Nicolas Charles
email: nicolas.charles@univ-orleans.fr

Yan Chen

Romain Augier

Charles Gumiaux

Wei Lin

Michel Faure

Patrick Monié

Flavien Choulet

Fuyuan Wu

Rixiang Zhu

Qingchen Wang

Palaeomagnetic constraints from granodioritic plutons (Jiaodong Peninsula): New insights on Late Mesozoic continental extension in eastern Asia

Keywords: Eastern China, Late Mesozoic extension, palaeomagnetism, granitic pluton, magnetic remanent age, Cretaceous, Shandong Peninsula

come

during the Late Jurassic-Early Cretaceous (e.g., [START_REF] Ren | Late Mesozoic and Cenozoic rifting and its dynamic setting in Eastern China and adjacent areas[END_REF][START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF]Wu et al., 2005a;[START_REF] Wang | Late Mesozoic volcanism in the Great Xing'an Range (NE China): Timing and implications for the dynamic setting of NE Asia[END_REF][START_REF] Lin | Mesozoic extensional tectonics in Eastern Asia : The South Liaodong Peninsula Metamorphic Core Complex (NE China)[END_REF]. Various models proposed to account for the geodynamical causes for this major extensional event are still highly debated (e.g., [START_REF] Meng | What drove late Mesozoic extension of the northern China-Mongolia tract?[END_REF][START_REF] Lin | Late Mesozoic extensional tectonics in the North China Block: a crustal response to subcontinental mantle removal?[END_REF][START_REF] Zhai | Mesozoic sub-continental lithospheric thinning under eastern Asia[END_REF]: (1) back-arc extension induced from Palaeo-Pacific plate subduction (e.g., [START_REF] Watson | Plate tectonic history, basin development and petroleum source rock deposition onshore China[END_REF][START_REF] Li | Cretaceous magmatism and lithospheric extension in southeast China[END_REF][START_REF] Zhou | Origin of Late Mesozoic igneous rocks in southeastern China: implications for lithospheric subduction and underplating of mafic magmas[END_REF][START_REF] Ren | Late Mesozoic and Cenozoic rifting and its dynamic setting in Eastern China and adjacent areas[END_REF], (2) post-orogenic collapse [START_REF] Meng | What drove late Mesozoic extension of the northern China-Mongolia tract?[END_REF], (3) lithospheric delamination [START_REF] Lin | Late Mesozoic extensional tectonics in the North China Block: a crustal response to subcontinental mantle removal?[END_REF], or (4) mantle plume [START_REF] Okada | Plume-related sedimentary basins in East Asia during the Cretaceous[END_REF].

The Late Mesozoic to Cenozoic extensional event affected eastern Asia composed of a set of continental blocks amalgamated during previous orogenies (e.g., [START_REF] Mattauer | Tectonics of Qinling belt: Build-up and evolution of eastern Asia[END_REF][START_REF] Kusky | Tectonic evolution of the North China Block: from orogen to craton to orogen[END_REF]. The knowledge of kinematic histories of these blocks during Late Mesozoic times can provide essential informations to better understand relations between rigid blocks movement of and development of major extensional crustal structures. In fact, according to palaeomagnetic and structural studies, [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF] and [START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF] have proposed that clockwise (CW) rotation of the Eastern Liaoning-Korean block (ELKB) with respect to the North China Block (NCB) was accommodated by the exhumation of South

Liaodong and Yiwulüshan MCCs and the opening of Songliao-Xialiaohe basin (see Fig. 1 in [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF]. Moreover, the Euler pole of this CW rotation is located in the southern end of Bohai bay [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF]. Such a CW rotation is also invoked by [START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF] for the easternmost and southeastern parts of the Jiaodong Peninsula (Shandong Province, NE China) to the south of the ELKB and the Bohai bay, since Early Cretaceous. As a result, [START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF] assume that the Jiaodong Peninsula cannot be considered as a rigid part of the NCB. However, an other palaeomagnetic study carried out in the peninsula showed that the northern part of the Jiaodong did not experience any rotation nor latitudinal displacement since Cretaceous times with respect to NCB [START_REF] Gilder | Tectonic evolution of of the Tancheng-Lujiang (Tan-Lu) fault via Middle Triassic to Early Cenozoic palaeomagnetic data[END_REF]. These discrepancies have major consequences on the understanding of the mechanism of the Mesozoic extension in the eastern Asia and thus, need to be clarified. The aim of the present paper is to better palaeomagnetically constrain the kinematics of the Jiaodong Peninsula with well-dated ferromagnetic granodioritic plutons (i.e. the Weideshan and Haiyang massifs; Fig. 1 and Table 1). This palaeomagnetic study has been carried out with careful structural and AMS analyses of both plutons and country rocks. New palaeomagnetic and time constraints are thus provided to better understand the Late Mesozoic extension of Eastern Asia.

Geological background

Jiaodong Peninsula

The Jiaodong Peninsula, in Shandong Province (NE China), is located along the southeastern margin of the North China Block (NCB; Fig. 1a). This area is delimited to the west by the NNE-trending crustal-scale Tancheng-Lujiang (or Tan-Lu) fault (Fig. 1a), and to the south by the Sulu ultrahigh-pressure metamorphic belt which was tectonically active during Triassic times (e.g., [START_REF] Faure | Where is the North China-South China block boundary in eastern China?[END_REF][START_REF] Faure | Exhumation tectonics of the ultrahigh-pressure metamorphic rocks in the Qinling orogen in east China: New petrological-structural-radiometric insights from the Shandong Peninsula[END_REF]. To the northwest of the Jiaodong Peninsula, Precambrian basement rocks are uncomformably covered by Mesozoic to Cenozoic sedimentary and volcanic series (Fig. 1b; e.g., [START_REF] Sbgmr | Regional geology of the Shandong Province: Geological Memoirs of Ministry of Geology and Mineral Resources 26[END_REF]. Precambrian basement rocks consist of (1) Late Archaean gneisses, granulites, amphibolites and micaschists (Jiaodong Group), unconformably overlain by (2) Palaeoproterozoic multimodal volcanic and sedimentary rocks metamorphosed to amphibolite-granulite facies (Jinshan and Fengzishan Groups) and ( 3) Neoproterozoic (Sinian) metasedimentary rocks (Penglai Group).

Mesozoic rocks are mainly composed of Cretaceous volcano-sedimentary rocks present in extensional intracontinental sedimentary basins (e.g., the Jiaolai basin). Cretaceous stratigraphic series consist into three volcano-sedimentary groups (Laiyang Group dated from ~138 to 126 Ma, Qingshan Group dated from ~126 to 100 Ma, and Wangshi Group dated to the Late Cretaceous K 2 ; [START_REF] Yu | Study of Early Cretaceous angiosperm spores in north China[END_REF][START_REF] Liu | Typical gold deposits in Jiaodong[END_REF][START_REF] Qiu | 40 Ar/ 39 Ar dating for volcanic rocks of Qingshan formation in Jiaolai basin, eastern Shandong province: a case study of the Fenlingshan volcanic apparatus in Wulian County[END_REF]Song and Wang, 2003;[START_REF] Yan | Mantle xenoliths from Late Cretaceous basalt in eastern Shandong Province: New constraint on the timing of lithospheric thinning in eastern China[END_REF][START_REF] Zhang | Cretaceous deformation history of the middle Tan-Lu fault zone in Shandong Province, eastern China[END_REF]. These latter series are mainly exposed in the Jiaolai extensional basin [START_REF] Zhang | Cretaceous deformation history of the middle Tan-Lu fault zone in Shandong Province, eastern China[END_REF] and were already sampled for palaeomagnetic studies [START_REF] Gilder | Tectonic evolution of of the Tancheng-Lujiang (Tan-Lu) fault via Middle Triassic to Early Cenozoic palaeomagnetic data[END_REF][START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF]. Both Precambrian basement rocks and most sedimentary series are intruded by large volume of Late Jurassic to Early Cretaceous plutonic rocks (Fig. 1b). They are divided into two well-defined temporal suites [START_REF] Sang | The origin and evolution of the Linglong granites[END_REF][START_REF] Sbgmr | Regional geology of the Shandong Province: Geological Memoirs of Ministry of Geology and Mineral Resources 26[END_REF][START_REF] Wang | Constraints on crustal evolution and gold metallogeny in the Northwestern Jiaodong Peninsula, China, from SHRIMP U-Pb zircon studies of granitoids[END_REF] [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF][START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF]. Indeed, the Late Jurassic Linglong monzogranite is associated with migmatites, and both of them constitute the Linglong Metamorphic Core Complex (MCC) exhumed below the Linglong detachment fault (LDF) before 134 Ma (Fig. 1b;[START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]. Besides, the Early Cretaceous Guojialing granodiorite is a synkinematic pluton emplaced below the Guojialing shear zone (GSZ) at around 124 Ma (Fig. 1b;[START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]. Granitic rocks also occur as unstrained intrusive plutons into the upper crust and the sedimentary cover (e.g., Late Jurassic Luanjiahe pluton, Early Cretaceous Haiyang and Weideshan plutons). Emplacement and cooling ages of these granitic rocks are particularly well constrained in the region (e.g., [START_REF] Wang | Constraints on crustal evolution and gold metallogeny in the Northwestern Jiaodong Peninsula, China, from SHRIMP U-Pb zircon studies of granitoids[END_REF] and provide more accurate time-constrained samples for palaeomagnetic aims than surrounding country sedimentary rocks.

Weideshan and Haiyang granitic plutons

Both the Weideshan and Haiyang plutons are composed of pinkish-reddish granodiorite with a high proportion of pink K-feldspar, and show porphyritic textures with large microcline megacrysts and medium to coarse-grained groundmass [START_REF] Sbgmr | Regional geology of the Shandong Province: Geological Memoirs of Ministry of Geology and Mineral Resources 26[END_REF][START_REF] Yang | The geology-geochemistry of gold deposits of the greenstone belt in Jiaodong district[END_REF]. Graphic and miarolitic textures with quartz crystals aggregates in the miarolitic cavities are commonly observed (e.g., [START_REF] Sbgmr | Regional geology of the Shandong Province: Geological Memoirs of Ministry of Geology and Mineral Resources 26[END_REF][START_REF] Wu | Weideshan plutonic complex, an example of magmatic pulsating crystallization[END_REF][START_REF] Zhou | Tectonics, granitoids and Mesozoic gold deposits in East Shandong, China[END_REF]. The main accessory minerals include magnetite, apatite, zircon and titanite.

Geochemical analyses of these plutons yielded 87 Sr/ 86 Sr ratios ranging from 0.7050 to 0.7076 [START_REF] Li | Geochemistry of Jiaodong gold deposits[END_REF] highlighting a significant upper mantle contribution [START_REF] Song | Characteristics of dioritic inclusions in Weideshan superunit and its magma origin in Jiaonan area[END_REF].

The Weideshan massif is an elliptic granodioritic pluton (~520 km²) located in the easternmost part of Jiaodong Peninsula (Figs. 1b and2a), to the north of Rongcheng city and to the east of the Muping-Jimo fault (Fig. 1b). U/Pb analyses on zircon yielded an age of 118 ± 2 Ma [START_REF] Zhang | Tectono-chronologic constraints on a Mesozoic slip and thrust belt in the eastern Jiaodong Peninsula[END_REF]. In the field, granodioritic rocks appear devoid of macroscopic deformation and present isotropic textures. Structural investigations along its contact with the country rocks show clear intrusive relationships either devoid of subsequent penetrative deformation or brittle faulting. Anisotropy of magnetic susceptibility (AMS) study has been carried out to access to planar and linear structural fabrics and showed that magnetic foliation trajectories design concentric patterns [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]Fig. 2a). Moreover, magnetic foliations are mainly subhorizontal to gently-dipping at the pluton-scale (Fig. 2c). Magnetic lineations are scattered and present sub-horizontal plunges (Fig. 2c). Corrected anisotropy degree P J [START_REF] Jelinek | Characterization of the magnetic fabric of rocks[END_REF] shows low values (mean value of P J for the pluton = 1.123) which denotes that granitic rocks of the Weideshan pluton did not experience significant deformation since the AMS acquisition (Fig. 2e; [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]. Gravity data point to the existence of a main feeder zone located below the pluton centre where the Bouguer anomaly value is the lowest (i.e. -16 mGal; [START_REF] Ma | The comprehensive interpretation report of the gravity and magmatic data in Shandong Province[END_REF]. Structural and geophysical data imply that the pluton did not experience any rotation along a horizontal axis with respect to the country rocks. [START_REF] Yu | Study of Early Cretaceous angiosperm spores in north China[END_REF][START_REF] Liu | Typical gold deposits in Jiaodong[END_REF]. This age discrepancy motivated the geochronological study of these rocks with both U/Pb and 40 Ar/ 39 Ar methods on zircon and biotite to recognise emplacement and cooling ages of the Haiyang pluton. Thanks to good outcrop exposures, intrusive contact with Early Cretaceous layers has been observed (Figs. 1b and2b). There, horizontal to gently dipping strata indicates that sedimentary cover has not been significantly tilted during and more importantly after the emplacement of the pluton. Intrusion induced a metamorphic aureole with development of hornfels in the country rocks. Neither ductile nor brittle deformation has been observed along the contact between pluton and country rocks.

AMS investigations show concentric pattern for magnetic foliation trajectories accompanied with subhorizontal magnetic lineations and foliations [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]Figs. 2b and 2d). Low values of the corrected anisotropy degree P J (mean value of P J for the pluton = 1.080) imply that granitic rocks of the Haiyang pluton did not experience significative post-solidus strain since the AMS acquisition (Fig. 2e; [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]. Bouguer anomaly values involve a main feeder zone centered on the pluton cartographic outline (i.e. -14 mGal; [START_REF] Ma | The comprehensive interpretation report of the gravity and magmatic data in Shandong Province[END_REF]. Similarly to the Weideshan pluton, structural and geophysical evidences show that the Haiyang pluton has not undergone any rotation along a horizontal axis.

Geochronological study of the Haiyang pluton

3.

Results

U/Pb analyses on zircons

Analysed zircons display a homogeneous population of >100µm sized grains in length (Fig. 3a). According to the typology proposed by [START_REF] Pupin | Zircon and granite petrology[END_REF], zircons with a rather prismatic habit are characteristic of high temperatures (between 750-900°C). After data reduction (GLITTER software; [START_REF] Jackson | The application of laser ablation inductively coupled plasma-mass spectrometry (LA-ICP-MS) to in situ U-Pb zircon geochronology[END_REF] and common Pb corrections [START_REF] Andersen | Correction of common lead in U-Pb analyses that do not report 204 Pb[END_REF], isotopic ratios were plotted into an inverted concordia diagram [START_REF] Tera | U-Th-Pb systematics in three Apollo 14 basalts and the problem of initial Pb in lunar rocks[END_REF]. Eight analyses (Table 2) yielded a lower intercept age of 117.8 ± 2.7 Ma (Fig. 3b) interpreted as the cooling below 800°C (i.e. [START_REF] Spear | Metamorphic phase equilibria and pressure-temperature time paths[END_REF][START_REF] Cherniak | Pb diffusion in zircon[END_REF] shortly after granodiorite crystallisation. More details on isotopic measurements are given in Table 2 (115.7 ± 1.2 Ma). This age is interpreted as the time elapsed since the rock sample cooled below the closure temperature of the K/Ar isotopic system for biotite settled at 300 ± 30°C [START_REF] Villa | Isotopic closures[END_REF].

Palaeomagnetic study on granodioritic plutons of Jiaodong Peninsula

With respects to their bulk chemistry [START_REF] Sbgmr | Regional geology of the Shandong Province: Geological Memoirs of Ministry of Geology and Mineral Resources 26[END_REF] the Weideshan and Haiyang plutons are characterised by the relative abundance of ferromagnetic minerals, such as magnetite. Their unstrained character together with their intrusive relationships into undeformed Lower Cretaceous strata, make them highly suitable for accurate palaeomagnetic study. Furthermore, magnetic remanent ages accurately constrained by a critical compilation of geochronological data completed by new acquired time constraints, allow to obtain new kinematic constraints of Jiaodong Peninsula since Cretaceous times.

Palaeomagnetic sampling

Palaeomagnetic samples were collected from 27 sites, 13 from the Weideshan pluton and 14 sites from the Haiyang pluton (Table 1 andFigs. 2a and2b). These sites have been chosen based on their geographic location so as to obtain a representative spatial distribution for each pluton (Figs. 2a and2b). Sampling was carried out with a portable gasoline drill and the cores were oriented with magnetic and solar compasses, when it was possible. The average difference between magnetic and solar declinations is about -6.5° ± 2.7° and -6.8° ± 2.5° for the Weideshan and Haiyang plutons, respectively. As a result, an azimuth measurement has been corrected to cores that have been only magnetically oriented. Cores were cut into standard specimens (2.5 cm in diameter and 2.2 cm in length).

Laboratory measurements

Magnetic remanent carrier(s) has already been determined in the two plutons for a detailed AMS study [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF]. Magnetic mineralogy was investigated by two methods:

(1) thermomagnetic measurements carried out at the Institut des Sciences de la Terre d'Orléans (ISTO) and (2) hysteresis loops obtained at the Institut de Physique du Globe de Paris (IPGP). Thermal magnetic (Curie point) experiments were performed with an © AGICO KLY-3S kappabridge susceptometer coupled with a CS3 furnace. Magnetic hysteresis curves

were measured with a magnetic inductometer.

Both thermal and alternating magnetic field (AF) demagnetisations were applied to clean the magnetic remanence at ISTO and IPGP. AF demagnetisation was done using a 2G in-line three-axis AF demagnetiser and the magnetic remanence was measured using a 2G three-axis DC SQUID magnetometer at IPGP. Other samples were demagnetised with a laboratory built furnace and magnetic remanence was measured with an © AGICO JR-5A

automatic spinner magnetometer at ISTO. Inclinations and declinations obtained by progressive demagnetisations were plotted on orthogonal vector diagrams (Zijderveld, 1967).

Magnetic remanence directions were determined by principal component analysis [START_REF] Kirschvink | The least squares line and the analysis of palaeomagnetic data[END_REF]. Site-mean directions were computed with Fisher statistics (i.e. [START_REF] Fisher | Dispersion on a sphere[END_REF].

Palaeomagnetic software packages of [START_REF] Cogné | A Macintosh TM application for treating paleomagnetic data and making plate reconstructions[END_REF] and R. Enkin (unpublished) were used for the data analysis.

Magnetic mineralogy

Representative results of the magnetic mineralogical experiments are presented in Granodioritic rocks of the Weideshan pluton present unimodal NRM intensities and magnetic susceptibilities ranging from 100 to 1100 mA.m -1 and 10 to 70 x 10 -3 SI with averages of 343.9 mA.m -1 and 33.2 x 10 -3 SI, respectively. Such high NRM values ensure to perform demagnetisation and to analyse magnetic remanence of these granodioritic rocks.

Typical thermal and AF demagnetisation responses of granodioritic rocks from the Weideshan pluton are presented in Figures 5a and5b. Overall, the magnetic remanence decays linearly from 150°C to 630°C for thermal demagnetisation and from 4 mT to 100 mT for AF one, with well-defined magnetic component of northward to north-northeastward declinations and downward inclinations. Site-mean directions and the corresponding statistical parameters are listed in Table 1 andFigure 5e. It is noteworthy that three sites (i.e.

KW01, KW17 and KW39) show significantly different directions with others rather due to probably the close vicinity to the contact zone which may have experienced a higher cooling rate and border deformation effect (specified by * in Table 1). Moreover, the mean direction of site KW14 may be influenced by the secular variation due to weak number of samples (only three; specified by *** in Table 1). Consequently, these four sites are excluded from site-mean direction calculation for the Weideshan pluton (Table 1). Thus a mean direction was calculated for the Weideshan pluton from nine out thirteen sites: D = 11.3°, I = 61.0°, k = 59.5 and α 95 = 6.7 (Table 1 and Fig. 5e).

Haiyang pluton

Natural remanent magnetisation (NRM) intensities for granitic rocks of the Haiyang pluton range from 30 to 1200 mA.m -1 with average of 375.5 mA.m -1 . Magnetic susceptibility values range between 25 and 60000 x 10 -6 SI with average of 23200 x 10 -6 SI, which is typical of ferromagnetic granitoids [START_REF] Bouchez | Granite is never isotropic: an introduction to AMS studies in granitic rocks[END_REF][START_REF] Bouchez | Anisotropie de susceptibilité magnétique et fabriques des granites[END_REF]. The rather high NRM values allow to perform magnetic remanence measurements and, then, to isolate the palaeomagnetic directions acquired during pluton cooling. specified by ** in Table 1). A possible origin of scattered directions is discussed in Section 5.1. Consequently, mean directions were computed by Fisher statistics from eleven out fourteen sites. Mean direction calculated for the Haiyang pluton is: D = 1.5°, I = 57.0°, k = 60.2 and α 95 = 6.3° (Fig. 5f and Table 1).

Discussion

Thermal and AF demagnetisations have successfully isolated magnetic characteristic components of high temperature and high AF field, mainly carried by MD magnetite with probably small amount of maghemite and hematite. Cluster pluton-mean directions have been obtained from well consistent sites with k > 30 (i.e. 59.5 and 60.2 for Weideshan and Haiyang plutons, respectively), α 95 < 10° (i.e. 6.7° and 6.3° for Weideshan and Haiyang plutons, respectively). Haiyang and Weideshan plutons show equivalent mean directions with an insignificant angular difference of 6.8° ± 8.9° leading to calculate a mean direction for both plutons on the basis of twenty sites: D = 6.6°, I = 59.8°, k = 51.1 and α 95 = 4.6° (Fig. 5g and Table 1). Prior to discuss tectonic significations of these new palaeomagnetic data from granodioritic rocks of Jiaodong Peninsula, reliability and age of magnetisation are discussed below.

Reliability and age of the magnetic remanence

For the Haiyang pluton, new acquired age data yielded a narrow ~2.5 Ma time difference between U/Pb age on zircon and 40 Ar/ 39 Ar age on biotite which is well in line with an emplacement in the upper crust. This difference is interpreted as the time elapsed during cooling from about 800 ± 50°C down to about 300 ± 30°C, according to the mineral closure temperatures [START_REF] Harrison | Diffusion of 40 Ar in biotite: temperature, pressure and compositional effects[END_REF][START_REF] Villa | Isotopic closures[END_REF][START_REF] Cherniak | Pb diffusion in zircon[END_REF]. 6) in order to estimate tentatively the magnetic remanence age of the Haiyang pluton (Fig. 6).

Using magnetic unblocking temperatures for MD magnetite between 560 and 585°C, a magnetic remanence age of about 116.0 ± 1.8 Ma for the Haiyang pluton is proposed. As far as the Weideshan pluton is concerned, 118 ± 2 Ma U/Pb age on zircon [START_REF] Zhang | Tectono-chronologic constraints on a Mesozoic slip and thrust belt in the eastern Jiaodong Peninsula[END_REF] appears amazingly strictly coeval with the Haiyang pluton (117.8 ± 2.7 Ma). A main magmatic pulse is thus observed in the whole region supported by the geochronological data (e.g., [START_REF] Wang | Constraints on crustal evolution and gold metallogeny in the Northwestern Jiaodong Peninsula, China, from SHRIMP U-Pb zircon studies of granitoids[END_REF]Wu et al., 2005a), the obvious petrographic and geochemical similarities of the plutons [START_REF] Sbgmr | Regional geology of the Shandong Province: Geological Memoirs of Ministry of Geology and Mineral Resources 26[END_REF]. While 40 Ar/ 39 Ar ages is currently lacking for the Weideshan pluton, it is tempting to assume similar fast cooling rates of the same order; the involved magnetic remanence age for the Weideshan pluton would also be settled at 116.0 ± 1.8 Ma (Fig. 6).

Age of magnetisation for both plutons is thus consistent with the Cretaceous Normal Superchron (CNS) period between ~118 and 83 Ma [START_REF] Gradstein | International Commission on Stratigraphy (ICS)[END_REF] and the absence of reverse polarities in our palaeomagnetic collection. Moreover, the magnetic unblocking temperature interval considered for magnetite (560-585°C) and the fast cooling rate, involve a duration of the remanent acquisition of about 120,000 years. Consequently, we assume that secular variation is broadly averaged out for sites of both plutons. However, seven palaeomagnetic sites (i.e. KW01, KW14, KW17, KW39, KH50, KH57 and KH58) reveal dispersed directions and have been excluded for mean calculation (Table 1). In fact, palaeomagnetic sampling that covers a large area within plutons allows to access to different structural levels since the elevation variations reach several hundred meters. Three excluded sites (i.e. KW01, KW14 and KW39; specified by * in Table 1) are located in close vicinity to the pluton envelope with respect to country rocks associated with a presumed fast and more heterogeneous cooling (<10,000 yr), impeding to average the secular variation. Site KW14

(specified by *** in Table 1) has been excluded due to weak number of samples (3 cores)

impeding to average the secular variation too. The last three excluded sites (i.e. KH50, KH57

and KH58; specified by ** in Table 1) are poorly statistically defined with high α 95 and low k values. Besides, structural and AMS studies have shown that both plutons have not experienced any significant deformation subsequently to their emplacement [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF], nor any rotation along a horizontal axis.

The characteristic magnetic remanence isolated from granodiorite of both plutons can be confidently considered as primary thermal magnetisations and, therefore, used for tectonic applications. Two palaeomagnetic poles have been thus calculated for the Weideshan pluton 7).

These three poles, measured from andesitic rocks of the Qingshan Group, are located in the southern and easternmost parts of Jiaodong Peninsula, close to the Muping-Jimo fault (MJF; see Fig. 2 in [START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF]. According to their study, age of remanence for these three poles spans between ~125-118 Ma. Thus such northeastward declinations have led the authors to suggest a relative clockwise (CW) rotation of the southern and easternmost parts of Jiaodong Peninsula with respect to the other sites from the same peninsula and NCB as well, since mid-Early Cretaceous [START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF]. However, our new palaeomagnetic results from two granodioritic plutons located to the east of MJF present rather consistent directions, for inclination and declination, with those from the west of MJF, but significantly different from these three previous discussed poles (Table 4 and Fig. 7). This difference can be probably explained by two main reasons: (1) the short magnetisation time and (2) the vicinity of the palaeomagnetic sampling sites with respect to main fault zones. In addition, these three poles are derived from volcanics of the Qingshan Group whose cooling ages are constrained by 40 Ar/ 39 Ar means to 125-118 [START_REF] Qiu | 40 Ar/ 39 Ar dating for volcanic rocks of Qingshan formation in Jiaolai basin, eastern Shandong province: a case study of the Fenlingshan volcanic apparatus in Wulian County[END_REF]Song and Wang, 2003;[START_REF] Zhang | Cretaceous deformation history of the middle Tan-Lu fault zone in Shandong Province, eastern China[END_REF].

Since sampled rocks of these poles (specified in Table 4 by *) are andesites, it is highly questionable if secular variations have been averaged from these palaeomagnetic measurements to ensure an axial-centred-dipolar field as the hypothesis for the application of the palaeomagnetism to the plate tectonics. Furthermore, two of these latter three results are obtained from zones close to the Muping-Jimo Fault (MJF; Fig. 7b) whose motion has been recently constrained at 122.8 ± 0.3 Ma [START_REF] Zhang | Tectono-chronologic constraints on a Mesozoic slip and thrust belt in the eastern Jiaodong Peninsula[END_REF]Fig. 1b) by 40 Ar/ 39 Ar analyses on synkinematic biotites. Deformation (e.g., motion over the fault zone) then appears of coeval with andesite emplacement (~125-118 [START_REF] Ma | The comprehensive interpretation report of the gravity and magmatic data in Shandong Province[END_REF][START_REF] Qiu | 40 Ar/ 39 Ar dating for volcanic rocks of Qingshan formation in Jiaolai basin, eastern Shandong province: a case study of the Fenlingshan volcanic apparatus in Wulian County[END_REF]Song and Wang, 2003;[START_REF] Zhang | Cretaceous deformation history of the middle Tan-Lu fault zone in Shandong Province, eastern China[END_REF]. Consequently, these zones may have experienced local faulting/deformation-related rotations along the MJF. Moreover, the remained rotated palaeomagnetic pole (also from andesite) is issued from a small basin located in the easternmost part of the peninsula, to the north of the Weideshan pluton [START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF]Figs. 1b and 7b). Field observations revealed that the scarce outcrops are often highly weathered and are affected by brittle faulting likely to produce local yet important rotations. Furthermore, the initial geometry of the bedding may be also questionable as these sediments are thought to be syntectonic.

According to (1) a good consistency between most of the palaeomagnetic results from both sides of the MJF, (2) questions on secular variations and local rotations, these three poles may not be applied to Jiaodong Peninsula-scaled tectonics. Consequently, the good consistency among all poles from the Jiaodong Peninsula (except for three discussed above) illustrate that there is no significant palaeomagnetic evidence for a relative rotation or movement within the Jiaodong Peninsula and therefore, a Cretaceous age-mean palaeomagnetic pole has been calculated at λ (K 1-2 ) = 84.7°N, Φ (K 1-2 ) = 191.3°E, A 95 = 5.3° with n = 12 (Table 4 and Fig. 7a). This observation confirms also the Cretaceous stationary kinematic nature of China blocks (NCB and SCB; [START_REF] Enkin | A Cretaceous pole from South China and the Mesozoic hairpin turn of the Eurasian apparent polar wander path[END_REF].

Relative motions between rigid blocks and tectonic implications for eastern Asia

Eastern Asia appears as a landmass formed by the welding of several blocks (e.g., North China block (NCB) and the South China block (SCB)) and experienced a multiphase tectonic evolution as argued by geological evidences and palaeomagnetic studies (e.g., [START_REF] Mattauer | Tectonics of Qinling belt: Build-up and evolution of eastern Asia[END_REF][START_REF] Zhao | Paleomagnetic constraints on the collision and rotation of north and south China[END_REF][START_REF] Sun | Indosinides in China and the consumption of eastern paleotethys[END_REF][START_REF] Enkin | Paleomagnetic constraints on the geodynamic history of the major blocks of China from the Permian to the Present[END_REF][START_REF] Okay | Tectonics of an ultrahigh-pressure metamorphic terrane: The Dabieshan-Tongbaishan orogen[END_REF][START_REF] Gilder | Timing of the north-south China collision from new middle to late Mesozoic paleomagnetic data from the North China Block[END_REF][START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF][START_REF] Zhai | Mesozoic sub-continental lithospheric thinning under eastern Asia[END_REF]. During Late Mesozoic times, this region was marked by extensional tectonics and lithospheric thinning (e.g., [START_REF] Menzies | Palaeozoic and Cenozoic lithoprobes and the loss of >120 km of Archean lithosphere, Sino-Korean Craton, China[END_REF][START_REF] Davis | Mesozoic deformation and plutonism in the Yunmeng Shan: A Chinese metamorphic core complex north of Beijing, China[END_REF][START_REF] Ren | Late Mesozoic and Cenozoic rifting and its dynamic setting in Eastern China and adjacent areas[END_REF][START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF][START_REF] Lin | Late Mesozoic extensional tectonics in the North China Block: a crustal response to subcontinental mantle removal?[END_REF][START_REF] Zhai | Mesozoic sub-continental lithospheric thinning under eastern Asia[END_REF][START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF]. The knowledge of block movements coupled with geological evidence is of prime importance to better understand their links with continental extension. Indeed, relative motions between blocks seem to play a major role in the development of extensional crustal structures like basins or Metamorphic Core Complexes (MCCs; e.g., [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF][START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF]. For instance, relative Cretaceous CW rotation (22.5 ± 10.2°) of the Eastern Liaoning-Korean block (ELKB) with respect to the NCB may be related to the northeast China and southeast Russia sedimentary basins development (e.g., Songliao, Xialiaohe basins) and alternatively, to the South Liaodong and Yiwulüshan MCCs exhumation, [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF][START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF]. Comparison of palaeomagnetic results from the Jiaodong Peninsula with the surrounding blocks may help to recognise the relative movement and tectonic history of this region (e.g., [START_REF] Gilder | Tectonic evolution of of the Tancheng-Lujiang (Tan-Lu) fault via Middle Triassic to Early Cenozoic palaeomagnetic data[END_REF][START_REF] Huang | Palaeomagnetism of Cretaceous rocks in the Jiaodong Peninsula, eastern China: Insights into blocks rotations and neotectonic deformation in eastern Asia[END_REF].

Poles used to discuss the relative motions between Jiaodong Peninsula (JP), Eastern Liaoning-Korean block (ELKB) and North China block (NCB) are presented in Table 4 and Figure 8a.

Palaeomagnetic studies in the ELKB yield five poles for the Early Cretaceous (K 1 ) and Upper Cretaceous (K 2 ) which exhibit consistent palaeomagnetic directions (Figs. 8a and8b).

Thus a Cretaceous (K 1-2 ) pole for ELKB was computed at λ (K 1-2 ) = 60.4°N, Φ (K 1-2 ) = 202.4°E, A 95 = 8.5° with n = 5 (see [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF] for details). Palaeomagnetic data available for the NCB, particularly for Cretaceous age, are compiled in several syntheses (e.g., [START_REF] Enkin | Paleomagnetic constraints on the geodynamic history of the major blocks of China from the Permian to the Present[END_REF][START_REF] Gilder | Timing of the north-south China collision from new middle to late Mesozoic paleomagnetic data from the North China Block[END_REF][START_REF] Yang | New Mesozoic apparent polar wander path for south China: Tectonic consequence[END_REF]. One of their main conclusions is that NCB experienced a stationary period during the Cretaceous and did not experience a palaeomagnetically observable internal relative displacement since the Early Cretaceous (see Table 4 and Fig. 8b for their small confidence error and coherent declination, respectively). The NCB Cretaceous (K 1-2 ) pole used for the following comparison is at λ (K 1- aligned on the small circle centred on the sampling site (JP), indicating the negligible latitudinal motion among them since the Cretaceous. The detailed comparison between the JP pole and the NCB poles reveals a statistically insignificant angular difference of 5.3° ± 7.4°.

This angular difference presents a relative latitudinal motion of -0.1° ± 5.9° and a relative rotation of -6.8° ± 9.5° between these two tectonic units at the geographic reference of 37°N and 121°E (Fig. 8a). It infers that the JP did not experienced any significant movement with respect to NCB since Cretaceous though some structures may be locally reactivated by Cenozoic tectonics due to India-Asia collision, implying that the JP can be considered as a rigid part of NCB at least since Cretaceous. Our results are in accordance with previous study of [START_REF] Gilder | Tectonic evolution of of the Tancheng-Lujiang (Tan-Lu) fault via Middle Triassic to Early Cenozoic palaeomagnetic data[END_REF]. Moreover, located to the east of the Tan-Lu fault, the JP shows the consistent palaeomagnetic results with those from the western part, indicating that no any significant latitudinal displacement has taken place along the Tan-Lu fault since Cretaceous.

In other words, our results indicate that the sinistral motion along the Tan-Lu fault involved by some authors (e.g., [START_REF] Xu | Formation and evolution of the Tancheng-Lujiang wrench fault system: A major shear system to the northwest of the Pacific Ocean[END_REF] should have occurred prior to Early Cretaceous.

Besides, the comparison of the ELKB pole with that of the NCB exhibits a significant angular difference of 19.1° ± 10.0°, corresponding a significant relative rotation 25.9° ± 13.5°, but an insignificant relative latitudinal movement of 2.7° ± 8.0° at the geographic reference of 42°N and 125°E. [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF] proposed that this CW rotation of the ELKB with respect to the NCB induced the development of northeast China and southeast Russia basins (Fig. 8b). [START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF] put forward the role played by the South Liaodong and the Yiwulüshan MCCs to accommodate the CW rotation of the ELKB (Fig. 8b). Based on available absolute time constraints of footwall mylonitic rocks within South Liaodong and Yiwulüshan MCCs at 130-110 Ma, Liu et al. (2005) proposed that most of the CW rotation took place during Early Cretaceous times. Such an accommodation by MCCs exhumation of crustal blocks rotation has been also recognised in the Aegean region [START_REF] Brun | Kinematics of the Southern Rhodope Core Complex (North Greece)[END_REF]. [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF] proposed that rotation (Euler) pole for ELKB with respect to NCB lies on the southern border of Bohai Bay and immediately to the northwest of the Jiaodong Peninsula (Fig. 8b). Amount of crustal extension thus increases progressively from the Bohai Bay toward the northeast. In the Jiaodong Peninsula, the Guojialing synkinematic pluton was emplaced coevally with ELKB CW rotation (i.e. ~124 Ma, Charles, 2010;[START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF] and is located very close to the rotation pole (Figs. 7b and8b). It is noteworthy that this synkinematic pluton reveals a weaker amount of crustal stretching compared to MCC exhumation, which can explain the insignificant rotation within the Jiaodong Peninsula during Late Early Cretaceous.

This new palaeomagnetic study reveals that the Cretaceous extension in the whole

Eastern Asia area appears heterogeneously expressed in time and in space but the proposition of a geodynamic model for this event is beyond the scope of the paper. For instance, geological structures observed and described in the Liaodong and Jiaodong Peninsulas do not reveal the same strain amount undergone by the crust through time (e.g., MCC, synkinematic plutons or sedimentary basins). The ELKB (including Liaodong Peninsula), which has suffered a significant CW rotation with respect to NCB, exhibits a Late Early Cretaceous MCCs exhumation (South Liaodong and Yiwulüshan, ~120-110 Ma; [START_REF] Darby | The newly discovered Waziyu metamorphic core complex, Yiwulüshan, western Liaoning Province, NW China[END_REF][START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF][START_REF] Yang | Rapid exhumation and cooling of the Liaonan metamorphic core complex: Inferences from 40 Ar/ 39 Ar thermochronology and implications for Late Mesozoic extension in the eastern North China Craton[END_REF][START_REF] Lin | Mesozoic extensional tectonics in Eastern Asia : The South Liaodong Peninsula Metamorphic Core Complex (NE China)[END_REF], synkinematic pluton emplacement (Laohushan, ~122 Ma;[START_REF] Lin | Polyphase Mesozoic tectonics in the eastern part of the North China Blocks: insights from the Liaoning Peninsula massif (NE China)[END_REF] and basin infilling [START_REF] Lin | Tectonic implications of new Late Cretaceous paleomagnetic constraints from Eastern Liaoning Peninsula, NE China[END_REF] coeval with the rotation. The Jiaodong Peninsula is characterised by the exhumation of the Linglong MCC during Early Cretaceous times (~143-134 [START_REF] Ma | The comprehensive interpretation report of the gravity and magmatic data in Shandong Province[END_REF][START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF][START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF], then followed by syntectonic emplacement of granitic plutons (e.g, Guojialing pluton, ~124 Ma).

No rotation has been documented for Jiaodong Peninsula since the late Early Cretaceous and coeval features can be regarded as simple unstrained and intrusive plutons such as the Weideshan and the Haiyang plutons (~118 Ma ;[START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF][START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF]. Further investigations are clearly needed to derive a reliable proposition for the mechanisms responsible of the Late Mesozoic extensional event in the whole eastern Asia.

Conclusion

Cretaceous kinematic evolution of the Jiaodong Peninsula closely related to the Mesozoic extension of eastern Asia has been palaeomagnetically studied on two granitic plutons whose internal structure (i.e. AMS data) and structural relationships with the country rocks are well observed. Furthermore, U/Pb and 40 Ar/ 39 Ar radiochronologic methods were performed to accurately constrain the magnetic remanent age and should be regarded as a major feed-back advantage in using granitic rocks for palaeomagnetic study. The multidomain (MD) magnetite has been characterised as the principal magnetic remanent carrier with probably small amount of maghemite and hematite. Cooling rate estimates from geochronological data yielded a magnetic remanent age of ~116 ± 2 Ma, during a relatively fast cooling between ~800°C and ~300°C. The observations of the solo normal magnetic polarity for this palaeomagnetic collection and of the consistent magnetic remanent age with the Cretaceous Normal Superchron (CNS, [START_REF] Gradstein | International Commission on Stratigraphy (ICS)[END_REF] argue that the characteristic magnetic remanence may be considered as primary. Detailed field observations on the contact between the pluton and country rocks and available AMS data [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF] reveal neither any subsequent deformation, nor the rotation along a horizontal axis of the plutons.

Furthermore, twenty palaeomagnetic sites from these two plutons show indistinguishable magnetic directions. Among 12 out 15 Cretaceous palaeomagnetic poles, including two from this study, from the Jiaodong Peninsula and on both sides of Tan-Lu and Muping-Jimo faults are statistically consistent, therefore, the deformation within the peninsula may be negligible.

The remaining three excluded poles are probably due to the absence of secular variation average or/and the vicinity of fault zones for the palaeomagnetic sampling sites, so they can not be applied to the Peninsula-scaled tectonics. Comparison between the Cretaceous palaeomagnetic results of the Jiaodong Peninsula and the NCB data set reveal no significative difference. This result indicates that the Jiaodong Peninsula is rigidly part of NCB since, at least, the Cretaceous as suggested by [START_REF] Gilder | Tectonic evolution of of the Tancheng-Lujiang (Tan-Lu) fault via Middle Triassic to Early Cenozoic palaeomagnetic data[END_REF]. In addition, these new palaeomagnetic results confirm that the relative CW rotation of Eastern Liaoning-Korea Block (ELKB) with respect to NCB does not affect the Jiaodong Peninsula. This palaeomagnetic study revealed that the extensional event is heterogeneously expressed in time and space in the Jiaodong Peninsula. For instance, this extension is characterised by Early Cretaceous exhumation of MCC (i.e. Linglong; ~143-134 [START_REF] Ma | The comprehensive interpretation report of the gravity and magmatic data in Shandong Province[END_REF][START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF][START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF] followed by emplacement of synkinematic granitic pluton (i.e. Guojialing; 124 Ma) and finally late Early Cretaceous intrusion of unstrained granodioritic plutons (i.e. Weideshan and Haiyang; ~118 Ma). Conversely, in the Liaodong Peninsula, the extension is marked by the opening of the Songliao basin coeval with the emplacement of granitic plutons (e.g., Jiuliancheng pluton, ~156 Ma; [START_REF] Wu | Initial constraints on the timing of granitic magmatism in North Korea using U-Pb zircon geochronology[END_REF], then is followed by the exhumation of MCCs (South Liaodong and Yiwulüshan MCCs, ~120-110 Ma; [START_REF] Yang | Rapid exhumation and cooling of the Liaonan metamorphic core complex: Inferences from 40 Ar/ 39 Ar thermochronology and implications for Late Mesozoic extension in the eastern North China Craton[END_REF][START_REF] Lin | Mesozoic extensional tectonics in Eastern Asia : The South Liaodong Peninsula Metamorphic Core Complex (NE China)[END_REF] which accommodated CW rotation of the ELKB.

separation. Pr. Stuart Gilder and an anonymous reviewer are both greatly acknowledged for their helpful comments to improve our manuscript.

ISOPLOT 3.1 [START_REF] Ludwig | User's manual for ISOPLOT/EX, version 3[END_REF]. Errors of individual analyses are based on counting statistics and are given at the 1σ level. The detailed procedure can be found in [START_REF] Xie | In situ simultaneous determination of trace elements, U-Pb and Lu-Hf isotopes in zircon and baddeleyite[END_REF]. 15 runs. System blanks were evaluated every three experiments. They were in the range of 2 x 10 -12 cm 3 for 40 Ar, 1.5 x 10 -14 cm 3 for 39 Ar, 3 x 10 -15 cm 3 for 38 Ar, 6 x 10 -14 cm 3 for 37 Ar and 7

x 10 -15 cm 3 for 36 Ar.

For all analyses, mass discrimination was calculated on the basis of an 40 Ar/ 36 Ar ratio of 289.0 ± 2.1 for the MAP 215-50 mass spectrometer. For each individual age reported in Table 3, usual isotope corrections including blanks, mass discrimination, radioactive decay of 37 Ar and 39 Ar and irradiation-induced mass interference were applied. Errors reported for individual steps take into account the errors introduced by these corrections as well as the analytical errors on signals. The uncertainty on the J-factor was propagated in the calculation of the error on the total age of each sample, equivalent to a K/Ar age. All ages are given at the one standard deviation (1σ) confidence level.

Zijderveld, J.D.A., 1967. AC demagnetisation of rocks: analysis of results. In: Collinson, D.W., Creer, K.M., Runcorn, S.K. (eds), Methods on Paleomagnetism. Elsevier, 245-286. [START_REF] Mazukabzov | Structure and age of the metamorphic core complex of the Burgutui ridge (Southwestern Transbaikal region)[END_REF][START_REF] Donskaya | Age and evolution of late Mesozoic metamorphic core complexes in southern Siberia and northern Mongolia[END_REF], ED:

Figure captions

Ereendavaa [START_REF] Daoudene | The Ereendavaa Range (north-eastern Mongolia): an additional argument for Mesozoic extension throughout eastern Asia[END_REF], GD: Gudaoling [START_REF] Charles | Mechanisms of Mesozoic continental extension in East Asia[END_REF], HG: Hongzhen [START_REF] Luo | The research of metamorphic core complex of Hongzhen-Anqing area, Lower Yangtze[END_REF], LL: Linglong [START_REF] Charles | Metamorphic Core Complex vs. Synkinematic pluton in continental extension setting: Insights from key structures (Shandong Province, eastern China)[END_REF], LS: Lushan [START_REF] Lin | Tectonics of SE China, new insights from the Lushan massif (Jangxi Province)[END_REF], LU: Luotian [START_REF] Eide | 40 Ar/ 39 Ar geochronology and exhumation of high-pressure to ultrahigh-pressure metamorphic rocks in east-central China[END_REF][START_REF] Hacker | U/Pb zircon ages constrain the architecture of the ultrahigh-pressure Qinling-Dabie orogen, China[END_REF][START_REF] Faure | Tectonics of the Dabieshan (E. China) and possible exhumation mechanism of ultra high-pressure rocks[END_REF], SL: South Liaodong [START_REF] Liu | The Liaonan metamorphic core complex, southeastern Liaoning Province, North China: A likely contributor to Cretaceous rotation of eastern Liaoning, Korea and contiguous areas[END_REF], YA: Yagan-Onch Hayrhan [START_REF] Zheng | The apparent polar wander path for the North China Block since the Jurassic[END_REF][START_REF] Webb | Occurrence, age, and implications of the Yagan-Onch Hayrhan metamorphic core complex, southern Mongolia[END_REF], YU: Yunmengshan [START_REF] Davis | Mesozoic deformation and plutonism in the Yunmeng Shan: A Chinese metamorphic core complex north of Beijing, China[END_REF], YW: Yiwulüshan [START_REF] Darby | The newly discovered Waziyu metamorphic core complex, Yiwulüshan, western Liaoning Province, NW China[END_REF], ZA: Zagan [START_REF] Donskaya | Age and evolution of late Mesozoic metamorphic core complexes in southern Siberia and northern Mongolia[END_REF].

Tables Table 1:

  Palaeomagnetic constraints from granodioritic plutons (Jiaodong Peninsula): New insights on Late Mesozoic continental extension in eastern Asia. Nicolas Charles, Yan Chen, Romain Augier, Charles Gumiaux, Wei Lin, Michel Faure, Patrick Monié, Flavien Choulet, Fuyuan Wu, Rixiang Zhu, et al. To cite this version: Nicolas Charles, Yan Chen, Romain Augier, Charles Gumiaux, Wei Lin, et al.. Palaeomagnetic constraints from granodioritic plutons (Jiaodong Peninsula): New insights on Late Mesozoic continental extension in eastern Asia.. Physics of the Earth and Planetary Interiors, 2011, 187 (3-4), pp.276-291. 10.1016/j.pepi.2011.05.006. insu-00599546

Figure 4 .(

 4 Figure 4. Curie temperature measurements for all five measured samples show sharp drop of

Figures

  Figures 5c and 5d present representative behaviours of measured specimens for

  (λ = 80.1°N, Φ = 180.1°E, A 95 = 9.0° with N = 9) and Haiyang pluton (λ = 86.5°N, Φ = 161.1°E, A 95 = 8.9° with N = 11), respectively (

  Single crystals of biotite were extracted from sample KH283 and progressively degassed using a laser probe. Minerals separated from these two samples were hand-picked under a binocular microscope directly on the rock sample or from the 1000-500 µm fraction after crushing and sieving. Separates were washed in distilled water, acetone and ethanol before the packaging in individual aluminium foil packets for irradiation. Samples were irradiated in the McMaster nuclear reactor (Canada) together with aliquots of MMHb-1 flux monitors (520.4 ± 1.7 Ma;[START_REF] Samson | Calibration of the interlaboratory 40 Ar/ 39 Ar dating standard, MMhb-1[END_REF]. The samples were irradiated in several batches. The effects of interfering reactions with Ca and K were evaluated using the following correction factors obtained during a previous irradiation[START_REF] Mcdougall | Geochronology and thermochronology by 40 Ar/ 39 Ar method, 2 nd Ed[END_REF]: ( 40 Ar/ 39 Ar) K = 0.0156; ( 39 Ar/ 37 Ar) Ca = 0.000651; ( 36 Ar/ 37 Ar) Ca = 0.000254. Single crystals were degassed by means of a defocused continuous 50WCO 2 laser with a beam diameter at least twice the size of the heated grain. After heating and gas cleaning, argon was introduced in a MAP215-250 mass spectrometer with a Nier ion source and a Johnston MM1 electron multiplier, at the Géosciences Montpellier geochronology laboratory. Each analysis involved 5 min for gas extraction and cleaning and 8 min for data acquisition by peak switching from argon mass 40 to 36. Isotopic compositions were estimated by regression on

Figure 1 :

 1 Figure 1: (a) Tectonic sketch of eastern Asia. Sulu UHP B.: Sulu Ultra-High Pressure Belt;

Figure 2 :

 2 Figure 2: (a) and (b) Structural maps of Weideshan and Haiyang plutons with the location of

Figure 3 :

 3 Figure 3: (a) CL images of representative zircons. Shown are locations of analytical points.

Figure 4 :

 4 Figure 4: Results of the magnetic mineralogical investigation for Weideshan and Haiyang

Figure 5 :

 5 Figure 5: Magnetic remanent measurements and palaeomagnetic results from the Weideshan

Figure 6 :

 6 Figure 6: Temperature/time diagram showing the inferred cooling path for both plutons and

Figure 7 :

 7 Figure 7: (a) Equal-area projection of available Cretaceous palaeomagnetic poles for the

Figure 8 :

 8 Figure 8: (a) Equal-area projection of mean Cretaceous poles for Jiaodong Peninsula (JP),

  GPS position of the sampling sites and palaeomagnetic directions for Weideshan and Haiyang plutons. Abbreviations: n, number of samples taken into statistic calculation; N, number of measured samples; D, I, declination and inclination, respectively; k: the precision parameter; α 95 : the radius that the mean direction lies within 95% confidence. The sites in italic are excluded from mean calculations (*: site located close to the pluton border, **: site poorly statistically defined with high α 95 and low k values, ***: site with weak sampling).

  Fig.2a

  Fig.2b

  The Haiyang granodioritic pluton (~480 km²) is located in the southwest part of Jiaodong Peninsula, close to Rushan and Haiyang cities, and east of the Muping-Jimo fault (Figs.1b and 2b). Available Jurassic Rb-Sr ages available for this pluton

	SBGMR, 1991) do not make geological sense since these granodioritic rocks are clearly
	intrusive into Early Cretaceous sediments (i.e. Laiyang Group) leading to consider an
	intrusion age younger than ~126 Ma (consistent age of volcanic rocks of the Laiyang Group;

  Magneticremanence was acquired by the MD magnetite grains during this fast cooling history of the Haiyang intrusion and the age of magnetisation must, therefore, lie between 117.8 ± 2.7 Ma and 115.4 ± 1.2 Ma. Direct dating of magnetic remanence by hornblende 40 Ar/ 39 Ar analysis

whose closure temperature lies in the same temperature range unluckily failed. Assuming a linear decrease of temperature, age data are plotted on a temperature/time (T-t) diagram (Fig.

  Table 4 and Fig. 7a). Fifteen palaeomagnetic poles, including results of this study, are available for the Jiaodong Peninsula in Table 4 and Figures 7a with corresponding declinations presented in Figure 7b. Among them, three poles from Huang et al. (2007) are distinguishable by abnormal northeastward declination with respect to the others (specified in Table 4 by *; see also Fig.

	5.2. Is there Cretaceous rotation(s) within Jiaodong Peninsula?

  Table 2 for details). (d) Biotite 40 Ar/ 39 Ar age spectrum of the Haiyang granodiorite (KH283). Errors are given with 1σ. See Table 3 for details. TFA = Total Fusion Age. bi : biotite.

Table 2 :

 2 Details of U/Pb analyses on zircons of the Haiyang granodioritic pluton after data reduction and common Pb corrections. Uncertainties are at 1σ, including the error related to calibration.

Table 3 :

 3 Details of 40 Ar/ 39 Ar of analyses performed on biotites of the Haiyang granodioritic pluton.

Table 4 :

 4 Compilation of Cretaceous palaeomagnetic data of North and eastern China.

	122°E
	Weihai

Abbreviations: N, the number of sites; Slat (Plat), the latitude of sampling site (pole); Slong (Plong), the longitude of sampling site (pole); A 95 , the radius that mean pole lies within 95% confidence, in degree. The palaeomagnetic poles denoted by * are eliminated from the average.

57.0 60.2 6.3 Mean of both plutons 20 6.6 59.8 51.1 4.6

	Spots	U	Th	Geographic coordinates 232 Th/ 238 U 206 Pb 207 Pb/ 235 U		±1σ	206 Pb/ 238 U	±1σ	207 Pb/ 206 Pb	±1σ	206 Pb/ 238 U
	Site (ppm) (ppm)	Longitude (°E) (ppm)	Latitude (°N)	n/N	D	I	k	α 95	Age (Ma) ±1σ
	7-056/1 498.2 957.9	1.92278	41.4	0.12624	0.0021	0.01880	0.0004	0.04867	0.0005	120 ± 3
	Weideshan pluton 7-056/2 102.7 301.3	2.93445	8.8	0.12176	0.0036	0.01827	0.0004	0.04830	0.0006	117 ± 3
	7-056/3 93.9	223.1	2.37640	8.8	0.13592	0.0044	0.01897	0.0005	0.05195	0.0008	121 ± 3
	KW01* 7-056/4 508.4 825.7	122.420 1.62414	51.3	37.200 0.15009	0/6 0.0027	0.01856	excluded 0.0004	0.05862	0.0006	119 ± 3
	KW10 7-056/5 481.1 1457.8	122.410 3.03045	57.2	37.230 0.14080	4/4 0.0026 11.3 60.5 0.01852	33.6 16.1 0.0004 0.05513	0.0005	118 ± 3
	KW13 7-056/6 370.2 1302.5	122.384 3.51842	44.5	37.273 0.14004	5/5 0.0028 31.0 65.1 0.01805	435.1 3.7 0.0004 0.05626	0.0005	115 ± 3
	KW14*** 7-056/7 307.9 1676.5	122.426 5.44479	43.8	37.296 0.18468	0/3 0.0057	0.01908	excluded 0.0005	0.07020	0.0010	122 ± 3
	KW17* 7-056/8 315.5 1938.1	122.440 6.14211	44.0	37.340 0.12892	0/4 0.0076	0.01796	excluded 0.0006	0.05204	0.0017	115 ± 4
	KW20			122.340		37.250	5/6 359.9 57.7	40.1 12.2
	KW23 able 2			122.275		37.287	5/5	2.2 56.8	79.3 8.6
	KW30			122.330		37.230	3/3	12.6 53.2	35.8 20.9
	KW32			122.462		37.246	4/6	1.8 47.6	17.1 22.8
	KW36			122.501		37.294	3/5	26.1 55.9	44.3 18.7
	KW39*		122.452		37.195	0/6			excluded
	KW40			122.520		37.180	4/6	11.5 73.5	70.8 11.0
	KW45			122.268		37.157	4/4	12.9 74.9	40.7 14.6
	Mean						9	11.3 61.0	59.5 6.7
	Haiyang pluton
	KH50**		121.170		36.850	0/5			excluded
	KH54			121.130		36.870	6/6 337.9 59.3	56.5	9.0
	KH57**		121.159		36.896	0/5			excluded
	KH58**		121.140		36.890	0/5			excluded
	KH60			121.140		36.890	4/6	16.1 49.1	90.3 9.7
	KH67			121.076		36.834	5/6	19.7 64.3	188.8 5.6
	KH68			121.212		36.837	4/5	3.1 59.6	41.2 14.5
	KH69			121.290		36.870	5/5	11.0 49.8	344.2 4.1
	KH70			121.267		36.893	5/5 359.8 53.7	26.7 15.1
	KH74			121.220		36.930	4/6	10.3 47.3	110.0 8.8
	KH281		121.379		36.911	6/6	40.7 73.8	61.7 8.6
	KH286		121.369		36.858	3/5 346.9 56.4	804.1 4.4
	KH290		121.360		36.809	5/6 347.7 70.1	34.3 13.2
	KH294		121.301		36.832	6/6 349.7 53.2	196.2 4.8
	Mean 1.5 Table 1 11

) = 79.5°N, Φ (K 1-2 ) = 198.7°E, A 95 = 5.2° with n = 7 (Table4).Three Cretaceous palaeomagnetic poles from NCB, the Jiaodong Peninsula and Eastern Liaoning-Korea block are presented on Figure8a. At the first order, they are all

Acknowledgements

This study benefited from a grant in the frame of the French national "Failles-Fluides-Flux" project supported by the INSU-CNRS. The field work was also financed by Chinese projects KZCX2-YW-Q05-05-03, NSFC (90714007 and 40730315). The first author benefited a scholarship grant from the French Minister of Education and Research. We present our special thanks to M. Le Goff and B. Henry for their help during magnetic measurements at the Institut de Physique du Globe de Paris, Chen Ke, Chu Yang and Wang Jun for their assistance on the field. P. Benoist (ISTO) is acknowledged for mineral

Appendix A. U/Pb analyses on zircon

Zircon crystals were obtained from crushed rock using a combination of heavy liquid and magnetic separation techniques at the Institut des Sciences de la Terre d'Orléans (ISTO).

Individual crystals were hand-picked and mounted in the epoxy resin. Experiments were carried out at the (Q)-ICP-MS laboratory of the Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing. An Agilent 7500a quadruple (Q)-ICPMS was used for simultaneous determination of zircon U/Pb age and trace elements with an attached GeoLas Plus 193 nm excimer ArF laser-ablation system. All the gas lines were purged for over 1 h prior to each analytical session to reduce Pb on the surface to 204 Pb <50 cps in the gas blank.

The measurements were carried out using time resolved analysis operating in a fast, peak hopping sequence in DUAL detector mode. Rawcount rates for 29 Si, 204 Pb, 206 Pb, 207 Pb, 208 Pb, 232 Th and 238 U were collected for age determination. 202 Hg is usually <10 cps in the gas blank, therefore the contribution of 204 Hg to 204 Pb is negligible and is not considered further. The integration time for four Pb isotopes was ~60 ms, whereas for the other isotopes (including 29 Si, 232 Th and 238 U), it was 30 ms. Data were acquired over 30 s with the laser off and 40 s with the laser on, giving ca. 340 (= 170 reading/replicate x2 sweeps) mass scans for a penetration depth of ca. 20 µm. U, Th and Pb concentrations were calibrated by using 29 Si as internal standard and NIST SRM 610 as external standard. 207 Pb/ 206 Pb and 206 Pb/ 238 U ratios were calculated using GLITTER 4.0 [START_REF] Jackson | The application of laser ablation inductively coupled plasma-mass spectrometry (LA-ICP-MS) to in situ U-Pb zircon geochronology[END_REF], which was then corrected using the zircon 91500 as external standard. The 207 Pb/ 235 U ratio was calculated from the values of 207 Pb/ 206 Pb and 206 Pb/ 238 U. Common Pb was corrected according to the method proposed by [START_REF] Andersen | Correction of common lead in U-Pb analyses that do not report 204 Pb[END_REF]. The 206 Pb/ 238 U weighted ages and concordia plot were processed using


Julien Bonnel
email: julien.bonnel@gipsa-lab.grenoble-inp.fr

Barbara Nicolas
email: barbara.nicolas@gipsa-lab.grenoble-inp.fr

Jerome I Mars
email: jerome.mars@gipsa-lab.grenoble-inp.fr

Dominique Fattaccioli
email: dominique.fattaccioli@dga.defense.gouv.fr

Source localisation in deep water using waveguide invariant distribution

When recorded on an horizontal array, the acoustic intensity of a broadband source presents a striation pattern due to interferences between modes. This pattern is a robust feature of waveguide propagation and is described by a scalar parameter called waveguide invariant. In classical shallow water configurations, this invariant is close to 1 and has been used to perform source localization or to study the environment. For deep water waveguides, the invariant varies and no straightforward methods exist to take benefit of it. Indeed, in deep water many modes contributes to the acoustic intensity and the invariant depends on different modal contributions at different frequencies. This paper proposes a study of the invariant in deep water in a propagation channel. It is viewed as a three dimensional distribution depending on frequency, on central mode and on number of modes considered. Then, it is shown that the invariant distribution can be used to perform range estimation of the source in deep water. To compute it, a priori knowledge of the environment is required. However, the proposed localization method is robust to realistic errors on the environment knowledge as it uses mean of the invariant distribution.

Introduction

In oceanic waveguides, an acoustic field can be viewed as a coherent sum of modes. It is described by the modal propagation theory [START_REF] Jensen | Computational ocean acoustics[END_REF]. When one considers a broadband source and a field recorded on an horizontal array, interferences will naturally appear in the domain r -f (where r represents the radial distance between source and frequency and f the frequency). The formation of the interference structure and their localization in the r -f plane depends on the waveguide properties : the sound speed profile in the water and sediments and the source localization. This interference structure is summarized by a quantity β called waveguide invariant and introduced by Chuprov [START_REF] Chuprov | Interference structure of a sound field in a layered ocean[END_REF]. It is a single scalar parameter that takes into account the dispersive nature of the oceanic waveguide. Indeed, when plotted versus range r and pulsation f , the sound field intensity I(r, f ) exhibits a striation patterns. The slope of this striations df dr is related to the frequency and the source/receiver range through the waveguide invariant by [START_REF] Brekhovskikh | Fundamentals of ocean acoustics[END_REF]

β = r f df dr . ( 1
)
In an isovelocity waveguide with a perfectly reflecting bottom, the waveguide invariant is actually invariant and is equal to 1. For more complicated waveguides with arbitrary sound speed profile, β can vary when distance and/or frequency vary. In shallow water waveguides, the variations of β are often minor, and the waveguide invariant represents an attractive simple feature of the propagation which can even be used in an environment that varies in range and azimuth [START_REF] D'spain | Application of waveguide invariants to analysis of spectrograms from shallow water environments that vary in range and azimuth[END_REF]. It has been applied for various purposes, such as moving the focal range in a time-reversal experiment [START_REF] Hodgkiss | A long-range and variable focus phase-conjugation experiment in shallow water[END_REF], source range estimation [START_REF] Thode | Source ranging with minimal environmental information using a virtual receiver and waveguide invariant theory[END_REF], geoacoustic characterization [START_REF] Heaney | Rapid geoacoustic characterization for limiting environmental uncertainty for sonar system performance prediction. Impact of littoral environmental variability of acoustic predictions and sonar performance[END_REF] or source motion compensation [START_REF] Yang | Motion compensation for adaptive horizontal line array processing[END_REF]. However, it has rarely been con-sidered in deep water waveguides, were only very low frequencies are dispersive.

In this paper, we consider the waveguide invariant for very low frequency propagation (less than 150 Hz) in deep water (more than 1500 m). We first show that it is no longer invariant and has to be considered as a distribution. Then, we use that distribution to device an estimation of the source/receiver range. Finally, we apply this estimation on simulated deep water data.

2 The waveguide invariant in a range independent deep water waveguide

In this section, we will first provide a quick reminder on how the waveguide invariant is computed. Then, we will use it to show that it can be problematic when applied to deep water propagation.

Waveguide invariant calculation

In any range independent waveguide, the acoustic intensity described by modal theory is given by [START_REF] Jensen | Computational ocean acoustics[END_REF]:

I(r, f ) = Q ∑ n ψ m (z s , f )ψ m (z r , f ) e irkrm(f ) √ rk rm (f ) 2 , ( 2
)
where Q is a constant depending on the environment, ψ m (z, f ) is depth function of mode m, r is the source-receiver range and k rm the horizontal wavenumber. The waveguide invariant characterizes the interference pattern for which acoustical intensity is constant. Thus, it is computed by differentiating the acoustical intensity

I(r, f ) = cst ⇒ δf δr = - ∂I ∂r / ∂I ∂f . ( 3
)
To approximate the the quantity ∂I ∂r and ∂I ∂f , one has to assume a functional relationship between the group slowness and the phase slowness. It allows to expand S n g , the group slowness of mode n, in Taylor series at order 1 in the neighborhood of S g , the mean group slowness of a group of modes:

S n g = S g + dS g dS p (S n p -S p ) (4)
where S n p is the phase slowness of mode n and S p the mean phase slowness of a group of modes. Some straightforward algebra allows to obtain the final result [START_REF] Chuprov | Interference structure of a sound field in a layered ocean[END_REF]:

δr δf = - r f dS g dS p , ( 5
)
with waveguide invariant β defined as:

1 β ≡ - dS g dS p . ( 6
)

Analysis of the waveguide invariant using simulated deep water waveguide

Equation ( 4) is the most critical step of the waveguide invariant definition. As we are doing an order 1 Taylor series, we obtain a local result (in term of frequency). Moreover, the mean group and phase slowness of a group of mode have been defined. This brings questions about that group of modes. What is its size? Which modes are considered? For shallow water propagation, these are not critical question as the functional relationship between phase and group slownesses does not change much between different propagating modes. It is not the case for deep water propagation where there is a big number of propagating modes, which have changing properties in term of phase and group slownesses.

To illustrate it, we consider a typical Mediterranean sound speed profile represented in figure 1. Water depth is 2500 m, and there is a propagation channel at D = 100 m. We consider an impulsive source with a flat spectrum between 0 and 130 Hz, and a 238 element horizontal array for reception. Both source and receivers are in the channel, and are separated by a radial distance R = 30 km. Simulation was computed using the simulator MOCTESUMA [START_REF] Cristol | Multistatic acoustic reverberation through compressed modal representations of sonar pulses in shallow marine environments[END_REF][START_REF] Fattaccioli | Modal representations of transient sound pulses in deep and shallow environments, with investigations about detailed space and time correlation of propagated waves[END_REF]. Figure 2 presents the simulated recorded signal on the array in the distance-frequency r -f domain. Note that the first hydrophone of the array has been take as the distance origin r = 0. One can clearly see the striation pattern of the field. Each striation seems to have a constant slope. We will see that it is not strictly true.

To show the variations of β, phase and group slownesses of modes 30 and 35 are computed in this environment for frequencies from 35 to 130 Hz. Figure 3 presents the functional relationship between these slownesses. One can see that the relationship depends on the frequency and on the mode number. It confirms the idea that, in deep water, waveguide invariant is a local notion ; and that as pointed in [START_REF] Brekhovskikh | Fundamentals of ocean acoustics[END_REF], equation ( 5) is valid only for a group of modes.

As in [START_REF] Rouseff | Modeling the waveguide invariant as a distribution[END_REF], we will now consider the waveguide invariant as a distribution. The idea of our distribution is to take into account the approximations brought by the Taylor expansion of the group slowness in equation ( 4). As mean slownesses are computed over a group of modes, we consider β as a 3 dimensional distribution depending on :

• the number of modes over which the mean slownesses are computed,

• the number of the central mode of this group,

• the frequency at which the invariant is considered.

Next section will show how to compute the invariant as a 3 dimensional distribution, and that it can easily be applied to perform range estimation using an horizontal array of receivers in deep water. 3 The source localization scheme

Computation of the invariant as a 3 dimensional distribution

To compute the invariant distribution, we apply the following method:

1. Follow a ridge line along a striation (local maximum or minimum).

2. Interpolate this curve with a second order polynomial. This interpolation smooth the curve to get ride of the sampling stair effect and of the eventual noise. The invariant is now noted β(N, m, f ). The quantity N is the number of modes in the group that we use to compute β(N, m, f ), m is the central mode of this group, and f is the frequency at which it is computed. Note that thanks to interpolation, a single distance r corresponds to each frequency f .

Range estimation

By combining equations ( 5) and ( 6), an estimated range R(N, m, f ) can be computed for each value of β(N, m, f ):

R(N, m, f ) = β(N, m, f ) f δr δf , ( 7
)
where δr δf is the slope of the striation computed for every frequency/distance pair thanks to the interpolation.

The final estimated range R is obtained by averaging the distribution R(N, m, f )

R = mean R(N, m, f ), ( 8
)
for every N , m, f verifying

• f min < f < f max , with f min and f max the minimum and maximum frequencies of the striation,

• N is odd (so that the considered group of mode has a central mode) and 3 < N < M max , where N max is the highest mode at frequency f min ,

• N -1 2 < m < N max -N -1 2 .

Application to simulated data

The source localization scheme is applied on the simulated data presented in section 2.2. We follow the ridge between 89 and 109 Hz. Figure 4 presents the result. We see that even for that simple simulation, the slope of the striation -and thus the invariant-is not constant. We compute the invariant distribution for each frequency of the green curve. Figure 5 presents a 2 dimensional slice of this distribution, with parameter N constrained to 7. Figure 6 presents the corresponding estimated range distribution. As the slope of the considered striation is not strictly constant, the link between that two distributions is not linear. The range is estimated by averaging the full 3 dimensional distribution, and the result is R = 32 km. This is a satisfying result as

R = R ± 6.6%. (9)
The same treatment has been applied for several striations, and also for another dataset presented in figure 7. This dataset was computed in the same Mediterranean environment but with a 40 km range. These are good results as each time the estimation error is less than 10%. The last estimation (striation around 60 Hz in the 30 km dataset) is voluntary not as good as the others. Indeed, invariant computation based on equation 3 assumes to follow a constant line striation. That is impossible in a realistic case because of noise and hydrophone calibration. To overcome this, the method follow a maximum (or minimum line) which is assumed to be locally constant. This condition is less respected for this last estimation than for the others, which explains that the result is less accurate than the others. As the proposed method is based on a 3 dimensional mean of the invariant distribution it should be robust to environmental mismatch. To test it, the method is applied on simulated data coming from two new datasets, but the invariant distribution is computed using the celerity profile presented in figure 1. This corresponds to two realistic scenario where there is an error on the environment:

• the propagation channel is 10 m shallower than what we think.

• there is an undetected surface duct profile, resulting in a first propagation channel at depth 50 m.

Each time, the range estimation is good and the estimation error is about 5%. The error of the environment have been filtered out by the 3 dimensional mean of the invariant.

Conclusion

This article proposes to consider the waveguide invariant as a 3 dimensional distribution, depending on the frequency and the modes that are used to compute it. It is mainly useful for deep water environment. Indeed, when recorded on an horizontal array, the acoustical intensity presents an interference pattern, and the slope of the striation is not constant for deep water propagation. Then, this article proposes a simple method to estimate the source distance from an horizontal array based on the invariant. This method is validated on simulations. It requires knowledge of the environment. However, as it is based on a 3 dimensional mean of the waveguide invariant, it is robust to realistic errors on environmental knowledge.

Figure 1 :

 1 Figure 1: Sound speed profile used for Mediterranean deep water simulation

Figure 2 :Figure 3 : 3 .

 233 Figure 2: Simulated signal in the Mediterranean environment for a 30 km range

Final range estimations are :•

  Dataset with r = 40 km : striation around 105 Hz : r = 38.0 km, striation around 95 Hz : r = 42.2 km, striation around 95 Hz : r = 41.2 km.

Figure 4 :Figure 5 :

 45 Figure 4: Ridge line along a striation from the Mediterranean data : raw data (blue) and interpolation (green)

Figure 6 :Figure 7 :

 67 Figure 6: Estimated range distribution slice N = 7 for the Mediterranean data and the striation around 105 Hz

Acknowledgments

The authors would like to thanks Thales Underwater System and particularly X. Cristol and B. Chalindar for provinding the simulated dataset computed with MOCTESUMA.