Isabelle Deniau

Jean-Robert Disnar
email: jean-robert.disnar@univ-orleans.fr

François Baudin

Jean-Pierre Houzay

Characterization of organic matter in the Oligocene (Chattian) turbiditic fine grained deposits, offshore Angola

Keywords: South-East Atlantic, Turbidites, Kerogen, On-line pyrolysis, Biomarkers, Palynofacies

Variations in the abundance and gross composition of the organic matter (OM) content of fine grained turbiditic Chattian sediments off Angola were explored with Rock-Eval pyrolysis of 216 core samples from 4 wells. The study revealed that the OM of the sediments is immature but is present in sufficient abundance [> 1% total organic carbon (TOC)] and with a good enough quality [hydrogen index (HI) around 185 mg hydrocarbon/g TOC on average] to constitute a potential source rock. A deeper insight into the composition of the OM was provided by the extraction and analysis of the kerogen from 15 samples. These kerogen samples were submitted to visual inspection and the counting of organic particles under the microscope (palynofacies analysis), elemental analysis and pyrolysis-gas chromatographymass spectrometry. Although individually the various observations did not provide evidence clearly diagnostic of the origin of the OM, they confirm that it has a mixed marine-terrestrial origin, a notable part of the terrestrial contribution very probably originating from leaves. The good buoyancy of the OM and its richness in lipid components might explain the spreading of terrestrial OM over a large area and its relatively good oil potential.

Introduction

Deep water, sandy marine sediments have drawn the attention of the scientific community, for both economic and academic reasons. First, siliciclastic turbidites represent huge volumes of sediments which have a particular interest for hydrocarbon production, as deep water sandy facies may be potential oil reservoirs [START_REF] Fetter | Petrographic and seismic evidence for the depositional setting of giant turbidite reservoirs and the paleogeographic evolution of Campos Basin, offshore Brazil[END_REF]Mansurberg et al., 2008). Second, they are of interest for understanding sediment transport processes from continent to deep sea basins. In contrast to sandy facies, fine grained turbidites have been less studied although they are particularly well developed as levees of submarine fan valleys [START_REF] Stow | Depositional processes of black shales in deep water[END_REF]. They are also closely associated with the turbiditic sandy reservoirs and can act as seals in a petroleum system. These marine shales are also of importance for the transfer and deposition of OM from continental margins to deep-sea basins by turbidity currents. However, turbidite emplacement is discontinuous and produces layers of sediment in which OM quantity, type and diagenesis are heterogeneous [START_REF] Cowie | Elemental and major biochemical changes across an oxidation front in a relict turbidite: An oxygen effect[END_REF][START_REF] Meyers | Effects of turbidity flows on organic matter accumulation, sulfate reduction, and methane generation in deep-sea sediments on the Iberia Abyssal Plain[END_REF][START_REF] Watanabe | Characterization of organic matter in the Miocene turbidites and hemipelagic mudstones in the Niigata oil field, central Japan[END_REF][START_REF] Lindblom | Organic geochemistry of lipids in marine sediments in the Canary Basin: Implications for origin and accumulation of organic matter[END_REF].

Organic geochemists have made various attempts to quantify the terrestrial contribution to the organic content of marine sediments (see [START_REF] Benner | new Approaches in Marine Organic Biogeochemistry; a Tribute to the Life and Science of John I. Hedges[END_REF][START_REF] Burdige | Burial of terrestrial organic matter in marine sediments; a re-assessment[END_REF] for a review). These studies show that the terrestrial component of the organic content of marine sediments may be significant even in remote regions of the ocean. The terrestrial organic carbon contribution in marine sediments related to deep sea fans, has, however, only given rise to a relatively limited amount of work. Nevertheless, recent work by [START_REF] Holtvoeth | Quaternary supply of terrigenous organic matter to the Congo deep-sea fan (ODP site 1075): implications for equatorial African paleoclimate[END_REF] and [START_REF] Cooke | Tracking soil organic carbon transport to continental margin sediments using soil-specific hopanoid biomarkers: A case study from the Congo fan (ODP site 1075)[END_REF] explored delivery of land-derived organic matter to near-modern Congo fan sediments.

In the present study, detailed organic geochemical characterization has been conducted with a set of samples from the Chattian interval off Angola in order to determine the proportion and state of preservation of terrestrial OM in these facies.

Geological setting

The Congo basin is the central of a series of continental shelf basins extending in West Africa from the edge of the Niger delta in Cameroon to the Walvis ridge near the Angola-Namibia border. This string of basins together forms the "Aptian salt basin" [START_REF] Edwards | Nine major play types recognized in salt basin[END_REF]) and as such were recently grouped as the West-Central Coastal Province [START_REF] Brownfield | Geology and total petroleum systems of the West-Central Coastal Province (7203)[END_REF]. The sedimentary series of the Congo basin shows the three classical tectonic stages of passive margins: pre-rift continental deposits (Late Proterozoic to Late Jurassic), syn-rift lacustrine deposits (Lower Cretaceous), and post-rift unit with a large accumulation of salt (middle Aptian) covered by a thick marine succession (Albian to present).

Since the LateCretaceous, terrigenous sediments have predominantly accumulated because of (i) the global fall in sea level during the Cenozoic, (ii) on-land structural deformation and uplift increasing bed-load transport in rivers and (iii) a humid climate in the drainage area. All these factors resulted in the initiation of large turbiditic systems from the Eocene to the present.

At the Eocene-Oligocene transition, a major erosional event known as the base Oligocene unconformity has been identified [START_REF] Séranne | Seismic super-units and post-rift evolution of the continental passive margin of southern Gabon[END_REF]. This erosive surface, well-developed on the shelf, may be the result of global sea level fall in response to the onset of Antarctic glaciation. This major unconformity is overlain by the overall westwards progradation of deltaic sediments of the Malembo Formation (Oligocene to present). The discharge from an ancestral Congo-Zaire river associated with short-term sea level changes induced the deposition of extensive turbiditic facies. Oligocene deposits are primarily composed of channel-levee complexes stacked vertically and laterally. The Chattian turbiditic system is characterized by well developed external levees, built during sediment transportation inside the channel axis (by-passing of coarse sediments), as the mud fraction is progressively dragged out of the turbiditic flow and feeds plumes and overspills (Lopez, 2001;[START_REF] Broucke | The role of deformation processes on the geometry of mud-dominated turbiditic systems, Oligocene and Lower-Middle Miocene of the Lower Congo basin (West African Margin)[END_REF].

They were deposited as amalgamated or stacked turbidite channels and sand bodies in delta, pro-delta and basin-floor-fan settings and are interbedded with marine shales.

The post-salt units of the Congo were gradually deformed by gravity tectonics in response to the differential subsidence of the basin and to the increase in sediment load. Since the middle of the 1990s, oil industry has had much interest in the central transitional domain because Oligocene to Miocene turbiditic deposits provide the reservoirs for huge offshore discoveries (Fig. 2). For instance, the Girassol oil field lies 150 km off the coast of Angola, between Soyo and Luanda, beneath 1400 m water depth (Fig. 2). This field, discovered in April 1996 and developed since 1998, is made up of extremely permeable Oligocene sand reservoirs, containing high quality crude oil and resulting in good well productivity. Since this discovery, several similar Oligocene sandy reservoirs have been discovered and developed, such as the Cravo and Lirio oil fields (Fig. 2). Most of these sandy reservoirs correspond to the filling of submarine channels and are interbedded with marine shales acting as seals.

Characterization of the OM preserved in these shaly facies was the goal of the present study.

To the best of our knowledge, information about the nature of the OM in these shaly facies is lacking. We provide here insight into its nature and distribution of OM in the Chattian turbiditic fine-grained deposits, which represent huge volumes of sediment.

Material and methods

Samples

Fine-grained sediments (216 samples in toto) were collected from cored intervals in four wells (Table 1). A total of 240 m were sampled, with an average spacing between samples varying from 0.6 to 1.7 m. The shaly and silty samples were hard, thickly laminated and with a texture indicative of low permeability, suggesting minimum risk of contamination by the oil-based mud used during drilling. In the laboratory, the samples were cleaned to eliminate surfaces that could have been affected by alteration or oil impregnation. Chips were cut and dried in an oven at 75 °C for 24 h, before being pulverized in a rotating disc mill.

Methods

Information on the quantity, type and thermal maturity of the bulk OM was obtained from Rock-Eval pyrolysis using both an Oil Show Analyser and a Rock-Eval 6 Turbo device (Vinci Technologies) under standard conditions (for details see [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications. Partie I[END_REF][START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications. Partie III[END_REF][START_REF] Peters | Guidelines for evaluating petroleum source rock using programmed pyrolysis[END_REF][START_REF] Lafargue | Rock-Eval 6 applications in hydrocarbon exploration, production and soils contamination studies[END_REF][START_REF] Béhar | Rock-Eval 6 Technology: performances and developments[END_REF]. TOC content (wt%), HI, mg hydrocarbon (HC)/g TOC), oxygen index (OI, mg CO 2 /g TOC) and T max (°C) were determined.

For bitumen extraction the powdered bulk rock was extracted with a mixture of MeOH/CH 2 Cl 2 (1:1 v/v 3 x 15 min each) at room temperature using an ultrasonic vat at 65 Hz). The bitumen was recovered by centrifugation and the solvent was evaporated under vacuum.

The kerogen was isolated from 15 samples selected to cover the larger range of palynofacies composition as possible among the numerous samples studied by Pirious (2006). These kerogens were isolated using the classical procedure described by [START_REF] Durand | Procedures for kerogen isolations[END_REF], including extraction with organic solvent after acid attack, under the same conditions as for bitumen extraction. The elemental composition (C, H, O, N, S, Fe) of the kerogen concentrates was determined as follows: thermal conductivity measurements were used for C, H and N, while coulometry was used for O measurements. S was determined via oxidation followed by quantitative coulometry of SO 2 ; for Fe, demineralization was carried out with a mixture of HNO 3 and H 2 SO 4 and Fe content was determined using atomic absorption spectrophometry.

Gas chromatography-mass spectrometry (GC-MS) analysis was performed with a Thermo-Finnigan TRACE-Polaris GCQ gas chromatograph-mass spectrometer. The gas chromatograph was fitted with an Rtx™-5Sil MS column (30 m x 0.32 mm i.d., 0.25 μm film thickness) with a 5 m guard column. The GC oven temperature programme was 40 °C (1 min) to 120 °C at 30 °C/min, to 300 °C (hold 30 min) at 3 °C/min. Samples were injected splitless, with the injector temperature set at 280 °C. He was the carrier gas. The mass spectrometer was operated in the electron ionization (EI) mode at 70 eV ionization energy and scanned from m/z 50 to 650. Compounds were tentatively assigned by comparison with library (NIST) mass spectra and relative retention times. Curie point pyrolysis (CuPy)-GC-MS) was performed with a Fischer GSG flash pyrolyser directly coupled to the GC-MS system, He being used as carrier gas. The temperature for the ferromagnetic cylinder was 650 °C, which allows cracking of the bulk macromolecular structure of the OM. The analyses were carried out under the same conditions as above except that the temperature programme was 50°C (hold 5 min) to 300°C (hold 10 min) at 4°C/min. Thermochemolysis was carried out as described by [START_REF] Disnar | Assessment of peat quality by molecular and bulk geochemical analysis; application to the Holocene record of the Chautagne marsh (Haute Savoie, France)[END_REF]. Briefly, ca. 25 mg of dried and crushed kerogen were introduced in SVL® screw cap glass tubes with 50 μL of a standard solution of heptylbenzoïc acid (21 μg/100 μl MeOH) and 100 μL tetramethyl ammonium hydroxide solution (TMAH; 25 % in MeOH). The tubes were placed open in an oven at 75 °C for 3 to 5 h to evaporate excess MeOH, cooled and closed under vacuum. They were then placed vertically in a sand bath heated at 250 °C and left for 20 min. After cooling in ambient air, 1.5 ml (Et) 2 O was introduced to extract the pyrolysis products. After solvent evaporation, the extract was diluted in 50 or 100 μl CH 2 Cl 2 and analysed using GC-MS under the same conditions as described above.

The samples selected for kerogen concentrates were also prepared for palynofacies analysis using a Zeiss Axoplan microscope under different illumination modes. Neither oxidation nor the ultrasonic probe were used during processing as all the particles were of interest in this study and might have been selectively destroyed by such procedures. To better differentiate and count small as well as large particles, the residue was separated into two fractions. One was mounted directly on to a glass microscope slide and the other on to a slide after sieving through a 10 µm sieve. Between 565 and 1050 particles were counted using a grid-ocular in order to obtain relative percentages of surface covered by each type of particle.

The classification of palynofacies has always been rather subjective [START_REF] Tyson | Sedimentary Organic Matter: Organic Facies and Palynofacies[END_REF]. For this study, a classification was chosen in order to obtain as much palaeoenvironment information as possible from palynofacies analysis. The following particles were identified during these quantitative investigations:

-Phytoclasts comprising all opaque and translucent land plant debris. Different subgroups could be observed within this fraction and were distinguished during counting.

-Sporomorphs comprising land derived pollen grains and spores.

-Fungal debris comprising all the filamentous segmented particles and sclerotes.

-Marine algae comprising dinoflagellate cysts and other organic-walled algae.

-Foraminiferal linings comprising the chitinous membranes of foraminifera.

-Amorphous OM (AOM) comprising all particulate organic components that appear structureless at the scale of light microscopy. Fluorescence microscopy was used to characterize the preservational state of the AOM.

Results and interpretations

Although attention was taken during sampling to avoid contamination, the results of Rock-Eval pyrolysis clearly indicated that the oil-based mud used during drilling had partly impregnated the shales and silts. Free hydrocarbons were systematically detected in all samples with quantities around 3 mg/g of rock, as measured from the S 1 peak of the Rock-Eval instrument.

Because the Chattian rocks are immature with respect to oil generation (see below), the hydrocarbon content clearly represented contamination and should not be taken into account in the characterization of the autochthonous OM. A set of 14 samples from the GIR 117 well was extracted with a mixture of MeOH/CH 2 Cl 2 and re-analyzed using Rock-Eval pyrolysis. This treatment resulted in the near complete disappearance (>91%) of the S 1 peak, without any change in the other parameters (S 2 , T max ). As the S 1 peak is included in the determination of the organic carbon content, the TOC value was corrected for the carbon associated with the S 1 peak and the HI value was recalculated for all 216 analysed samples.

Seven samples of drilling-mud used in the different wells were prepared as for kerogen isolation. They did not produce any insoluble residue, providing evidence that the oil used in the mud did not alter the kerogen of the Chattian samples. In consequence, our samples can be used for organic geochemical purposes after solvent extraction and kerogen preparation.

Bulk organic geochemical data

Rock-Eval pyrolysis

For the four wells, the mean TOC contents (%) are: GIR 117, 1.3%; GIR IS3, 1.0%; LIR 1, 1.4%; CRA 1, 1.6% (Table 2). Some of the individual samples are very rich with a TOC content reaching 4%. In addition, TOC concentrations are variable with depth without any stratigraphic trend in the evolution of the organic richness, except for the GIR IS3 well, where a constant decrease in TOC is noted from the top to the base of the studied interval.

An assessment of the degree of thermal maturity reached by the Chattian rocks offshore Angola is provided by T max values which vary between 400 and 437 °C (average 427 °C) for the four wells; Table 2). Due to the relatively small thickness of studied intervals (25 to 120 m; Table 1), the large range in T max cannot be assigned to differences in OM maturity and is rather due to differences in OM composition, including differences in the amount of recycled and oxidized OM (see Discussion). In any case, the Chattian fine-grained sediments are immature with respect to oil generation and so have not experienced high temperature during burial.

Consequently, the type of OM could be estimated from pyrolysis data on the basis of the HI parameter. According to the range of values (31-381 mg HC/g TOC; Table 2), the OM could be classified as a mixture of Types III and II (higher HI values). The four wells display similar average values around 185 mg HC/g TOC (Table 2).

Elemental analysis

Elemental analysis data obtained from kerogens (Table 3) first show that they are rather pure, the sum of organic elements being always above 87%, with a maximum value of 97%, except for samples GIR117-28 and GIR IS3-4 in which the total recovery is only 58%, due to a high amount of non destroyed minerals. The TOC content comprises between 8 and 49.2% and both S and Fe are two major elements present in the concentrates, meaning that pyrite (FeS 2 ) is mixed with the OM.

Using the formula of pyrite, it is possible to calculate the total pyrite content in the concentrates and consequently the organic sulfur content [START_REF] Durand | Elemental analysis of kerogens[END_REF]. The results in Table 3 show that the organic sulfur content comprises between 0 and 8% and that the pyrite content is as high as 73%. The dilution effect from the pyrite has no direct influence on the measurement of other elements like C, H, O and N. Thus, the elemental analysis data can be corrected for the pyrite content and the H/C and O/C atomic ratios derived for the kerogens are effectively representative of their composition (Table 3). The H/C values are relatively homogeneous and range from 1.08 to 1.23, thereby indicating medium aliphaticity for the kerogen. When plotted in a H/C-HI diagram, the data points for the kerogens appear well grouped around an average HI value of 345 mg HC/g TOC, on the general trend defined by [START_REF] Espitalié | Méthode rapide de caractérisation de roches mères, de leur potentiel pétrolier et de leur degré d'évolution[END_REF] for such materials. In contrast, the O/C values vary between 0.21 and 0.58 and are poorly correlated with the OI values of the concentrates, which are nearly constant with an average value around 32 mg CO 2 /g TOC. This discrepancy between O/C and OI values might be related to the presence of mineral residues, which affects the determination of the organically-bound oxygen content. In addition, the flash pyrolysis results that show a very low proportion of oxygen-compounds in the kerogen (see below) also support an overestimation of the oxygen from elemental analysis. The N/C ratio values, which are between 0.047 and 0.027 (i.e. C/N values between 21 and 37) are typical for terrestrial OM [START_REF] Meyers | Preservation of elemental and isotopic source identification of sedimentary organic matter[END_REF]. However, any algal contribution has very probably been highly degraded in the water column and the sediment, with a preferential degradation of nitrogenous components [START_REF] Wakeham | Molecular indicators of diagenetic status in marine organic matter[END_REF]. The low molecular weight nitrogenous compounds preferentially released during degradation might have been lost in the water column or simply during the kerogen extraction, thereby excluding the use of such elemental analysis data by way of the very classical Corg/Ntot parameter so appreciated by soil scientists.

The much higher HI values found for the isolated kerogens than for bulk sediments (av. 345 vs 185 mg HC/g TOC) might be explained by a classical mineral matrix effect [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications. Partie III[END_REF].

Using both van Krevelen and modified van Krevelen (HI-OI) diagrams (Fig. 3), the OM of the Chattian fine-grained sediments off Angola appears as a mixture of Type II and III according to [START_REF] Tissot | Petroleum Formation and Occurrence[END_REF]. This confirms the origin of the OM deduced from the analysis of the bulk samples.

Palynofacies

Palynofacies analysis indicates a strong predominance of AOM (AOM) which represents between 70 and 96% of the total particles (Figs. 4a andb). The structured particles consist almost exclusively of higher plant debris at different states of diagenetic alteration. Different types can be recognized from yellow coloured cuticles with well preserved cellular structures (Fig. 4c) to dark brown and gelified ligno-cellulose debris (Fig. 4d). Ligno-cellulose particles represent between 2 and 30% (ave. 18%) of total particles. Cuticles are present at a low level (0.5 to 9%), whereas sporomorphs are rare (< 2%). Fungal remains are frequent and may represent a few percent in some samples. Marine palynomorphs and foraminiferal linings are scarce.

The AOM particles can be classified into two main groups. The first is composed of highly fluorescent granular AOM (Figs. 4e andf). The second is represented by AOM particles generally presenting sharp contours with remains of plant fibres (Fig. 4g). This type of AOM shows a lower fluorescence, some granules included in the AOM fluorescing whereas the fibre and the remainder of the AOM show a dull fluorescence (Fig. 4h). Variations in AOM appearance probably highlight the origin of the OM (marine vs. terrestrial) but also the role of the depositional setting, i.e. intensity of oxidation.

Molecular data (Py-GC/MS and thermochemolysis with TMAH)

Representative pyrograms are displayed in Fig. 5. The distributions are dominated by the peaks from n-alkane/n-alkene doublets. However, the latter can be more clearly appreciated using selective ion detection (Fig. 6). The py-GC/MS traces also show various other compounds belonging to the following series (Fig. 5; Table 4): isoprenoids, phenols, alkylbenzenes, alkylnaphtalenes, indanes/indenes and hopanes. The distributions and the origins of the various compounds are discussed in the following section.

In a study of the kerogen from turbidites of the Madeira abyssal plain, [START_REF] Hoefs | Changes in kerogen composition across an oxidation front in Madeira Abyssal Plain turbidites as revealed by pyrolysis GC-MS[END_REF] found series of pyrolysis products very similar to those found in the present study, namely, nalkane/n-alkene doublet distributions extending over a large carbon number range (<10 to >30), alkylphenols, alkylbenzenes and other alkylated aromatics, plus isoprenoids. Except for the phenols assumed to be derived from the lignin of terrestrial plants, the authors hypothesized that all the other compounds derived from algal biopolymers. In this way, the nalkanes/n-alkenes and the isoprenoids were attributed to algaenan. The latter comprises non hydrolyzable and highly aliphatic macromolecules biosynthesized by certain (micro)algae, especially Chlorophyceae and Eustigmatophyceae, and is a component of their outer cell walls [START_REF] Largeau | Formation of Botryococcus braunii kerogens. Comparative study of immature Torbanite and of the extant alga Botryococcus braunii[END_REF][START_REF] Goth | Origin of Messel Oil Shale kerogen[END_REF][START_REF] Derenne | Similar morphological and chemical variations of Gloeocapsomorpha prisca in Ordovician sediments and cultured Botryococcus braunii as a response to changes in salinity[END_REF][START_REF] De Leeuw | A review of macromolecular compounds that comprise living organisms and their role in kerogen, coal and petroleum formation[END_REF][START_REF] Gelin | Novel, resistant microalgal polyethers: An important sink of organic carbon in the marine environment[END_REF]. In fact, most or even all the pyrolysis products identified can have various sources and are thus hardly diagnostic of an unequivocal biological origin. For example n-alkane/n-alkene distributions extending beyond C 30 can also originate from cutan, a biopolymer in the cuticles of certain plants, especially species thought to be adapted of coping with severe drought [START_REF] Boom | Cutan, a common aliphatic biopolymer in cuticles of drought-adapted plants[END_REF]. However, other hypotheses cannot be excluded, such as a heterogeneous polymer formed in soil by clay-catalyzed polymerization of fatty acids with humic colloids [START_REF] Almendros | A structural study of alkyl polymers in soil after perborate degradation of humin[END_REF][START_REF] Almendros | Preservation of aliphatic macromolecules in soil humins[END_REF].

Aromatic compounds are dominated by benzene and its alkylated counterparts with up to 22 C atoms in the side chain, but with a prevalence of short chain alkylbenzenes (C 0 with C 2 ).

Polyaromatic compounds are dominated by a series of naphthalenes from C 0 to C 4 and anthracenes and/or phenanthrenes from C 0 to C 2 . A series of alkyl indanes and alkyl indenes are also present. The origin of all these aromatic compounds is still poorly defined. They are likely to originate from acyclic hydrocarbon cyclization and aromatization during pyrolysis [START_REF] Largeau | Pyrolysis of immature Torbanite and the resistant biopolymer (PRB A) isolated from extant alga Botryococcus braunii. Mechanism of formation and structure of Torbanite[END_REF]. They can also come from pre-existing aromatic entities in the kerogen.

For example, complex mixtures of polyaromatic hydrocarbons (PAHs) observed in the pyrolysates of various kerogens were regarded as coming from highly resistant polyaromatic nuclei-containing algal biomacromolecules selectively preserved in the kerogen [START_REF] Horsfield | On the atypical petroleum generating characteristics of alginite in the Cambrian alum shale[END_REF][START_REF] Derenne | First example of an algaenan yielding an aromatic-rich pyrolysate. Possible geochemical implications on marine kerogen formation[END_REF]. This second assumption which might account for the great diversity of PAHs in the pyrolysates, is supported by the presence of important deformation bands of groupings adjacent to C=C bond, evidenced by Fourier transformed infra-red analysis of the kerogens (data not shown).

Alkylated phenols (up to C 5 ) are the only oxygenated compounds detected in the pyrograms.

Their abundance falls rapidly beyond the C 2 compounds. Phenolic compounds with short chains are generally regarded as products of the thermal breakdown of lignin (e.g., Saiz-Jimenez and de [START_REF] Saiz-Jimenez | Lignin pyrolysis products: their structures and their significance as biomarkers[END_REF]) and are thus regarded as indices of a terrestrial contribution to the kerogen of the Chattian facies, in agreement with the palynofacies observations. This hypothesis can hardly hold for the phenols with a side chain longer than C 3 since the lignin monomers have a phenylpropane skeleton. However, these higher homologues are present inonly low amounts, are not discussed further. The distributions of the polyaromatic compounds and phenolic compounds show few differences among the various samples.

Polycyclic alkanes of the hopane type in the range C 27 to C 30 -C 31 were detected in all the pyrolysates, but in very variable abundance. Hopanes assure the rigidity in the structure of the walls of bacteria and are thus regarded as bacterial markers (e.g., [START_REF] Ourisson | The hopanoids: palaeochemistry and biochemistry of a group of natural products[END_REF].

In the present case, a notable contribution to the OM from terrestrial plant material is supported by the thermochemolysis of two samples which effectively provided very nice distributions of n-fatty acid methyl esters (FAMEs: Fig. 7) highly dominated by even numbered FAs in the range C 22 to C 34 , with a mode at n-C 28 . These FAMEs are also accompanied by: (i) lower but still notable proportions of n-alkanes in the range C 28 to C 39 , with a mode at n-C 31 to n-C 33 but only with a very slight predominance of odd-numbered homologues over their even counterparts, and (ii) even lower but still non negligible amounts of α, ω diacids (as dimethylesters), with a predominance at n-C 31 to n-C 33 and a marked even/odd predominance above n-C 27 . All these compounds are assumed to be typical components of the waxes that coat the outer cuticle of subaerial plant tissues and help protect them from various external physical and biological agents. Such long chain FA distributions have recently been found in coastal lagoon sediments near Rio de Janeiro in Brazil [START_REF] Zink | Organic matter composition in the sediment of three Brazilian coastal lagoons -District of Macaé, Rio de Janeiro (Brazil)[END_REF]. The authors assumed a terrestrial plant origin but without providing any identification of its source. The higher plant origin of the compounds is also supported by the presence, among the low molecular weight thermochemolytic products, of vanillic acid, in addition of its demethoxylated counterpart, p-HO benzoic acid (compounds being under the form of methyl esters/methyl ethers). Vanillic acid is a typical degradation product of gymnosperm lignin but might also derive from the syringyl moieties of angiosperm lignin via demethoxylation. The absence of the corresponding aldehyde and ketone (i.e. vanillin and acetovanillone) suggests that the lignin suffered extensive oxidative alteration before burial.

Discussion

OM maturity

A low degree of maturity for the Chattian sediments is evidenced by both T max values and the colour of organic particles. Average T max values vary between 426 and 428 °C in the four wells, with a deviation between 3.53 and 5.26 °C. The wider range of measured T max values (400-437 °C) for all the samples is attributed partly to analytical errors but mostly to variations in the composition of the OM (OM) depending on its type and on the extent of the alteration it suffered during early diagenesis . In any case, average T max values below 430 °C indicate that the OM did not experience high temperatures during burial and is thus uniformly immature with respect to oil generation. This maturity assessment is also supported by the light colour of the AOM, the strong fluorescence of certain AOM flakes and the yellow colour of cuticles observed in palynofacies. This low degree of maturity is in agreement with the moderate burial of the Chattian sediments in that part of the Angola margin, as the studied intervals experienced around 1500 m of overburden sedimentation.

OM type and origin

An important question that remains open is the origin of the OM or more precisely the relative importance of the marine and terrestrial fractions since we have evidence that both contributed to the Chattian sedimentation off Angola.

Rock-Eval pyrolysis data and elemental analysis (Tables 2 and3, Fig. 3) indicate an OM type varying between a mixed Type II and III, with highly variable but often extensive alteration of the original input as evidenced by some low HI values (whose significance is exaggerated by the strong mineral matrix effect evidenced by comparison of the HI values of the bulk sediments with those of the corresponding kerogens). Palynofacies preparations are largely dominated by AOM which is usually related to marine sources, although some terrestrial organic particles may also be "amorphised" [START_REF] Tyson | Sedimentary Organic Matter: Organic Facies and Palynofacies[END_REF][START_REF] Marchand | The composition of sedimentary organic matter in relation to the dynamic features of a mangrove-fringed coast in French Guiana[END_REF].

Fluorescence investigations can help differentiate highly fluorescencent well preserved marine OM from oxidised low fluorescent particlesof terrestrial or marine origin. Here, both fluorescent and non-fluorescent AOM co-exist.

At the present state of knowledge the various families of compounds released by on-line pyrolysis data are not adequately diagnostic to allow a clear biological origin to be assigned forthe OM. For example, the n-alkane/n-alkene doublets can be attributed to various kinds of resistant biopolymers of different origin (See section 4.3). However, at least two other factors should be considered: (i) the presence of well preserved labile material (of marine and/or terrestrial origin) which adds its own signature to that of the refractory constituents; e.g. for the marine material this is supported by recent work which clearly demonstrated that algal material can be preserved after settling to great depth in well-oxygenated waters [START_REF] Bertrand | Organic-rich sediments in ventilated deep-sea environments: relationship to climate, sea level, and trophic changes[END_REF][START_REF] Pichevin | Organic matter accumulation and preservation controls in a deep-sea modern environment: an example from Namibian Slope sediments[END_REF]; (ii) the exact nature of the terrestrial input, whose composition and quality can which differ greatly whether it is inherited from soils (and thus is mostly made up of residual humic substances) or whether it is mostly made up of woody debris (rich in cellulose and lignin) or of leaves (rich in waxy components). In an attempt at overcoming these various difficulties, we suggest interpreting the n-alkane/n-alkene distributions, obtained from kerogen pyrolysis, in the same way as is classically done for the free hydrocarbon distributions where it has been assumed for a long-time that the low and high molecular weight ends derive from marine and terrestrial contributions, respectively. For the terrestrial part, the assumption is supported here by the classical odd/even n-alkane/n-alkene predominance among the longer chains (i.e. ≥ n-C 27 ) and additionally by the high molecular weight FAs, n-alkanes and diacids evidenced by thermochemolysis. Nevertheless, this assumption remains difficult to verify. Indeed, following our assumption, sample GIR IS3-29, the pyrolysis of which only revealed low molecular weight n-alkanes/n-alkenes (predominance at C 11 -C 17 ; Fig. 6) does not differ markedly from the other samples, namely, it is rather poor in TOC (0.6%) but has a "normal" HI value of 222 mg HC.g-1 TOC and, above all, it also released notable amounts of phenolic compounds, aromatics and isoprenoids upon pyrolysis (Table 4). Its palynomorph composition is made up of dark brown to black rounded amorphous particles. So, with the support of other data, the globally bimodal n-alkane/nalkene distributions produced upon pyrolysis of the studied kerogens provide evidence that they are made up of variable contributions of marine and terrestrial OM, the first estimate made from these distributions having to be corrected for other factors, beginning with the degree of early diagenetic alteration suffered by both contributions.

Organic matter sedimentation and implications for deep offshore source rocks

The ca. ~ 1% TOC of the Chattian sediments off Angola is significantly greater than the average TOC of 0.2% measured for Recent deep sea sediments [START_REF] Mciver | Hydrocarbon occurrences from JOIDES Deep Sea Drilling Project cores[END_REF] but is typically in the range determined by Premuzic et al. (1982) for margin sediments. This rather high value is evidently related to the high delivery of OM associated with the sedimentary systems on this part of the West African continental margin.

The studied area received both pelagic and terrigeneous inputs. The terrigeneous flux was delivered by the ancestor of the Congo-Zaire River and its submarine canyon system, whereas the autochthonous pelagic sedimentation was driven by the activity of the plume related to fluvial discharge. Depending on the intensity of one or the other of these sources, the delivery of OM has evidently varied both spatially and temporally during the Chattian. Part of this variability may result from the turbidite phenomenon itself, but the primary cause of the differences in OM concentration is probably the degree of export of OM to the sea floor and the changes in its post-depositional preservation.

This variability would suggest a mixing of OM sources with variable intensity of oxidation. Such a feature is expected for turbiditic facies in which a terrestrial contribution is mixed with the marine autochthonous OM [START_REF] Watanabe | Characterization of organic matter in the Miocene turbidites and hemipelagic mudstones in the Niigata oil field, central Japan[END_REF][START_REF] Lindblom | Organic geochemistry of lipids in marine sediments in the Canary Basin: Implications for origin and accumulation of organic matter[END_REF]. As supported by the high molecular weight n-alkane/n-alkene doublets and FAMEs evidenced respectively by py-GC/MS and thermochemolysis, the terrestrial OM seems abundant and rich in leaf-derived fragments. These particles, including cuticles, have good buoyancy and may thus be transported far from their source where they can mix with marine planktonic OM, leading to a homogenisation of the OM content.

The variation in organic input is primarily controlled by the depositional setting (distance from the channel, rate of sedimentation, intensity of sea surface primary productivity). After its deposition this organic mixture is still homogenised by diagenetic alteration, which is mainly oxidative, but can also develop under reducing condition as evidenced by the amount of pyrite and the proportion of hopanoids revealed by py-GC/MS.

Conclusions

TOC is the primary parameter for source rock appraisal, with a threshold of 1 wt% at the immature stage for potential source rocks. With an average TOC value which exceeds this threshold in almost all the studied samples (and nearly up to 4% TOC), the Chattian finegrained turbiditic series off Angola appear to be potential source rocks. Such TOC values also indicate that, despite long distance transportation and/or deposition at rather great water depth, sediment emplacement occurred under conditions suitable for OM preservation. The high sedimentation rate during the Chattian (200m/Ma according to [START_REF] Broucke | The role of deformation processes on the geometry of mud-dominated turbiditic systems, Oligocene and Lower-Middle Miocene of the Lower Congo basin (West African Margin)[END_REF] allowed rapid burial of the sedimentary succession and the preservation of its OM content.

The second prerequisite for good petroleum source rock is the quality of the OM. Bulk geochemical parameters (S 2 , HI) as well as elemental analysis results indicate that the Chattian organic-rich facies have moderate oil to gas-prone potential.

Although the sedimentary overburden was not sufficient to reach the beginning of the oil window in this part of the margin, deep water shaly facies associated with sandy reservoirs should be considered as potential source rocks where they experience a higher thermal evolution in other parts of the margin.

Captions of figures and tables

Table 1

-Selected wells and number of samples of the Chattian fine-grained sediments from Angola margin.

Table 2

-Bulk geochemical parameters from Rock-Eval pyrolysis of 216 samples from Chattian finegrained sediments from Angola margin.

Table 3

-Bulk elemental analysis of 15 kerogen concentrates selected fromdifferent wells, corrected elemental analysis from pyrite and ash contents and calculated atomic ratios.

Table 4

-Relative abundance of main compounds in pyrolysates of 15 kerogen concentrates a . Time (min.) of the main compounds identified in pyrolysates of the 15 selected kerogen concentrates. The abundance is calculated as the surface of the identified compound (+ its isomers) normalized to the most intense peak of the series. nd : not determined.

                   L L L L L  Wells

  Fig. 1 illustrates a schematic cross section of the Southern Congo basin (offshore Angola), showing the salt tectonic style with (i) an upslope extensional domain characterized by normal faults, (ii) a central transitional domain characterized by huge diapirs, and (iii) a downslope compressional domain of thrust faults and accumulation of salt near the transition between the continental and oceanic crusts.

Fig. 1 .

 1 Fig.1. Schematic cross-section of Angola margin redrawn from Philippe (1999). The wells are located between two ridges of salt in the outer part of the extension domain of the margin.The studied stratigraphic interval corresponds to upper Oligocene turbiditic facies.

Fig. 2 .

 2 Fig. 2. General map showing location of block 17 offshore Angola and distribution of the oil fields in Oligocene and Miocene turbiditic facies. The wells come from the Girassol, Lirio and Cravo oilfields.

Fig. 3 .

 3 Fig. 3. Location of 15 kerogens extracted from selected samples of different wells in a van Krevelen diagram and modified van Krevelen diagram (HI-OI) indicating a mixture of Type II and III OM in the Chattian fine-grained sediments of the Angola margin.

Fig. 4 .

 4 Fig. 4. Microphotographs of palynofacies of the Chattian fine-grained sediments of the Angola margin. a: Amorphous organic matter (AOM) with small ligno-cellulose debris (sample GIR 117-28 -TOC: 1.62% -HI: 248 mg HC/g TOC). b: AOM with small black woody needle, large orange ligno-cellulose debris and numerous pyritic crystals (sample GIR IS3-4 -TOC: 1.74% -HI: 241 mg HC/g TOC). c: Cuticle fragments with stomatal outline. d: partly-gelified ligno-cellulosic debris on which vegetal fibre are still discernible. e: gelified AOM. f: same view under UV showing intense fluorescence. g: "amorphised" higher plant debris (plant fibres are still visible) and h: same view under UV showing dull fluorescence (scale bar 100 µm for all photomicrographs).

Fig. 5 .

 5 Fig. 5. Examples of partial reconstituted Total Ion Current (TIC) chromatogram of pyrolysates of selected kerogen concentrates from Chattian sediments off Angola.

Fig. 6 .

 6 Fig. 6. Distribution of n-alkanes and n-alkenes issued from the pyrolysates of kerogen extracted from selected samples of the different studied wells.

Fig. 7 .

 7 Fig. 7. Example of partial reconstituted chromatogram of products of thermochemolysis with TMAH of a kerogen sample from the Chattian off Angola (sample GIR 117-23). Black circles and "L" designate Fatty acid methyl esters (FAMEs) and lignin-derived compounds, respectively.

  Selected wells and number of samples studied in the Chattian fine-grained sediments of the Angola margin. Bulk geochemical parameters issued from Rock-Eval pyrolysis of 216 samples from the Chattian fine-grained sediments of the Angola margin. Bulk elemental analyses of 15 kerogen concentrates selected in the different wells, corrected elemental analyses from the pyrite and ash contents and calculated atomic ratios.

	Number of	Thickness of studied	Average spacing
	studied samples 70 31 72 43 Number of studied samples [average] TOC (%) 70 0.39-3.99 [1.30] 31 0.54-1.74 [1.00] 72 0.58-2.54 [1.40] 43 0.82-3.71 [1.57] S tot Fe FeS 2 S org * 25.08 2.38 11.82 1.01 23.69 22.62 48.47 0.00 62.25 5.91 29.24 2.51 0.00 1.14 0.35 0.035 interval (m) between samples (m) Girassol 117 (GIR117) 121.4 1.73 Girassol IS3 (GIRIS3) 24.5 0.79 Lirio 1 (LIR1) 69.0 0.96 Cravo 1 (CRA1) 25.5 0.59 Tab. 1 -Wells Tmax (°C) [average] HI (mg HC/g TOC) [average] Girassol 117 (GIR 117) 400-435 [427] 57-381 [189] Girassol IS3 (GIR IS3) 414-434 [426] 88-317 [186] Lirio 1 (LIR 1) 404-437 [428] 34-331 [180] Cravo 1 (CRA 1) 404-434 [427] 31-337 [181] Tab. 2 -Samples C H O N C H O N Sorg H/C O/C N/C GIR117-3 GIR117-23 47.07 4.26 17.13 1.71 6.96 5.72 12.26 0.42 66.68 6.03 24.27 2.42 0.60 1.09 0.27 0.031 GIR117-28 28.84 2.84 13.16 1.15 6.06 5.97 12.79 0.00 62.71 6.18 28.61 2.50 0.00 1.18 0.34 0.034 GIR117-32 8.53 0.86 6.58 0.46 42.90 31.99 68.55 6.34 37.46 3.78 28.90 2.02 27.84 1.21 0.58 0.046 GIR117-53 49.20 4.45 17.16 1.95 5.42 3.25 6.96 1.71 66.07 5.98 23.04 2.62 2.29 1.09 0.26 0.034 GIRIS3-4 26.05 2.43 13.59 0.95 24.51 20.59 44.12 0.98 59.21 5.52 30.89 2.16 2.22 1.12 0.39 0.031 GIRIS3-11 27.27 2.52 12.72 0.91 11.85 3.06 6.56 8.35 52.67 4.87 24.57 1.76 16.13 1.11 0.35 0.029 GIRIS3-29 12.82 1.31 7.82 0.68 40.24 34.15 73.18 1.21 53.77 5.49 32.80 2.85 5.08 1.23 0.46 0.045 LIR1-3 33.30 3.16 11.56 1.76 19.63 17.15 36.75 0.03 66.85 6.34 23.21 3.53 0.06 1.14 0.26 0.045 LIR1-21 23.51 2.32 9.14 1.28 28.56 23.42 50.19 1.79 61.8 6.10 24.02 3.36 4.72 1.18 0.29 0.047 LIR1-55 47.56 4.37 13.23 2.23 12.50 9.63 20.64 1.49 69.04 6.34 19.21 3.24 2.17 1.10 0.21 0.040 LIR1-63 34.91 3.14 13.33 1.69 17.18 14.28 30.60 0.86 64.73 5.82 24.72 3.13 1.59 1.08 0.29 0.041 CRA1-2 21.18 2.10 10.61 1.02 31.01 26.24 56.23 1.02 58.95 5.84 29.53 2.84 2.84 1.19 0.38 0.041 CRA1-24 37.54 3.57 13.07 1.66 16.46 15.84 33.94 0.00 67.23 6.39 23.41 2.97 0.00 1.14 0.26 0.038 CRA1-35 19.99 1.92 11.04 1.10 27.58 21.79 46.69 2.68 54.43 5.23 30.06 3.00 7.29 1.15 0.41 0.047 GIR IS3 LIR 1 m/z Compound Samples 3 23 28 32 53 4 11 29 3 21 55 63 2 78 Benzene 100 50 58 60 54 88 92 100 44 81 37 59 86 92 Toluene (C 1 -benzene) 79 40 57 56 43 76 78 60 45 74 48 66 74 94 Phenol 43 27 35 28 30 38 53 21 28 27 24 29 34 106 C 2 -benzenes 94 55 70 77 62 100 100 72 56 62 44 24 100 108 C 1 -phenol 55 48 58 38 49 67 72 34 27 51 51 47 50 116 Indene 10 6 8 9 7 9 12 9 6 9 6 6 6 118 Indane 7 3 4 5 5 7 9 5 3 5 3 4 4 120 C 3 -benzenes 52 27 30 33 35 45 50 57 32 48 15 24 27 122 C 2 -phenols 32 13 33 22 24 37 46 22 21 34 34 27 25 128 Naphtalene 5 2 5 4 nd 5 7 4 3 5 4 3 5 130 C 1 -indenes 7 5 6 5 10 13 11 10 5 19 8 2 9 132 C 1 -indanes 14 8 10 8 6 14 15 7 5 8 8 5 6 134 C 4 -benzenes 24 12 12 15 15 22 23 24 8 13 7 13 16 136 C 3 -phenols 8 7 7 8 5 13 21 11 8 5 13 12 9 142 C 1 -naphtalenes 5 3 5 8 5 6 15 2 9 21 6 4 3 144 C 2 -indenes 16 11 12 8 14 24 28 24 12 30 7 10 17 146 C 2 -indanes 4 3 3 6 3 5 9 6 2 7 13 4 3 150 C 4 -phenols 4 4 4 3 7 5 4 nd 12 15 47 1 8 156 C 2 -naphtalenes 26 17 18 12 13 5 9 10 26 31 15 11 12 158 C 3 -indenes 3 6 4 3 3 5 6 9 12 20 5 1 6 160 C 3 -indanes nd 4 nd nd 4 nd nd nd 3 6 7 2 4 166 Fluorene nd nd nd nd nd nd nd nd 2 2 7 nd 1 170 C 3 -naphtalenes 21 15 16 7 7 12 15 7 25 22 17 7 9 178 Phenanthrene/anthracene 2 1 2 1 1 nd nd nd 4 1 2 1 0 180 C 1 -fluorene nd nd nd nd nd nd nd nd 3 5 2 3 4 184 C 4 -naphtalenes 4 4 4 3 3 2 4 1 2 3 3 nd 3 192 C 1 -phenanthrene/anthracene 2 3 3 2 3 3 6 nd 7 2 6 3 3 206 C 2 -phenanthrene/anthracene 3 8 5 nd 4 4 4 nd 4 2 6 3 3 n-Alcanes (C 9 -C 33 ) 59 100 100 100 100 45 87 26 100 87 100 100 64 n-Alcenes (C 12 -C 28 ) 60 81 89 75 81 46 67 25 98 100 83 98 78 Pristenes 11 12 11 9 10 5 7 4 15 13 8 12 10 Hopanes (C 27 -C 29 ) 8 14 15 21 20 3 11 1 11 11 46 17 15 Tab. 3 -GIR 117 Tab. 4 -Relative abundance	CRA 1 24 100 45 38 43 50 5 3 21 26 4 7 6 13 13 10 19 4 6 14 7 3 1 12 1 5 3 4 6 80 83 8 11	35 94 68 41 68 63 8 5 33 31 7 11 8 19 14 7 22 4 nd 19 6 4 2 13 2 5 2 6 9 90 100 9 19

Acknowledgements

This project was supported, in part, by financial support from the INSU GDR Marges Program and by a post-doctoral grant from TOTAL (for I.D.). TOTAL is acknowledged for allowing sampling of their wells and for granting permission to publish this article.We also acknowledge M. Hatton and F. Savignac for their analytical help and A. Lethiers for having transforming our crude drawings into intelligible illustrations. We are also grateful to P. A Meyers and an anonymous referee for their constructive reviews of the manuscript.


Bertrand Gottin
email: bertrand.gottin@gipsa-lab.grenoble-inp.fr

Cornel Ioana

Jocelyn Chanussot

Guy D ' Urso

Thierry Espilit

Enis Cetin

Detection and Localization of Transient Sources: Comparative Study of Complex-Lag Distribution Concept Versus Wavelets and Spectrogram-Based Methods

The detection and localization of transient signals is nowadays a typical point of interest when we consider the multitude of existing transient sources, such as electrical and mechanical systems, underwater environments, audio domain, seismic data, and so forth. In such fields, transients carry out a lot of information. They can correspond to a large amount of phenomena issued from the studied problem and important to analyze (anomalies and perturbations, natural sources, environmental singularities, ...). They usually occur randomly as brief and sudden signals, such as partial discharges in electrical cables and transformers tanks. Therefore, motivated by advanced and accurate analysis, efficient tools of transients detection and localization are of great utility. Higher order statistics, wavelets and spectrogram distributions are well known methods which proved their efficiency to detect and localize transients independently to one another. However, in the case of a signal composed by several transients physically related and with important energy gap between them, the tools previously mentioned could not detect efficiently all the transients of the whole signal. Recently, the generalized complex time distribution concept has been introduced. This distribution offers access to highly concentrated representation of any phase derivative order of a signal. In this paper, we use this improved phase analysis tool to define a new concept to detect and localize dependant transients taking regard to the phase break they cause and not their amplitude. ROC curves are calculated to analyze and compare the performances of the proposed methods.

Introduction

Transient signals can be globally defined as impulsive or very short duration signals often with oscillations. These signals exist in many different applications and systems from underwater acoustic [START_REF] Michalopoulou | Underwater transient signal processing: marine mammal identification, localization, and source signal deconvolution[END_REF] with opportunity sources to audio signals with sound attacks [START_REF] D U X B U R Y | S e p a r a t i o no f transient information in musical audio using multiresolution analysis techniques[END_REF] or electrical systems with partial discharges and faults in cables [START_REF] Leonard | On-line location of partial discharges in an electrical accessory of an underground power distribution network[END_REF]. The main processing done on this type of signal aims to detect them and localize their source in multisensor configuration. One issue point concerning transient signals is that they are typically defined on very few samples and are difficult to modelize by asymptotic approach. Consequently, their characterization remains a very challenging goal of increasing interest. Until now, several signal processing tools such as Higher-Order Statistics (HOS), Wavelets (WVT) coefficients, and wellknown Time-Frequency analysis by Spectrogram are widely used to perform the goals of detection and localization. The HOS are very suitable to detect transients drowned in additive white Gaussian noise. The HOS give high value statistics for non-Gaussian components as transients, whereas all Gaussian parts of the signal have higher-order statistics very closed to zero [START_REF] Dp | Higher Order Statistics for Signal Processing[END_REF][START_REF] Ravier | Combining an adapted wavelet analysis with fourth-order statistics for transient detection[END_REF]. Wavelet coefficients and spectrogram are based on energy criteria and give high energetical value signatures where transient components occur [START_REF] Papandreou-Suppappola | Applications in Time-Frequency Signal Processing[END_REF][START_REF] Quinquis | A few practical applications of wavelet packets[END_REF]. One of the common points of these approaches is the use of energy coefficients to prove the transients detection. Alternatively, the transients could be detected via the changes of signal instantaneous phase. Hence, one promising approach consists of analyzing the instantaneous phase derivatives of the studied signal and to detect the transitions as the signal parts which provide significant values of phase derivatives at several orders. Recently, for phase derivatives analysis, the time-frequency distribution based on complex lag arguments has been introduced [START_REF] Lj | Time-frequency distributions with complex argument[END_REF][START_REF] C O R N U | Time-frequency distributions with generalized complex lag argument[END_REF]. This distribution is able to reduce inner interferences terms which appear when studying nonlinear TF components. It also offers access to an instantaneous law representation of any phase derivative order.

Unlikely to HOS, WVT, and Spectrogram, this concept of complex-lag distribution focuses directly on the phase law of the signal with no regard to its amplitude. Therefore, in a task of transients detection, when the amplitude of the transient is too low, the statistics obtained by HOS (Skewness, Kurtosis or fourth-order cumulant) as well as WVT coefficients could be too weak and not enough significant for detection. In the same way, the spectrogram distribution could miss low amplitude transients masked by other transients or components of strong amplitude. Using the complex-lag distribution, despite low amplitude signal, the phase break due to the transient remains significant and visible for detection in the "Time-Phase Derivatives" representation plane.

In this paper, we consider transient signals from a multipath configuration. This configuration is encountered in many application areas such as electrocardiograms (EKG), underwater acoustic, seismic, and so forth. A good illustrative case of such configuration signals is the one of signals from reflectometry studies using Gaussian pulse transmission in electric power cables. This reflectometry signal is composed by the emitted pulse and several other pulses due to some reflections on particular cable spots (end of cable, junction points, faults points) [START_REF] Ravot | Method and device for analyzing electric cable networks[END_REF]. These reflection pulses come back to the cable reflectometry transmission point with some delay, amplitude attenuation, and phase shift.

The paper is organized as follows. In Section 2 apresentation of the complex time distribution concept is done. The capability of each method to deal with full detection and localization of transients contained in typical reflectometry signalsispresentedinSection 3.W econcludeinSection 4.

Time-Frequency Distribution Based on Complex Lag Arguments

The concept of complex lag distributions has been introduced in [START_REF] Lj | Time-frequency distributions with complex argument[END_REF] as a way for inner interferences reduction with respect to Wigner distribution. Recently, this concept has been generalized in order to focus on arbitrary instantaneous phase derivative of a signal [START_REF] C O R N U | Time-frequency distributions with generalized complex lag argument[END_REF]. Let us consider the signal defined as

s(t) = A • e jφ(t) . ( 1
)
The case of A depending of t can also be addressed since the effect of slowly varying amplitude is "visible" on the instantaneous phase. Otherwise, after a signal normalization, we can consider A = 1.

-1+ 1

Re

Im

Figure 1: Lag coefficients taken on the real axis.

In order to better understand the concept of complex lag distribution and its generalization, applied on such a signal, let us introduce the very well-known Wigner distribution with appropriate analysis of its moment and lags definition.

The Wigner Ville Distribution.

The Wigner Ville distribution of a signal s(t) is by definition [START_REF] Cohen | Time-Frequency Analysis[END_REF]:

WVD(t, ω) = F τ ⎡ ⎢ ⎢ ⎢ ⎣ Mwv(t,τ) s t + τ 2 s * t - τ 2 ⎤ ⎥ ⎥ ⎥ ⎦ . ( 2
)
This corresponds to the Fourier transform, with respect to the lag variable τ, of a higher-order moment denoted M wv (t, τ). As illustrated in Figure 1,t h i sm o m e n ti sc a l c ulated using two lag coefficients taken on the real axis.

For a signal defined in (1), the expression of the moment becomes M wv (t, τ) = e j[φ(t+τ/2)-φ(t-τ/2)] .

(

) 3
Let us express the signal phase law in terms of Taylor series expansion:

φ t + τ 2 = +φ ′ (t) τ 2 1 1! + φ (2) (t) τ 2 2 2 2! + φ (3) (t) τ 3 2 3 3! + ••• , φ t - τ 2 =-φ ′ (t) τ 2 1 1! + φ (2) (t) τ 2 2 2 2! -φ (3) (t) τ 3 2 3 3! + ••• . ( 4
)
Using the derivation results above, the expression (3) becomes

M wv (t, τ) = e jφ ′ (t)τ × e j[φ (3) (t)(τ 3 /2 2 3!)+••• ] . ( 5
)
By substituting ( 5)in(2), we obtain a new analytical expression [START_REF] Papandreou-Suppappola | Applications in Time-Frequency Signal Processing[END_REF] of the WVD defining it as an ideally concentrated representation of the Instantaneous Frequency Law (IFL), but degraded because of the convolution with a spreading factor.

WVD(t, ω) = δ ω -φ ′ (t) * ω F τ e jQwv(t,τ) ,( 6
)
where Q wv is the spread function defined as (5) (t) τ 5 2 4 5! + φ (7) (t) From this spread function expression, it is easy to understand that the concentration of the WV representation for a chirp signal (polynomial phase law of second order) will be optimal in so far as all φ's derivates terms in Q wv will be equal to zero.

Q wv (t, τ) = φ (3) (t) τ 3 2 2 3! + φ
τ 7 2 6 7! ••• . ( 7

The Complex-Time Distribution.

The Complex-Time distribution of a signal s(t) is by definition [START_REF] Lj | Time-frequency distributions with complex argument[END_REF]:

CTD(t, ω) = F τ ⎡ ⎢ ⎢ ⎢ ⎣ Mct(t,τ) s t + τ 4 s * t - τ 4 s -j t + j τ 4 s j t -j τ 4 ⎤ ⎥ ⎥ ⎥ ⎦ . ( 8
)
In the same way as WVD, this corresponds to the Fourier transform, with respect of the lag variable τ,o fah i g h e r order moment denoted M ct (t, τ). As illustrated in Figure 2, this moment is in this case of order four and calculated using two lag coefficients taken on the real axis as well as on the imaginary axis, hence the concept of "complex-time arguments emerges".

Following the same frame of analysis described in Section 2.1 leads to a new expression of CTD defined with the same form as [START_REF] Papandreou-Suppappola | Applications in Time-Frequency Signal Processing[END_REF]. The spread function for this distribution is [START_REF] Lj | Time-frequency distributions with complex argument[END_REF] Q ct (t, τ) = φ (5) (t) τ 5 4 4 5! + φ (9) (t) τ 9 4 8 9! + φ (13) (t) τ 13 4 12 13!

••• . ( 9
)
Defining a distribution using well-chosen "complex-lag" arguments (+ j and -j on the imaginary axis) involves a significant decreasing of the spread factor. The first term of Q ct (t, τ) is of the fifth order. The terms of phase derivatives of order 3, 7, 11, ... are completely eliminated and all remaining terms are much more reduced with respect to the ones in the Wigner distribution. The Complex Time Distribution improves the concentration of the IFL representation comparing to the one obtained by Wigner Distribution (Figure 4). In the case of a nonlinear and rapidly varying TF structure, the inner interferences are strongly reduced.

The Generalized Complex-time Distribution.

Recently, a generalization of the concept of CTD has been defined [START_REF] C O R N U | Time-frequency distributions with generalized complex lag argument[END_REF]. The starting point of this generalization procedure was the Cauchy's integral formula [START_REF] Rudin | Real and Complex Analysis[END_REF]. Using this theorem, it is possible to compute the Kth order derivative of the instantaneous phase as

φ (K) (t) = K! 2πj γ φ(z) (z -t) K+1 dz. ( 10
)
This relation shows the interest of the complex time concept: the Kth order derivate of function φ at instant t (which could correspond to one signal singularity) can be computed as the complex integral over the integration path γ defined, in the complex plane, around this point. Applying the theory of Cauchy's integral theorem [START_REF] Rudin | Real and Complex Analysis[END_REF] and considering a circle as integration path, the expression (10)becomes [START_REF] C O R N U | Time-frequency distributions with generalized complex lag argument[END_REF]

φ (K) (t) = K! 2πτ K 2π 0 φ t + τe jθ e -jKθ dθ. ( 11
)
As illustrated in Figure 3, the discrete version of (11)i s defined for θ = 2πp/N and p = 0, ..., N -1, where N is the number of discrete values of the angle θ (expression (12)).

φ (K) (t) = K! Nτ K N-1 p=0 φ t + τe j(2πp/N) e -j(2πpK/N) + ε, ( 12
)
where ε is the discretization error.

Using the property of the unitary roots ω N,p = e j2πp/N and the variable change τ ← K τ(K!/N ), the expression ( 12) becomes (cf. Appendix A)

N-1 p=0 φ ⎛ ⎝ t + ω N,p K τ K! N ⎞ ⎠ ω N-K N,p = φ (K) (t)τ + Q(t, τ), ( 13
)
where Q is the spread function defined as [START_REF] C O R N U | Time-frequency distributions with generalized complex lag argument[END_REF]:

Q(t, τ) = N +∞ r=1 φ (Nr+K) (t) τ Nr/K+1 (Nr + K)! K! N Nr/K+1 . ( 14
)
As indicated by (13)a n d( 14), the sum of the phase samples defined in the complex coordinates (left side of ( 13)) is linear depending on τ if the φ's derivates of orders greater than N + K are0.Inordertoexploitthispropertywe define the generalized complex-lag moment (GCM) of s as the operation leading to (13):

GCM K N [s](t, τ) = N-1 p=0 s ω N-K N,p ⎛ ⎝ t + ω N,p K τ K! N ⎞ ⎠ = e jφ (K) (t)τ+ jQ(t,τ) . ( 15
)
The computation of GCMs implies the evaluation of signal samples at complex coordinates. This is achieved using the analytical continuation of a signal defined as [START_REF] C O R N U | Time-frequency distributions with generalized complex lag argument[END_REF] s t

+ jm = +∞ -∞ S f e -2πm f e j2πft df , (16)
where S( f ) is the Fourier transform of signal s. Taking the Fourier transform of GCM with respect to τ, we define the generalized complex-lag distribution (GCD): (t,τ) .

GCD K N [s](t, ω) = F τ GCM K N [s](t, τ) = δ ω -φ (K) (t) * ω F τ e jQ
(

) 17
As stated by this definition, the Kth order distribution of the signal, obtained for N complex-lags, highly concentrates the energy around the Kth order derivate of the phase law. This concentration is optimal if the φ'sderivatesofordersgreater than N +K are 0, exactly like in the case of chirps represented by Wigner distribution. The general definition (17)l e a d st oal a r g en u m b e ro f TF Representations (TFRs), part of them well known in literature. For example, for K = 1; N = 2 the WVD is obtained (Section 2.1), whereas the case K = 1; N = 4 corresponds to the complex-time distribution (CTD) (Section 2.2). In [START_REF] Ravier | Combining an adapted wavelet analysis with fourth-order statistics for transient detection[END_REF], we have shown that increasing the number of complex lags leads to an attenuation of inner interferences due to the time-frequency nonlinearity. This is illustrated by the example in Figure 4 for the following test signal having a transient-behavior TF structure:

s 1 (t) = e j(3cos(πt)+cos(3πt)+(2/3) cos(6πt)+(1/3) cos(9πt)) . ( 18
)
We remark the better concentration of time-frequency energy in the case of GCD 1 6 than in the case of the other TFRs. This is analytically proved by the spread function expression ( 14) and illustrated by the example in Figure 4.

The next example (Figure 5) points out the derivability property of GCD. The signal s 2 used in this example is a train of three frequency modulations (FM) corrupted by some additive noise (SNR = 35 dB). The three FM have short duration (two linear FM s 2-1 and s 2-3 in phase opposition defined on 128 samples and one parabolic FM s 2-2 defined on 64 samples) compared to the analysis time frame (1510 samples). Their analytical expressions are given by:

s 2-1 (t) = e jφ2-1(t) = e j(c0+c1t+c2t 2 ) , s 2-2 (t) = e jφ2-2(t) = e j(d0+d1t+d2t 2 +d3t 3 ) , s 2-3 (t) = s * 2-1 (t) = e jφ2-3(t) = e -j(c0+c1t+c2t 2 ) . ( 19
)
Table 1: Expressions of the first and second phase derivatives.

First φ's derivative Second φ's derivative φ ′ 2-1 (t) = c 1 +2c 2 tφ (2) 2-1 (t) = 2c 2 φ ′ 2-2 (t) = d 1 +2d 2 t +3d 3 t 2 φ (2) 2-2 (t) = 2d 2 +6d 3 t φ ′ 2-3 (t) =-c 1 -2c 2 tφ (2) 2-3 (t) =-2c 2
Such signal could correspond to a received signal from two different radars using linear and parabolic FM waveforms, respectively (Figure 5(a)). The derivability capability of GCD is used here to enable the characterization of transient natures which would be more difficult using just timefrequency representation because of confusing TF signatures. As shown in Figure 5(b) and expressed in Ta b l e 1 , the two linear FM have their well-expected TF structure whereas the transient parabolic FM looks like a chirp in the TF plane. This is because of the short duration effect on the large frame of analysis. We can observe that the linear and parabolic shapes of the FM are not easily distinguishable. The parabolic shape appears to be linear. Without a priori knowledge about the signal, the TF representation alone leads to consider three transients of chirp nature which is actually wrong. To point out the true nature of the parabolic FM, the GCD 2 6 is used. As analytically proved by the equations in Ta b l e 1 , considering the second-order phase derivative allows to stationarize the linear FM and gives a linear signature for the parabolic FM (Figure 5(c)).

The TF rate representation plane avoids the previous confusion.

Transients Detection by Wavelets and

Higher-Order Statistics, Spectrogram and Complex-Lag Distribution

In this section, we compare the phase derivative capabilities of GCD, in the context of transients detection, with the wavelet transform and time-frequency based methods. The signal used in this section comes from a reflectometry study using a Gaussian pulse emission in electric cables.

Reflectometry is used to analyze and control anomalies in electric cables. A signal is emitted at one extremity of the cable and during its propagation when there is a fault at some point of the cable, a part of the emitted signal, amplitude is reflected due to the impedance discontinuity at this point. The reflected part of the signal comes back to the cable extremity point. The aim of the reflectometry study is to detect and localize the faults by analyzing the signal at the emission point after its propagation and reflections.

In our case, a Gaussian pulse is used as emission signal at the entry point of a cable network. The cable network corresponds here to two different cables separated by a junction (Figure 6). During the propagation in the line "cable 1-junction-cable 2", the pulse is reflected at junction points P2 and P5, faults points if there are some, and end of line point P6. As illustrated in Figure 7, the overall analyzed signal is composed by the original emitted pulse and the reflected pulses affected by some delays, amplitude attenuation, and phase shifts. Figure 8 explains more precisely how the different transients composing the signal are matched to the propagation of the emitted pulse in the cable network physical system.

Method of Higher-Order Statistics and Wavelets.

Higherorder statistics measure the non-Gaussianity of a component. Therefore, for a signal composed of transients with additive white Gaussian noise, HOS allow to detect them by giving high values for transient components that are over-Gaussian and putting to zero the rest of the signal corresponding with Gaussian noise. The fourth-order cumulant and the Kurtosis (normalized version of 4th-order cumulant) are the HOS usually considered in time or in frequency. The 4th-order cumulant and time Kurtosis use the signal in time domain and are obtained via estimators calculated from the signal samples and affected by bias and variance [START_REF] Dp | Higher Order Statistics for Signal Processing[END_REF]. Their frequency versions use estimators calculated with the signal samples in frequency domain, that is,

X(ν) = TF[x(t)].
The wavelet transform is widely used for transients detection. Using Discrete Wavelet Transform, the signal is decomposed in a time-scale plane segmented in blocks and representing a wavelets basis. Each block is a basis vector and corresponds to the main wavelet affected by some delay and scale factor. The representation obtained by DWT is formed by wavelet coefficients which are the projections of the signal on each basis vector. DWT proves its efficiency in transients detection in so far as wavelets signals are usually very similar to transient signals. Consequently, the DWT distribution gives higher inner products between transients and wavelet basis vectors. Transients are then detected and located where higher energetical wavelet coefficients are obtained in the distribution.

In 1996, an efficient method blending the advantages of both HOS and wavelets was proposed to detect and localize transient signals with improved performances [START_REF] Ravier | Combining an adapted wavelet analysis with fourth-order statistics for transient detection[END_REF][START_REF] Ravier | Using Malvar wavelets for transient detection[END_REF][START_REF] Ravier | Wavelet packets and de-noising based on higher-order-statistics for transient detection[END_REF]. This method is based on the adapted segmentation of the time-frequency plane by Malvar wavelets. The criteria used to fit the wavelets for the TF plane segmentation are based on HOS, especially the 4th-order cumulant. The final detection curve corresponds to this 4th-order cumulant calculated at each time-delay regions resulted from the final wavelet distribution optimally segmented in terms of Gaussian and non-Gaussian signal components. This curve will be then normalized for the performance analysis by ROC curves.

Figures 9 and10 show the good performances of this method to detect efficiently transient components embedded in white Gaussian noise with very low SNR. The two strongest transients of the signal noised at SNR = -8db are as well detected as in the case SNR = 30 db. However, the limitation of this method, in terms of simultaneous detection of all the transitions contained in the signal, is illustrated as well. With a signal to noise ratio of 30 db, the four transients of the signal are all well visible but, as well as the case SNR = -8 db for which the transients are highly drowned in the noise, they cannot be detected in the same time in an efficient way. This is because of the amplitude difference between the two highest pulses (from emission and end of line reflection in the cable network) and the two lowest ones. From Figure 8, we know that the two lowest pulses result from the presence of a junction separating the two cables. As shown in Figure 9, the detection curve gives for the two low pulses HOS quasi equal to zero relatively with the highest HOS from the other pulses. Consequently, such a detection result leads to consider that we have only the emitted pulse and its reflection at the end of the line but with no presence of a junction. The physical system is therefore badly interpreted. As illustrated in Figure 11, in terms of ROC curves (cf. Appendix B), for a threshold equal to zero the detection and false alarm probabilities (Pd and Pfa) are equal to one, but for a threshold higher than zero of just one calculation step equal to 0.01, the Pd cannot be more than 50%. However, the Pfa is always quasi equal to zero, even when SNR is -8db.

Method of Spectrogram.

In order to eliminate the limitations of the Fourier Transform when analyzing nonstationary signals, the idea is to consider the signal as locally stationary on an adapted time frame. The principle is to make the Fourier analysis of successive time blocks of the signal weighted by a time frame (Uniform, Hanning, Hamming ...). This is equivalent to express the signal by a set of basis functions localized both in time and frequency [START_REF] Cohen | Time-Frequency Analysis[END_REF]:

STFT (h) x t, f = R x(θ)h * t, f (θ)dθ = R x(θ)h * (θ -t)e -j2πfθ dθ. ( 20
)
The above expression corresponds to the inner product between signal x(t) and the basis functions h t, f (θ) = h(θt)e -j2πfθ . The representation resulting from relation (20)i sn a m e dt h eS h o r tT i m eF o u r i e r Transform-STFT. According to its definition, the STFT

P1 P2 P3 P4 P5 P6 Cable 1 Junction Cable 2 T 1 T 2 T 3 T 4
Emitted signal at P1

Total signal obtained at P1 after propagation and reflections of a signal is a complex values representation. For this its square modulus is generally represented and used as a traditional Time-Frequency (TF) distribution. This distribution is named the Spectrogram [START_REF] Cohen | Time-Frequency Analysis[END_REF]. The spectrogram as well as the STFT considers actually the whole nonstationary signal as a succession of short quasi-stationary signals defined on the time domain of the weighting frame h(u). These TF representations are limited by the Heisenberg uncertainty principle [START_REF] Cohen | Time-Frequency Analysis[END_REF]: where ∆t and ∆ f are respectively the distribution resolution in time and frequency. This uncertainty principle means that the distribution cannot have in the same time good resolution in time and frequency. These two parameters are antagonistic and there is always a trade-off to do between time and frequency resolutions. The methodology of detection based on spectrogram consists in calculating the spectrogram distribution of the reflectometry signal (Figure 7). Our study focuses on detection and localization of transient signals which are, by definition, short in time and large band in frequency. The spectrogram is consequently calculated using a very short frame h(u) in order to have a good resolution in time resulting in a limited resolution in frequency which is not actually problematic (Figure 12). A good resolution in time is moreover suitable and necessary in so far as two transient components of the signal can be very close. The final detection curve DC Spectro (t) corresponds to the curve of maxima of each column of the obtained distribution. This curve will be then normalized for the performance analysis by ROC curves. As illustrated in Figure 12(a), the spectrogram of the reflectometry signal gives high energetical values where transients occur, and we can note that the energy signatures of the two small energy transients (samples 600 and 1450) are not visible. Thus, for SNR = 30 db (the signal is the same as Figure 9(a)) the detection curve represented in Figure 12(b) shows that the two strongest pulses are well detected whereas the two lowest ones are not significantly detected. Only the low transient at sample 600 gives a very weak detection signature. As well as in Section 3.1 , the two low pulses, associated with the presence of a junction in the system, have an amplitude much weaker than the other pulses. Their energy is consequently masked in the distribution by the higher energy of the other transients.

∆t • ∆ f ≥ 1 4π , (∆t) 2 = +∞ -∞ t 2 |x(t)| 2 dt +∞ -∞ |x(t)| 2 dt , ∆ f 2 = +∞ -∞ f 2 |X( f )| 2 df +∞ -∞ X f 2 df , ( 21
This detection based on spectrogram, as well as the adapted wavelets analysis (Section 3.1) , has the advantage to be robust to noise. For SNR = -8 db, the signal in time (the same as Figure 10(a)) is highly corrupted and the transients contained in the signal are strongly drowned in the noise. As illustrated in Figure 13(a), the distribution obtained by spectrogram still gives well visible high energetical signatures for the two strongest pulses. The energy of the two low pulses remains masked in the distribution and the noise is spread on the whole TF plane. The noisy detection curve represented in Figure 13(b) leads to a good detection of the two main transients, whereas the two weak pulses are totally drowned in the noise. This second method has the same efficiency limitations as the one presented in Section 3.1 in terms of simultaneous detection of all the transitions contained in the signal. As illustrated in Figure 14,i nt e r m so fR O Cc u r v e s (cf. Appendix B), for a threshold equal to zero the Pd and the Pfa are equal to one. For a threshold higher than zero of just one calculation step equal to 0.01, the two main pulses and the first small one are detected. The Pd is consequently of 75%. However, for higher threshold the Pd cannot be more than 50% because only the two main pulses can be detected.

Method of Complex-Lag Distribution. The Generalized

Complex-time Distribution (GCD) presented in Section 2.3 represents an efficient phase analysis tool. Using directly the signal samples, enables to give highly concentrated representations of any instantaneous phase derivatives, independently of the signal amplitude. Using this concept, another methodology of transients detection and localization can be defined. This methodology is based on the analysis of the signal instantaneous phase. It allows to detect efficiently and in the same time one or several transients contained in a signal. The transients are detected via the phase discontinuity they cause in the instantaneous phase and consequently with no regard to their amplitude. In this section, the explained methodology is applied on the same reflectometry signal as before. Let us analyze the theoretical phase law of this signal. As shown in Figure 15(a), signal s is composed of four transients with two of them having a very low amplitude comparing to the others. This disparity "strong amplitudeweak amplitude" between the transients of the signal makes difficult the simultaneous detection of all of them, as explained in Sections 3.1 and 3.2 . The two weakest pulses do not give significant signatures and are missed in detection by classical tools (HOS, wavelets, spectrogram). This limitation in detection leads to a bad interpretation of the system which will be considered as a cable line without junction. According to Figure 15(b), analyzing the phase of the signal proves its interest in so far as the four phase breaks due to the four transients have the same importance. The analytic signal associated to the real values signal s is used here.

Figure 16(a) represents the theoretical instantaneous frequency law (IFL) of s obtained by first derivation of the theoretical instantaneous phase law. As illustrated in Figure 16(b), the GCD calculates, using directly the signal samples, the representation of the IFL. This representation is very well concentrated around the theoretical law to analyze. The final detection curve DC GCD (t) used in this methodology comes from the curve of positions in frequency of the maximum value of each column of the obtained distribution. As the distribution obtained by GCD is well concentrated around the IFL, the argmax curve obtained is also very close to the theoretical law. The modulus of the argmax curve defines the detection curve represented in Figure 16(c). On this curve (22), all the transients are well detected with the same importance via pulse signatures resulting from the derivation of the phase discontinuities. This curve will be normalized for the performance analysis by ROC curves. Unlikely to adapted wavelets and spectrogram analysis, phase analysis is sensitive to noise. Adapted wavelets and spectrogram can detect only the two main pulses giving consequently probability of detection not more than 50% but a false alarm rate always quasi equal to zero even for very low SNR. On the opposite, detection by phase analysis allows to detect, on the same signal realization, all the transients of the signal in spite of their differences. But, as shown by Figures 17 and18, for decreasing SNR, the sensitivity to noise of the phase analysis involves an increasing false alarm rate. However, in terms of detection performances illustrated by ROC curves Figure 19, this method allows to reach probabilities of detection which remain equal to one until very high threshold and for very low false alarm rates (cf. Appendix B).

DC GCD (t) = Arg max f GCD t, f . ( 22
)

Conclusion

In this paper, three methods for transients detection and localization have been presented in the case of an analyzed signal composed of several transients. All the transients contained in the signal result from a common physical phenomenon but are marked by strong differences between one another, in terms of difference of amplitude or phase shift. In such a context, a method by phase analysis using the tool of Generalized Complex-time Distribution proves its advantages. In terms of phase, the transient signatures by phase breaks remain all in the same way more significant than the ones obtained by consideration of energetical or statistical criteria. ROC curves illustrating and comparing the performances of the different methods lead to consider the phase analysis method as more suitable, as long as the SNR is reasonable. For very low SNR, spectrogram, wavelets, and HOS methods remain preferable in spite of their efficiency limitation to detect all the transients of the signal. This work about detection and localization performances represents the first step of a more complete study concerning the analysis of transients. A next step would be to characterize the transient signal. It means to be able to match a transient and its characteristics to a particular phenomenon. The Generalized Complex-time Distribution tool offers access to any derivative order of the phase law of a signal. Using this derivability property could lead to characterization via analysis of the successive phase derivates of a signal composed of several transients of different nature.

Appendices A. Derivation of the Spread Function Expression

The final spread function expression is obtained as follows.

We start by considering the discrete version of Cauchy's integral formula expressed in Section 2.3 equation [START_REF] Rudin | Real and Complex Analysis[END_REF]. By using the unitary roots notation ω N,p = e j2πp/N , this equation becomes

φ (K) (t) Nτ K K! = N-1 p=0 φ t + ω N,p τ ω -K N,p + ε Nτ K K! . (A.1)
The unitary roots ω N,p verify the following property:

B. ROC Curves Calculation

The detection and false alarm probabilities Pd and Pfa are calculated through a statistical estimation based on a significant set of realizations of the noised signal. Let N be the number of realizations, it is equal to 100 for our study.

(i) Detection probability P d : each realization of the signal with additive random noise is composed of T transients at known instants. On the detection curve resulted from the chosen method, we calculate q i corresponding to the number of good detections for each realization i and for a set threshold η.T h e detection criteria (i.e., statistic of detection >η )a r e checked on each block of the T transients, which enables a number of maximum T good detections per realization. The Detection Probability is estimated as

P d = N i=1 q i T × N . (B.6)
(ii) False alarm probability P fa : for each realization i, we calculate r i the number of detection curve samples (transients blocks excluded), where the detection criteria are checked. Let L be the length of the detection curve and L Block the length of each T transients time domain block. The False alarm Probability for a set threshold η is estimated as

P fa = N i=1 r i (L -T × L Block ) × N . (B.7)
Each point (P d ; P fa ) of the ROC curves is obtained for a set threshold value. The final ROC curves must be calculated on a significant number of points. The threshold value η is varyingherefrom0to1perstepof0.01, which involves ROC curves composed of 101 calculation points (P d ; P fa ).

C. GCD Calculation Cost

The GCD complexity depends on the time and frequency resolutions of the distribution, that is, the length of the signal and the range of lag τ. As expressed in equation (15), the GCM is obtained by calculating, for each τ,apr oductofN complex-lagged signals. Each of these signals is obtained by the analytical continuation (16) and involves the calculation cost of an Inverse Fast Fourier Transform (IFFT matlab algorithm). Once the GCM is obtained, the GCD is obtained by Fast Fourier Transform (FFT matlab algorithm) with respect to τ at each time sample (cf. equation ( 17)). The GCD calculation cost is the one of multiple FFT and IFFT. It remains fast.

Figure 2 :

 2 Figure 2: Lag coefficients taken on the real and imaginary axis.

Figure 3 :

 3 Figure 3: The complex lag coordinates.

Figure 4 :

 4 Figure 4: Inner interferences reduction property of GCD.

Figure 5 : 6 EURASIPFigure 6 :

 566 Figure 5: (a) Signal s 2 composed of two linear FM, one parabolic FM and additive noise (SNR = 35 dB); (b) GCD 1 2 of s 2 ;(c)GCD 2 6 of s 2 .

Figure 7 :

 7 Figure 7: Total signal obtained at the cable entry point P1.

Figure 8 :

 8 Figure8: T 1 = emitted pulse at P1; T 2 = pulse resulting from the reflection at junction point P2 (cable1/junction) of T 1 ; T 3 = pulse resulting from the reflection at end of line point P6 of T 1 -T 2 ; T 4 = pulse resulting from the reflection at P6 of the part of T 3 coming from its reflection at junction point P5 (cable2/junction).

Figure 9 :

 9 Figure 9: Detection of transient components by adapted wavelet analysis; SNR = 30 db.

Figure 10 :Figure 11 :

 1011 Figure 10: Detection of transient components by adapted wavelet analysis; SNR = -8db.

Figure 12 :Figure 13 :

 1213 Figure 12: (a) Spectrogram of the reflectometry signal noised at SNR = 30 db; (b) associated detection curve.

Figure 14 :Figure 15 :

 1415 Figure 14: ROC curve for detection by spectrogram for SNR = 30 db.

Figure 16

 16 Figure 16: (a) Theoretical instantaneous frequency law (IFL) of s; (b) IFL obtained by GCD for SNR = 30 db; (c) associated detection curve.

Figure 17

 17 Figure 17: (a) Signal s noised at SNR = 20 db; (b) IFL obtained by GCD; (c) associated detection curve.

Figure 18 Figure 19 :

 1819 Figure 18: (a) Signal s noised at SNR = 10 db; (b) IFL obtained by GCD; (c) associated detection curve.

4 )

 4 f k = 0 (mod N) 0, otherwise. (A.2)Using the Taylor series expansion of φ and the property expressed in (A.2), we obtain of (A.1)a n d( A.3), it follows that the error term expression is τ)Applying the variable change τ ← K τ(K!/N ) makes linear, comparing to variable τ, the Kth order phase derivative term. As a result of this variable change inserted in expression (A.4), the final expression of the spread function is obtained asQ(t, τ) = N

Acknowledgments

This work was supported by the French project PHC Pelikan. The signals of reflectometry come from a software simulating the propagation of Gaussian pulses in power cables, work realized by the Research and Development site of EDF in Paris with the participation of Guy D'Urso and Thierry Espilit.