Santo La Delfa
email: ladelfa@unict.it

Maria Giovanna Camuti

Giuseppe Patanè

A multidisciplinary study of the volcanoclastic deposit named "Chiancone", inland and offshore, in the frame of the evolution of volcanism in the Etna area

Keywords: Mt. Etna, Chiancone, Morphology, Vertical Electrical Soundings

In this work a detailed study about the volcanoclastic deposit pyroclastic cone called Chiancone is carried out by 98 Vertical Electrical Soundings (VES) and by analysis of the surface morphology made by two Digital Elevation Model (DEM) with high resolution. The results seem to confirm what has been suggested by other researchers and shed some light on some dark aspects of this volcanoclastic deposit. In particular it was found that the Chiancone is quite a complex geologic formation, whose deposition took place in several steps, filling in depressions of various conductive substrate to depths of at least 300 m below the sea level. The conductive substrate generally shows a higher resistivity than a clay one and is strongly tectonized and eroded. The multibeam surveys carried out offshore in the Ionian Sea, allow us to idetify the geolithological nature of the substrate, never before detected in the past. They show the existence of an outcrop of a large cone-shaped submarine pyroclastic body, about 600 m deep, covered, near the coast, by Chiancone deposits. The authors believe therefore that this pyroclastic body may form "inland" the conductive substrate of the formations outcropping in the studied area (lava and Chiancone) and representing the product of the demolition of one or more existing volcanic apparatuses prior to the Trifoglietto and Mongibello, which was given, in the past, the name of Ancient Etna.

Introduction

In the past, Etna has been the subject of countless geophysical and geo-structural surveys and can be considered one of the most studied volcanoes. It arose and grew along the eastern edge of Sicily, at the crossing point of two regional tectonic structures (Fig. 1), which have aided magmatic migration within the crust and the formation of eruptive apparatuses with a central rising axis, over a period of at least 150 ky B.P.. In this time interval several volcanic edifices have developed, overlaying and juxtaposing on the ruins of previous ones. The deep geodynamic causes which determined the successive periods of growth and demolition of the various eruptive apparatuses are still not clear, but nonetheless the vast depressions called calderas represent the proof of a past dynamism, at times violent, in the Etna area [START_REF] Bousquet | The tectonics and geodynamics of Mt. Etna: synthesis and interpretation of geological and geophysical data[END_REF][START_REF] Branca | Eruptions of Mt. Etna during the past 3,200 years: a precise compilation integrating the historical and stratigraphic records[END_REF]. There were probably many types of phenomena which caused the partial destruction of the various volcanoes and they developed over longer or shorter time frames. They were followed by the removal in various different ways of the volcaniclastics deposit as well as its depositing in the eastern sector of Etna. The volcanoclastics formed several fans of which the best known and most studied, in that they still outcrop, are those called Chiancone (CH). This is a formation mostly constituted of an alluvial fan-shaped body made up of debris flows and fluvial deposits, with differing degrees of compactedness, which occupy an area of about 25 km 2 according to [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] or about 40 km 2 according to [START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF]. The CH has a maximum outcrop thickness of 30 m which can be seen along the Ionian coast (Olmo cliff) and a general slope of 3-4° eastwards.

The various calculations of its real volume and its extension are still under way and are based on data which are sometimes unpublished hydrogeological surveys searching for water, and various types of onshore and offshore geophysical surveys. The study of this geoformation is of notable scientific importance to explain the origin and evolution of the Valle del Bove, an ancient collapse depression formed following the destruction of various alkaline centres which existed before the Mongibello (AA [START_REF] Vv | Carta Geologica del Monte Etna[END_REF].

According to [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] there are 4 opening phases of the Valle del Bove; the first two happened about 50 ky B.P. and 35 ky B.P. and the lahars situated in the Macchia stream between Milo and Moscarello can be attributed to them.

According to [START_REF] Duncan | The quaternary stratigraphy of mount Etna, sicily: the effects of differing palaeoenvironments on styles of volcanism[END_REF], Guest el al., (1984), [START_REF] Chester | The Pyroclastic deposits of Mount Etna volcano, Sicily[END_REF], these lahars are probably associated with the lower tephra (AA [START_REF] Vv | Carta Geologica del Monte Etna[END_REF], dated around 26-20 ky B.P. and linked to the first event of the Valle del Bove opening. The formation of the Chiancone in a strict sense, is instead associated to the last two phases which occurred in the time frames 14-10 ky B.P. and 8-3 ky B.P.. [START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF] hold that the Chiancone was deposited during the life-span of the Ellittico volcano (40-15 ky B.P.) and so, the first event of the opening of the Valle del Bove was much older than 5 ky B.P. as proposed by [START_REF] Romano | Succession of the volcanic activity in the Etnean area[END_REF], Cortese et al. (1988). In this work the authors studied the three-dimensional geostructural set-up of the pyroclastic body which crop out at medium-low altitudes of the eastern sector of Etna by studying the interpretation of 98 Vertical Electrical Soundings. This sector has been analysed in the past by other researchers using various other geological and geophysical methodologies. The analysis carried out with geo-electrics shows the existence under the CH of a conductive body which, in the past, researchers attributed to sedimentary rocks. The interpretation of the apparent resistivity curves and the geostructural surveys suggest that this conductor emerges in the southern zone of the area under study and is dislocated by various faults and should instead be associated to some pyroclastic formations mostly cropping out along the surface of fault between Acireale and S. Tecla.

The authors hold that this pyroclastic formation, older than the CH, might have been produced by the demolition of the volcanic apparatuses which pre-existed the Ancient Alkaline Centres and the Trifoglietto, in agreement with [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF].

Tectonic and geomorphological features.

On the lower eastern side of Etna the largest volcaniclastic sequence in this sector of Sicily is exposed, whose geolithological and structural evolution is directly linked to the dynamism of the volcanism and tectonics. This deposit, mainly made up of volcaniclasts, which are called the Chiancone (CH), was deposited in a large depression generated at the meeting point of two huge structures on a regional scale, oriented NNE-SSW (Messina-Giardini fault system) and NNW-SSE (Ibleo-Maltese escarpment) (Fig. 1), [START_REF] Cristofolini | Integrazione di dati geologici, geofisici e petrologici per la stesura di un profilo crostale in corrispondenza dell[END_REF][START_REF] Lentini | The geology of the Mt. Etna basement[END_REF][START_REF] Monaco | Tettonica estensionale nel settore orientale del Monte Etna (Sicilia): dati morfotettonici e sismici[END_REF]Lanzafame et al., 1996). It lies in an area bordered on the South by the S. Tecla fault oriented NW-SE (Fig. 2, faults a) and to the North by the Macchia-Riposto fault (Fig. 2, fault i) oriented NE-SW, which are two seismically very active volcanic-tectonic structures [START_REF] Patanè | Evolution of volcanism around the eastern sector of Mt. Etna, inland and offshore, in the structural framework of eastern Sicily[END_REF][START_REF] Montalto | Recent seismicity of Mount Etna: implications for flank instability[END_REF][START_REF] Delfa | Changing in crust mechanical behaviour due to raising magma: A fracturing model of SE flank of Mt Etna (Sicily)[END_REF]. Moreover the CH is cut across by various faults oriented NNW-SSE named the Timpe Fault System (TFS) (Fig. 2,faults c,d,e,f,l,m,n,p,q) considered to be the inland extension of the Ibleo-Maltese escarpment; the TFS to the North shows mostly fragile type deformations and also creep to the South (Stewart et al., 1993 a b;[START_REF] Delfa | Changing in crust mechanical behaviour due to raising magma: A fracturing model of SE flank of Mt Etna (Sicily)[END_REF]. Along the southern edge of the CH, a fault outcrops oriented NE-SW (Fig. 2,b) which has been historically active [START_REF] Patanè | Evolution of volcanism around the eastern sector of Mt. Etna, inland and offshore, in the structural framework of eastern Sicily[END_REF], but does not appear on the geological map of Etna (AA. [START_REF] Vv | Carta Geologica del Monte Etna[END_REF]. On the Northern side two larger mesoseismic areas have been found (Fig. 2, r,s) [START_REF] Baratta | I terremoti d'Italia[END_REF], whose seismic-genetic axes are linked, respectively, to a non-outcropping structure oriented roughly E-W (Fig. 2,g) and to the Macchia-Riposto fault (Fig. 2,i); this latter is shown by the alignment of morphological escarpments, by the unusual orientation of a lava flow and by checking the flow direction, roughly NE-SW, of the Macchia torrent (Fig. 2, i; Fig. 3). Finally, the NE extremity of the CH shows two morphological lineaments oriented NNW-SSE and NNE-SSW (Fig. 2, h,o), formed by small escarpments and by sudden changes in the streams aligned along the two above-mentioned directions. From a geological and genetic point of view [START_REF] Rittmann | Structure and evolution of Mount Etna[END_REF], [START_REF] Mcguire | Evolution of the Etna volcano: information from the southern wall of the Valle de1 Bove caldera[END_REF][START_REF] Romano | Succession of the volcanic activity in the Etnean area[END_REF], considered the CH to have been formed by mud flows associated with phreatomagmatic or phreatic explosive activity. According to [START_REF] Kieffer | Origine explosive de la Valle del Bove (Etna, Sicilie)[END_REF][START_REF] Kieffer | Les depots detritiques et pyroclastiques du versant oriental de l'Etna[END_REF][START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF], the CH is mainly an alluvial deposit located on the top of mud flows; Di [START_REF] Di Grande | Lineamenti Geologici dell'area del 'Chiancone' (Etna)[END_REF] describe this formation as an alluvial deposit more or less compacted and stratified. [START_REF] Guest | The Valle del Bove, Mount Etna: its origin and relation to the stratigraphy and structure of the Volcano[END_REF] and [START_REF] Borgia | Importance of gravitational spreading in the tectonic and volcanic evolution of Mount Etna[END_REF], consider this deposit as the product of erosion and landslides or dyke intrusion and lava overload which generate gravitational instability, respectively. Along the Northern edge of the CH, the most ancient volcanics which outcrop are the lavas of the Ancient Alkalic Centres (AAC) which lie under the lower tuffs correlated to the Trifoglietto (AA.VV., 1979), the lavas of the Ancient Mongibello (Ellittico), the pyroclasts of the CH and the historical lavas (Fig. 3). To the West, along the Northern edge of the Moscarello fault (Fig. 2,c Tecla fault (Fig. 2,a) and along the escarpment East of Acireale, do the lavas associated with the AAC outcrop again.

On top of these latter lie the lavas, the lower tuffs and lahars of theTrifoglietto Unit (AA. [START_REF] Vv | Carta Geologica del Monte Etna[END_REF], which are covered on the top of the Recent and historical lavas of the Mongibello (Fig. 3). In the Digital Elevation Model (DEM) shown in figure 4, two morphologically different zones can be distinguished which have been indicated as A and B. Zone A is included in a triangular area whose corners are at the town of S. Venerina in the West and, in the East at two points corresponding to the mouth of the Fago stream and at the southern-most tip of the town of Torre Archirafi, respectively. This area is particularly disturbed by faults oriented NNW-SSE which outline Graben structures (Fig. 2: faults c and d, e and f) and Horsts (Fig. 2: faults d and e) and is affected by various lava flows from the Recent Mongibello (Fig. 3). To the East instead, there are some morphological structures , which have been described as a result of faulting, like the sudden change in direction of the talweg of the Fago stream (Fig. 2, fault l; Fig. 4) and the escarpment at Praiola (Fig. 2, fault m). Furthermore, Zone A has a hydrographic network made up of a few sub parallel streams oriented mostly WNW-ESE. The talwegs show several interruptions in their stream probably caused by tectonic disturbances and the various rectilinear stretches of drainage are often less than 2 km long. The water courses break off suddenly to the WNW cut off by theTFS, or near a stream oriented about E-W, which crosses the whole CH and flows out north of Priaola Piccola (Fig. 4). Slightly further north, oriented almost E-W, a watershed separates the drainage lines of Zone A from those of Zone B, which is also triangular in shape (Fig. 4). In this latter zone there are many more small water courses which fan out and are generally oriented from WSW to ENE. In the eastern zone of Zone B, some stretches of drainage are often rectilinear, short, oriented E-W, WNW-ESE and NE-SW and are probably associated to tectonic structures oriented in the same directions. Along the coast, between Torre Archirafi and Riposto, the mouths of the various water courses are at sea level. However, about 300 metres south of Praiola, the easternmost tip of the talweg of a perched stream is at a height of 8 m a.s.l.. The largest value for this difference in height is about 20 m in a perched stream located not far from Praiola Piccola, where the Olmo cliff (Fig. 3) is at its highest. Further north the difference in height of another perched stream reduces and reaches a height of about 8 m. According to Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF] the settling of the CH took place in two different events. In the first one the volcaniclastic deposit filled the trough south of a morphologic sedimentary high. In the second period, the volcaniclastic deposit filled another trough located north of this morphologic high where it reached its maximum thickness (Fig. 4, B Zone).

Lithological features of the Chiancone

Various detailed surveys have been carried out on the granulometry of the volcaniclasts, on the means of transportation and on the ages of some volcaniclastic deposits of the CH, through radiocarbon analysis (Kieffer,1985;Chester el al., 1987;Calvari andGroppelli,1996;[START_REF] Calvari | A comprehensive model for the opening of the Valle del Bove depression and hazard evaluation for the eastern flank of Etna volcano[END_REF]. [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] distinguishes between three main facies in the section of the Olmo cliff which runs along the Ionian Sea coast (Fig. 3): a basal conglomerate (facies 1a) formed by little blunt lava blocks more than 1 m 3 in size, immersed in an abundant and compact reddish-brown matrix; this lithological facies which is considered as a layering of different lahars, emerges for a few metres above sea level between Riposto and Torre Archirafi, and even reaches a thickness of 3 metres to the south of this second town; 1 km south of Torre Archirafi it dips progressively below the overlying layer and re-emerges north of Praiola afterwards; over this lies a conglomerate (facies 2a) made up of heterometrical lithological elements (from a few centimetres to several metres in diameter), well blunt and immersed in an abundant matrix made up of yellowish ash; this outcrops mainly south of Torre Archirafi as far as Praiola, and its thickness varies from 15 to 20 m and has a fluvial type of stratification; on the roof of this second facies lies a darker conglomerate (facies 3a) made up of black and grey but rarely yellowish volcanic ashes, which contain mainly rounded pebbles of different sizes from a few centimetres to a few decimetres and, sometimes, there are also some big blocks. This is formed by numerous stratified fluvial deposits and is 10 m thick, 1 km north of Praiola; it is considered the result of the demolition and the subsequent deposition of the underlying formation which, to the West, crop out at higher elevations. [START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF] have studied the CH sequence in detail from the stratigraphic, sedimentological and petrographic points of view and have distinguished between 5 lithofacies: basal mud flow (lithofacies 1), debris flows (lithofacies 2a), fluvial floods (lithofacies 2b), fluvial clast coarse grained (lithofacies 3), fluvial sand, silt, clay fine grained and scattered lava clast (lithofacies 4), and pyroclastic fall and flow (lithofacies 5). The lithofacies 1 and 2a, the lithofacies 2b and 3, the lithofacies 4 and 5, correspond respectively to the first, second and third facies proposed by [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF]. Overall, if the thicknesses of the various layers are considered, the basal mud flow is only 5% of the exposed lithologies. However this lithofacies, even though it outcrops discontinuously along the Olmo cliff, shows a much greater thickness than the overlying ones, as the results of the geo-electric survey illustrated in paragraph 5 show.

Vertical Electrical Soundings

To obtain a detailed definition of the relationships between the various types of volcanic products and the sedimentary basement, some geo-electric surveys and original sampling carried out in the Chiancone area by some geophysics companies like the C.G.G., the C.M.P., the Universities of Catania, Milan and Palermo and Milan's Geolab, respectively, were taken into consideration. Overall the apparent resistivity curves of 98 Vertical Electrical Soundings (VES) were analysed, which cover almost uniformly the area under study (Fig. 5). The VES were carried out using the classic type of electrode configuration as proposed by Schlumberger, with a maximum distance between current electrodes of 4000 meters. These soundings were performed after considering the morphology, geology and structural lineaments of the area to minimize the effects due to strong lateral variation of resistivity, as well as to respect the theoretical geometry configuration of the assemblage. The apparent resistivity curves have been grouped according to the classification criteria of [START_REF] Hummell | Der scheinbare spezifischer widerstand[END_REF] into six types, referred to as K, HK, KQ, QQ, KQQ and KHAK (Fig. 6,a).

Their features are typically those found in the Etna area and correspond to layers of more or less high resistivity, i.e. volcanic products, superimposed on a conductive layer generally identified as sedimentary basement. The apparent resistivity data were processed to obtain single-dimensional models (1D) of the real resistivity distribution and thickness of each VES using [START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF] analytical technique. It consists of generating the best fit of experimental values of apparent resistivity, thus giving a theoretical curve to which the best model of electric layers with various resistivity and thickness are associated (Fig. 6,b). Equivalent models are calculated through the theorem of constant transversal resistivity within the limits defined by the theoretical curve and that of the apparent resistivity. The resistivity anomalies found in the experimental curve were then eliminated as they could mistakenly be interpreted as layers but which do not actually exist. This was done through calculations of a matrix of coefficients, resulting from a combination of resistivity and thickness. This procedure leads to a considerable reduction of error for the thickness, and makes possible a comparison between the electrical and geological stratigraphy. In the area under study the lavas represent the least common lithotype, while strong layers of both primary pyroclastic deposits and reworked volcanoclastics outcrop.

The resistivity of the lavas varies according to the degree of alteration, fracturing, porosity and the presence or absence of water. The pyroclastic and volcanoclastic deposits show differing resistivities according to their granulometry (from 1 mm to a few metres or more), to their mineralogical make-up, to the degree of alteration and of humidity. The huge variability both laterally and in depth of the physio-chemical conditions of the two lithotypes in the sites where the VES were carried out, influenced the values of resistivity derived from the interpretation of the experimental electrical curves. To correlate the various lithotypes to different classes of resistivity, a cumulative curve was preferred which expresses as a percentage the number of times the resistivity values were measured (Fig. 7 The lowest values are associated to a fine grain matrix, mostly on larger lithological elements with a higher degree of humidity. This class of resistivity thus corresponds to the second facies proposed by [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF]. The third class of resistivity (320 > 140 ohm m), is mostly associated to the deepest layer of the CH, where there are abundant fine grain volcaniclasts with high humidity. This class corresponds to the first facies of [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] and of [START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF], less superficial, which discontinuously outcrops along the Olmo cliff. However the interval of resistivity of the third class can also correspond to the more superficial layers of ash fall and reshuffled volcaniclasts which correspond to Kieffer's third facies (1985). Finally, the fourth class of resistivity ( < 140 ohm m), is associated to a deeper conductive formation (Table I) which, according to some researchers, corresponds to sedimentary rocks like, for example, clay (Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF]. The correspondence between the various classes of resistivity and the above-mentioned lithofacies was obtained by means of carrying out calibration VES readings near boreholes up to 300 m deep, of which there are many in the area under study and of which the detailed stratigraphy is well-known. VES readings taken near faults and deep incisions in the Chiancone and the surrounding area were also taken into account. Here there is a clear stratigraphy of the Chiancone which was used to calibrate and correlate correctly the resistivity parameter with the various lithofacies. Moreover, the optimal area distribution of the VES, which take readings in numerous geologically significant points, makes the extrapolation of the relationships between lithofacies and classes of resisitivity to the whole area under study reliable. The methodology used is the most common and the most appropriate one for exploring the geolithological features of an area, in this case the Chiancone, using the results of the apparent resisitivity soundings in situ. Obviously the transfer of the geo-electrical information to the various lithological and structural aspects of the area under study was carried out also bearing in mind the relationships of the overlapping and juxtaposition of the different outcropping geological formations, and that the various classes of resisitivity have been recognised in volcanic formations other than that of the Chiancone, as can be seen from the geological map of Mount Etna (AA. [START_REF] Vv | Carta Geologica del Monte Etna[END_REF] and other detailed scientific studies.

Analysis of the results

To further analyse the geometric relationships which exist between the various classes of resistivity and thus, the relationships of overlaying and juxtaposition of the lithofacies associated to these classes, the spatial trend of resistivity and thicknesses relative to the 73 VES was represented through five profiles. Four mostly cut the CH in a NNE-SSW (A,B,C) and ENE-WSW direction (D), the fifth (E) in direction NNE-SSW cuts across the S.Tecla fault and only marginally includes the CH (Fig. 5). Along profile A (Fig. 8), the CH is mostly represented by the 3 rd class of resistivity values which lies directly on the conductive substratum (4 th class). Proceeding from the SSW to the NNE, in the first half of the profile, the 1 st class, attributed to the lavas, lies directly on the conductive substratum or on the 3 rd class of resistivity. It is on this latter, in the second half of the profile,that the CH terrains attributed to the 2 nd class mostly lie, but they are not very thick, generally less than 40 metres. The trend of the roof of the conductive substratum (4 th class) is very variable and shows a sharp downward dislocation towards the NNE of the profile. In profile B (Fig. 8), oriented like the previous one, the 3 rd and 4 th classes show the same overlapping observed in profile A. In the second half of the profile, towards the NNE, the 4 th class (conductive substratum) is noticeably dislocated downwards and its roof reaches a depth of 160 m b.s.l.. The roof of the 3 rd class however, stays above sea level and on it lie the electrostrata belonging to the other three classes of resistivity. The 2 nd class is more frequent and shows thicker electrostata compared to those of profile A. The northernmost tip of profile C (VES 1-9; Fig. 8) includes the remains of an ancient volcanic apparatus located north of Macchia [START_REF] Cristofolini | Morphologic evidence for ancient volcanic centres and indications of magma reservoirs underneath Mt. Etna, Sicily[END_REF]) made up of alternating lavas and pyroclasts (1 st , 2 nd and 3 rd classes of resistivity) which lie on the conductive (4 th resistivity class) most probably formed by sedimentary terrains. The roof of the conductive dips down towards the SSW to a depth of about 400 m b.s.l.. This depth varies a lot and remains below sea level at the town of S. Venerina (Fig. 8), where the zone of lava invasion starts (Fig. 3). To the South, the conductive rises above sea level. The 3 rd class of resistivity, associated with the deepest facies of the CH, shows a greater thickness along the tract of the profile between Macchia and S. Venerina (VES 10-22), to the SSW of VES22, the thickness of the electrostratum relative to this class is notably reduced and in many VES results does not appear. The CH terrains associated to the 2 nd class are more frequent and, in many VES, are thicker than those found in profiles A and B, when they lie directly on the conductive stratum. Profile D (Fig. 8), goes from WSW to ENE and its conductive substratum has a very variable depth. Between the two classes of resistivity associated with the CH, the most common is the 3 rd , while the 2 nd narrows noticeable from west to east. The lavas, 1 st class, sporadically appear at the WSW tip of the profile. Lastly, profile E cuts across the S. Tecla fault (Fig. 8). NE of this structural discontinuity the 1 st , 2 nd , 3 rd and 4 th classes of resistivity become prevalent, the latter of these classes outcrops at the base of the escarpment of the above mentioned fault. To the SW again the three above mentioned classes prevail showing, however, much greater thicknesses. The 3 rd class is less frequent and along with the 2 nd class are probably associable to different pyroclasts from those outcropping in the CH, ascribable to the Trifoglietto (AA. VV., 1979).

After having correlated the four classes of resistivity to the lithological facies along these profiles, the relationship which exists between these latter in three-dimensional space was studied. For this the VES results not yet included, because they were too far from the previously analysed five profiles, were taken into consideration. Later three 2D and 3D sketches were reconstructed (Fig. 9:1,2,3). In particular the figure 9:1 shows the surface morphology of the studied area in which lavas and Kieffer's 2 nd and 3 rd facies outcrop [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF]; the deepest level of the CH (Kieffer's 1 st facies, 1985) and the basal conductive formation are respectively shown on figure 9:2,3. Moreover, on both the 2D and the 3D sketches, the outcropping and two buried active faults linked to the 18-19 July 1865, 27 July 1979 [START_REF] Baratta | I terremoti d'Italia[END_REF], earthquakes and the morphological alignments are shown (see also Fig. 2). The 3D sketch in figure 9 clearly

shows that the flow of the pyroclasts and lavas happened along two main directions indicated by the grey arrows (Fig. 9:1). The morphological trend of the surface of the deepest layer of the CH (Fig. 9:2), mirrors that of the surface topography and so the flow of pyroclasts associated to this facies follows that of the lithofacies nearer the surface. This where there are abundant fine grain volcaniclasts, shows very variable thicknesses due to the very uneven morphology of the conductive substratum. This facies reaches up to 400 metres thick west of Torre Archirafi (Fig. 10,B), and is full of morphological lows and narrows from N to S. It is dislocated by most of the faults found on the surface (continuous lines) and by others which do not crop out (dashed line) hypothesized on the basis of the rather complex trend of the two surfaces which distinguish its top and bottom (Fig. 10). At the Olmo cliff it outcrops discontinuously with a thickness of a few metres and [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF][START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF]. Finally, the conductive substratum (4 th class of resistivity) shows a rather rough surface formed by horsts, with very steep walls and grabens that are at times deep and narrow. This morphology, in our opinion, was determined by intense erosion, which has operated over a long time period, modelling it together with the volcani-tectonic activity in this area. The conductive substratum rises up again near the North of Torre Archirafi, Praiola North forming morpho-structural steps (Fig. 10,E

Conclusions

The detailed geo-lithological and electrical study of the volcanic formations (lavas and pyroclasts) cropping out in the triangle formed by the towns of S.Tecla, S.Venerina and Giarre-Riposto, was fundamental to improving our knowledge of the evolution of volcanism in the Etna area. The geo-lithological, sedimentological and textural analyses of the CH's uppermost layers which crop out, clearly show that it was formed during successive episodes with various modes of genesis and transport of the volcaniclasts [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF][START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF] and its formation required overall a period of 40-50 ky. Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF], during an onshore magnetic soundings found a morphological high, oriented E-W, of the sedimentary substratum. This high borders two troughs: the southern shallower one was filled by the CH mostly masked by the Mongibello lavas, while the northern one, which is deeper, seems to have been filled in later. The geoelectric soundings carried out in this study confirms what Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF] state.

However, it highlights that the conductive substratum (which is sedimentary according to Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF] shows a rather complex morphology determined by strong erosive processes and by frequent faulting, which give it ever deeper horsts and grabens going from South to North (Fig. 10 3) in our opinion, contributed to the above-mentioned depositing of the pyroclastic flows. Lastly, the overlapping found between the body of Zone B and that of Zone A is highlighted by the geometry of the offshore border between the two bodies (Fig. 11), which seems to be further confirmation of what has been previously asserted.

The slower water run-off towards the sea, moreover, determined a reduction in erosion in Zone A, which is slowly uplifting. So along the coast, between the mouth of the Fago stream and Praiola Piccola (Fig. 4) where the Olmo cliff has quite a high wall, there are several talwegs of watercourses, perched at different heights above sea level. The drastic reduction in erosion must have begun about 13,000±2,600 years BP, if we consider the maximum difference in height of 20 metres of one of the watercourses and an average uplift (rise) of 1.5 mm/year (between 1.0 and 2.1 mm/year according to [START_REF] Monaco | Tettonica estensionale nel settore orientale del Monte Etna (Sicilia): dati morfotettonici e sismici[END_REF]. According to [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] between 14,180 years (age determined by C 14 ) and 12,000/10,000 years BP, the Ellittico Crater was formed, the penultimate phase in the opening of the Valle del Bove occurred together with the subsequent deposition of the outcropping lowest CH levels. The above mentioned time span overlays, surprisingly, the one found during the drastic decreasing in the erosive phenomena. It is plausible, therefore, that the different hydrographies of Zones A, B and to the South of the Fago stream were determined by a strong volcanic-tectonic dynamism of Etna in this period. North of Praiola Piccola the coast becomes quite low, probably due to the fact that the volcanic-tectonic phenomena, and subsidence caused by the great thickness of the CH, work against the general uplifting of the area. A crucial point in this study was that of determining the geo-lithological nature of the conductive substratum which is often considered a non-volcanic sedimentary deposit (Bousquet et al., 1998;[START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF], probably clay. In all the geognostic surveys deeper than 200 m whose results are known to the authors, no clay has ever been found, only pyroclasts. Moreover, the interpretation of the experimental curves of apparent resistivity rarely give values around 2 ohm m or, however, very low ones typical of clays. In any case, resistivity values lower than 10 ohm m can also be attributed to levels of fine grain pyroclasts, which are much altered and/or very humid. An unhoped for solution to this problem comes from the offshore multibeam surveys, carried out near the Ionian coast of Etna, within the "Campagna Oceanografica 1908, Progetto V36 Etna", in which researchers from the Universities of Rome, Calabria, Palermo and Catania (INGV) took part and whose results were partially published in a degree thesis written for the Department of Geological Science of the University of Catania (Cavallaro, 2005[START_REF] Cavallaro | Correlazione tra gli elementi morfologico strutturali subaerei e subaquei della costa ionica etnea[END_REF]. Figure 11 clearly shows that offshore there is a geological fan-shaped body, indicated as C, underlying the CH and with a main flow axis independent of those indicated as a and b onshore. In the same zone, Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF] found a volcaniclast body which they considered still part of the CH (Fig. 12). In particular the pyroclasts in Zone B, to the South lie on the geological body C as can be seen from the reciprocal contact between the two offshore formations. Instead to the North, again offshore, they cover the sedimentary rocks, partially filling a morphotectonic depression oriented NE-SW (Riposto Canyon). The CH in Zone A and the lavas further south cover most of the body C onshore. In our opinion, the pyroclasts of the body C are genetically different from those of the CH, because they lie from a stratigraphical point of view under this latter and show a sharply lower resistivity, less than 140 ohm m.

Such a low value can be associated to a much altered pyroclastic formation, rich in fluids and probably, very old. This latter statement is supported by the consideration that, onshore, the surface morphology of the body C, identified in this study as the conductive basement, is very rough due to erosion and tectonic phenomena which have been operating for a long time. Finally, the top of the conductive substratum found under the southern edge of Zone A (Fig. 9,H) and situated slightly further north of Pozzillo, corresponds, in our opinion, to the top of the pyroclastic body C (Fig. 11). It was this latter that guided the flow of the CH's volcaniclasts towards Zones B and A. To the South it also controlled the flow of the lava towards S.Tecla. Offshore, the pyroclastic body C reached a depth of 500 m b.s.l. banking up against the ancient underwater volcanic apparatus [START_REF] Patanè | Evolution of volcanism around the eastern sector of Mt. Etna, inland and offshore, in the structural framework of eastern Sicily[END_REF]. This bank can be seen in the differences in the sea-floor morphology around the edges of bodies C and D (Fig. 11,D).

On the basis of these considerations, various conclusions can be drawn.

In the CH's pyroclasts there are various lithological facies which were generated 50-40 Ky B.P., in particular violent geodynamic manifestations of the volcano (various opening episodes of the Valle del Bove, strong hydromagmatic, strombolian explosions, etc.).

In various phases, the CH's volcaniclasts have occupied some Grabens of the pyroclastic body C, which is in southern zone of the area studied (La [START_REF] Delfa | Changing in crust mechanical behaviour due to raising magma: A fracturing model of SE flank of Mt Etna (Sicily)[END_REF]. In the northern zone, a tectonic depression as deep as 400 m and oriented WNW-ESE (Fig. 9:3), accommodated most of the pyroclastic flows. The North face of this ditch, towards Giarre-Riposto seems to be made up of non-volcanic sedimentary rocks, while the South face is formed by the pyroclasts from the geological body C, as the offshore multibeam surveys show (Fig. 11).

The CH's pyroclasts poured into the sea reaching a maximum depth of 100 m b.s.l. (surveys by the "Istituto Idrografico della Marina", 1993), near Torre Archirafi, overcoming two morpho-tectonic steps: the sedimentary one North of Torre Archirafi (Fig. 9:3; Fig. 10,E) and the pyroclastic one (body C), South of Torre Archirafi, near Praiola Piccola (Fig. 4). In accordance with our results, [START_REF] Loddo | Application of gravity and dipole geoelectrics in the volcanic area of Mt. Etna Sicily[END_REF] using deep dipole measurements found a rise in the non-eruptive substratum in the same area along the coast.

Considering the presence of morpho-structural steps near the coast and the different depth of the base of the CH to the west (about 300 m, below the sea level) and to the east, off the coast line (at about 100 m below the sea level) of these steps, the possibility that the impact of the mass of the detritus, although quite modest, falling into the Ionian Sea may have caused a tsunami wave [START_REF] Pareschi | Large submarine landslides offshore Mt. Etna[END_REF] big enough to affect the central-eastern Mediterranean about 8 000 years B.P. is excluded, in our opinion. Indeed, as is known, the CH's pyroclasts were deposited in various successive episodes, over quite a long time frame (40-50 ky) and considering the particular morphology of the conductive substratum in the area under study, they mostly filled the various existing depressions to the west of these steps, interacting only a bit with the waters of the Ionian Sea.

Instead, in our opinion, what is much more interesting is the existence of the pyroclastic body C which extends out into the Ionian as far as 5 km from the coast and which forms, onshore, most of the conductive substratum under the CH.

This latter, on the basis of the morpho-structural and geo-electrical analyses carried out, is completely distinct from the underlying fan-shaped geological body indicated as C. The authors of this study hold that it may have had a similar origin to the CH and that it is the product of the demolition of one or more eruptive apparatuses which pre-existed the Trifoglietto volcano. In fact, according to [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] between 150,000 and 70,000 ky B.P. there was an "Etna ancient" which evolved in two phases. During the first phase which lasted about 50 ky, a vast, quite flat apparatus built up followed by its partial destruction, as a system of calderas formed. During the second phase, which lasted about 30 ky, a large new apparatus built up with much more explosive activity and later it was partially destroyed. Following this violent activity many lahars were deposited on the eastern side of the volcano, outcropping along the Acireale and S.

Tecla Timpas due to very frequent uprising and faulting phenomena in the area under study. The lithotype outcropping around S.Tecla (Fig. 5) and found also in profile E of figure 8 to the South of Pozzillo, with resistivity lower than 140 ohmm (4 th class), seems mostly to be made up of these lahars. These make up the conductive substratum of the CH, corresponding to the 4 th class of resistivity ( < 140 ohmm). It is therefore possible that the fan-shaped structure found in the Ionian Sea was formed between 100,000 and 70,000 ky B.P. from the product of the demolition of the "Etna ancient". This time span also ties in with the dating carried out by [START_REF] Corsaro | Paleo-environmental and volcano-tectonic evolution of the southeastern flank of Mt. Etna during the last 225 ka inferred from the volcanic succession of the 'Timpe', Acireale, Sicily[END_REF], on various pyroclastic bodies outcropping in the Acireale and S. Tecla Timpa, which are aged between 125 and 65 ky B.P. Finally, the underwater volcanic apparatus in front of the Ionian coast of Etna [START_REF] Patanè | Evolution of volcanism around the eastern sector of Mt. Etna, inland and offshore, in the structural framework of eastern Sicily[END_REF]) acted as a impediment to the pyroclasts flux of the body C (Fig. 10; Fig. 11), which partially overlaid the western flank of this apparatus. Thus it is held that it may be the same age the "Etna ancient" or that it may, however, be older than 100 ky B.P.

Figure captions

Fig. 1 -Sketch map of Sicily: the main regional structures are shown of the eastern edge of the island and the area of Etnean volcanism (A); the inset A shows the studied area (in grey) and the location of the Chiancone (CH).

Fig. 2 -Structural framework of eastern sector of Etna: a,b,c,d,e,f,m,n,p,q are dip-slip normal faults with a little strikeslip movements; l is a strike-slip fault; g is a buried fault, linked to mesoseismic area r of 18-19 th July 1865 earthquake, i is Macchia-Riposto fault linked to mesoseismic area s of 27 th July 1879 earthquake [START_REF] Baratta | I terremoti d'Italia[END_REF]; h,o lineament with structural meanings. The Digital Elevation Model (DEM) 3D, has been made using a topographic map with 1:2000 scale. [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF]; the white dashed line separates the two zones with different run-off.

(2) Morphology on the horizontal plane and in 3D of the surface of the Chiancone's lithological facies 1a which is deeper and outcrops [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] show the trend at depth of the various electrostrata and of the lithotypes correlated to these latter as follow: 1 st class, electrostratum with 1200 ohm m correlated to the lavas; 2 nd class, electrostrata with 1200> 320 ohm m and 320> 140 ohm m, correlated to [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] lithofacies 2a and 3a, respectively deeper and more superficial; cl.3, electrostratum with 320> 140 ohm m correlated to [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] deepest lithofacies indicated with 1a; 4 th class, electrostratum with <140 ohm m corresponding to the conductive substratum. 1) Remains of an eruptive apparatus located in the North-western sector of the area under study; 2) SR, Sedimentary Rocks; SR?, Sedimentary Rocks or very altered and/or humid volcaniclasts; 3) faults hypothesized on the basis of the morphology of the surfaces which border the electrostrata; a-m, outcropping faults. The locations indicated in the sections are those located along the coast and easily allow correlation of the variations of the geometry of the electrostrata in N-S and E-W directions. Fig. 11 -A and B are the areas of outcropping of the Chiancone inland and offshore; a and b (white arrows) represent the main lines of flow off of the Chiancone's volcaniclasts and of the overlaying lavas; C is the area of offshore outcropping of the pyroclastic body underlying the Chiancone and the white arrow represents the corresponding main line of off flow of the volcanoclastic body in the area C; this line, inland, is that of the underground watershed H shown in Fig. 9; D represents the NW flank of the underwater volcano [START_REF] Patanè | Evolution of volcanism around the eastern sector of Mt. Etna, inland and offshore, in the structural framework of eastern Sicily[END_REF] which is 100,000 years B.P. older; SR Sedimentary Rocks.

Fig. 12 -Inland and offshore deposits of the Chiancone according to [START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF], Bousquet et al. (1998), Del [START_REF] Negro | Ground and marine magnetic surveys of the lower eastern flank of Etna volcano (Italy)[END_REF]. Second facies [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] cl 3a

First facies [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF][START_REF] Calvari | Relevance of the Chiancone volcanoclastic deposit in the recent history of Etna Volcano (Italy)[END_REF] Third facies [START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] cl 4a

Deeper Conductive Formation ρ>1200 ρ< 140

320> >140 ρ 1200> >320 ρ CHIANCONE

On the basis of our considerations, various conclusions can be drawn.

In the CH's pyroclasts there are various lithological facies which were generated 50-40 Ky B.P., in particular violent geodynamic manifestations of the volcano (various opening episodes of the Valle del Bove, strong hydromagmatic, strombolian explosions, etc.).

In various phases, the CH's volcaniclasts have occupied some Grabens of the pyroclastic body C, which is in southern zone of the area studied (La [START_REF] Delfa | Changing in crust mechanical behaviour due to raising magma: A fracturing model of SE flank of Mt Etna (Sicily)[END_REF]. In the northern zone, a tectonic depression as deep as 400 m and oriented WNW-ESE (Fig. 9:3), accommodated most of the pyroclastic flows. The North face of this ditch, towards Giarre-Riposto seems to be made up of non-volcanic sedimentary rocks, while the South face is formed by the pyroclasts from the geological body C, as the offshore multibeam surveys show (Fig. 11).

The CH's pyroclasts poured into the sea reaching a maximum depth of 100 m b.s.l. (surveys by the "Istituto Idrografico della Marina", 1993), near Torre Archirafi, overcoming two morpho-tectonic steps: the sedimentary one North of Torre Archirafi (Fig. 9:3; Fig. 10,E) and the pyroclastic one (body C), South of Torre Archirafi, near Praiola Piccola (Fig. 4). In accordance with our results, [START_REF] Loddo | Application of gravity and dipole geoelectrics in the volcanic area of Mt. Etna Sicily[END_REF] using deep dipole measurements found a rise in the non-eruptive substratum in the same area along the coast.

Considering the presence of morpho-structural steps near the coast and the different depth of the base of the CH to the west (about 300 m, below the sea level) and to the east, off the coast line (at about 100 m below the sea level) of these steps, the possibility that the impact of the mass of the detritus, although quite modest, falling into the Ionian Sea may have caused a tsunami wave [START_REF] Pareschi | Large submarine landslides offshore Mt. Etna[END_REF] big enough to affect the central-eastern Mediterranean about 8 000 years B.P. is excluded, in our opinion. Indeed, as is known, the CH's pyroclasts were deposited in various successive episodes, over quite a long time frame (40-50 ky) and considering the particular morphology of the conductive substratum in the area under study, they mostly filled the various existing depressions to the west of these steps, interacting only a bit with the waters of the Ionian Sea.

  ; Fig 3), the Trifoglietto Unit lavas outcrop, which lie under the pyroclasts of the Ancient and Recent Mongibello, the CH and historical lavas (Fig. 3). To the South between S.Venerina and S.Tecla there are mostly outcrops of the Recent Mongibello and the historical lavas. Only along the S.

  m. The first one ( 1200 ohm m), shows quite a low frequency percentage, and has higher values for compact and/or fractured, porous lavas with low humidity; these are generally Recent Mongibello lavas. The average and low resistivity values are associated instead with Recent or Ancient Mongibello lavas which are deeper, altered and more humid. The second class of values (1200 > 320 ohm m), is found in the most superficial zone of the CH. The highest values of resistivity are attributed to the presence of lava blocks bigger than 1 m 3 , very frequent, immersed in a fine grain matrix.

  movement seems to be conditioned by the unique morphology of the conductive substratum which shows some quite deep depressions west of Torre Archirafi and of Pozzillo, separated by a watershed oriented roughly E-W located slightly north of Pozzillo (Fig.9:3,H). These depressions were filled by the deepest facies of the CH which climb over the morpho-tectonic steps near and under the present day coastline (Fig 9:3) pouring out into the Ionian Sea. Overall, there is a total inversion of the morphological relief and a complete masking of the morphology of the conductive substratum. A detailed analysis of the relationship of the overlaying and juxtaposition between the four electrostrata was carried out considering seven sections: four are oriented in a roughly N-S direction (Fig.9:2,3; Fig.10,A,B,C,D) and three in a roughly E-W direction (Fig.9:2,3; Fig.10,E,F,G). The traces of these sections were also transferred to geological maps (Fig.3) so as to correlate spatially the various electrostratigraphs and the corresponding outcropping volcanological formations. The lavas (1 st class of resistivity) are in the southern zone (Fig.3; Fig.10,B,C,D) and the western one (Fig.3; Fig.10,E,F,G) of the area under study, lying mostly on the CH and at times on the conductive substratum; these become thicker towards the W and generally fill the morphological lows. The stratigraphic sequences revealed in the three wells located respectively at S.Venerina (350 m a.s.l.), 1 km to the SW and 2 Km SE of the same location, are in accordance with the interpretation of the VES, clearly show that the lava covering is generally thin, little more than 80 metres thick, and that below it the CH extends down for at least another 60 metres. The facies of the CH, made up of fine grain, large volcaniclasts (2 nd class of resistivity), becomes ever thicker (from a few tens to a hundred or so metres) from East to West and mostly outcrops in the area under study and along the coast (Olmo cliff). It is often dislocated by faults as shown in the various sections of figure10. The deepest facies of the CH (3 rd class of resistivity),

  ,F) and outcrops near S.Tecla dislocated by a fault (Fig.9:3, fault b; Fig.10,B, fault b). Quite a deep excavation carried out in this latter location showed the presence of a very fine grain, dark red pyroclastic formation, (dark red because it is very oxidized), with a resistivity of less than 140 ohm m associable to the conductive substratum of an indefinite thickness, at least in this area. In accordance with DelNegro and Napoli (2001), under the less thick CH, to the NW of Pozzillo and near S.Venerina (Fig.10,B,C,D), the substratum rises up again. This latter, according to the two researchers, should be formed by clay. In our opinion, instead, it may be formed by the above-mentioned pyroclastic formation with low resistivity, represented by the 4 th class which extends towards the North.

  ,B,C,D). Moreover the CH's surface morphological characteristics analysed with a DEM show the existence of two zones A and B which are susbstantially different. Zone A is affected by relatively few sub parallel streams where the water run-off was interrupted or drastically reduced due to lava invasions uphill of these watercourses. The notable dynamics of the faults oriented NNW-SSE and the pyroclasts deposited later further North, in area B, where the flow lines are more numerous, must have also played a determining role: the dynamics of the faults contributed to deviating the water drainage, both the pyroclasts and the lava flows partially invaded the hydrographic network of the valleys in Zone A, filling it up. This succession of events suggests the hypothesis that the surface pyroclastic facies of the CH were laid down at different times, first in the South and later in the North. A similar conclusion can be reached by observing that the eastern side of the Etna Caldera shows a breach to the SE (AA.VV. 1979). It is indeed possible that the various pyroclastic flows were channelled through this opening towards the SSE and were deposited first in Zone A, in front of the opening itself and later in Zone B. Moreover the conductive substratum morphology, affected by various grabens North and South of the watershed H oriented ESE-WNW (Fig 9:

Fig. 3 -Fig. 4 -

 34 Fig. 3 -Geological sketch of the outcropping volcanic products, according to AA.VV. (1979): (1) lavas, tephra and tuffaceous sediments from the Ancient Alkalic Centres; (2) lower tuffs (Trifoglietto Unit); (3) products of the Ellittico and Leone eruptive centres (Ancient Mongibello). (4) Chiancone; (5) a) historical lavas of the Recent Mongibello, b) not dated Ancient and Recent Mongibello lavas; (6) recent alluvium; (f) fault; (l) structural lineaments; the electrostratigraphic sections A-G (see Fig 9 and 10) are drawn on the geological map to aid correlation between lithotypes and electrostrata. Fig. 4 -Hydrography of the area where the Chiancone outcrops; a watershed separates zone A, where the watercourses are oriented WNW-ESE, from zone B, where the watercourses flow more in an ENE-WSW direction.

Fig. 5 -

 5 Fig. 5 -The map shows the location of the 98 Vertical Electrical Soundings (VES) and of the resistivity profiles A,B,C,D,E.

Fig. 6 -

 6 Fig. 6 -(a) Types of apparent resistivity curves; (b) best fit of experimental values of apparent resistivity (left) and best model of electric layers associated (right) by using the technique of Marquard (1963).

Fig. 7 -

 7 Fig. 7 -Cumulative curve of resistivity values obtained through the interpretation of experimental curves; it shows basically four classes of values with different frequency percentages (see text).

Fig. 8 -

 8 Fig.8-The letters A,B,C,D,E and the numbers from 1 to 98 indicate, respectively, the resistivity profiles and the VES shown on the map in Fig.5; 1 st , 2 nd , 3 rd , 4 th classes are the resistivity classes obtained using the cumulative curve in Fig.7. Fig. 9 -(1) Morphology of the area under study, on the horizontal plane and in 3D: a,b,c,d,e,f,i,l,m,n traces of outcropping faults; g buried fault; h,o, structural lineaments (see also Fig.2); A and B, main lines of run-off of the most superficial lithofacies of the Chiancone indicated in the text as 2a and 3a[START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF]; the white dashed line separates the two zones with different run-off. (2) Morphology on the horizontal plane and in 3D of the surface of the Chiancone's lithological facies 1a which is deeper and outcrops[START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] correlated to the 3 rd resistivity class (Fig.8): the lower case letters a,b, etc. have the same meaning as those shown in point (1); A,B,C,D,E,F,G, are the traces of the sections shown in Fig.10. (3) Morphology on the horizontal plane and in 3D of the surface of the conductive body which is considered the basal formation with 4 th resistivity class (Fig.8): the lower case letters a,b, etc. have the same meaning as those shown in point (1); the two horizontal surfaces (2 and 3) shows the traces of the sections A,B,C,D,E,F,G; the horizontal surface (3a) show the traces of the various types of fault and structural lineaments (in white).

  Fig. 9 -(1) Morphology of the area under study, on the horizontal plane and in 3D: a,b,c,d,e,f,i,l,m,n traces of outcropping faults; g buried fault; h,o, structural lineaments (see also Fig.2); A and B, main lines of run-off of the most superficial lithofacies of the Chiancone indicated in the text as 2a and 3a[START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF]; the white dashed line separates the two zones with different run-off. (2) Morphology on the horizontal plane and in 3D of the surface of the Chiancone's lithological facies 1a which is deeper and outcrops[START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] correlated to the 3 rd resistivity class (Fig.8): the lower case letters a,b, etc. have the same meaning as those shown in point (1); A,B,C,D,E,F,G, are the traces of the sections shown in Fig.10. (3) Morphology on the horizontal plane and in 3D of the surface of the conductive body which is considered the basal formation with 4 th resistivity class (Fig.8): the lower case letters a,b, etc. have the same meaning as those shown in point (1); the two horizontal surfaces (2 and 3) shows the traces of the sections A,B,C,D,E,F,G; the horizontal surface (3a) show the traces of the various types of fault and structural lineaments (in white).

Fig. 10 -

 10 Fig.10-The sections oriented roughly N-S (A,B,C,D) and E-W (E,F,G) show the trend at depth of the various electrostrata and of the lithotypes correlated to these latter as follow: 1 st class, electrostratum with 1200 ohm m correlated to the lavas; 2 nd class, electrostrata with 1200> 320 ohm m and 320> 140 ohm m, correlated to[START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] lithofacies 2a and 3a, respectively deeper and more superficial; cl.3, electrostratum with 320> 140 ohm m correlated to[START_REF] Kieffer | Evolution structurale et dynamique d'un grand volcan polyge´nique: stades d'edification et activite´ actuelle de l'Etna (Sicile)[END_REF] deepest lithofacies indicated with 1a; 4 th class, electrostratum with <140 ohm m

  Figure2

  Figure10

Table I -

 I Correlation between resistivity classes and lithofacies.

Acknowledgements

The authors wish to thank Angus Duncan for his suggestions and constructive reviews which improved the quality of the paper. Mark Jellinek is thanked for his comments and for the editorial handling of this paper.


Alexandre Chotard

Anne Auger

Nikolaus Hansen

Cumulative Step-size Adaptation on Linear Functions

Keywords: CSA, cumulative path, evolution path, evolution strategies, step-size adaptation

The CSA-ES is an Evolution Strategy with Cumulative Step size Adaptation, where the step size is adapted measuring the length of a so-called cumulative path. The cumulative path is a combination of the previous steps realized by the algorithm, where the importance of each step decreases with time. This article studies the CSA-ES on composites of strictly increasing functions with affine linear functions through the investigation of its underlying Markov chains. Rigorous results on the change and the variation of the step size are derived with and without cumulation. The step-size diverges geometrically fast in most cases. Furthermore, the influence of the cumulation parameter is studied.

Introduction

Evolution strategies (ESs) are continuous stochastic optimization algorithms searching for the minimum of a real valued function f : R n → R. In the (1, λ)-ES, in each iteration, λ new children are generated from a single parent point X ∈ R n by adding a random Gaussian vector to the parent, X ∈ R n → X + σN (0, C) .

Here, σ ∈ R * + is called step-size and C is a covariance matrix. The best of the λ children, i.e. the one with the lowest f -value, becomes the parent of the next iteration. To achieve reasonably fast convergence, step size and covariance matrix have to be adapted throughout the iterations of the algorithm. In this paper, C is the identity and we investigate the so-called Cumulative Step-size Adaptation (CSA), which is used to adapt the step-size in the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [START_REF] Ostermeier | Step-size adaptation based on non-local use of selection information[END_REF][START_REF] Hansen | Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation[END_REF]. In CSA, a cumulative path is introduced, which is a combination of all steps the algorithm has made, where the importance of a step decreases exponentially with time. Arnold and Beyer studied the behavior of CSA on sphere, cigar and ridge functions [START_REF] Arnold | Performance analysis of evolutionary optimization with cumulative step length adaptation[END_REF][START_REF] Arnold | On the behaviour of evolution strategies optimising cigar functions[END_REF][START_REF] Arnold | Cumulative step length adaptation on ridge functions[END_REF][START_REF] Arnold | Evolution strategies with cumulative step length adaptation on the noisy parabolic ridge[END_REF] and on dynamical optimization problems where the optimum moves randomly [START_REF] Arnold | Random dynamics optimum tracking with evolution strategies[END_REF] or linearly [START_REF] Arnold | Optimum tracking with evolution strategies[END_REF]. Arnold also studied the behaviour of a (1, λ)-ES on linear functions with linear constraint [START_REF] Arnold | On the behaviour of the (1,λ)-es for a simple constrained problem[END_REF].

In this paper, we study the behaviour of the (1, λ)-CSA-ES on composites of strictly increasing functions with affine linear functions, e.g. f : x → exp(x 2 -2). Because the CSA-ES is invariant under translation, under change of an orthonormal basis (rotation and reflection), and under strictly increasing transformations of the f -value, we investigate, w.l.o.g., f : x → x 1 . Linear functions model the situation when the current parent is far (here infinitely far) from the optimum of a smooth function. To be far from the optimum means that the distance to the optimum is large, relative to the step-size σ. This situation is undesirable and threatens premature convergence. The situation should be handled well, by increasing step widths, by any search algorithm (and is not handled well by the (1, 2)-σSA-ES [START_REF] Hansen | An analysis of mutative σ-self-adaptation on linear fitness functions[END_REF]). Solving linear functions is also very useful to prove convergence independently of the initial state on more general function classes.

In Section 2 we introduce the (1, λ)-CSA-ES, and some of its characteristics on linear functions. In Sections 3 and 4 we study ln(σ t ) without and with cumulation, respectively. Section 5 presents an analysis of the variance of the logarithm of the stepsize and in Section 6 we summarize our results.

Notations In this paper, we denote t the iteration or time index, n the search space dimension, N (0, 1) a standard normal distribution, i.e. a normal distribution with mean zero and standard deviation 1. The multivariate normal distribution with mean vector zero and covariance matrix identity will be denoted N (0, I n ), the i th order statistic of λ standard normal distributions N i:λ , and

Ψ i:λ its distribution. If x = (x 1 , • • • , x n ) ∈ R n is a vector, then [x] i will be its value on the i th dimension, that is [x] i = x i . A random variable X distributed according to a law L will be denoted X ∼ L. 2 The (1, λ)-CSA-ES
We denote with X t the parent at the t th iteration. From the parent point X t , λ children are generated: Y t,i = X t +σ t ξ t,i with i ∈ [ [1, λ]], and ξ t,i ∼ N (0, I n ), (ξ t,i ) i∈[ [1,λ]] i.i.d. Due to the (1, λ) selection scheme, from these children, the one minimizing the function f is selected: X t+1 = argmin{f (Y ), Y ∈ {Y t,1 , ..., Y t,λ }}. This latter equation implicitly defines the random variable ξ t as

X t+1 = X t + σ t ξ t . (1)
In order to adapt the step-size, the cumulative path is defined as

p t+1 = (1 -c)p t + c(2 -c) ξ t (2)
with 0 < c ≤ 1. The constant 1/c represents the life span of the information contained in p t , as after 1/c generations p t is multiplied by a factor that approaches 1/e ≈ 0.37 for c → 0 from below (indeed (1-c) 1/c ≤ exp(-1)). The typical value for c is between 1/ √ n and 1/n. We will consider that p 0 ∼ N (0, I n ) as it makes the algorithm easier to analyze.

The normalization constant c(2 -c) in front of ξ t in Eq. ( 2) is chosen so that under random selection and if p t is distributed according to N (0, I n ) then also p t+1 follows N (0, I n ). Hence the length of the path can be compared to the expected length of N (0, I n ) representing the expected length under random selection.

The step-size update rule increases the step-size if the length of the path is larger than the length under random selection and decreases it if the length is shorter than under random selection:

σ t+1 = σ t exp c d σ p t+1 E( N (0, I n ) ) -1
where the damping parameter d σ determines how much the step-size can change and is set to d σ = 1. A simplification of the update considers the squared length of the path [START_REF] Arnold | Random dynamics optimum tracking with evolution strategies[END_REF]:

σ t+1 = σ t exp c 2d σ p t+1 2 n -1 . ( 3
)
This rule is easier to analyse and we will use it throughout the paper.

Preliminary results on linear functions. Selection on the linear function,

f (x) = [x] 1 , is determined by [X t ] 1 + σ t [ξ t ] 1 ≤ [X t ] 1 + σ t ξ t,i 1 for all i which is equivalent to [ξ t ] 1 ≤ ξ t,i 1
for all i where by definition ξ t,i 1 is distributed according to N (0, 1).

Therefore the first coordinate of the selected step is distributed according to N 1:λ and all others coordinates are distributed according to N (0, 1), i.e. selection does not bias the distribution along the coordinates 2, . . . , n. Overall we have the following result.

Lemma 1. On the linear function f (x) = x 1 , the selected steps (ξ t ) t∈N of the (1, λ)-ES are i.i.d. and distributed according to the vector ξ :

= (N 1:λ , N 2 , . . . , N n ) where N i ∼ N (0, 1) for i ≥ 2.
Because the selected steps ξ t are i.i.d. the path defined in Eq. 2 is an autonomous Markov chain, that we will denote P = (p t ) t∈N . Note that if the distribution of the selected step depended on (X t , σ t ) as it is generally the case on non-linear functions, then the path alone would not be a Markov Chain, however (X t , σ t , p t ) would be an autonomous Markov Chain. In order to study whether the (1, λ)-CSA-ES diverges geometrically, we investigate the log of the step-size change, whose formula can be immediately deduced from Eq. 3:

ln σ t+1 σ t = c 2d σ p t+1 2 n -1 (4)
By summing up this equation from 0 to t -1 we obtain

1 t ln σ t σ 0 = c 2d σ 1 t t k=1 p k 2 n -1 . (5)
We are interested to know whether 1 t ln(σ t /σ 0 ) converges to a constant. In case this constant is positive this will prove that the (1, λ)-CSA-ES diverges geometrically. We recognize thanks to (5) that this quantity is equal to the sum of t terms divided by t that suggests the use of the law of large numbers to prove convergence of (5). We will start by investigating the case without cumulation c = 1 (Section 3) and then the case with cumulation (Section 4).

Divergence rate of (1, λ)-CSA-ES without cumulation

In this section we study the (1, λ)-CSA-ES without cumulation, i.e. c = 1. In this case, the path always equals to the selected step, i.e. for all t, we have p t+1 = ξ t . We have proven in Lemma 1 that ξ t are i.i.d. according to ξ. This allows us to use the standard law of large numbers to find the limit of 1 t ln(σ t /σ 0 ) as well as compute the expected log-step-size change.

Proposition 1. Let ∆ σ := 1 2dσn E N 2 1:λ -1 . On linear functions, the (1, λ)-CSA- ES without cumulation satisfies (i) almost surely lim t→∞ 1 t ln (σ t /σ 0 ) = ∆ σ , and (ii) for all t ∈ N, E(ln(σ t+1 /σ t )) = ∆ σ .
Proof. We have identified in Lemma 1 that the first coordinate of ξ t is distributed according to N 1:λ and the other coordinates according to N (0, 1), hence

E ξ t 2 = E [ξ t ] 1 2 + n i=2 E [ξ t ] 2 i = E N 2 1:λ + n -1. Therefore E ξ t 2 /n -1 = (E N 2 1:λ -1)/n.
By applying this to Eq. ( 4), we deduce that E(ln

(σ t+1 /σ t ) = 1/(2d σ n)(E(N 2 1:λ ) -1). Furthermore, as E(N 2 1:λ ) ≤ E((λN (0, 1)) 2 ) = λ 2 < ∞, we have E( ξ t
2 ) < ∞. The sequence ( ξ t 2 ) t∈N being i.i.d according to Lemma 1, and being integrable as we just showed, we can apply the strong law of large numbers on Eq. ( 5). We obtain

1 t ln σ t σ 0 = 1 2d σ 1 t t-1 k=0 ξ k 2 n -1 a.s. -→ t→∞ 1 2d σ E ξ • 2 n -1 = 1 2d σ n E N 2 1:λ -1
The proposition reveals that the sign of E N 2 1:λ -1 determines whether the stepsize diverges to infinity. In the following, we show that E N 2 1:λ increases in λ for λ ≥ 2 and that the (1, λ)-ES diverges for λ ≥ 3. For λ = 1 and λ = 2, the step-size follows a random walk on the log-scale. [1,λ]] be independent random variables, distributed according to N (0, 1), and N i:λ the i th order statistic of

Lemma 2. Let (N i ) i∈[
(N i ) i∈[[1,λ]] . Then E N 2 1:1 = E N 2 1:2 = 1. In addition, for all λ ≥ 2, E N 2 1:λ+1 > E N 2 1:λ .
Proof. (see [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF] for the full proof) The idea of the proof is to use the symmetry of the normal distribution to show that for two random variables U ∼ Ψ 1:λ+1 and V ∼ Ψ 1:λ , for every event E 1 where U 2 < V 2 , there exists another event

E 2 counterbalancing the effect of E 1 , i.e E2 (u 2 -v 2 )f U,V (u, v) du dv = E1 (v 2 -u 2 )f U,V (u, v) du dv, with f U,V
the joint density of the couple (U, V ). We then have E N 2 1:λ+1 ≥ E N 2 1:λ . As there is a non-negligible set of events E 3 , distinct of E 1 and E 2 , where

U 2 > V 2 , we have E(N 2 1:λ+1 ) > E(N 2 1:λ ). For λ = 1, N 1:1 ∼ N (0, 1) so E(N 2 1:1 ) = 1. For λ = 2 we have E(N 2 1:2 + N 2 2:2 ) = 2E(N (0, 1) 2 ) = 2, and since the normal distribution is symmetric E(N 2 1:2 ) = E(N 2 2:2 ), hence E(N 2 1:2 ) = 1.
We can now link Proposition 1 and Lemma 2 into the following theorem:

Theorem 1. On linear functions, for λ ≥ 3, the step-size of the (1, λ)-CSA-ES without cumulation (c = 1) diverges geometrically almost surely and in expectation at the rate 1/(2d σ n)(E(N 2 1:λ ) -1), i.e.

1 t ln σ t σ 0 a.s. -→ t→∞ E ln σ t+1 σ t = 1 2d σ n E N 2 1:λ -1 . (6)
For λ = 1 and λ = 2, without cumulation, the logarithm of the step-size does an additive unbiased random walk i.e. ln σ t+1 = ln σ t + W t where E[W t ] = 0. More precisely W t ∼ 1/(2d σ )(χ 2 n /n -1) for λ = 1, and W t ∼ 1/(2d σ )((N 2 1:2 + χ 2 n-1 )/n -1) for λ = 2, where χ 2 k stands for the chi-squared distribution with k degree of freedom.

Proof. For λ > 2, from Lemma 2 we know that E(N 2 1:λ ) > E(N 2 1:2 ) = 1. Therefore E(N 2 1:λ ) -1 > 0, hence Eq. ( 6) is strictly positive, and with Proposition 1 we get that the step-size diverges geometrically almost surely at the rate 1/(2d σ )(E(N 2 1:λ ) -1). With Eq. 4 we have ln(σ t+1 ) = ln(σ t ) + W t , with W t = 1/(2d σ )( ξ t 2 /n -1). For λ = 1 and λ = 2, according to Lemma 2, E(W t ) = 0. Hence ln(σ t ) does an additive unbiased random walk. Furthermore

ξ 2 = N 2 1:λ + χ 2 n-1 , so for λ = 1, since N 1:1 = N (0, 1), ξ 2 = χ 2 n .
In [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF] we extend this result on the step-size to |[X t ] 1 |, which diverges geometrically almost surely at the same rate.

Divergence rate of (1, λ)-CSA-ES with cumulation

We are now investigating the (1, λ)-CSA-ES with cumulation, i.e. 0 < c < 1. The path P is then a Markov chain and contrary to the case where c = 1 we cannot apply a LLN for independent variables to Eq. ( 5) in order to prove the almost sure geometric divergence. However LLN for Markov chains exist as well, provided the Markov chain satisfies some stability properties: in particular, if the Markov chain P is ϕ-irreducible, that is, there exists a measure ϕ such that every Borel set A of R n with ϕ(A) > 0 has a positive probability to be reached in a finite number of steps by P starting from any p 0 ∈ R n . In addition, the chain P needs to be (i) positive, that is the chain admits an invariant probability measure π, i.e., for any borelian A, π(A) = R n P (x, A)π(A) with P (x, A) being the probability to transition in one time step from x into A, and (ii) Harris recurrent which means for any borelian A such that ϕ(A) > 0, the chain P visits A an infinite number of times with probability one. Under those conditions, P satisfies a LLN, more precisely: Lemma 3. [11, 17.0.1] Suppose that P is a positive Harris chain with invariant probability measure π, and let g be a π-integrable function such that

π(|g|) = R n |g(x)|π(dx) < ∞. Then 1/t t k=1 g(p k ) a.s -→ t→∞ π(g).
The path P satisfies the conditions of Lemma 3 and exhibits an invariant measure [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF]. By a recurrence on Eq. ( 2) we see that the path follows the following equation

p t = (1 -c) t p 0 + c(2 -c) t-1 k=0 (1 -c) k ξ t-1-k i.i.d. . (7)
For i = 1, [ξ t ] i ∼ N (0, 1) and, as also [p 0 ] i ∼ N (0, 1), by recurrence [p t ] i ∼ N (0, 1) for all t ∈ N. For i = 1 with cumulation (c < 1), the influence of [p 0 ] 1 vanishes with (1 -c) t . Furthermore, as from Lemma 1 the sequence ([ξ t ] 1 ]) t∈N is independent, we get by applying the Kolgomorov's three series theorem that the series

t-1 k=0 (1 -c) k ξ t-1-k 1 converges almost surely.
Therefore, the first component of the path becomes distributed as the random variable

[p ∞ ] 1 = c(2 -c) ∞ k=0 (1 - c) k [ξ k ] 1 (by re-indexing the variable ξ t-1-k in ξ k , as the sequence (ξ t ) t∈N is i.i.d.).
We now obtain geometric divergence of the step-size and get an explicit estimate of the expression of the divergence rate.

Theorem 2. The step-size of the (1, λ)-CSA-ES with λ ≥ 2 diverges geometrically fast if c < 1 or λ ≥ 3. Almost surely and in expectation we have for 0 < c ≤ 1,

1 t ln σ t σ 0 -→ t→∞ 1 2d σ n 2(1 -c) E (N 1:λ ) 2 + c E N 2 1:λ -1
>0 for λ≥3 and for λ=2 and c<1

.

Proof. For proving almost sure convergence of ln(σ t /σ 0 )/t we need to use the LLN for Markov chain. We refer to [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF] for the proof that P satisfies the right assumptions. We now focus on the convergence in expectation. From Eq. ( 4) we have

E(ln(σ t+1 /σ t )) = c/(2d σ )(E( p t+1 2 )/n -1), so E( p t+1 2 ) = E( n i=1 p t+1 2
i ) is the term we have to analyse. From Eq. ( 7) and its conclusions we get that for  = 1 [p t ] j ∼ N (0, 1), so

E( n j=1 p t+1 2 j ) = E( p t+1 2 1
) + (n -1). When t goes to infinity, the influence of [p 0 ] 1 in this equation goes to 0 with (1 -c) t+1 , so we can remove it when taking the limit:

lim t→∞ E p t+1 2 1 = lim t→∞ E c(2 -c) t i=0 (1 -c) i ξ t-i 1 2 (9)
We will now develop the sum with the square, such that we have either a product

ξ t-i 1 ξ t-j 1 with i = j, or ξ t-j 2
1 . This way, we can separate the variables by using Lemma 1 with the independence of ξ i over time. To do so, we use the development formula

( n i=1 a n ) 2 = 2 n i=1 n j=i+1 a i a j + n i=1 a 2 i .
We take the limit of

E( p t+1 2 1
) and find that it is equal to

lim t→∞ c(2-c)      2 t i=0 t j=i+1 (1-c) i+j E ξ t-i 1 ξ t-j 1 =E[ξ t-i ] 1 E[ξ t-j ] 1 =E[N 1:λ ] 2 + t i=0 (1-c) 2i E ξ t-i 2 1 =E[N 2 1:λ ]      (10)
Now the expected value does not depend on i or j, so what is left is to calculate

t i=0 t j=i+1 (1 -c) i+j and t i=0 (1 -c) 2i . We have t i=0 t j=i+1 (1 -c) i+j = t i=0 (1 -c) 2i+1 1-(1-c) t-i 1-(1-c
) and when we separates this sum in two, the right hand side goes to 0 for t → ∞. Therefore, the left hand side converges to

lim t→∞ t i=0 (1 - c) 2i+1 /c, which is equal to lim t→∞ (1 -c)/c t i=0 (1 -c) 2i . And t i=0 (1 -c) 2i is equal to (1 -(1 -c) 2t+2 )/(1 -(1 -c) 2
), which converges to 1/(c(2 -c)). So, by inserting this in Eq. ( 10) we get that E p t+1

2 1 -→ t→∞ 2 1-c c E (N 1:λ ) 2 + E N 2 1:
λ , which gives us the right hand side of Eq. [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF].

By summing E(ln(σ i+1 /σ i )) for i = 0, . . . , t -1 and dividing by t we have the Cesaro mean 1/tE(ln(σ t /σ 0 )) that converges to the same value that E(ln(σ t+1 /σ t )) converges to when t goes to infinity. Therefore we have in expectation Eq. [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF].

According to Lemma 2, for λ = 2, E(N 2 1:2 ) = 1, so the RHS of Eq. ( 8) is equal to

(1 -c)/(d σ n)E(N 1:2 ) 2 .
The expected value of N 1:2 is strictly negative, so the previous expression is strictly positive. Furthermore, according to Lemma 2, E(N 2 1:λ ) increases with λ, as does E(N 1:2 ) 2 . Therefore we have geometric divergence for λ ≥ 2.

From Eq. ( 1) we see that the behavior of the step-size and of (X t ) t∈N are directly related. Geometric divergence of the step-size, as shown in Theorem 2, means that also the movements in search space and the improvements on affine linear functions f increase geometrically fast. Therefore, as we showed in Theorem 2 geometric divergence for the step-size when λ ≥ 2 and c < 1, or when λ ≥ 3, we expect geometric divergence on the first dimension of (X t ) t∈N (the first dimension being the only dimension with selection pressure). Analyzing (X t ) t∈N with cumulation requires to study a double Markov chain, which is left to possible future research.

5 Study of the variations of ln (σ t+1 /σ t )

The proof of Theorem 2 shows that the step size increase converges to the right hand side of Eq. ( 8), for t → ∞. When the dimension increases this increment goes to zero, which also suggests that it becomes more likely that σ t+1 is smaller than σ t . To analyze this behavior, we study the variance of ln (σ t+1 /σ t ) as a function of c and the dimension.

Theorem 3. The variance of ln (σ t+1 /σ t ) equals to Var ln

σ t+1 σ t = c 2 4d 2 σ n 2 E p t+1 4 1 -E p t+1 2 1 2 + 2(n -1) . (11) Furthermore, E p t+1 2 1 -→ t→∞ E N 2 1:λ + 2-2c c E (N 1:λ ) 2 and with a = 1 -c lim t→∞ E p t+1 4 1 = (1 -a 2 ) 2 1 -a 4 (k 4 + k 31 + k 22 + k 211 + k 1111 ) , (12)
where Fig. 1: ln(σ t /σ 0 ) against t. The different curves represent the quantiles of a set of 5.10 3 + 1 samples, more precisely the 10 i -quantile and the 1 -10 -i -quantile for i from 1 to 4; and the median. We have n = 20 and λ = 8.

k 4 = E N 4 1:λ , k 31 = 4 a(1+a+2a 2 ) 1-a 3 E N 3 1:λ E (N 1:λ ), k 22 = 6 a 2 1-a 2 E N 2 1:λ 2 , k 211 = 12 a 3 (1+2a+3a 2 ) (1-a 2 )(1-a 3 ) E N 2 1:λ E(N 1:λ ) 2 and k 1111 = 24 a 6 (1-a)(1-a 2 )(1-a 3 ) E (N 1:λ ) 4 . 0
(n 2α + n)/n 3α . Taking α = 1/3 is a critical value where the relative standard deviation converges to 1/( √ 2E(N 1:λ ) 2 ). On the other hand, lower values of α makes the relative standard deviation diverge with n (1-3α)/2 .

Summary

We investigate throughout this paper the (1, λ)-CSA-ES on affine linear functions composed with strictly increasing transformations. We find, in Theorem 2, the limit distribution for ln(σ t /σ 0 )/t and rigorously prove the desired behaviour of σ with λ ≥ 3 for any c, and with λ = 2 and cumulation (0 < c < 1): the step-size diverges geometrically fast. In contrast, without cumulation (c = 1) and with λ = 2, a random walk on ln(σ) occurs, like for the (1, 2)-σSA-ES [START_REF] Hansen | An analysis of mutative σ-self-adaptation on linear fitness functions[END_REF] (and also for the same symmetry reason). We derive an expression for the variance of the step-size increment. On linear functions when c = 1/n α , for α ≥ 0 (α = 0 meaning c constant) and for n → ∞ the standard deviation is about (n 2α + n)/n 3α times larger than the step-size increment. From this follows that keeping c < 1/n 1/3 ensures that the standard deviation of ln(σ t+1 /σ t ) becomes negligible compared to ln(σ t+1 /σ t ) when the dimensions goes to infinity. That means, the signal to noise ratio goes to zero, giving the algorithm strong stability. The result confirms that even the largest default cumulation parameter c = 1/ √ n is a stable choice. The curves were plotted using Eq. ( 11) and Eq. ( 12). On the left, curves for (right to left) n = 2, 20, 200 and 2000. On the right, different curves for (top to bottom) c = 1, 0.5, 0.2, 1/(1 + n 1/4 ), 1/(1 + n 1/3 ), 1/(1 + n 1/2 ) and 1/(1 + n).

Fig. 2 :

 2 Fig.2: Standard deviation of ln (σ t+1 /σ t ) relatively to its expectation. Here λ = 8. The curves were plotted using Eq. (11) and Eq. (12). On the left, curves for (right to left) n = 2, 20, 200 and 2000. On the right, different curves for (top to bottom) c = 1, 0.5, 0.2, 1/(1 + n 1/4 ), 1/(1 + n 1/3 ), 1/(1 + n 1/2 ) and 1/(1 + n).

Acknowledgments

This work was partially supported by the ANR-2010-COSI-002 grant (SIMINOLE) of the French National Research Agency and the ANR COSINUS project ANR-08-COSI-007-12.

Proof.

Var ln

)-E( p t+1 2 )

2

(13) The first part of Var( p t+1

2 ), E( p t+1 4 ), is equal to E((

). We develop it along the dimensions such that we can use the independence of

The other part left is E( p t+1 2 ) 2 , which we develop along the dimensions to get

1 ) 2 + 2(n -1), which we insert into Eq. (13) to get Eq. [START_REF] Meyn | Markov chains and stochastic stability[END_REF].

The development of

) is the same than the one done in the proof of Theorem 2. We refer to [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF] for the development of E( p t+1 4 1 ), since limits of space in the paper prevents us to present it here.

Figure 1 shows the time evolution of ln(σ t /σ 0 ) for 5001 runs and c = 1 (left) and c = 1/ √ n (right). By comparing Figure 1a and Figure 1b we observe smaller variations of ln(σ t /σ 0 ) with the smaller value of c.

Figure 2 shows the relative standard deviation of ln (σ t+1 /σ t ) (i.e. the standard deviation divided by its expected value). Lowering c, as shown in the left, decreases the relative standard deviation. To get a value below one, c must be smaller for larger dimension. In agreement with Theorem 3, In Figure 2, right, the relative standard deviation increases like √ n with the dimension for constant c (three increasing curves). A careful study [START_REF] Chotard | Cumulative step-size adaptation on linear functions[END_REF] of the variance equation of Theorem 3 shows that for the choice of c = 1/(1 + n α ), if α > 1/3 the relative standard deviation converges to 0 with