence issues which could be included in a future work.

We have proven that the inversion does not fail due to miscomputation of the gradient. Indeed, the gradients may indicate higher ill-posedness due to removal of the phase from synthetic data [START_REF] Karabas | On the use of adjoint methods for refractivity estimation in the troposphere[END_REF]. Therefore, we are interested in visualizing the extent of non-convexity in the inverse problem. In the next section, this is performed for two different 1-D parametrizations to give an idea about the complexity of inversion, which may give an idea about the sensitivity to initial guess with increasing range of propagation.

D. Results: cost function landscape

Since the method is validated, we suspect that the difficulty to invert originates from the problem (7) itself. We have evidence that the difficulty increases with range according to Figure 4 so sensitivity to initial guess must also increase and more local minima are expected to appear on a rougher optimization landscape.

In Figure 6, we plot the directional cost function landscape for M 1 ((7) for model [START_REF] Goldhirsh | Propagation factor errors due to the assumption of lateral homogeneity[END_REF]) for three ranges. The increasing complexity with range can be observed as the convexity of the landscapes decreases with R. Note that M OBJ is located at α = 0 and M IG at α = 1 on this landscape. The functional [START_REF]TERPEM User Guide[END_REF] between M IG and M OBJ becomes nonconvex with R. Thus M IG is more likely to be attracted to emerging local minima in its neighbourhood with gradient based optimization method.

Attention must be paid to the dimensionality of the problem when interpreting Figure 6 and7. There are infinitely many 1-D line cuts in 151-D hypercube and indications of nonlinearity and ill-posedness can be hidden at some other line cuts. For example at 10 km, ill-posedness of ( 7) is obvious in M INV of Figure 4c but J is quasiconvex in 6b for α ∈ [0, 1]. Nevertheless, one can deduce from Figure 6 that better initial guess is needed to successfully invert at long ranges. Otherwise, the optimization gets trapped at a local minimum as landscape over 151-D hypercube is expectedly more rough. Also note that the landscapes with the standard atmosphere as the initial guess (not presented here) are similar to Figure 6. Comparison of the gradient ∇m 0 J FD computed with finite differences (FD) method vs. ∇m 0 J AM computed with adjoint model (AM). Relative L2 error of these estimations are 13.2%, 3.2% and 1.8% respectively.

The analysis is enriched with the landscape drawn at the direction of the gradient according to [START_REF] Tabrikian | Estimating tropospheric refractivity parameters from radar sea clutter[END_REF] as given in Figure 7. Note that M IG is located at α = 0 on this landscape. The functional [START_REF]TERPEM User Guide[END_REF] has more local minima on this 1-D view when compared to Figure 6. Again, it is more difficult to invert at long ranges because the optimization landscape becomes more nonconvex. In the next section, we expand our analysis with inversions by using different initial guesses.

E. Results: impact of initial guess

Sensitivity of optimization to initial guess may appear when a nonlinear problem is solved with gradient-based methods. A good initial guess can significantly improve inversions by diminishing the undesirable effects of nonlinearity and illposedness. In this section, the aim is to check if we can observe some improvement with different initial guesses and if the conclusions made in the results section are reasonable for other initial guesses as well.

In Figure 8, we present the inverted parameters in altitude for different initial guesses at R = 10 km, in order to show examples of how an initial guess could improve or worsen the inversion. Firstly in Figure 8a, M IG is set at closer to M OBJ than in Figure 4c and the inversion improves. Without showing more figures, we comment that a detailed study of this case revealed that M INV exhibits further self-regularization with the further improvement of M IG towards M OBJ and the gradient information is retrieved more smoothly. Next, Figures 8b and8d show that a ducting condition in the initial guess significantly improves the inversion of a ducting condition, even if both initial guesses are linear with altitude. Another scenario is simulated in Figure 8c. It mimics conditions for which short-time history or meteorological input is available as initial guess. The inversion managed to track the assumed temporal variation nearly-successfully. Note that M IG and M OBJ have the same duct strength in in this example, Figure 8c.

F. Results: noise impact and robustness test

It is necessary to show that the inversions are robust to noise (see II-C for details). The motivation is to see if the conclusions made without noise are reasonable in the presence of perturbations on data. If our system is robust, it should be insensitive to small perturbations or precision errors on data; this increases confidence in the inversion results. Robustness test is performed at R = 10 km with different levels of noise τ on measurements according to [START_REF] Douvenot | A duct mapping method using least squares support vector machines[END_REF].

In Figure 9, we present the inverted parameters in altitude for different noise level τ at R = 10 km. The inversions naturally become more ill-posed with the addition of noise on measurement data. The mean of the inversions seem to follow M INV | τ =0 given in Figure 8a but dispersion of the parameters grows with τ . In this example, the inversion is resilient to 10% of perturbation on data as far as capturing of the gradient of M OBJ is concerned. The inversion deviates progressively from the attraction basin of M INV | τ =0 as τ > 10%.

V. CONCLUSION

This paper presents our attempts to predict the ambient refractive index in the troposphere. The goal is to provide real-time situational awareness for modern seaborne and airborne electromagnetic emitters under anomalous propagation conditions. The adjoint method properly addresses the highdimensionality of the refractivity inversion for such modern detection systems and it is the backbone method on which our inversion strategy will be built. For this reason, its validation has received emphasis at the initial stage of development [START_REF] Karabas | On the use of adjoint methods for refractivity estimation in the troposphere[END_REF]. We attach importance to upgrade the forward model to tighten the control on inversion accuracy for this ill-posed inverse problem in real-world scenarios. Despite being hypothetical for the implications, we advocate the use of the wideangle approximation of the parabolic equation for adjointbased refractivity inversions because the method is available at no additional cost when compared to that of NAPE. In this paper, we show how to derive the adjoint WAPE, explain how to construct the adjoint code, and validate the adjoint model by using a finite-difference approach. Thanks to the validation of the gradient, we are sure that it is the ill-posedness of the regarded inverse problem that leads to failure of inversion. We explain this failure with nonconvexity of directional cost function landscapes and high-dimensionality of the problem. Since the gradient-based method which we use is prone to be attracted to local minimum during iterations, the increasing sensitivity to initial guess with range can be inferred from the cost function landscape. This latter is seen to lose convexity with distance, as the problem becomes more nonlinear.

The parameter study shows the potential use of this method as a gradient retrieval system. In this study, we have obtained acceptable results until around 10 km range. We have also verified the validity of our assessments using different initial guesses. The ducting initial conditions are found to work better as initial guesses for the regarded objective refractivity profile. The proposed method is shown to be robust for small Gaussian perturbations in the measurements.

More detailed analyses are necessary to assess the performance prediction quality of the emitters. Nevertheless, it is inspiring to obtain some sort of gradient retrieval with our validated basic strategy. We have to note that the problem is expectedly more ill-posed in real world scenarios due to measurement and modeling errors. Our next step will be to investigate multiscaling strategies [START_REF] Diouane | A parallel evolution strategy for an earth imaging problem in geophysics[END_REF] and regularization techniques [START_REF] Chavent | Nonlinear least squares for inverse problems: theoretical foundations and step-by-step guide for applications[END_REF] to deal with inversion at long-distance and more realistic high-dimensional scenarios.

Fig. 1 .

 1 Fig. 1. An illustration of the bistatic configuration.

  Synthetically-generated measured field.

  Simulated field for initial guess.

  Absolute error between simulated and measured fields.

Fig. 3 .

 3 Fig. 3. Propagating fields at different range R.

Fig. 4 .

 4 Fig. 4. Inverted parameters at different range R.

Fig. 5 .

 5 Fig.5. Validation of the gradient for the first descend iteration. Comparison of the gradient ∇m 0 J FD computed with finite differences (FD) method vs. ∇m 0 J AM computed with adjoint model (AM). Relative L2 error of these estimations are 13.2%, 3.2% and 1.8% respectively.

  R = 60 km.

Fig. 6 .

 6 Fig.6. Obtained landscape for the function α → J(m 1 (α)) at different range R, see[START_REF] Goldhirsh | Propagation factor errors due to the assumption of lateral homogeneity[END_REF].

Fig. 7 .

 7 Fig. 7. Obtained landscape for the function α → J(m 2 (α)) at different range R, see (24).

Fig. 8 .Fig. 9 .

 89 Fig. 8. Inversion at R = 10 km for different initial guesses.

ACKNOWLEDGMENT

This work is part of activities of ENAC/ISAE-SUPAERO/ONERA research federation.


Keywords: Combating Economic and Financial Crimes, Prevention, Financial Conduct Authority, France Public Prosecutor's Office, French Criminal Law

widely to include a range of offenses. These include, but not limited to, fraud, bribery and corruption, insider dealing market abuse and, money laundering. This article offers a detailed overview of the rules regarding criminal investigations into financial-economic crimes in the French Legal system. These rules have turned out to be fundamental to the effective protection of France and the European Union's financial interests. Further, this study is enriched with cross-sectional essays that deal with more general issues, such as data protection and the future of investigations, preventing, combating and punishing economic crimes.

Tools to Combat Economic-Financial Crimes in French Criminal Justice System

Erfan Babakhani [START_REF] Blanchot | Infraction en matière économique et financière[END_REF] Doctoral student in criminal law and member of the research academy of Law, Paris-Nanterre University Jimmy Harang [START_REF] Cutajar | Le volet répressif de la loi sur la lutte contre la fraude fiscale et la grande délinquance financière[END_REF] LLM in criminal law and Assistant Professor, Lille University

SUMMARY

The present article provides an overview of the operational models deployed by a number of agencies and bodies in France, with a focus on investigating and prosecuting economic-financial crimes. In addition to the police and the criminal procedure, we study France Public Prosecutor's Office, which is in charge of tracking major economic and financial crimes.

The traditional method of combating financial crimes, such as money laundering, is to use prescriptive legislation. A new idea is that risky concepts may be applied to understanding the phenomenon of economic crime and devising strategies to minimize them. As noted by the European Commission, the range of crimes at the national level under which the "protection of the financial interests of the European Union" could fall is broad and varied across the Member States. In France, an economic crime is said to include any offense involving fraud, misconduct in or misuse of information related to a financial market. The term "to include" means that financial crime can be interpreted

INTRODUCTION

The globalization of economic crime related to the development of information technology and communication has created new opportunities for offenders at national and international levels, threatening economic stability, political and social companies, and serving as an obstacle to the country's development and economic growth. These offenses are defined by the French magistrate Jean-Claude Marin as "all illegal activities whose essential characteristics are that they take place in the context of economic activity, develop in structured organizations, private or public, do not resort to violence or force and that they require knowledge and knowledge specific to the business community, resulting in an ever greater need for specialization of the bodies responsible for prosecutions and investigations Since that time, the idea which governing the adoption of further reforms in this area has been the specialization of actors responsible for the repression of economic and financial offenses. The specialization of magistrates allows them to have a better knowledge of both the repressive texts to be applied and the fraudulent techniques used by criminals (I). This specialization was accompanied by a reinforcement of the technical and legal means at the disposal of the investigators, with the aim of identifying and effectively repressing these offenses (II). However, some shortcomings have been noted in practice (III).

I.THE SPECIALIZATION OF CRIMINAL JUSTICE A) Specialization of criminal courts

The specialization of criminal justice has led the French legislature to specialize certain jurisdictions and judicial actors.

In the jurisdictional organization of the 1970s, the specialized jurisdictions created by the law of 6 August 1975 appeared to be very innovative. However, two temperaments are to be noted. First, this law had not created a new category of jurisdiction but was only awarded a skill, within the jurisdiction of each court of appeal, to one or more courts. Then, it had not created a compulsory jurisdiction of these specialized courts, because their jurisdiction was only secondary to that of ordinary courts.

This specialization movement continued as the legislator reformed. But as it proved to be effective, this specialization may have seemed an unnecessary complexity. The law of 6 December 2013, therefore, came to revise the rules of division of powers between the ordinary courts and the competent courts in economic and financial matters. One is on the fight against tax evasion and the large economic and financial crime, while the other establishes a financial attorney Republic. This specialized prosecutor's function setting in motion public action against perpetrators offenses that are NOMIC and financial only and their prosecution in the criminal courts. From now on, in economic and financial issues, several jurisdictions coexist. Ordinary law jurisdictions still have a role to play, but are often competing with each other, either by the Paris court (and the financial prosecutor), which is a ordinary law jurisdiction with special powers, or by interregional jurisdictions (JIRS). There are also two economic and financial centers in the courts of Bastia and Nanterre, but they are not addressed during this presentation, because of their secondary nature and intimate link to the local context.

First, the prosecutor of the Republic Financial, located in Paris, has national jurisdiction for the prosecution of all fractions in economic and financial issues committed in France. Regarding insider trading and the offense of manipulation of courts, this jurisdiction is exclusive [START_REF] Dulin | le rôle du parquet dans le choix de la sanction des infractions économiques et financière[END_REF] , that is to say, that this specialized prosecutor, as well as the Paris court, shall have exclusive jurisdiction over the prosecution, investigation and adjudication of these offenses. For the perpetrators of tax offenses of small scale, the prosecutor of the financial Republic can continue only if the tax authorities have filed a complaint before [START_REF] Jean | La criminalité organisée : droit français, droit international et droit comparé[END_REF] . Finally, for all other offenses in economic and financial issues, the competency of the public prosecutor is concurrent with that of other jurisdictions. This will concern, for example, the crimes of breach of probity, unlawful influence on votes, tax fraud (value added tax), since they are highly complex [START_REF] Frison-Roche | la justice pénale face à la délinquance économique et financière[END_REF] , as well as bribery or trading in influence involving a public official of a foreign state or an international organization, fraudulent evasion of the establishment or payment of the tax, as well as, that of laundering these criminal offenses [START_REF]Nouveaux enjeux de la délinquance économique et financière[END_REF] .

In the event of competition between the public prosecutor (and the Paris court) and the other prosecutors (or courts) concerning the prosecution or the judgment of an infringement, a mechanism of divestment is provided for by the code of procedure criminal. The decision to divest can be appealed, which is processed quickly by the higher court (in a few days).

Then, in addition to this financial prosecutor, there are specialized interregional jurisdictions (JIRS). They were created by the law of 9 th March 2004 and replaced the first specialized jurisdictions created by the 1975 law. There are eight of them in France and siege in Bordeaux, Fort-de-France, Lille, Lyon, Marseille, Nancy, Paris, and Rennes 8 . They are responsible for most of the criminal offenses in economic and financial matters, provided they are of "Great Complexity" [START_REF][END_REF] . Some people prosecuted before these specialized courts have attempted to have the proceedings annulled on the ground that the case in question did not have the character of "great complexity". However, this argument is doomed to failure because in a pragmatic concern to secure procedures, the Court of Cassation (the highest court in France) ruled that the parties could not challenge the character of "great complexity" before the judge 10 . In fact, the assessment of this condition is the sole responsibility of the judicial authority and cannot be contested by the parties.

The jurisdiction of these specialized interregional jurisdictions extends to several courts of appeal. Specialized magistrates are assigned and they benefit

B) Specialization of judicial actors

Handling the improvements of dossiers of great complexity in economic and financial issues went through the provision of the jurisdictions of new means, in particular human. In response to an old and recurring demand of the practice, the law of July 2, 1998 came to create the function of specialized assistants whose role is to assist in their task which is the specialized magistrates. These specialized assistants are usually appointed officials who have particular expertise in certain areas. These areas can be legal (e.g. monetary and financial law, or tax law) or technical (e.g. accounting,). They have full access to the file and may attend the interrogations, the search, and all investigative actions organized by magistrates. Their study is a precious help to the day for the judicial authority. However, they have no jurisdictional power. Prior to the exercise of their activity, the specialized assistants lend oath.

Given the complexity of the files and the significant workload of the training firms in particular, the use of the service of these specialized assistants is frequent.

The specialization of the French criminal justice system makes it possible to implement specific tools designed to reinforce repressive efficiency.

II.THE SEARCH FOR REPRESSIVE EFFICIENCY

A) Expanded investigative means

The Code of Criminal Procedure (CCP) devotes a whole chapter to the broad powers of investigation conferred on police officers and magistrates in economic and financial issues. These means of the investigation come from those used in organized crime. Until the law of 6 December 2013, only certain offenses could give rise to this extension of investigative powers. Since the law of 6 December 2013, the legislature has considerably extended, or even generalized, certain special powers have been provided for organized crime in economic and financial crime. This extension has been justified because this delinquency has been considerably organized, complex and internationalized in recent years. However, not all investigative acts are possible for all economic and financial offenses. Indeed, the legislator wished to make a gradation of these means of investigation according to the type of offense concerned.

In general, the investigative powers conferred on investigators in economic and financial matters are exorbitant, since they are also used in the fight against organized crime. As such, investigators can conduct physical surveillance, which can be conducted throughout the national territory, and infiltration (delinquent networks, for example). Infiltration is the surveillance of people suspected of committing a crime by posing them as one of their co-perpetrators, accomplices or concealers. Investigators will then be authorized to tapping phones, sounding or fixing images in certain places or vehicles, as well as collecting computer data [START_REF]Court of Cassation[END_REF] . These powers are in particular provided for the offenses of banking or tax evasion, which can be difficult to establish by traditional means. In addition, during a judicial inquiry, the judge of liberty and detention, may in order to guarantee the payment of the fines incurred as well as, if necessary, the compensation of the victims, order provisional measures on the property, movable or immovable, divided or undivided, of the Charged Person.

The law of 6 December 2013 on the fight against tax evasion and the large economic and financial crimes also provided, in the interests of effective law enforcement, the possibility of ordering a placement in custody for offenses of art economic and financial under the same conditions as in the case of organized crime, i.e. for a maximum of 96 hours and with possible postponement of the intervention of the lawyer for a maximum of 48 hours in the event of particular circumstances justifying it 12 . However, the French 11 POTASZKIN (T.), L'éclatement de la procédure pénale ; Vers un nouvel ordre procédural pénale, Lextenso, Paris, 2014, p. 181. 12 Ibid.

Constitutional Council has ruled these provisions contrary to the Constitution. It has in fact held that some of the offenses, such as bribery, trading in influence and tax and customs fraud, are offenses which are not in themselves capable of impairing the security, the dignity or the life of people. It, therefore, considers that the use of custody according to special terms constitute a disproportionate interference with fundamental rights of people to the aim pursued by the legislature 13 . Thus, the legislator has the possibility of conferring investigative powers on investigators, provided that this does not violate the fundamental rights of suspects.

B) A shift in the burden of proof

The law of 6 November 2013 resulted in performing a shift in the burden of proof regarding money laundering. In fact, under French law, money laundering is the facilitation, by any means, of the false justification of the origin of the property or the income of the offender who has given him a direct or indirect profit. Money laundering also means providing assistance to a placement, concealment or conversion of the direct or indirect proceeds of an offense. However, this offense is extremely difficult to prove, since it is necessary for the public prosecutor to provide evidence of the criminal origin of the laundered funds. Hence, the interest of the new article 324-1-1 of the penal code, does provide that property or income is presumed to be the direct or indirect result of a crime or offense where the material, legal or financial conditions of the investment, concealment or conversion transaction cannot have any other justification than to conceal the origin or beneficial owner of these goods or revenues. Thus, since certain disturbing elements on the origin of the funds are brought by the prosecuting party, it is then up to the person prosecuted to demonstrate the lawfulness of their origin. This adjustment of the burden of proof will thus make it possible to reinforce the effectiveness of the repression of the offense of money laundering.

C) Towards a "contractualization" of criminal law

Since the French authorities wish to avoid impunity for the perpetrators of offenses, the legislator has developed alternative measures to criminal prosecution as such. The implementation of these alternatives is entrusted to the public prosecutor.

On the one hand, as long as the public action has not been set in motion, the public prosecutor may propose, directly or through an authorized person, a criminal composition to a natural person who acknowledges having committed one or more offenses punishable by a fine or imprisonment of up to five years. This criminal composition may consist in the payment of a fine, in the delivery of property to the State, or in the obligation not to leave French territory for a maximum period of six months. In some cases, the criminal composition will have to be homologated by a judge of the seat. In any case, if the penal composition is fully executed by the offender, the prosecution is extinguished.

On the other hand, the law of December 9, 2016 created the public interest judiciary agreement (CJIP). The conclusion of a CJIP involves the payment to the Public Treasury of a public interest fine, the amount of which is calculated in a proportionate manner to the benefits from the deficiencies found, within a limit of 30% of the average annual turnover recorded during last three years. The amount of the sanction is certainly limited, though the ceiling is still very high. The idea is to ensure effective repression of the offense by encouraging companies to conclude such an agreement rather than undergo a lengthy trial whose outcome will be uncertain, both for them and for the public prosecutor. Originally planned for corruption, trading in influence and the laundering of tax fraud, this convention has been allowed for tax evasion since autumn 2018.

These two possibilities offered to the Prosecutor of the Republic, who is the prosecutor, give him very great power. The intention of the legislator was thus to encourage large fraudulent companies to cooperate with the authorities when an offense is committed by providing them with a lesser penalty than that incurred if there was a trial. These possibilities lead French law towards a contractualization of criminal law in economic and financial matters, to the extent that the agreement of the author of the facts is necessary to implement them.

III. THE DISADVANTAGES OF THE CURRENT SYSTEM

A) A sense of impunity

Economic and financial offenses are often difficult to detect and prosecute. The convictions of the perpetrators of such offenses obviously exist, but some escape conviction because of the lack of evidence. This could give rise to a feeling of impunity in public opinion. Indeed, there seems to be a two-speed justice. The first is very fast, severe, and concerns many offenses "daily ". These offenses are simple and the evidentiary issues rarely arise as the facts often appear blatant. The second concerns economic and financial offenses where judicial time is much longer, because the nature of these offenses requires a thorough investigation and important means of investigation. The question of proof arises almost systematically. And the number of relaxes, because of lack of evidence or due to a defect in the procedure, is automatically larger. As a result, public opinion generally thinks that offenders "in white collar" are less severely punished than ordinary offenders. Moreover, since these people are often very well integrated into society, stiff prison sentences are rarely pronounced, which amplifies this feeling. But this difference of fact between ordinary and economic and financial offenses is due to the very specific nature of these offenses, which are very difficult to prove and therefore to pursue.

Moreover, many prosecutions are initiated not only by the public prosecutor but also by anti-corruption associations. In fact, article 2-23 of the Code of Criminal Procedure provides for the possibility that associations approved and declared for at least five years to become civil parties, and thus to set in motion public action. Here again, this state of affairs reinforces the idea that the public authorities are not doing enough to combat economic and financial infringements. However, this possibility should be seen as an effective and dissuasive tool for combating economic and financial crime.

B) A sometimes inappropriate penology

The main deficiency of the current system is the penology that is not necessarily always adapted to the reality of the offense. Indeed, while the penalties have been aggravated by the reforms in recent decades, the development of the contractualization of criminal law has the effect of punishing the perpetrators of offenses fairly weakly. For example, some large corporations who commit economic and financial offenses may be tempted to pay for the contingent amount they may have to pay to the state to avoid a criminal trial. This possibility obviously does not concern small businesses or natural persons. However, the risk does exist for large companies.

  from the technical support of certain assistants. The National School of the Judiciary (ENM) has set up specific training courses in the form of practical internships in specialized interregional jurisdictions or deepening knowledge in criminal economic and financial law. Otherwise, the assessment of the skills of the magistrates who have to deal with criminal procedures in economic and financial issues is often casuistic, depending on the training and experience of each one.

  [START_REF] Bonis-Garçon | Infraction en matière économique et financière[END_REF] ". The new opportunities created by the business have been seized by offenders, who have significantly more complex apprehension of this new criminal phenomenon. This complexity has led to many difficulties for the state to effectively enforce the rules of criminal

	law as economic and financial offenses are often hidden, concealed or even
	imperceptible to the authorities. Then in the absence of sufficient evidence, the
	relaxation	is	then	obvious for		these	offenders	"in	white
	collar". Es D deficiencies have been observed in handling cases in economic and
	financial matters of great complexity. Fraudulent practices used by criminals
	are becoming		more	sophisticated and extend	over	the	whole
	territory and more frequently outside	France via	the Internet.	This led
	French legislator to put in place various mechanisms to ensure effective
	repressions of these offenses. The first manifestation of the reaction of the
	legislator is the law of 6 August 1975, which resulted in establishing
	specialized courts	in	economic	and	financial	matters
	whose territorial jurisdiction was expanded.

babakhani_erfan@yahoo.com

jimmy.harang@lopste.net

François-Xavier Dulin, Le rôle du parquet dans le choix de la sanction des infractions économiques et financière, Aj pénal, 2019, Paris, p.3.

Code of Criminal Procedure, Article 705-1.

Book of Tax Procedures, Article L. 228.

The "great complexity" is assessed in particular with regard to the number of perpetrators, accomplices or victims of the offense and the geographical jurisdiction on which they extend.

Code of Criminal Procedure, Article 705.

Constitutional Counsel, 4 December 2013, No. 2013-679 DC.