Philippe Thomas
email: philippe.thomas.2008@orange.fr

Remy Billon

Cyril Hazif-Thomas

Narrative analysis in Alzheimer's disease

Keywords: Alzheimer's Disease, semiotics, semantics, textuality

Background. Alzheimer's Disease (AD) is a neurodegenerative illness, which occurs with increasing frequency as people age. While progressive memory impairment is the upfront element associated with the disease, other neurocogni-tive troubles are also associated with it, such as language impairment which can degenerate into aphasia. Language disorders interfere and worsen the functioning of memory.

Aim of the study. To evaluate semantic and textual impairment in AD patients.

Methods. The current study involved 151 AD patients undergoing consultation at Brest University Hospital. Certain sociodemographic data (sex, age, cultural levels) were collected, as well as results from the following neuropsychological tests: Folstein (MMSE); Dubois's 5-word test; Dubois's frontal assessment test battery (fluencies); Cornell's scale for depression; and Barbizet's test ( "The Lion's tale"). All were subject to textual analysis. Our sample of demented patients included 102 females and 49 males of average age 80.3 ± 6.91.

Results. All the tests, including the number of items recalled much later in the Barbizet's test, showed impairment, all the more by Folstein's test being altered. The demented patients' formal fluency was less impaired than their semantical lexical fluency (scored respectively 5.74 ± 1.09 versus 4.41 ± 2.19; t = 5.60, p< 0.01). The demented cohort exhibited more intrusions (n= 36) than inversions in the delayed recollection of the Lion's Tale, both for items and the episodes in which they occurred (n= 19). The regressive PLS analysis showed that, to explain the overall scores relating to "The Lion's Tale", calculated later, only attainment of lexical fluency had any notable influence (Regression coefficient CR = 0.224) or, more accessorily, the cultural level (CR = 0.12).

Conclusion. AD patients' proficiency in tests of category fluency and their cultural levels have effects on narrativity.

Introduction

Alzheimer's Disease (AD) is a neurodegenerative illness which occurs with increas-ing frequency along with advancing age (P. [START_REF] Thomas | Language Disorders in Alzheimer's Disease and Phatic Function[END_REF]. While progressive memory loss is the upfront element associated with AD, other neuro-psycho-cognitive troubles are also associated with it, such as lan-guage impairment which can degenerate into aphasia [START_REF] Gewirth | Altered patterns of word associations in dementia and aphasia[END_REF].

Language dysfunction is one of the most significant symptoms affecting AD patients' ability to narrate a story [START_REF] Hassenstab | Neuropsychological markers of cognitive decline in persons with Alzheimer disease neuropathology[END_REF]. In the earliest stages, the influence is slight, but it increases over time through foreseeable phases. Memory dysfunction also interferes with the patient's abilities to use language (Reilly, Rodri-guez, Lamy, & Neils-Strunjas, 2010). The patient's judgment and ability to conduct his/her own life also have a broad impact on their language difficulties. Put another way, disorganized phasic patterns and verbal abilities often limited by aphasia, reflect a deconstruction of a patient's personal meaning, which is itself essential for recall, as well for social interactions with others, and maintaining memories. An inconsistent story of one's life weakens one's self-esteem and ability to monitor daily life, a condition that leads to behavior disorders (P. [START_REF] Thomas | Language Disorders in Alzheimer's Disease and Phatic Function[END_REF]. Knowing the importance of both these components could help the individual prepare as well as possible for the consequences on his/her social life. We have tried, in this monocentric prospective observation and study, to evaluate the semantic and narrative dysfunction in AD.

Methods

The cohort we studied were elderly demented patients living at home with AD, who had been followed for at least three months by gerontopsychiatrists for cognitive, behavioral, and memory dysfunction. During a gerontopsychiatric interview, the patients and accompanying family were informed completely and loyally and in understandable language about how the study would be carried out. With the agreement of the patient and his/her assistants, the results of the neuropsychological tests were computerrecorded anonymously. The study protocol was ethically reviewed and approved by the Hospital of Brest ethics committee. Computing procedures for data capture and processing followed the recommendations of the French National Commission on Computer Technology and Freedom. Criteria were determined for inclusion based on the AD patients (MMSE >10) [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF] having given written consent, together with their family assistants, to take part in this study validated by the Hospital of Brest ethics committee. Only those patients still capable of understanding the instructions for the Barbizet's test, showing symptoms of Alzheimer dementia-i.e. with-out massive memory impairment and capable of asserting their consent and assent during the test and fit and able in the eyes of the Law-were included. A patient's inability to understand information relating to the study, his/her inability to choose whether or not to take part, and the possible opposition of the family member ac-companying him/her were decisive criteria for exclusion. The size of the population studied reflects the number of demented persons followed over the period of this one-year study.

Semantic memory is the memory necessary for language manipulation, and is not very vulnerable to amnesia [START_REF] Tulving | Organization of memory: quo vadis? The cognitive neurosciences[END_REF]. To evaluate semantic and textual disorders in the cohort of AD patients, we routinely recorded the results of tests measuring memory, as well as noting some sociodemographic data. These were sex; age; and cultural level as estimated by the patient's number of years of schooling: 1 = illiterate ; 2 = can read and write ; 3 = six years schooling ; 4 = 11 years schooling ; and 5 = more than 11 years of schooling. The following are the tests customarily carried out on patients under consultation for AD: Folstein Test (MMSE) [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF]); Dubois's 5-word test (D5WT) (B. [START_REF] Dubois | L'épreuve des cinq mots. Fiche technique[END_REF]; formal fluency (number of animals mentioned in one minute) semantic lexical flu-ency (number of words beginning with L named in one minute); (B. [START_REF] Dubois | L'épreuve des cinq mots. Fiche technique[END_REF]; DFB (Dubois's Frontal Battery ) (B. [START_REF] Dubois | The FAB: a Frontal Assessment Battery at bedside[END_REF]; Cornell's scale (depression) [START_REF] Alexopoulos | Cornell Scale for Depression in Dementia[END_REF]; and Barbizet's test [START_REF] Barbizet | L'histoire du lion (considérations sur la fabulation)[END_REF] aka "The Lion's Tale". The formal fluency test uses mechanisms of analogy between memorized objects, while semantic lexi-cal fluency refers to the use of vocabulary stored in memory, which is vulnerable in Alzheimer's disease.

"The Lion's Tale" (presented in Table 1) must be read by the examiner slowly. The subject must be informed that it is a long story, the point of which is to evaluate his memory capacity. He must be told it is not a matter of repeating the story word for word, but just recalling its main thrust and as many details as possible. Barbizet recommended an immediate application, an immediate recall, and then a delayed recall. In practice, in the present framework, the text was read out at the beginning of the consultation, followed by an immediate recall, and then a delayed recall to-wards the end of the consultation--that is, about one hour later.

Table 1 "The Lion's Tale" [START_REF] Barbizet | L'histoire du lion (considérations sur la fabulation)[END_REF] [START_REF] Tenenhaus | PLS path model-ing[END_REF]. If a classic linear model is used, collinearity between explanatory variables makes the model unstable, and the results uncertain.

Results

One hundred and fifty-one patients, living at home on their own or with their families, were included in the study. There were no cases recorded of refusal to participate in the study, or to having the results observed and computerized anonymously. The cohort of demented patients was made up of 102 females and 49 males of an average age of 80.3 years ± 6.91. The studied population showed more cases of intrusion (n= 36) than inversion in the later narration, whether items or episodes combining the two (n= 19). Nearly all the inversions were by patients whose MMSE was below 19, and the intrusions were by patients whose MMSE was below 15 (Table 3). Admittedly, intrusions are specific to AD pathology, where we find their greatest occurrence. However, these results could suggest that intrusions are essentially linked to executive problems concerning response control. In respect to patients' dysexecutive syndrome (DFT < 13; n= 77), it is to be noted that, overall, inversions occur in the more deficient group (Table 4). It is also to be noted that inversions are observed to be more precocious when the disease has caused a deterioration of textuality, and hindered the pragmatics of communication. MMSE for the executive syndrome sub-group (n= 77) was 11.9 ± 2.9 against 20.9 ± 2.6 for the less seriously affected group (n= 74). The correlation matrix for the various parameters showed many correlations, leading to our choice of PLS for regression analysis. The factor studied is the over-all score for Barbizet's test, with the model including the variables of the previous figure. Formal fluency, with a coefficient of 0.224, remains the characteristic that most influences the overall score. The higher it is, the greater the success in the test; this influence is almost three times that of the lexical memory. Then, we will retain cultural level (coefficient = 0.12) and, more accessorily, age (coefficient = 0.11); all the others are less interesting.

Discussion

We are presenting here the results of a preliminary study on textuality in AD, based on consultation with patients in an old-age psychiatric department. The very nature of the department explains the age of those recruited. The patients' worsening MMSE made it hard for them to understand the text, and even the value of their consent. The aid given by their family or accompanying person was then precious, and all the tests were carried out under the specialized consultation of our establishment.

The executive disorders studied by the DFB test and cognitive disorders estimated by the MMSE are strongly correlated. Also it is difficult to know which verbal problems resulted specifically from which factors. We chose a threshold of 13 for two reasons: one, so that the sub-populations were balanced, and two, for a clear cut-off point showing whether or not there was a dysexecutive syndrome connected to frontal disorders. The presence of such a dysexecutive syndrome, in case of late depression (with or without dementia), impacts on the patient's cognitive status [START_REF] Hazif-Thomas | Syndrome dysexécutif et dé-pression tardive[END_REF]. The depressive elderly person frequently presents changes in verbal fluencies and deficits of planning con-nected to frontal disorders [START_REF] Elliott | Abnormal response to negative feedback in unipolar depression: evidence for a diagnosis specific impairment[END_REF]. The scores on the Cornell scale were weak in the population we studied. Depression is common when dementia progresses, especially when the elderly person lives in a nursing home [START_REF] Alexopoulos | Cornell Scale for Depression in Dementia[END_REF]. The elderly included in this study lived at home. The scores on the scale of Cornell presented here are low (Table 2: 11.3 ± 4,5, while the depression threshold value is >8).

We shall note in this study the important progression of the narrative disorders observed in the scores on the delayed recall in "The Lion's Tale", all the more degraded that the cognitive disorders are installed. Intrusions and inversions can result from cognitive disorders as well as from related executive disorders. Formal fluencies weigh in in a significant but modest way on the model of regression (CR = 0.224), so that with a lower score on the cultural level (CR= 0.117), the cognitive disorders of the dementia, as well as the linguistic disorders, are being classically modulated [START_REF] Arroyo-Anllo | Lexical priming in Alzheimer's disease and aphasia[END_REF][START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF].

Although the PLS model is more influenced by formal fluency, the lexical se-mantic fluencies are more altered than the formal fluency (Table 2). Finding a word beginning with a letter ("L" in French, "S" in English, for reasons of frequency of use of this letter in the beginning of a word) requires a strategy of recovery based on the abstract lexical representation [START_REF] Lee | Langage et Maladie d'Alzheimer : Analyse Multidimensionnelle d'un Discours Pathologique[END_REF]. The degradation of formal fluency triggers a more important disorganization of the functions of language. More-over, the analog mechanisms mobilized by demented patients seem to be more concerned and better conserved in Alzheimer's disease, judging from the results observed with formal fluency. Memory, together with judgment, build a meaning of life conductive to a coherent self-narrative account [START_REF] Ricoeur | Soi-même comme un autre[END_REF].

Disorders of judgment and memory seem to be allied in Alzheimer's disease. Wording and the building of sentences are altered in this disease, but the analog operation seems to be preserved longer. The figures evoked for an analog mechanism mobilize old emotions, and from these emerge ancient memories buried deep in the patient's mind [START_REF] Brainerd | Fuzzy-Trace theory and lifespan Cognitive Development[END_REF][START_REF] Thomas | La reconstruction de la signification dans la démence[END_REF]. The concrete items with strong emotional connotation in "The Lion's Tale" (A woman, a child, a lion …) are better recalled than the more secondary features of the story (For a long time, fixedly …); even there, emotion appears clearly as the "power steering" of the reason and the memory.

Conclusion

Lexical semantic fluency is more altered than formal fluency in the Alzheimer's disease. The degradation of formal fluency testifies to a more important disorganization of the functions of the language. Formal fluencies are less impaired (Imaging or Gist memory process) than Semantic lexical Fluency (Verbatim process) [START_REF] Brainerd | Fuzzy-trace theory : Dual account in memory, reasoning, and cognitive neuroscience[END_REF]. The impairments of fluencies have an impact on the memorization of a text, in parallel with the development of cognitive disorders. The patients' cultural level also influences his/her narrative skills, which remain sensitive to the emotional aspect of a story when assessed in memory testing. This emotional mobilization is probably an interesting way to motivate patients in speech therapy, for the prevention of damage to, or maintenance of, lexical and textual capacities in patients with Alzheimer's disease.

  . Text and sequential organization These data are then compared with that of the neurosychological tests. Statistics were established with a SAS program by a bio-statistician, Dr. R. Billon. The tests selected were Chi2, Student (unpaired), Pearson's correlations, and Partial Least Squares regression, chosen to reduce risks of collinearity

	Episode	Text
	Escape	A lion named Sultan escapes from its cage, due to
		the door having been left open by a careless guard
	Crowd movement The crowd of visitors, which is numerous on
		Sunday, runs away towards the nearby buildings
	Woman and child A woman, dressed in blue, who was holding her one-
		year-old child in her arms, drops him. The lion
		seizes him.

Deal

The woman, in tears, retraces her steps, and begs the lion to return her young

Resolution

The animal looks at her for a long time, fixedly, and finally releases the child without having done him the slightest bit of harm.

So this short text, with 22 sub-items learned sequentially, is to be recalled intrusions and inversions.

  Table2reprises the main results. The average characteristics did not differ statistically between the males and the females. The tests showed that the formal fluency of the demented patients was less impaired than their semantical lexical fluency

	(scored respectively 5.74 ± 1. 09 versus 4.41 ± 2. 19;
	t= 5.60, p< 0.01).
	Ag	Cultural	MMSE	DFB
	e	level
	80.3 ± 6.9	2.82 ±	16.5 ± 5.1	11.3 ± 4.5
		0.50
	Cornell	D5WT	Semantic lexical	Formal
			Fluency	Fluency
	10.6 ± 4.2	4.58 ± 1.1	4.41 ± 2.2	5.7 ± 1.1

Table 2 Sociodemographic data and results in the diverse tests (N= 151). DFB= Dubois's Frontal Battery; D5WT = Dubois's 5 words test.

Table 3

 3

	Percentage of Inversions (n= 19) and intrusions (n= 36) according to the MMSE.
	Chi2 with Yates's correction (p< 0.05) comparing the number of intrusions and of
	inversions; and for each items, the level of MMSE (cut-off MMSE >14)
	MMSE> 19 N= 50	MMSE : 19-15 N= 42	MMSE 14-11 N= 39	MMSE = 10 N= 21

immediately, and repeated later. The elements can be organized, according to the author of the test, into five episodes: escape, crowd movement, woman and child, deal, resolution. Barbizet's test reveals what things from the tale are forgotten, and notes


Louis De Barros
email: louis.debarros@ujf-grenoble.fr

Michel Dietrich

Perturbations of the seismic reflectivity of a fluid-saturated depth-dependent poro-elastic medium

Keywords: Wave propagation, poro-elastic medium, Fréchet derivatives

Analytical formulas are derived to compute the first-order effects produced by plane inhomogeneities on the point source seismic response of a fluid-filled stratified porous medium. The derivation is achieved by a perturbation analysis of the poro-elastic wave equations in the planewave domain using the Born approximation. This approach yields the Fréchet derivatives of the P -SV -and SH-wave responses in terms of the Green's functions of the unperturbed medium. The accuracy and stability of the derived operators are checked by comparing, in the time-distance domain, differential seismograms computed from these analytical expressions with complete solutions obtained by introducing discrete perturbations into the model properties. For vertical and horizontal point forces, it is found that the Fréchet derivative approach is remarkably accurate for small and localized perturbations of the medium properties which are consistent with the Born approximation requirements. Furthermore, the first-order formulation appears to be stable at all source-receiver offsets. The porosity, consolidation parameter, solid density and mineral shear modulus emerge as the most sensitive parameters in forward and inverse modeling problems. Finally, the Amplitude-Versus-Angle response of a thin layer shows strong coupling effects between several model parameters.

I. INTRODUCTION

The evaluation of the sensitivity of a seismic wave field to small perturbations of the material properties is a classical issue of seismology which arises in the solution of forward and inverse scattering problems [START_REF] Aki | Quantitative seismology[END_REF][START_REF] Tarantola | Inversion of seismic reflection data in the acoustic approximation[END_REF]. Sensitivity operators are mainly useful for the optimal design of field or laboratory experiments, for the interpretation of time-lapse monitoring surveys, and for the development of imaging and linearized inversion techniques. In particular, sensitivity operators play a central role in least-square inversion schemes using gradient techniques. All these applications, especially the latter, call for fast and effective numerical computation methods of the sensitivity operators. Indeed, the most intuitive approach to compute the perturbational or differential seismograms in a structure described by N parameters is to use a finite perturbation scheme that demands N + 1 forward modeling computations. This approach rapidly becomes prohibitive as the number of parameters increases. The problem can be solved more elegantly by first deriving the so-called Fréchet derivatives of the seismic wave fields with respect to material properties. Although restricted to first-order effects only, this procedure makes it possible to efficiently predict small changes of the seismic response resulting from slight modifications of the material properties. Furthermore, this solution can be implemented by solving only one large forward problem.

One of the most significant contributions in this area is the work of [START_REF] Tarantola | Inversion of seismic reflection data in the acoustic approximation[END_REF] who applied a first-order perturbation analysis to the elastodynamic wave equation to derive a series of general formulas relating the scattered wave field to heterogeneities in an arbitrarily complex elastic medium. The same approach was subsequently used in a more restrictive sense for layered media in the plane-wave domain by [START_REF] Pan | Full-waveform inversion of plane-wave seismograms in stratified acoustic media: Theory and feasability[END_REF] in the acoustic case and by [START_REF] Dietrich | Perturbation of the plane-wave reflectivity of a depth-dependent elastic medium by weak inhomogeneities[END_REF] in the P -SV case. The Fréchet derivatives obtained in these different cases are all expressed as combinations of the incident wave field generated by the seismic source at the scatterer location and the Green's functions between the scatterer and receiver locations. The structure of these expressions underlines the fact that the scattered waves due to perturbations in the material properties (usually density and acoustic or elastic parameters) can be interpreted as a wave field generated by a set of secondary body forces coincident with the heterogeneities, and determined by complex interactions between the incident waves and the medium perturbations. This structure is met with all problems of acoustic, seismic or electromagnetic wave propagation in weakly inhomogeneous media, and will also be found in more complex situations such as anisotropic or poro-elastic media.

The poro-elastic model, which is the subject of this paper, involves more parameters than the visco-elastic case, but on the other hand, the wave velocities, attenuation and dispersion characteristics are computed from the medium's intrinsic properties without having to resort to empirical relationships. Since the pioneering work of [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. i. low-frequency range, ii. higher frequency range[END_REF], many authors [START_REF] Auriault | Dynamics of porous saturated media, checking of the generalized law of Darcy[END_REF][START_REF] Dutta | Attenuation and dispersion of compressionnal waves in fluid-filled porous rocks with partial gas saturation (White model) -Part I: Biot theory[END_REF]Johnson et al., 1994, for example) have introduced improvements of the poro-elastodynamic equations, either by averaging or by integrating techniques. The [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. i. low-frequency range, ii. higher frequency range[END_REF] theory and its applications is still a field of active research, as demonstrated by the large number of current publications devoted to the subject (see, e.g., [START_REF] Trifunac | Biot Centennial -Earthquake Engineering[END_REF]. The forward problem, i.e., the computation of synthetic seismograms in poro-elastic media has been solved in different configurations and with several techniques [START_REF] Carcione | Wave propagation in anisotropic, saturated porous media: Planewave theory and numerical simulation[END_REF][START_REF] Dai | Wave propagation in heteregeneous porous media: A velocity-stress, finite-difference method[END_REF][START_REF] Garambois | Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[END_REF][START_REF] Haartsen | Electroseismic waves from point sources in layered media[END_REF].

However, the inverse problem has only been rarely addressed, and to our knowledge, it has never been without first estimating the wave velocities [START_REF] Berryman | Estimating rock porosity and fluid saturation using only seismic velocities[END_REF][START_REF] Chotiros | An inversion for Biot parameters in a water-saturated sand[END_REF][START_REF] Spikes | Rock physics model-based inversion[END_REF]. Yet, inversion algorithms can provide useful information on the material properties, notably permeability and porosity which are the most important parameters to characterize porous media.

The main objective of this work is to extend the methodology used in the elastic case [START_REF] Dietrich | Perturbation of the plane-wave reflectivity of a depth-dependent elastic medium by weak inhomogeneities[END_REF] to obtain the Fréchet derivatives for stratified poro-elastic media. We consider here a depth-dependent, fluid-saturated porous medium representing reservoir rocks or sedimentary layers. The computation of the point source seismic response of the layered structure is carried out by combining the Generalized Reflection and Transmission Matrix Method [START_REF] Kennett | Seismic Wave Propagation in Stratified Media[END_REF] with the discrete wavenumber technique [START_REF] Bouchon | A simple method to calculate Green's functions for elastic layered media[END_REF]. This combination was already implemented by [START_REF] Garambois | Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[END_REF] for the numerical simulation of the coupled seismic and electromagnetic wave propagation in porous media by using the work of [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. In the following sections, we first present the governing equations and constitutive parameters for porous materials before expressing the wave propagation equations for depth-dependent media. Next, we develop the analytical computation of the Fréchet derivatives in the frequency-ray parameter domain for the P -SV -and SH-wave cases. Finally, we check the accuracy of the sensitivity operators obtained in the time-distance domain, both in an infinite medium and in a complex seismic model. We conclude with the sensitivity of the seismic waveforms with respect to the different model parameters.

II. WAVE PROPAGATION IN STRATIFIED POROUS MEDIA

A. Governing equations

Assuming a e -iωt dependence, where t is time and ω the angular frequency, [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF][START_REF] Pride | Relationships between seismic and hydrological properties, chapter 8, Hydrogeophysics[END_REF] rewrote [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. i. low-frequency range, ii. higher frequency range[END_REF] equations of poroelasticity in the form

                   ∇ • τ = -ω 2 (ρ u + ρ f w) τ = [ K U ∇ • u + C ∇ • w ] I + G [ ∇u + (∇u) T -2/3 (∇ • (u I)) ] -P = C ∇ • u + M ∇ • w -∇P = -ω 2 ρ f u -ω 2 ρ w , (1)
where u and w are the solid average displacement and the relative fluid-to-solid displacement, respectively. More precisely, defining u s and u f as the displacements of the solid and fluid phases of a porous continuum, we can write u ≃ u s and w = φ(u s -u f ), where φ is the porosity. P represents the interstitial pressure and τ is the 3×3 stress tensor. ρ denotes the density of the porous medium which is related to the fluid density ρ f , solid density ρ s and porosity φ via the relationship

ρ = (1 -φ) ρ s + φ ρ f .
(2)

The undrained bulk modulus K U is defined under the condition w = 0. G is the shear modulus of a drained or an undrained medium as it is independent of the fluid properties [START_REF] Gassmann | On the elasticity of porous media ( Über die elastizität poröser medien)[END_REF]. The fluid storage coefficient M represents the amount of fluid a sample can accumulate at constant sample volume. Biot's C modulus is a mechanical parameter describing the variation of the fluid pressure due to a change of the sample volume in an undrained medium. At low frequencies, these parameters as well as the Lamé parameter λ u defined below are real, frequency-independent and can be expressed in terms of the drained modulus K D , porosity φ, mineral modulus of the grains K s and fluid modulus K f [START_REF] Gassmann | On the elasticity of porous media ( Über die elastizität poröser medien)[END_REF]:

K U = φ K D + 1 -(1 + φ) K D K s K f φ (1 + ∆) , λ U = K U - 2 3 G C = 1 - K D K s K f φ (1 + ∆) , M = K f φ (1 + ∆) (3) with ∆ = 1 -φ φ K f K s 1 - K D (1 -φ) K s .
It is also possible to link the bulk properties K D and G to the porosity and constitutive mineral properties via empirical relationships derived from experimental results [START_REF] Bemer | Geomechanical log deduced from porosity and mineralogical content[END_REF][START_REF] Pride | Relationships between seismic and hydrological properties, chapter 8, Hydrogeophysics[END_REF]:

K D = K s 1 -φ 1 + c s φ and G = G s 1 -φ 1 + 3 c s φ/2 . ( 4
)
Equations ( 4) have the merit of being very simple and introduce only two additional parameters, namely, the shear modulus of the grains G s and the consolidation parameter c s . The latter mainly depends on the cementing properties of the grains, but also on the pore shape.

The value of the consolidation parameter c s is typically between 2 to 20 in a consolidated medium, and can be very much greater than 20 in an unconsolidated soil.

Finally, the wave attenuation is explained by Darcy's law which uses a complex, frequency-dependent dynamic permeability [START_REF] Johnson | Probing porous media with first and second sound[END_REF]:

ρ = i η ω k(ω) with k(ω) = k 0 / 1 -i 4 n J ω ω c -i ω ω c . (5)
The dynamic permeability k(ω) tends toward the hydrogeological (dc) permeability k 0 at low frequencies where viscous losses are dominant. It includes a correction accounting for the inertial effects at higher frequencies. These two domains are separated by the relaxation frequency

ω c = η ρ f F k 0 ( 6
)
where η is the viscosity of the fluid. Archie's law ( F = φ -m ) expresses the electrical formation factor F in terms of the porosity φ and cementing exponent m whose value is between 1 to 2 depending on the pore topology. Parameter n J is considered constant and equal to 8 to simplify the equations. We refer the reader to the work of [START_REF] Pride | Relationships between seismic and hydrological properties, chapter 8, Hydrogeophysics[END_REF] for more information on the parameters used in this study.

B. Coupled second-order equations for plane waves

The horizontally layered model lends itself to a number of analytical developments if one performs a plane wave decomposition of the wave fields represented by equations (1). This procedure involves a series of changes of variables and integral transforms which are described in detail by [START_REF] Kennett | Seismic Wave Propagation in Stratified Media[END_REF] in the elastic case. As in the elastic case, the introduction of new variables [START_REF] Hudson | A quantitative evaluation of seismic signals at teleseismic distances-i, radiation from point sources[END_REF] leads to a useful decomposition into P -SV -and SH-wave systems in cylindrical coordinates. By applying the whole sequence of transformations and arranging the terms, we find that the governing equations of the P -SV -wave system in depth-dependent poro-elastic media reduce to the following system of second-order equations in the angular frequency ω and ray parameter p domain:

                                         F 1z = ∂ ∂z (λ U + 2 G) ∂U ∂z + C ∂W ∂z -ω p (λ U V + C X) -ω p G ∂V ∂z + ω 2 [ ρ U -p 2 G U + ρ f W ] F 1r = ∂ ∂z G ∂V ∂z + ω p G U + ω p λ U ∂U ∂z + C ∂W ∂z + ω 2 [ ρ V -p 2 (λ U + 2 G) V -p 2 C X ] F 2z = ∂ ∂z C ∂U ∂z -ω p C V + M ∂W ∂z -ω p M X + ω 2 [ ρ f U + ρ W ] F 2r = ω p C ∂U ∂z + M ∂W ∂z + ω 2 [ ρ f V + ρ X -p 2 (C V + M X) ] . (7)
In these equations, U = U(z R , ω; z S ) and V = V (z R , ω; z S ) respectively denote the vertical and radial components of the solid displacements. Similarly, W = W (z R , ω; z S ) and X = X(z R , ω; z S ) respectively denote the vertical and radial components of the relative fluid-to-solid displacements. Variables z R and z S stand for the receiver and seismic source depths. Equations ( 7) are valid in the presence of body forces

F 1 = [F 1z (z S , ω), F 1r (z S , ω)] T
and F 2 = [F 2z (z S , ω), F 2r (z S , ω)] T defined by their vertical (z index) and radial (r index) components: force F 1 is applied on an average volume of porous medium and represents a stress discontinuity, while force F 2 is derived from the pressure gradient in the fluid.

We can then cast equations (7) in the form of a matrix differential equation as

L PSV Q PSV = F PSV , where Q PSV =          U V W X          and F PSV =          F 1z F 1r F 2z F 2r          . ( 8
)
L PSV is a differential operator given by

L PSV = ∂ ∂z M PSV 1 ∂ ∂z + ωp M PSV 2 -ωp [ M PSV 2 ] T ∂ ∂z + ω 2 ( M PSV 3 -p 2 M PSV 4 ) (9)
where the M PSV i , i = 1..4 are 4 × 4 matrices defined by

M PSV 1 =          λ U + 2 G 0 C 0 0 G 0 0 C 0 M 0 0 0 0 0          , M PSV 2 =          0 -λ U 0 -C G 0 0 0 0 -C 0 -M 0 0 0 0          , M PSV 3 =          ρ 0 ρ f 0 0 ρ 0 ρ f ρ f 0 ρ 0 0 ρ f 0 ρ          , M PSV 4 =          G 0 0 0 0 λ U + 2 G 0 C 0 0 0 0 0 C 0 M          . ( 10
)
Apart from the dimensions of the matrices, we may note that equations ( 9) and ( 10) are very similar to the expressions obtained in the elastic case.

Formally, equation ( 8) admits an integral solution for the displacement fields in terms of the Green's functions of the problem. For example, the vertical displacement U at depth z R and frequency ω is given by

U(z R , ω) = M [ G 1z 1z (z R , ω; z ′ ) F 1z (z ′ , ω) + G 2z 1z (z R , ω; z ′ ) F 2z (z ′ , ω) + G 1r 1z (z R , ω; z ′ ) F 1r (z ′ , ω) + G 2r 1z (z R , ω; z ′ ) F 2r (z ′ , ω) ] dz ′ , (11)
where G kl ij (z R , ω, z S ) is the Green's function corresponding to the displacement at depth z R of phase i (the values i = 1, 2 correspond to solid and relative fluid-to-solid motions, respectively) in direction j (z or r) generated by a harmonic point force F kl (z S , ω) (k = 1, 2) at depth z S in direction l (z or r). A total of 16 different Green's functions are needed to express the 4 displacements U, V , W and X in the P -SV -wave system (4 displacements × 4 forces).

The integrals of equation ( 11) are taken over the depths z' of a region M including the body forces F 1 and F 2 . In the case of a vertical point force at depth z S , the expressions of the forces become

     F 1z (z S , ω) = δ(z -z S ) S 1 (ω) F 1r (z S , ω) = 0 and      F 2z (z S , ω) = δ(z -z S ) S 2 (ω) F 2r (z S , ω) = 0 , (12)
where S 1 (ω) and S 2 (ω) are the Fourier transforms of the source time functions associated with forces F 1 and F 2 . Assuming that the amplitudes of both forces are similar ( F 1 ≃ F 2 ) [START_REF] Garambois | Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[END_REF], we take S(ω) = S 1 (ω) = S 2 (ω). The displacement fields for a vertical point force can then be written in simple forms with the Green's functions:

U(z R , ω; z S ) = [ G 1z 1z (z R , ω; z S ) + G 2z 1z (z R , ω; z S ) ] S(ω) V (z R , ω; z S ) = [ G 1z 1r (z R , ω; z S ) + G 2z 1r (z R , ω; z S ) ] S(ω) W (z R , ω; z S ) = [ G 1z 2z (z R , ω; z S ) + G 2z 2z (z R , ω; z S ) ] S(ω) X(z R , ω, z S ) = [ G 1z 2r (z R , ω, z S ) + G 2z 2r (z R , ω, z S ) ] S(ω) . (13)
The displacement fields corresponding to a horizontal point force and to an explosive point source are similarly defined. Explosions would be represented by Green's functions G kE ij (i = 1, 2; j = z, r; k = 1, 2) representing the radiation of an explosive point source E.

The SH case is treated in exactly the same way as the P -SV case. The corresponding second-order differential equations of motion are

     F 1t = ∂ ∂z G ∂T ∂z + ω 2 -p 2 G T + ρ T + ρ f Y F 2t = ω 2 [ ρ f T + ρ Y ] , (14)
where T = T (z R , ω; z S ) and Y = Y (z R , ω; z S ) stand for the transverse solid and fluid-to-solid displacements; F 1t (z S , ω) and F 2t (z S , ω) are the transverse components of body forces F 1 and F 2 .

As before, we can rewrite equations ( 14) in matrix form as

L SH Q SH = F SH , where Q SH =    T Y    and F SH =    F 1t F 2t    . (15)
Here, L SH is a linear operator defined as

L SH = ∂ ∂z M SH 1 ∂ ∂z + ω 2 ( M SH 2 -p 2 M SH 1 ) (16)
where the M SH i , i = 1..4, are 2 × 2 matrices defined by

M SH 1 =    G 0 0 0    , M SH 2 =    ρ ρ f ρ f ρ    . (17)

III. FR ÉCHET DERIVATIVES OF THE PLANE WAVE REFLECTIVITY A. Statement of the problem

The Fréchet derivatives are usually introduced by considering the forward problem of the wave propagation, in which a set of synthetic seismograms d is computed for an earth model m using the non-linear relationship d = f(m). [START_REF] Tarantola | Inversion of seismic reflection data in the acoustic approximation[END_REF] uses a Taylor series expansion to relate a small perturbation δm in the model parameters to a small perturbation δf in the wave field

f(m + δm) = f(m) + D δm + o ( δm 2 ) or δf = D δm ( 18
)
where D = ∂f / ∂m is the matrix of Fréchet derivatives.

Our aim is to compute the various Fréchet derivatives corresponding to slight modifications of the model parameters at a given depth. Considering for instance the density at depth z, this problem reduces, in the P -SV case, to finding analytical expressions for the quantities

A PSV 1 (z R , ω; z S |z) = ∂U(z R , ω; z S ) ∂ρ(z) A PSV 2 (z R , ω; z S |z) = ∂V (z R , ω; z S ) ∂ρ(z) A PSV 3 (z R , ω; z S |z) = ∂W (z R , ω; z S ) ∂ρ(z) A PSV 4 (z R , ω; z S |z) = ∂X(z R , ω; z S ) ∂ρ(z) . ( 19
)
We can similarly define the Fréchet derivatives

B PSV i , C PSV i , D PSV i , E PSV i , F PSV i and G PSV i , i = 1.
.4, for model parameters ρ f , ρ, C, M, λ U and G. This is the natural choice of parameters to carry out a perturbation analysis because of the linear dependence of these parameters with the wave equations. We also introduce the set of Fréchet derivatives ÂPSV c s which we will use in a second stage, and which are more convenient to use as physical parameters of the problem. The sensitivity operators are derived by following the procedure presented in [START_REF] Dietrich | Perturbation of the plane-wave reflectivity of a depth-dependent elastic medium by weak inhomogeneities[END_REF] for the elastic case.

i , BPSV i , ĈPSV i , DPSV i , ÊPSV i , F PSV i , ĜPSV i and ĤPSV i , i = 1..4, corresponding to model parameters ρ s , ρ f , k 0 , φ, K s , K f , G s and

B. Perturbation analysis

We first present, with some detail, the perturbation analysis for the P -SV case before addressing the simpler SH case. We consider small changes in the model parameters at a given depth z that result in small perturbations ∆Q PSV = [ δU, δV , δW , δX ] T of the seismic wave field and in a modified seismic response

Q ′ PSV = [ U ′ , V ′ , W ′ , X ′ ] T .
The latter can be written as

Q ′ PSV = Q PSV + ∆Q PSV ( 20
)
by assuming that the magnitudes of the scattered waves ∆Q PSV are much smaller than those of the primary waves Q PSV . Considering for instance the first component of the above vector equation

U ′ (z R , ω; z S ) = U(z R , ω; z S ) + δU(z R , ω; z S ) , (21)
we can write the scattered displacement δU as

δU(z R , ω; z S ) = M [ A PSV 1 (z R , ω; z S |z) δρ(z) + B PSV 1 (z R , ω; z S |z) δρ f (z) + C PSV 1 (z R , ω; z S |z) δ ρ(z) + D PSV 1 (z R , ω; z S |z) δC(z) + E PSV 1 (z R , ω; z S |z) δM(z) + F PSV 1 (z R , ω; z S |z) δλ U (z) + G PSV 1 (z R , ω; z S |z) δG(z) ] dz . ( 22
)
With the model parametrization adopted (i.e., a linear dependence of the model parameters with the wave equation), the perturbation analysis can mainly be done in symbolic form.

Indeed, the wave operator L PSV in the perturbed medium can be written as

L ′ PSV = L PSV + ∆L PSV , (23)
so that equation ( 8) becomes [START_REF] Hudson | The use of the Born approximation in seismic scattering problems[END_REF],

L PSV + ∆L PSV Q PSV + ∆Q PSV = F PSV . ( 24
)
We then use the single scattering (or Born) approximation to solve the above equation for

∆Q

PSV under the assumption (already used above) that

∆Q PSV ≪ Q PSV . Combining L PSV ∆Q PSV ≃ -∆L PSV Q PSV ≡ ∆F PSV (25)
This matrix equation shows that the scattered waves ∆Q PSV due to perturbations of the material properties can be interpreted as a wave field generated by secondary body forces ∆F PSV defined by the interaction of the incident waves with the heterogeneities (term

∆L PSV Q PSV ).
Moreover, this wave field propagates in the unperturbed medium represented by the wave operator L PSV . Consequently, equation ( 25) has for each of its components a solution similar to equation ( 11), by substituting ∆F PSV for F PSV and ∆Q PSV for Q PSV . Considering again the scattered displacement δU, we have

δU(z R , ω; z S ) = M [ G 1z 1z (z R , ω; z ′ ) δF 1z (z ′ , ω) + G 2z 1z (z R , ω; z ′ ) δF 2z (z ′ , ω)+ G 1r 1z (z R , ω; z ′ ) δF 1r (z ′ , ω) + G 2r 1z (z R , ω; z ′ ) δF 2r (z ′ , ω) ] dz ′ , (26)
where the secondary Born sources δF 1z , δF 1r , δF 2z , δF 2r are obtained from equation ( 7):

∆F PSV =          δF 1z δF 1r δF 2z δF 2r          =                              - ∂ ∂z (δλ U + 2 δG) ∂U ∂z + δC ∂W ∂z -ωp (δλ U V + δC X) + ωp δG ∂V ∂z -ω 2 δρ U -p 2 δG U + δρ f W - ∂ ∂z δG ∂V ∂z + ωp δG U -ωp δλ U ∂U ∂z + δC ∂W ∂z -ω 2 [ δρ V -p 2 (δλ U + 2 δG) V -p 2 δC X ] - ∂ ∂z δC ∂U ∂z -ωp V + δM ∂W ∂z -ωp X -ω 2 [ δρ f U + δ ρ W ] -ωp δC ∂U ∂z + δM ∂W ∂z -ω 2 [ δρ f V + δ ρ X -p 2 (δC V + δM X) ]                              . ( 27
)
By inserting these expressions into equation ( 26) and integrating by parts in order to separate the contributions in δρ, δρ f , δ ρ, δC, δM, δλ U and δG, we obtain an integral representation of the scattered wave field δU that we can directly identify with equation ( 22) to get the Fréchet

derivatives A PSV 1 , B PSV 1 , C PSV 1 , D PSV 1 , E PSV 1 , F PSV 1 and G PSV 1
corresponding to displacement U:

A PSV 1 = -ω 2 [ U G 1z 1z + V G 1r 1z ] B PSV 1 = -ω 2 [ W G 1z 1z + U G 2z 1z + V G 2r 1z ] C PSV 1 = -ω 2 [ W G 2z 1z + X G 2r 1z ] D PSV 1 = ∂W ∂z -ωp X ∂G 1z 1z ∂z -ωp G 1r 1z + ∂U ∂z -ωp V ∂G 2z 1z ∂z -ωp G 2r 1z E PSV 1 = ∂W ∂z -ωp X ∂G 2z 1z ∂z -ωp G 2r 1z F PSV 1 = ∂U ∂z -ωp V ∂G 1z 1z ∂z -ωp G 1r 1z G PSV 1 = ∂V ∂z + ωp U ∂G 1r 1z ∂z + ωp G 1z 1z + 2 ∂U ∂z ∂G 1z 1z ∂z + ω 2 p 2 V G 1r 1z . (28)
Here, U = U(z, ω; z S ), V = V (z, ω; z S ), W = W (z, ω; z S ) and X = X(z, ω; z S ) denote the incident wave fields at the model perturbations. These wave fields can be expressed in terms of Green's functions using equations ( 13) for a vertical point force, and similarly for a horizontal point force or for an explosive point source. Expressions G kl ij = G kl ij (z R , ω; z) represent the Green's functions conveying the scattered wave fields from the inhomogeneities to the receivers, as noted before. A total of 32 Green's functions (16 for up-going waves and another 16 for down-going waves) are to be computed to completely solve these equations.

The Fréchet derivatives for the radial displacement V are easily deduced from expressions (28) by changing G kl 1z to G kl 1r . In the same way, the Fréchet derivatives corresponding to the vertical and radial relative fluid-to-solid displacements are obtained by substituting G kl 2j for G kl 1j . As a first verification of the formulas, we note that the expressions corresponding to perturbations of parameters ρ, λ U and G (A

PSV i , F PSV i and G PSV i
, respectively) are in perfect agreement with the Fréchet derivatives formulas computed for the density ρ and Lamé parameters λ and µ in the elastic case [START_REF] Dietrich | Perturbation of the plane-wave reflectivity of a depth-dependent elastic medium by weak inhomogeneities[END_REF].

C. Fréchet derivatives for relevant parameters

As mentioned above, the set of 7 parameters (ρ, ρ f , ρ, C, M, λ U and G) used in the perturbation analysis was primarily chosen to considerably simplify the derivation of expressions (28). In practice, and as with the Lamé parameters in the elastic case, it is more convenient to consider model parameters which are easier to measure or estimate. We introduce here a new set of 8 parameters, namely ρ s , ρ f , k 0 , φ, K f , K s , G s and c s which are more naturally related to the solid and fluid phases. Furthermore, these parameters are independent from each other in terms of their mechanical or hydrological meaning. They are all real and frequency independent, contrary to ρ. Fluid viscosity is not considered here as the Fréchet derivatives with respect to the fluid viscosity and to the permeability are strictly proportional. We concentrate on the permeability by stressing that the comments made for the permeability will also be valid for the fluid viscosity.

To obtain the Fréchet derivatives corresponding to the new set of parameters from expressions (28), we construct the 8 × 7 Jacobian matrix J whose coefficients are formally defined by

J ij = ∂p i ∂p ′ j , (29)
where p i , i = 1..7, stands for one of the parameters ρ, ρ f , ρ, C, M, λ U or G, and where p ′ j , j = 1..8, represents one of the parameters ρ s , ρ f , k 0 , φ, K s , K f , G s or c s . The elements of matrix J are determined from equations (2) to (6).

The Fréchet derivatives with respect to the new set of parameters for the vertical component of the solid displacement U are then defined by

                      ÂPSV i BPSV i ĈPSV i DPSV i ÊPSV i F PSV i ĜPSV i ĤPSV i                       = J                    A PSV i B PSV i C PSV i D PSV i E PSV i F PSV i G PSV i                    . ( 30
)
The combinations involved in equation ( 30) significantly complicate the expressions of the Fréchet derivatives corresponding to the new set of parameters. For sake of clarity and simplification, we introduce the following quantities

Ω = n J -4 i ω ω c a = (1 + c s φ) K s + c s (1 -φ) K f b = φ 2 4 (2 + 3 c s ) G s a 2 -3 (2 + 3 c s φ) 2 K s (1 + c) K s 2 -c s K f 2 + (c s -1) K s K f c = φ 2 (2 + 3 c s φ) 2 K s (K s -K f ) 2 -4 a 2 G s . (31)
With these parameters, the final expressions of the Fréchet derivatives

ÂPSV 1 , BPSV 1 , ĈPSV 1 , DPSV 1 , ÊPSV 1 , F PSV 1 , ĜPSV 1 and ĤPSV 1 with respect to model parameters ρ s , ρ f , k 0 , φ, K s , K f , G s and c s are ÂPSV 1 = (1 -φ) A PSV 1 BPSV 1 = φ A PSV 1 + B PSV 1 + 2 + Ω Ω F C PSV 1 ĈPSV 1 = - i n J + 2 ω ω c η k 0 2 Ω ω C PSV 1 DPSV 1 = 1 a 2 φ 2 (2 + 3 c s φ) 2 a 2 φ 2 (ρ s -ρ f ) (2 + 3 c s φ) 2 A PSV 1 -2 a 2 φ 2 (2 + 3 c s ) G s G PSV 1 -(2 + 3 c s φ) 2 (1 + c s φ) 2 K s + c s (1 -2 φ -c s φ 2 ) K f K s K f E PSV 1 -φ 2 c s (1 + c s ) (2 + 3 c s φ) 2 (K s -K f ) K s K f D PSV 1 + 1 3 b F PSV 1 ÊPSV 1 = 1 -φ a 2 φ φ c s (1 + c s ) K f 2 D PSV 1 + c s (1 + c s φ) K f 2 E PSV 1 (32) + φ (1 + c s φ) K s 2 + c s (c s + φ) K f 2 + 2 c s (1 -φ) K s K f F PSV 1 F PSV 1 = 1 a 2 φ (1 + c s φ) K s 2 φ (1 + c s ) D PSV 1 + (1 + c s φ) E PSV 1 + φ (c s + φ) (1 + c s φ) -c s (1 -φ) 2 K s 2 F PSV 1 ĜPSV 1 = 2 1 -φ 2 + 3 c s φ - 2 3 F PSV 1 + G PSV 1 ĤPSV 1 = 1 -φ a 2 φ (2 + 3 c s φ) 2 (2 + 3 c s φ) 2 K s K f φ (K s -K f ) D PSV 1 -K f E PSV 1 -c F PSV 1 -6 a 2 φ 2 G s G PSV 1
We note that the Fréchet derivatives BPSV 1 and ĈPSV 1 with respect to fluid density ρ f and permeability k 0 are complex due to the role of these parameters in the attenuation and dispersion of seismic waves.

In addition, formulas (32) can be further simplified if source and receivers are located at the same depth z 0 = z R = z S . In this case, we can take advantage of the representation of the incident wave fields U, V , W and X in terms of the Green's functions (see equations 13), and use the reciprocity theorem:

G kl ij (z R , ω; z S ) = G ij kl (z S , ω; z R ) . ( 33
)
The number of Green's functions required to describe the wave propagation then reduces from 32 to 16. These simplifications are straightforward and are not developed here.

D. SH case

We follow the same procedure as above to derive the Fréchet derivatives of the solid displacement T and relative fluid-to-solid displacement Y in the SH case. We denote by

A SH i , B SH i , C SH i , D SH i , E SH i , F SH i
and G SH i , respectively, the Fréchet derivatives with respect to model parameters ρ, ρ f , ρ, M, K U , G and C, where subscript i = 1 refers to the solid displacement T and subscript i = 2 refers to the relative fluid-to-solid displacement Y . We also introduce the notations

ÂSH i , BSH i , ĈSH i , DSH i , ÊSH i , F SH i , ĜSH
A SH i = -ω 2 T G 1t 1t B SH i = -ω 2 [ Y G 1t 1t + T G 2t 1t ] C SH i = -ω 2 Y G 2t 1t D SH i = 0 E SH i = 0 F SH i = 0 G SH i = ∂T ∂z ∂G 1t 1t ∂z + ω 2 p 2 T G 1t 1t (34)
We note, as before, that the displacements T and Y can be expressed in terms of the Green's functions G jt it , where subscripts i = 1 and j = 1 relate to solid displacement and force, whereas subscripts i = 2 and j = 2 relate to relative fluid-to-solid displacement and force. Subscript t stands for the tangential displacement or force. The transformation of these expressions with the Jacobian matrix J defined in equation ( 29) finally yields:

ÂSH i = (1 -φ) A sh1 BSH i = φ A sh1 + B sh1 + 2 + Ω Ω F C sh1 ĈSH i = - (in J + 2 ω w c ) η k 2 0 Ω ω C sh1 DSH i = (ρ s -ρ f ) A sh1 -2 G s 2 + 3 c s (2 + 3 c s φ) 2 G sh1 ÊSH i = 0 F SH i = 0 ĜSH i = 2 1 -φ 2 + 3 c s φ G sh1 ĤSH i = -6 G s φ 1 -φ (2 + 3 c s φ) 2 G sh1 (35)
We note that the Fréchet derivatives with respect to K s and K f are zero since these parameters have no influence on shear waves.

IV. NUMERICAL SIMULATIONS AND ACCURACY TESTS

A. Fréchet derivatives vs discrete perturbations

As mentioned in the introduction, the Green's functions for layered media are computed with the Generalized Reflection and Transmission Matrix Method of [START_REF] Kennett | Seismic waves in a stratified half space[END_REF] which yields the plane-wave response in the frequency-ray parameter (or horizontal wavenumber) domain. In our numerical applications, the Fréchet derivatives are first calculated in the frequency-wavenumber domain before being transformed into the time-distance domain with the discrete wavenumber integration method [START_REF] Bouchon | A simple method to calculate Green's functions for elastic layered media[END_REF].

In order to test our analytical formulations and assess their limitations, we compare the differential seismograms computed with the first-order Fréchet derivative approach with TABLE I. Model parameters of the infinite medium used for the numerical tests of the Fréchet derivatives. Thus, considering for instance the vertical component of the solid displacement in the P -SV case, the partial derivative ∂U/∂p j with respect to parameter p j (j = 1..8, according to the parameter set considered) can be approximated by the following finite difference expression:

φ ( ) k 0 (m 2 ) ρ f (kg/m 3 ) ρ s (kg/m 3 ) K s (GP a) G s (GP a) K f (GP a)
∂U(z R , ω; z S ) ∂p j ≃ U p j +∆p j (z R , ω; z S ) -U p j (z R , ω; z S ) ∆p j , ( 36
)
where ∆p j represents a small perturbation of parameter p j .

The similarity between the seismograms computed with the two approaches indicated in the left-and right-hand sides of equation ( 36) is evaluated from the correlation coefficients between the traces.

B. Uniform medium

We first consider the simple case of small perturbations δρ s , δρ f , δk 0 , δφ, δK s , δK f , δG s and δc s within a thin slab embedded in an infinite uniform medium. The slab thickness is of the order of one twentieth of the dominant wavelength of the P -waves (i.e., 1 m). The amplitude of the relative perturbations ∆p j /p j is 10 % for each of the parameters considered.

Source and receivers are located at the same depth, 50 m above the model perturbation. The parameters of the uniform model are listed in table 1. The seismic response of the thin slab is considered as the reference for the comparisons with the Fréchet derivative seismograms.

The simulations shown in figure 1 include P f ast -, P slow -and S-waves whose computed velocities are respectively equal to 2250, 130 and 750 m/s at a frequency of 85 Hz. The slow P -waves are not visible, but three reflected waves (compressional P P , converted P S and SP , and shear SS) are easily identified in the four sections displayed in figure 1. It is seen that a small perturbation of the fluid modulus K f has no influence on shear waves. The same behavior is observed for the solid modulus K s (not shown). On the contrary, slight changes in the other parameters mainly generate SS reflections, as noted in particular for the porosity φ, mineral shear modulus G s and permeability k 0 in figure 1. We also observed that the differential seismograms are very similar for the following pairs of perturbations: i) consolidation parameter and porosity; ii) fluid and solid moduli; iii) fluid and solid densities.

In addition, we found that the correlation coefficients between the Fréchet derivative and discrete perturbation seismograms are greater than 99 % for most model parameters at all source-receiver offsets. The only exception concerns the Fréchet derivative with respect to the permeability which shows correlation coefficients between 60 and 95 % in the P -SV case, depending on the source-receiver offset. However, this operator appears more accurate in the SH case. By and large, the tests performed in a uniform medium validate our analytical expressions derived in sections III.C and III.D.

We further check the accuracy and stability of the first-order sensitivity operators by modifying the amplitude of the discrete perturbations, with the following results: i) the Fréchet derivatives with respect to parameters that only influence P -waves are more accurate than the Fréchet derivatives with respect to parameters that influence both P -and Swaves. ii) Strong perturbations of the solid and fluid moduli K s and K f do not produce any distortion of the waveforms, but merely result in a global increase of the amplitudes of the discrete perturbation seismograms. iii) For strong amplitude perturbations, the Fréchet derivatives are more stable at near offsets (i.e., for small angles of incidence) than at large offsets. We interpret this observation as being due to the nonlinearity inherent to large offsets where the wave fields interact more strongly with the subsurface structure. However, as an exception to this rule, the Fréchet derivatives with respect to permeability appear more stable at large source-receiver offsets. This may be explained by the fact that a perturbation of the permeability mainly influences the wave attenuation and dispersion and therefore has a stronger effect for longer travel paths. The deterioration of the seismograms correlation with decreasing k 0 (and corresponding decrease in wave attenuation), is another indication of the weak influence of the permeability on the seismograms. iv) For strong perturbations, the accuracy of the Fréchet derivatives deteriorates for specific offsets corresponding to critical angles. This reduced accuracy manifests itself by modifications of the relative amplitudes of P -and S-waves rather than by waveform changes. v) When checked against discrete perturbations of positive and negative amplitude of the same magnitude, the first-order approximations do not show exactly the same accuracy. In general, the Fréchet derivative seismograms obtained for positive perturbations display a better accuracy. vi) As a general rule, the first-order approximations appear remarkably accurate for amplitude perturbations up to 20 % in absolute value.

We now consider the robustness of the Fréchet derivative seismograms with respect to FIG. 1. Seismic sections corresponding to the first-order perturbation of the vertical displacement with respect to porosity φ, fluid modulus K f , mineral shear modulus G s and permeability k 0 in the uniform and infinite medium described in table 1. The seismic excitation is a vertical point force.

the thickness of the perturbed layer. In our uniform model, the wavelengths λ P and λ S corresponding to P -and S-waves are respectively equal to 26 m and 9 m at the dominant frequency of the Ricker wavelet used in the simulation. Our computations show that the Fréchet derivative seismograms are very well correlated with the discrete perturbation seismograms until the thickness of the perturbed layer reaches about 20% of the dominant wavelength λ P (that is, ≃ 5 m) for parameters K s and K f , and 20% of the dominant wavelength λ S (that is, ≃ 2 m) for all other parameters that influence both P -and S-waves. Thus, we observe that the Fréchet derivatives with respect to K s and K f are more robust than the other expressions with respect to departures from the "small and localized perturbation" assumption of the Born approximation.

C. Complex model

We used the 16-layer model shown in figure 2 to numerically check the stability and accuracy of the Fréchet derivative formulas in a more complex structure. In this model, the perturbed layer is at a depth of 50 m, source and receivers being located near the surface.

Figure 3 presents the seismic sections obtained with the Fréchet derivative and discrete perturbation methods for slight changes of the solid density ρ s and permeability k 0 . On the whole, the comparison of the waveforms obtained with both methods is very satisfactory in spite of some differences observed at small offsets for the perturbation of the permeability.

Figure 4 shows that the Fréchet derivative seismograms are remarkably accurate as long as the perturbation amplitude remains small. In this case, the maximum acceptable perturbation amplitude is approximatively 10 % of the model parameter value. When the perturbation amplitude is increased beyond this limit, the P -waveforms remain practically unchanged whereas the S-waveforms are distorted. In all cases, no variations in travel times are observed.

We also checked the behavior of the Fréchet derivative operators relative to the thickness of the perturbation layer, or equivalently, relative to the central frequency of the wavelet used in the computations. As for the uniform medium investigated before, we note that the first-order approximations remain very accurate as long as the layer thickness layer is lower than λ P /5 or λ S /5 depending on the model parameter considered. the most consolidated material, figure 5), the solid modulus K s has a stronger influence on the seismograms than the fluid modulus K f . On the contrary, for an unconsolidated medium (figure 6), the seismograms are mainly influenced by the fluid properties. With the model parameters used in this study, we find that the transition between these two regimes occurs except for the consolidation parameter c s which is equal to 20. The amplitude scale is non linear for sake of readability.

on the solid modulus, and vice-versa for a medium with low porosity. As a consequence (and confirmation of field observations), the P -waves are strongly influenced by the fluid properties when they propagate in a fluid-saturated and poorly consolidated medium with high porosity. This makes it possible to determine the fluid characteristics from the seismic waveforms if these favorable conditions are met. Conversely, the estimation of the fluid characteristics will be more difficult in consolidated or unsaturated or low porosity media.

Another consequence of the results shown in figures 5 and 6 is that φ and c s are the most attractive parameters to invert for in an inversion procedure, assuming that ρ s and G s can be estimated independently. result in similar modifications of the seismic response. An obvious consequence of parameter coupling is that it becomes difficult or even impossible to reliably estimate the model parameters in an inversion procedure [START_REF] Tarantola | A strategy for nonlinear elastic inversion of seismic reflection data[END_REF].

To look into this problem, we computed the plane wave responses corresponding to 10% perturbations of the model properties in the infinite medium described in table 1. Figure 7 shows the reflected energy for the eight model parameters as a function of angle of incidence at the perturbed layer, both for the P P and SS reflections. The smooth aspect of the curves is due to the intrinsic attenuation of the seismic waves propagating in the porous medium.

The peaks and troughs of the curves are explained by the strong variations of the reflection and transmission coefficients, as shown for instance by de la [START_REF] De La Cruz | Reflection and transmission of seismic waves at the boundaries of porous media[END_REF]. Some of these rapid variations are seen on the seismograms of figure 1 which were computed with the same model and source-receiver configuration.

The magnitudes of the seismic responses shown in figure 7 are consistent with the study presented in section V.A. SS reflections are about twice as large as P P reflections for perturbations of the solid and fluid densities. For SS reflections, the maximum value is reached for a perturbation in porosity φ, whereas G s , c s and ρ s produce perturbations of the same magnitude. For P P reflections, the most influential parameters are the solid density ρ s and porosity φ.

We also see in figure 7 that the radiation patterns associated with perturbations in ρ s , ρ f and k 0 on the one hand, and K s and K f on the other hand are exactly the same for the P P reflections. The same resemblance is observed for the SS reflections for the ρ s , ρ f and k 0 group of parameters, and for the G s , φ and c s group of parameters. We note in particular that permeability k 0 and densities ρ s and ρ f behave similarly despite their different roles in the constitutive equations. In all cases, the backscattered energy is maximum at normal incidence because of the shorter wave path and corresponding minimal attenuation.

pecially useful in full waveform inversion algorithms implemented with gradients techniques.

As a first step toward such an application, we evaluated the sensitivity of the seismic response of a poro-elastic medium with respect to each model parameter. We showed that the porosity and consolidation parameter are the most attractive parameters to invert for, whereas the permeability appears to be the most difficult parameter to determine. The solutions (yet to come) accounting for 3D heterogeneities in the propagation medium.

  our alternative set of model parameters ρ s , ρ f , k 0 , φ, K s , K f , G s and c s . The perturbation analysis leads to the following expressions of the Fréchet derivatives of the transverse solid displacement T :

  by introducing discrete perturbations in the medium properties.

FIG. 2

 2 FIG. 2. 16-layer model used for the numerical tests of the Fréchet derivative formulation in a stratified medium. The figure shows the distributions of the eight model parameters as a function of depth.

  FIG. 5. Normalized maximum amplitude of the Fréchet derivative seismograms for all solid force and displacement pairs. F z , F r and F t respectively denote the vertical, horizontal radial and horizontal tangential components of the forces acting on an average volume of porous medium. U, V and T stand for the vertical, horizontal radial and horizontal tangential components of the solid displacement. The model parameters are those given in table 1

  FIG.6. Same as figure5with consolidation parameter c s equal to 100.

  wave fields are more sensitive to the fluid bulk modulus than to the mineral bulk modulus, or inversely, according to porosity and consolidation parameter values. A multi-parameter inversion of backscattered energy looks challenging because of the strong coupling of several model parameters in a wide range of angles of incidence. Finally, this sensitivity study should prove useful for the interpretation of time-lapse monitoring surveys and for checking

Acknowledgments

We are grateful to Stéphane Garambois, Michel Bouchon, Jean-Louis Auriault, Patrick Rasolofosaon and Helle Pedersen for many helpful discussions in the course of this work.

The numerical applications were performed by using the computer facilities of the Grenoble Observatory. We thank the Associate Editor and two anonymous reviewers for their useful comments.

We derived the Fréchet derivatives of the seismic response of a depth-dependent porous medium. The Fréchet derivatives are analytically expressed in terms of the Green's functions of the propagation medium through a perturbation analysis of the poro-elastic wave equations expressed in the plane wave domain. Started with a primary set of seven model parameters chosen because of their linear relationship with the wave equations, the derivation was carried on with a secondary set of eight model parameters more convenient to use as physical parameters of the problem. The eight model parameters considered in our analysis are related to the fluid properties (density, bulk modulus), to the mineral properties (density, bulk modulus, shear modulus) and to the arrangement of the porous material (porosity, permeability and consolidation parameter). In the P -SV case, we derived Fréchet derivatives for 3 different sources (horizontal and vertical point forces and explosive point source), 4 displacement components and 8 model parameters. In the SH case, we obtained 12 expressions for 1 horizontal point force, 2 displacement components and 6 model parameters leading to non-zero Fréchet derivatives.

We checked the accuracy of these sensitivity operators in the time-distance domain by comparing the waveforms computed from the first-order expressions with seismograms obtained by introducing discrete perturbations in the medium properties. The numerical tests were carried out both in a homogeneous and in a more complex earth model excited by oriented point forces. By and large, we found that our analytical expressions of the Fréchet derivatives are remarkably accurate as long as the Born approximation assumptions are satisfied, that is, as long as the perturbations of the model parameters are weak and localized. However, as in other studies relying on the Born approximation, we showed that the first-order operators are robust enough to model parameter perturbations up to 20 % and layer thicknesses up to one fifth of the dominant wavelength. Furthermore, our formulation appears to be stable at all source-receiver offsets.

Due to their analytical formulation, the sensitivity operators derived in this paper will be es-


Gaëlle Loosli

TrIK-SVM : an alternative decomposition for kernel methods in Kreȋn spaces

The proposed work aims at proposing a alternative kernel decomposition in the context of kernel machines with indefinite kernels. The original paper of KSVM (SVM in Kreȋn spaces) uses the eigendecomposition, our proposition avoids this decompostion. We explain how it can help in designing an algorithm that won't require to compute the full kernel matrix. Finally we illustrate the good behavior of the proposed method compared to KSVM.

Introduction

Learning with indefinite kernels is a question that has been treated many times since the introduction of the hyperbolic tangent kernel (tanh) in the libsvm toolbox [START_REF] Hsuan-Tien | A Study on Sigmoid Kernels for SVM and the Training of non-PSD Kernels by SMO-type Methods[END_REF]. This particular kernel has never really shown its practical interest, but many other cases of indefinite kernels have been published, such as graph kernels or similarity based kernels [START_REF] Chen | Fusing similarities and kernels for classification[END_REF]. In those cases, the need for an indefinite kernel is driven by the application field and the cost of avoiding the indefiniteness can be huge in terms of accuracy [START_REF] Loosli | Study on the loss of information caused by the "positivation" of graph kernels for 3d shapes[END_REF]. On the theoretical side, [START_REF] Cheng | Learning with non-positive kernels[END_REF] have shown that learning with an indefinite kernel implies learning in reproducing kernel Kreȋn spaces (RKKS), in which basically the scalar product between a vector and itself can be negative. In [START_REF] Loosli | Learning SVM in Kreȋn Spaces[END_REF], the KSVM algorithm is proven to produce a valid solution in an RKKS, even though the optmization in those spaces is not well defined. We anchor our work in the same framework, and we propose an upgraded algorithm for KSVM, named TrIK-SVM.

Essentials of RKKS

A Kreȋn space is a vector space in which the dot product of two identical vectors can provide a negative value. A well-known example of a Kreȋn space is the complex space, in which this property is denoted by i 2 = -1. In the learning context, it has been shown that trying to optimize a quadratic program (for instance an SVM) using an indefinite kernel matrix is actually a stabilization problem in a Reproducing Kernel Kreȋn space [START_REF] Cheng | Learning with non-positive kernels[END_REF][START_REF] Loosli | Learning SVM in Kreȋn Spaces[END_REF].Those spaces are indefinite inner product spaces endowed with a Hilbertian topology without any requirement of positive-definiteness. Definition 2.1. Kreȋn space [START_REF] Ya | Linear Operators in Spaces with an Indefinite Metric[END_REF] An inner product space (K, ., . K) is a Kreȋn space if there exists two Hilbert spaces H+, H-spanning K, with f+ ∈ H+ and f-∈ H-,

such that ∀f ∈ K f = f+ ⊕ f- ∀f, g ∈ K f, g K = f+, g+ H + -f-, g-H - (1)
In a nutshell, we provide here the essential facts that can be found in details in the above-cited literature:

• If (H + , ., . ) and (H -, -., . ) are RKHS, K = H + + H -is an RKKS.

• There is a representer thoerem in a RKKS, and the decomposition of the reproducing kernel k = k + -k -consists of reproducing kernels in the associated RKHS. • Any symmetric indefinite kernel can be decomposed in k + and k -and associated to an RKKS • This decomposition is not unique Solving a quadratic program in an RKKS (K) is a stabilization problem that cannot be directly solved. However, it can be reformulated as an equivalent minimization problem [START_REF] Loosli | Learning SVM in Kreȋn Spaces[END_REF] lying in an associated RKHS (H) defined as the sum of the positive (H + ) and negative (H -) parts of the RKKS. The solution of the minimization problem lies in H and thus needs to be projected back to K. While the original paper shows the interest of KSVM considering the accuracy of the method on a large variety of problems, KSVM is not used in practice. Indeed, it suffers from several major practical drawbacks: it requires to compute the eigen-decompostion of the matrix (computation time and stability issues) and the final solution is full (the sparsity is lost during the projection) back to the RKKS. Recently, several papers dealing with indefinite matrices have noted those drawbacks and proposed alternative methods. Essentially, some paper are proposing minimization methods that ignore the stabilization problem [START_REF] Xu | Solving Indefinite Kernel Support Vector Machine with Difference of Convex Functions Programming[END_REF] and others rely on using kernel as features, which is a way to embed the RKKS in an RKHS [START_REF] Huang | Indefinite kernels in least squares support vector machines and principal component analysis[END_REF][START_REF] Siamak Mehrkanoon | Indefinite kernel spectral learning[END_REF]. The first category of methods are likely to provide an admissible but not optimal solution in the RKKS. The second category usually works pretty well but might hide some information from the negative part. One can note a major improvement over KSVM in terms of computational complexity, brought in [START_REF] Schleif | Indefinite Core Vector Machine[END_REF] in the form of a carefully approximated version, based on kernel approximation, partial eigen-decomposition and CVM algorithm. The final solution is also approximated in order to be sparser [START_REF] Schleif | Sparsification of Indefinite Learning Models[END_REF].

The stabilization problem We recall here the stabilization problems in both RKKS and RKHS to exhibit the place of the kernel decomposition that will be discussed later. Let {x i ∈ X , y i ∈ [-1, 1], ∀i ∈ [1, . . . , ]} be a binary training set, and τ an hyper-parameter for error penalization, the following problem aims at finding coefficients α i (∀i ∈ [1, . . . , ]) and a bias b such that the decision function

D(x) = sign(f (x) + b) with f (x) = i=1 α i y i k(x i , x).
The equivalent minimization problem is formulated by replacing f by f , such that f , f = f+ , f+ H+ + f-, f-H-and f = f+ + f-. Then the equivalent minimization problem to eq.2 becomes:

             stab f ∈K,b∈I R 1 2 f, f K s.t. i=1 max(0, ( 1
-y i (f (x i ) + b))) ≤ τ              min f ∈H,b∈I R 1 2 f , f H s.t. i=1 max(0, ( 1
-y i ( f (x i ) + b))) ≤ τ (2)
The dual can be calculated as a usual SVM and it lets the modified kernel matrix K = K + + K -appear :

     max α - 1 2 α Y (K + + K -)Y α + α 1 with α y = 0 and 0 ≤ αi ≤ C ∀i ∈ [1 . . . ] (3)
Y is the diagonal matrix of labels y. This problem provides a solution that needs to be projected in the Kreȋn space and all the previous computations are independent of the choice of the decomposition.

TrIK-SVM

Our new algorithm aims at solving eq.2 in a more efficient way than KSVM. First let's detail the KSVM decomposition [START_REF] Loosli | Learning SVM in Kreȋn Spaces[END_REF], which is naturally based on the eigen-decomposition of the kernel matrix. Then we give the new decomposition.

KSVM "flip" Decomposition Let U and D be respectively the eigenvectors and the diagonal matrix containing the eigenvalues ok the kernel matrix K, such that K = U DU . The modified semi-definite positive (SDP) kernel associated to K is defined as K = U |D|U where |U | contains the absolute values of U . In this case, K + = U max(D, 0)U and K -= -U min(D, 0)U .

TrIK-SVM "shift" Decomposition Admitting we know the most negative eigenvalue of the kernel matrix, denoted λ:

K = (K -λI) + λI (4)
with I the identity matrix. Checking that K + = K -λI and K -= -λI are both SDP matrices is straight forward. Then the modified SDP kernel becomes K = K -2λI.

Transition matrices Transition matrices are used to project either the kernel matrix or the solution back and forth between the RKKS and the RKHS.

KSV M T rIK -SV M P + = U + U + P -= U -U - P = P + -P - = U SU with S = sign(D) P + = (K -λI)K -1 = I -λK -1 P -= λK -1 P = P + -P -= I -2λK -1 (5)
Algorithm Based on the transition matrix, the algorithm of KSVM is very simple, it consists in projecting the indefinite kernel matrix K into the RKHS ( K), solve a regular SVM and project the obtained solution α back to the RKKS (α).

The most straight forward way to use the proposed decomposition is to apply it directly, similarely to the KSVM algorithm. However doing so suffers from severe numerical unstabilities. Instead we observe that the proposed decomposition can be applied in a rank-one fashion. This features makes it possible to design a solver that can compute the kernel elements on the fly and produce directly a solution in the RKKS.

TrIK-SVM algorithm

We propose here a dedicated solver that computes at each step both the solution in a RKHS and in the RKKS, so that gradient steps are performed in the minimization setting and optimality conditions are checked in the RKKS. The algorithm is based on a active set method [START_REF] Vishwanathan | SimpleSVM[END_REF]. The algorithm iteratively checks dual or primal constraints and updates the solution after each modification of the support vector set. The dual checking deals with bounds on α: in case of a bound violation, the corresponding point is removed from the set of support vectors. The primal checking deals with the good classification constraint: in case of a violation, it means that there remains some misclassified training examples and one is picked to be added to the set of support vectors. Algorithm 3.1 gives the global scheme in which we insert the indefiniteness treatment. Steps [1.] and [3.] are modified while step [2.] will not. In the following, we present in parallel the hard-margin case (HM) and the soft-margin case (SM). Bounded support vectors are subscripted (bsv) and free support vectors are subscripted (sv). Matrices G and G are the labeled versions of K and K (G = Y KY ).

Algorithm 1 TrIK-SVM

Require: y, C, K, λ, SV 1: converged ← f alse 2: while not(converged) do 3: Step [1.] The idea is to compute at each step both α (already done) and α. Originally, this step, restricted to examples that are in the set of current support vectors, consists in computing b (see [START_REF] Vishwanathan | SimpleSVM[END_REF] for details) and α. Introducing our kernel decomposition, and noting that only diagonal terms are different between G and G, we obtain that α sv = (I -2λG -1 sv,sv )α sv .

[1.] [ α, α, b] ← linearSystem(K, C,
(HM ) (SM ) Gsv,sv αsv + bysv = 1sv

Gsv,sv αsv + C Gsv,bsv

1 bsv + bysv = 1sv α = G-1 (1sv -bysv) α = G-1 (1sv -C Gsv,bsv -bysv) Gsv,svPsv,sv αsv + bysv = 1sv Gsv,svPsv,sv αsv + CG sv,bsv 1 bsv + bysv = 1sv αsv = Psv,sv αsv αsv = Psv,sv αsv (6)
Step [2.] This step aims at checking the dual admissibility, ie. checking that all αi ≥ 0 and projecting the solution in the admissible set if it is not. Since in the RKKS α is not constrained [START_REF] Loosli | Learning SVM in Kreȋn Spaces[END_REF], this step can only be performed in the RKHS.

Step [3.] This last step is performed once the current set of support vector gives an admissible solution in the dual. Its goal is to check the optimality in the primal by verifying that all non support vectors are well classified, which has to be true in the RKKS.

Using the modified steps in the original algorithm, we can compute values for K and K on the fly, or use a cache strategy for more efficiency. Since both solutions are computed on the SV set at each step, the output α can be sparse.

Experimental evaluation of TrIK-SVM

KSVM has already proven its interest, so the goal of the presented experiments is to check that TrIK-SVM behaves at least as well as KSVM. We took randomly generated dataset (based on make blobs in scikit-learn) in dimension 5 with 4 centers (2 are assigned to class -1, 2 are assigned to class 1, at random). Resulting problems may be almost separable and others very overlapping.

Sensitivity to λ The first point to check about our decomposition is linked to the fact that we claim that any value under the least eigenvalue of the spectrum will produce a valid decomposition. If it's true on the paper, one might wonder on the numerical effect. Figures 1 illustrate the stability of the solution in the RKKS independently of λ. We run TrIK-SVM on one randomly picked dataset, with different values of λ, starting from the true least eigenvalue up to 20 times this value. We report the α and α values for each λ. TrIKSVM vs KSVM The instabilities due to the eigen-decomposition in KSVM are more easily observed for datasets with many similar training points. In our experiments, it happens more often than for real datasets, since all training point are picked in a restricted space. Figure 1 present an illustrative result of the accuracy performance of TrIK-SVM and KSVM. We observe that TrIK-SVM is competitive over KSVM, which suffers from numerical instabilities. We also plot on figure 1 the corresponding training time for each, on loglog axis. At first sight the advantage of TrIK-SVM is not obvious, but we need to mention that here, KSVM is internally using libsvm which is fully optimized, while the current TrIK-SVM implementation is fully implemented in Python without any trick and pre-computes the complete kernel, so there's space for improvement.

Conclusion

This paper introduces a novel algorithm called TrIK-SVM that solves most of the KSVM limitations: it provides a sparse solution, that can be computed without pre-computing the full kernel matrix and thus that can benefit from advanced cache strategies. Since the hyper-parameter λ does not influence the quality of the solution neither the computation time, given that it is sufficiently negative, TrIK-SVM comes without additional parameter to tune compared to a classical SVM. Future work includes application to problems others than binary classification, including Multiple Kernel Setting, and a more efficient implementation of the proposed algorithm.

Figure 1 :

 1 Figure 1: Left: solutions in the RKHS (top) and in RKKS (bottom) for decreasing λ values, starting from the least computed eigenvalue. We observe that solutions are similar with a scale factor in RKHS but quasi-identical in the original RKKS. Right top: train and test accuracy on 30 toy datasets generated randomly, and of increasing sizes (from 100 to 3000 training examples). On this task, TrIK-SVM performs at least as well as KSVM. Right bottom: training time in loglog scale.