Giusy Emmanuele
email: gi.emmanuele@rfi.it

Massimiliano Ciaffi

Omar Garcia Crespillo

Alessandro Neri

Alessia Vennarini

Agostino Ruggeri

Juliette Marais

Susana Herranz De Andres

Jorge Ignacio

Iglesias Diaz

Daniel Molina Marinas

Agostino Ruggeri3

Ricardo Campo Cascallana

Antonio Águila Martínez-Casariego

Jose Conrado

Martinez Acevedo

Fabio Senesi

Salvatore Sabina

User Needs for the Development of New Methodologies and R&D Tools for Building a Railway Digital Map and for the Experimental Performance Evaluation of On-Board Subsystems

Keywords: Digital Map, Ground Truth, ERTMS, Railway Survey

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the last decade, many R&D projects have investigated the use of EGNSS navigation and sensors' technologies such as IMU, LIDAR and Camera in Automatic Train Protection (ATP) systems with the objective of improving the performance of the on-board train position and odometry functions and, possibly, reducing the costs of the whole signalling system. The use of Digital Maps in ATPs or ATOs is also assumed to play a crucial role in the evolution of CCS systems. The performance analysis of signalling systems is in most of previous work analysed with project dependent mechanisms and procedures that identify key points along each train trajectory as one dimensional trustable reference locations to compute the position errors. Furthermore, other types of on-board subsystems (e.g. braking or tilting subsystem1 ) require the use of "reference ground truth data" to perform verification activities.

Unfortunately, there is no agreed methodology for building "reference ground truth data" in railway applications; each project used their own mechanisms and procedures. In 2018, in the context of a R&D GSA project, UNISIG suppliers defined and developed a first Ground Truth methodology [START_REF] Sabina | Satellite Technology for Advanced Railway Signalling[END_REF]. It required the use of the track database and the availability of absolute and relative reference measurement systems associated with the lines for which the Ground Truth had to be built and, therefore, representing a huge investment. Indeed, the need of a track database is a very demanding requirement and the survey procedures for building and maintaining such databases are very expensive and time consuming.

In general, the methodologies for building a Ground Truth investigated up to now also suffers of other two limitations, i. The R&D RAILGAP project has three main objectives. The first one is the development of an innovative Ground Truth methodology and related toolset that overcome the above described constraints and limitations; they are based on the post-processing of recorded measurement data and provide reference ground truth data and related accuracies for different types of physical entities such as position, velocity, acceleration, and attitude. This Ground Truth does not require (a) installation of trackside equipment or modifications to existing trackside signalling subsystems and (b) a-priori knowledge of track databases. The second objective is to perform field surveys by using measured physical entities collected with trains during commercial services for building and maintaining Digital Maps. The third one is the use of Artificial Intelligence (AI) techniques to off-line process big amount of collected data to allow the extraction of the information required for building and maintaining Digital Maps. The RAILGAP measurement system used for both methodologies is composed of COTS sensors like GNSS antenna and receivers, cameras, IMUs, and LIDARs. This paper presents the user needs related to both methodologies and the expected benefits coming from the R&D RAILGAP project.

Ground Truth User Needs

Ground truth provides information on the knowledge of the truth of a measured or observed entity related to a specific application. In a measurement process or during a research activity, the knowledge of the truth value to be measured or of the truth about the hypothesis to be verified plays a key role. This truth knowledge is named "Ground Truth". This Ground Truth is obtained or computed by gathering the proper objective (provable) data, named "reference ground truth data" or simply "reference data". In the context of the R&D RAILGAP project, the Ground Truth concept has been defined as:

"Set of georeferenced data with known accuracy, precision, availability and reliability, built by means of a well described process, to be considered (a) a stable and true reference, suitable for the purpose of relative comparison and validation of other data sources in the railway domain according to established requirements and (b) the basis for developing other railway components such a high accurate Digital Map".

During the first year of the project, we examined different R&D initiatives related to (a) train positioning and odometry functions, (b) surveys of railway areas aimed at getting the geographical data for designing trackside signalling subsystems, and (c) digital maps for ATP and ATO systems. This analysis has allowed the identification of the types of "reference ground truth data" that might be required for performing the quantitative performance assessment of new products (e.g. kinematic sensors, GNSS receivers), subsystems (e.g. Odometry, Train Positioning, Braking, Doors Management), systems (e.g. ERTMS/ETCS, ATO) or survey processes (e.g. those used for designing signalling systems or building Digital Maps). The following examples are used for describing some results of the analysis of the RAILGAP Ground Truth user cases. Let us suppose that a train position function has to be characterized for correctly sizing the safe confidence interval under any possible environmental and operational conditions. This characterization has to consider the different sources of noise, biases and environmental conditions that can impact the performance or safety. Consequently, this activity cannot be only done by laboratory verification tests since experimental field data are essential to evaluate the effects of external influences such as, radio frequency interferences, vibrations, multipath, NLOS, adhesion factors between wheels and rail, and styles of guidance of the train drivers.

To reach this objective, the test facilities have to allow the collection of the measurement data and the execution of a comparative analysis w.r.t. related ground truth data. Thus, there are a Unit Under Test (UUT) which includes the Train Position function and the set of sensors required (e.g. wheel sensors, accelerometers -usually supplier dependent) for estimating the train position and Ground Truth facilities with their own sensors, i.e. IMU, LIDAR, Camera and GNSS Receiver. The UUT directly or indirectly via an external data record unit records the Estimated train Position EP(t) values, timestamped with GPS time, to allow offline post processing. Each EP(ti) value represents the front end position of the train Active Cabin at the time instant ti. On the other hand, the Ground Truth facilities are made up of both on-board and trackside components; the former is the measurement environment responsible for (a) collecting measured data timestamped with GPS time during each train ride and (b) sending them to trackside component; the latter contains tools for building the ground truth, i.e. sequence of reference data GT(tk). Each GT(tk) represents the truth front end position of the Active Cabin that the train had during the train movement at time instant k, projected at the track centreline on the rail plane. The Ground Truth facilities also produce the truth travelled distance values starting from the sequence of GT(tk) reference data. Therefore, the comparative analysis of EP(ts) values stored during a train ride "i" can be done by using the ground truth reference data related to the same train ride "i". Let us suppose that two different field test train rides have been executed, one at day "i" and the second one at day "k"; both started from the same location towards the same final location, and characterized by two different speed profiles. Figure 1 outlines the position errors occurred during the two train rides at the same relative time instants t1, t2 and t3 from the starting time.

Table 1 summarizes such position errors for the simple case of two train rides and three time instants only. The analysis of errors in time can allow the detection of correlations between the position errors and the related occurrence time instants; for example, GNSS technology can present unacceptable performance in some daily time windows due to the occurrence of low DOP values or time-dependent local multipath phenomena. On the other hand, the train position function has to constantly and safely compute the train confidence interval; therefore, the analysis of the position errors at any location has to confirm that the true train position always lie2 within the interval [minimum safe frond end position, maximum safe front end position]. Table 2 reports the position errors occurred during the train rides only at the two locations S1 (e.g. Danger Point location) and S2 (e.g. Imperative Signal location). It is evident that an accurate characterization of the train position function requires a statistical inference analysis based on many train rides, large numbers of time instants and locations so as to define and dimension the Error Models of Train Position and Odometry functions. The Ground Truth facilities have to provide support for these analysis and the Ground Truth methodology has to be defined to automatically recognize the route followed by the train during the field measurement campaigns (i.e. has to discriminate the track) and has to guarantee a robust behaviour with respect to events that influence the train position performance. The RAILGAP Ground Truth methodology and related toolset [START_REF] Railgap | User Requirements Document related to Digital Map and Reference Measurement Values & Procedures[END_REF] provides ground truth data for different types of physical entities such as position, velocity, acceleration, and attitude (i.e. pitch, roll and yaw) and facilities for efficiently supports error and performance analysis on signalling functions or on-board equipment.

Digital Map User Needs

The ERTMS principles for monitoring and controlling the safe movements of trains use two types of location based data [START_REF]ERTMS/ETCS System Requirements Specification[END_REF], location and profile data, which are referred on the track with respect to LRBGs. The design of any ERTMS/ETCS trackside signalling system requires [START_REF]ERTMS/ETCS System Requirements Specification[END_REF] the correct installations of reference points (in terms of balise locations in the track), and the setup of ERTMS/ETCS trackside configurations to dimension the trackside data to be sent to on-board, e.g. distances of fixed signalling targets 3 from LRBGs.

To this end, the trackside design documents would contain geographical data of fixed existing Points of Interests (POIs) with respect to the centrelines of tracks, e.g. signals, joints, toe of points. The geographical data are set 3 A signalling target is defined as the couple {target location and target speed} to which a train must decelerate before reaching the target location [START_REF]ERTMS/ETCS System Requirements Specification[END_REF]. e 3i = e 2i S1 S2 out as the results of trackside surveys (eventually carried out during the construction of the line if still valid) and are composed of two known components, i.e. POI Location and POI Gradient. The former is the POI longitudinal position along the track centreline; whereas, the POI Gradient is the grade or slope of the track segment adjacent to it, and defines the input data for the profile used by an on-board equipment for performing the ERTMS speed and distance monitoring [START_REF]ERTMS/ETCS System Requirements Specification[END_REF]. In consequence, all the POIs have a defined position along the track, according to the trackside surveys while the data regarding the type and location of the signalling targets is transmitted to the on-board constituents via balises located along the track or an RBC. Therefore, the final accuracy of the target location in the ERTMS/ETCS system depends on several factors that goes from the track survey precision, passing through the longitudinal accuracy of each balise location, up to the accuracy of the relative (rolling) distance between a fixed target and the last balise group (i.e. LRBG). With regard to the evolution of ERTMS/ETCS, the use of an ATO subsystem leads a Digital Maps to be a mandatory component. Furthermore, in the context of innovative train localizations that use technologies such as, for example, GNSS receivers eventually aided with an IMU or a combination of IMU and LIDAR, the use of a Digital Map can enable the exploitation of the track geometry for (a) computing a track constrained position solution with better location accuracy [START_REF] Neri | An Analytical Evaluation for Hazardous Failure Rate in a Satellite-based Train Positioning System with Reference to the ERTMS Train Control Systems[END_REF], [START_REF] Crespillo | Bayesian GNSS/IMU tight integration for precise railway navigation on track map[END_REF], (b) determine on which track the train is operating when the line consists of parallel tracks, (c) detecting the train passage over known trackside locations usable for reducing the train confidence interval, and (d) providing continuous input to update the trackside inventory. In consequence, the Digital Map could lead to define new POIs that provide additional information to increase or optimize the system performance. The RAILGAP Digital Map focuses on the description of the railway network infrastructure and addresses topics related to the following information:

• Railway infrastructure elements: in the RAILGAP Project, these elements consider the Commission Regulation (EU) No 1299/2014 [7] recommendations such as track geometry, gradient, curvatures, height of the check rail, cant, gauges;

• Trackside signaling elements: for example, signals, Eurobalises, marker boards, points, buffer stops, level crossings, fouling points;

• Track elements: for example, tunnels, bridges, viaducts, platforms, pole of overhead power line.

The main user needs behind the design and generation of the RAILGAP Digital Map are based on (see Figure 2):

• The post processing of imaging and ranging sensors' records (done by using cameras and LIDAR) aimed at detecting the railway infrastructure elements and estimating their relative positions w.r.t. the train position;

• The combination of the train location derived from sensors' data with the object's relative position and the absolute (geographic) object's location based on previous runs stored in the Digital Map to set/update the absolute position of each infrastructure element. The process of building the Digital Map is an incremental process that improves the accuracies of the Rail Net Elements by increasing the number of runs;

• The post processing of the GNSS records aimed at detecting the presence of local phenomena (multipath, EMI) [START_REF] Crespillo | Local GNSS Threat Detection Methods for Virtual Balise Placement in Railway Applications[END_REF] and estimating their temporal occurrence during a run;

• The combination of the train location derived from sensors measurements with the temporal occurrence of the events associated with the presence of immaterial objects to set/update the absolute (geographic) position of each of them as a function of the train mileage, based on the post processing of the GNSS records.

Conclusion

Even though different R&D projects used LIDAR, IMU and EGNSS, only some of them have done an accurate technology characterization in railway environments. In addition, the Ground Truth facilities developed up to now satisfied specific projects' needs only and required significant investments. Furthermore, no public information is available on methodologies for executing accurate and precise railway surveys and building Digital Maps, by using measured data collected with commercial trains.

The RAILGAP project aims at providing the characterizations of the LIDAR, IMU and EGNSS technologies in railway environment and the definitions of both methodologies (i.e. Ground Truth and Digital Map) to the R&D communities as project public deliverables. In particular, the RAILGAP Ground Truth provides "reference data along with their accuracy" for different types of quantities to be monitored (i.e. position, speed and acceleration) and does not require installation of trackside equipment and the a priori knowledge of the database of the lines.

The RAILGAP project objectives have been conceived from the analysis of the real needs of the project beneficiaries RFI and ADIF in the role of Infrastructure Managers, and Trenitalia in the role of the Railway Undertaking. The project started on Jan 2021; the first year has been focused on the user requirement phase related to Ground Truth and Digital Map and the identification and characterization of the measurement sensors to be installed. The installations of the sensors and the field test campaigns for recording measurements will start on 2022 both in Italy and Spain along with the technologies' characterizations. The last year (2023) will complete the tools' development and their verification.

RFI and Trenitalia have already planned the use of our research outputs during the deployment of the satellitebased Italian ERTMS solution; whereas, ADIF will use the project results for supporting the system requirements phase for Spanish ERTMS applications. In addition, from 2022, Hitachi Rail STS will apply the results of our research to the Hitachi Innovative Train Position under development.

  e. it provides (a) "reference ground truth data" for the train position only and (b) no information about the accuracy of the provided "reference ground truth data". On the other hand, with regard to the design of Digital Maps and trackside signalling subsystems, they are based on the knowledge of two types of data: (a) the longitudinal positions of Points Of Interests (POIs) (e.g. light signals, ETCS markers) with respect to the track centreline and (b) the grades or slopes of the track segments adjacent to them. Therefore, as a first step, such designs require the acquisition or the verification of the data related to POIs (i.e. nominal values of locations and gradients along with their related accuracies) and this causes extra costs and time-consuming surveying activities leading to delays in the commissioning of the lines.

Figure 1 :Table 2 :

 12 Figure 1: Truth and Estimated Travelled Distances and Related Position Errors

Figure 2 :

 2 Figure 2: Methodology for digital map generation, verification and validation

A train capable of tilting the bodies inwards in curves are called tilting trains

According to the required confidence level.

Acknowledgment

The RAILGAP project is funded by the H2020 EUSPA with GA N. 101004129.


Tom Henshall

David L Cook
email: david.cook@erdw.ethz.ch

Marion Garҫon

Maria Schönbächler

High-precision Strontium Isotope Analysis of Geological Samples by Thermal Ionisation Mass Spectroscopy

Keywords: TIMS, strontium isotopes, dynamic measurements

, our dynamic data reveal no 84 Sr variations in terrestrial rocks or standard material. The dataset indicates that, by the analytical procedure employed, 84 Sr variations in extra-terrestrial material larger than ~0.5ε are resolvable.

Introduction

Strontium has four stable isotopes: 84 Sr, 86 Sr, 87 Sr, and 88 Sr. The isotope 87 Sr varies due to ingrowth from the decay of 87 Rb (t1/2=48 Ga) and provides an important chronometer for geological processes that fractionate Rb from Sr [START_REF] Goldschmidt | The Principles of distribution of chemical elements in mineral and rocks[END_REF][START_REF] Hahn | Uber die moglichkeit geologischer altersbestimmungen rubidiumhaltiger mineralien un gesteine[END_REF][START_REF] Hahn | Geologische altersbestimmungen mit der strontiummethode[END_REF][START_REF] Papanastassiou | Initial strontium isotopic abundances and the resolution of small time differences in the formation of planetary objects[END_REF][START_REF] Mcculloch | Primitive 87 Sr/ 86 Sr from an Archaean barite and conjecture on the Earth's age and origin[END_REF]Hans et al., 2013a;[START_REF] Villa | IUPAC-IUGS recommendation of the halflife of 87 Rb[END_REF]. As well as chronological applications, Sr isotopic data are versatile tracers for geological and archaeological provenance studies, such as investigating historical continental weathering and ocean circulation (e.g., Frank, 2000) and forensic tracing of human and animal migration (e.g., [START_REF] Beard | Strontium isotope composition of skeletal material can determine the birth place and geographic mobility of humans and animals[END_REF]. For these reasons, Sr isotopes are widely applied in the fields of cosmochemistry, mantle geochemistry, palaeoceanography, and archaeology. In particular, the 87 Rb-87 Sr chronometer has proved to be a key tool in dating the volatile element depletion of rocky planets and asteroids (e.g., [START_REF] Papanastassiou | Initial strontium isotopic abundances and the resolution of small time differences in the formation of planetary objects[END_REF][START_REF] Mcculloch | Primitive 87 Sr/ 86 Sr from an Archaean barite and conjecture on the Earth's age and origin[END_REF].

Rubidium, a moderately volatile element, is more volatile than Sr (50 % condensation temperatures: Rb=800 K, Sr=1464 K (Lodders, 2003)), which led to the preferential loss of Rb over Sr during condensation and evaporation processes in the nascent solar system and subsequent fractionation of Rb/Sr ratios. In materials that were highly depleted in volatiles, extensive Rb loss led to minimal 87 Sr ingrowth. Accordingly, the timing of fractionation can be gauged by (i) directly measuring the 87 Sr/ 86 Sr ratio for samples devoid of Rb, which yields the initial 87 Sr/ 86 Sr ratio at the time the sample formed or (ii) measuring the 87 Rb/ 86 Sr and 87 Sr/ 86 Sr ratios in the material of interest to determine the initial 87 Sr/ 86 Sr ratio using the isochron approach. The slope of a 87 Rb- 87 Sr isochron provides the time information, while the y-axis intercept provides the initial 87 Sr/ 86 Sr ratio of the reservoir from which the sample originated. This initial 87 Sr/ 86 Sr ratio is important because it provides the root ratio for model age calculations.

Calcium-aluminium-rich inclusions (CAIs) in primitive meteorites are the oldest material identified in the solar system (e.g., [START_REF] Bouvier | The age of the Solar System redefined by the oldest Pb-Pb age of a meteoritic inclusion[END_REF]. They have highly refractory compositions and incorporated little Rb during their formation, which led to minimal 87 Sr ingrowth. Consequently, CAIs provide an estimate for the initial Sr isotopic composition of the solar system (e.g., [START_REF] Gray | The identification of early condensates from the solar nebula[END_REF]Hans et al., 2013a). The initial 87 Sr/ 86 Sr ratio of CAIs represent a baseline for 87 Sr evolution and calculations of model ages for volatile depletion in the inner solar system. Calcium-aluminium-rich inclusions, however, exhibit nucleosynthetic Sr isotopic variations that have been attributed to excesses in the p-process isotope 84 Sr [START_REF] Moynier | Planetary-scale strontium isotopic heterogeneity and the age of volatile depletion of early solar system materials[END_REF][START_REF] Paton | Identification of an 84 Sr-depleted carrier in primitive meteorites and implications for thermal processing in the solar protoplanetary disk[END_REF][START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF]. This nucleosynthetic variation is significant for the interpretation of 87 Sr/ 86 Sr ratios. Due to analytical issues concerning the correction of instrumental mass bias, it is not possible to accurately identify the anomalous Sr isotope using Sr isotopic data and traditional approaches alone. Because 87 Sr contains an unknown radiogenic component from 87 Rb decay, it cannot be used to correct for mass-bias effects. Therefore, 86 Sr and 88 Sr are used for mass-bias correction. Because of this, 84 Sr variations could in fact be due to variation in 88 Sr or 86 Sr propagating through the mass-bias correction and cause an apparent effect on 84 Sr (equation 1). Identifying the exact location of the variation ( 84 Sr or 88 Sr) is of tangible importance, as it may significantly alter our picture of early solar system volatile element evolution during planet formation. Nucleosynthetic 88 Sr excesses require an upward correction to measured 87 Sr/ 86 Sr ratios in CAIs (e.g., Hans et al., 2013a) and, consequentially, an earlier age for volatile depletion (i.e., Rb loss) by several Ma. These factors highlight the importance of a well-tested, reproducible method to measure Sr isotopes at high accuracy and precision.

In previous studies, Sr isotopic analyses of terrestrial rocks by thermal ionisation mass spectrometry (TIMS) yielded different results for dynamic and static acquisition modes. Despite the fact that all studies were performed on Thermo Scientific Triton TIMS instruments, static measurements of BCR-2 and BHVO-2 displayed a consistent negative offset in ε 84 Sr, relative to the NIST SRM 987 Sr standard [START_REF] Moynier | Planetary-scale strontium isotopic heterogeneity and the age of volatile depletion of early solar system materials[END_REF][START_REF] Paton | Identification of an 84 Sr-depleted carrier in primitive meteorites and implications for thermal processing in the solar protoplanetary disk[END_REF][START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF], whereas dynamic analyses did not display an offset (Hans et al., 2013a;[START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF]. Here, we address this discrepancy with farreaching implications for cosmochemistry. We collected an extensive dataset for selected terrestrial samples using both dynamic and static methods. Our samples included a range of terrestrial rocks (BCR-2, BHVO-2, AGV-2, AN-G, and several anorthosites) to test our methods for high-precision Sr isotopic measurements for a variety of matrix compositions. We evaluated pre-existing analytical procedures (Hans et al., 2013a) and adapted the method to improve the low yields for anorthitic sample matrices. We reproduced the consistent offset from the NIST SRM 987 Sr standard for static analyses and demonstrate that the offset is not observed when using a dynamic routine. We also show that instrumental mass-dependent fractionation does not cause the observed offset between static and dynamic values.

Sample Preparation

Reagents and materials

Hydrochloric and nitric acids were purified twice in Teflon stills, hydrofluoric acid once.

Boric acid was prepared by the addition of 99.995% pure Puratronic boric crystals (BH3O3) to 3.5M HNO3 to achieve a molarity of 0.45. Reagent grade H2O2 and Milli-Q grade (18.2 Ω) water (MQ) were used throughout this study.

Selected samples and sample digestion

The NIST SRM 987 Sr standard, four rock standards (the basalts BCR-2 and BHVO-2, the andesite AGV-2, and the anorthosite AN-G), two anorthite separates (AN-1 and AN-2) from anorthosites (from Sierra Nevada (California) and Adamello (Italy), respectively), and one additional whole rock anorthosite (JL-27; from Rum, Scotland) were measured in this study. The whole rock samples were crushed either with a hydraulic press or sizeable hammer and powdered with an agate pestle and mortar. For AN-1 and AN-2, anorthite separates were obtained by crushing, followed by magnetic separation and hand picking under an optical microscope. The anorthosite rock standard AN-G was chosen as a proxy for calcium-aluminium-rich inclusions because of their similar, Ca-rich sample matrices.

For each sample, 30 mg of powder were digested in 15 ml Savillex® vials at 100 o C overnight using a mixture of 0.3 ml HF and 0.1 ml HNO3. The sample mass was limited to 30 mg to prevent overloading the column. The samples were dried at 120 o C and redissolved in 1 ml of concentrated HNO3 and dried. This HNO3 dry down was repeated and was followed by fluxing in 1 ml concentrated HCl overnight. Finally, the samples were dried and redissolved in 2.5 M HCl. If necessary, samples were placed in an ultra-sonic bath before loading onto the ion exchange column.

Ion Exchange Column Chemistry

A two-stage procedure for the separation of Sr from the sample matrix was used and follows the procedure of Hans et al. (2013a) (Table 1). A Teflon column (resin reservoir dimensions: 18 cm length, 5 mm inner diameter) was loaded with 3.7 ml of BioRad ® cation resin AG50W-X8 (200-400 mesh). The resin was cleaned and preconditioned with 10 ml MQ followed by 20 ml 2.5M HCl. The sample was loaded in 1 ml 2.5M HCl and matrix elements were eluted with 17 ml 2.5M HCl before eluting Sr in 9 ml 2.5M HCl. The resin was then cleaned with 20 ml 6M HCl and reused up to 4 times. The sample elution was dried at 70 o C overnight and then redissolved in 0.5 ml of concentrated HNO3. This was dried to a very small drop and 1 ml of 3M HNO3 was added for loading onto the second column.

In order to further purify Sr, the Sr elution from the first column was passed through a Sr spec resin in a Teflon shrink-wrap column (resin reservoir dimensions: 25 mm length, 3 mm inner diameter). The column was filled with 0.2 ml resin, which was cleaned and preconditioned with 4 ml MQ, 4 ml 0.05M HNO3 and 4 ml 3M HNO3. After the sample was loaded, matrix elements were eluted with 4 ml 3M HNO3. Strontium was eluted in 3 ml 0.05M HNO3. The resin was not reused. The Sr fraction was dried at 120 o C, re-dissolved in 1 ml concentrated HNO3, and fluxed at 100 o C for one hour. The sample was left to cool and a few drops of H2O2 were added to destroy organics. Subsequently, the sample was dried at 80 o C. A few drops of 6M HCl were added and dried down again. Yields were >90% and blanks were typically a few hundred picograms. The amount of Sr in the samples ranged from 2 -11 μg and, therefore, no blank correction was necessary. A total of nine full procedural blanks were measured over the course of three years.

AN-G and boric acid

For the AN-G rock standard, an additional step was added to the dissolution procedure. This included boric acid to prevent the formation of fluorides [START_REF] Greenwood | Chemistry of the Elements[END_REF][START_REF] Yokoyama | Evaluation of the coprecipitation of incompatible trace elements with fluoride during silicate rock dissolution by acid digestion[END_REF] and keep Sr in solution, following the procedure outlined by [START_REF] Connelly | A method for purifying Lu and Hf for analyses by MC-ICP-MS using TODGA resin[END_REF]. After sample digestion, the samples were dried and then fluxed in concentrated HNO3 for 24 hours, after which a mixture of 1 ml 0.45 M boric acid in 3.5 M HNO3 and 1 ml concentrated HCl was added and fluxed for 24 hours. Samples were dried and fluxed in 6 M HCl for 24 hours. Samples were then dried and re-dissolved in 1 ml 2.5 M HCl for loading on the cation column. Because boric acid may affect the behaviour of elements on the resin, and therefore the Sr elution curve, a column calibration was carried out. This resulted in the Sr elution being extended to a total of 14 ml, now including 5 ml of the matrix elution (Table 1).

Mass Spectrometry

Filament loading and measurement procedure

All measurements were performed on a Triton Multicollector Thermal Ionisation Mass Spectrometer (TIMS) at ETH Zürich. The purified Sr was taken up in 10 -20 μL of 6M HCl and left to dissolve for an hour. This volume was varied to achieve 1 μL loads that contained ~ 500ng Sr, depending on the Sr content of the sample. One μL of the dissolved sample was combined with 1 μL Ta phosphate activator solution and mixed using a syringe and capillary tubing. The 2 μL solution was then loaded onto the center of a Re filament, which was then glowed to a dull-red for a few seconds until the deposit turned white. Prior to loading, filaments were outgassed at 4 A for two hours and 4.5 A for thirty minutes. All filament loads contained roughly the same amount of Sr, whether the sample material is BCR-2, NIST 987 or other. Processing 30 mg of BCR-2 results in sufficient Sr (~10.4 μg) for 20 filament loads.

The filaments were heated to 3500 mA (~1,500 o C) over two hours. Autofocusing and zoom optics were used to obtain flat and well-centred peaks. Masses from 84 -88 amu were measured with 20V on the 88 Sr signal. The electronic baseline was measured for 30s prior to each block of measurements and subtracted from all signal intensities. Amplifier rotation was used after every block (20 integrations) to average differences in relative amplifier gains. A two-line acquisition scheme (dynamic protocol) was employed in order to cancel out variations in Faraday cup efficiencies. One integration lasted 8.4 seconds. In order for the signal to decay to background levels between lines, delays of 4 seconds and 8 seconds were used after lines 1 and 2, respectively. These delays are sufficient that the signal decays to a level in the Faraday cups at which it does not affect the measurement of isotopes on the following line (Hans et al., 2013b). One two-line integration lasted ~30 seconds and one total sample measurement lasted up to 5 and a half hours. Generally, 15-30 blocks of 20 integrations were obtained during this time and a 2 sigma outlier rejection was applied. Levels of 85 Rb, monitored for interference correction, were negligible due to the efficient chemical separation procedure and the fact that remaining Rb in the filament load evaporates at a lower temperature than that at which Sr isotopes are measured. Sample measurements were interspersed with NIST SRM 987 Sr standard analyses. Sr intensities of standards and samples were corrected for gain, baseline and interferences online. The calculation of static and dynamic ratios was performed offline after measurement following equations (Eq.1 and Eq.2) detailed below. The data are presented as ε 84 Sr, which is defined as the deviation from the average 84 Sr/ 86 Sr ratio of the NIST SRM 987 standard of each analytical session expressed in parts per ten thousand (ε 84 Sr = [( 84 Sr/ 86 Sr )sample/( 84 Sr/ 86 Sr)standard-1] x 10 4 ).

Calculation of static ratios

Two different measurement routines (static and dynamic) were investigated as they have been shown to produce dissimilar results (Hans et al., 2013a,b;[START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF]. For static measurements, the signal intensity of 84,86,88 Sr (and 85 Rb) were analysed simultaneously and instrumental mass bias was corrected using the exponential law and [START_REF] Thirlwall | Long-term reproducibility of multicollector Sr and Nd isotope ratio analysis[END_REF]. Given the cup configuration (Table 2), the static ratio, ! "# %+ "# %& ' ",-,./ 0 , from acquisition line 1 can be written as:

! "# %% "# %& ' (#)* = 8.3752
× < C S0 × G S0 C U0 × G U0 > × < C UX × G UX C U0 × G U0 > b`
Using m84 = 83.91319; m86 = 85.90926 and m88 = 87.905612 [START_REF] Wang | The Ame2012 atomic mass evaluation[END_REF], φ is equal to -1.02337, hence the static ratio scales to ~

c de × f de × c gh × f gh (c ge × f ge ) i
. Assuming that the amplifier gains are properly calibrated and do not vary during a run (i.e. GH1 = GL1 = GH3 = 1), a simple sensitivity test shows that if faraday collector L1 deteriorates faster than faraday collectors H1 and H3 so that %& ' ratio will be shifted by -100 ppm (see [START_REF] Garçon | Factors influencing the precision and accuracy of Nd isotope measurements by thermal ionization mass spectrometry[END_REF] for more details on the calculation).

Long-term accuracy and precision of static ratios should thus degrade slowly through time as collector performance deteriorates.

Calculation of dynamic ratios

A two-line dynamic acquisition scheme was also used for the determination of 84 Sr/ 86 Sr ratios. Varying the magnet settings enabled the measurement of Sr isotopes in different faraday cups from line 1 to line 2 (

× < C S0 × G S0 C U0 × G U0 > 0r`
Because 𝜑 is ~ -1, faraday cup efficiencies and amplifier gains (i.e.,

c de × f de c ge × f ge ) almost divide out in the dynamic calculation of ! "# %+ "#
%& ', hence dynamic ratios are much less sensitive to faraday collector deteriorations and changes in amplifier gains than static ratios. Using the same sensitivity test as above for the static ratio, that is assuming that GH1 = GL1 = 1 and the faraday collectors degraded such that j ke j le = 0.999900, one can show that the dynamic ratio will only be shifted by +2.3 ppm [START_REF] Garçon | Factors influencing the precision and accuracy of Nd isotope measurements by thermal ionization mass spectrometry[END_REF]. In other words, the calculation of dynamic ratios mathematically cancels out ~97.7% of the component of imprecision and inaccuracy arising from collector inefficiencies and amplifier gain drift during measurement. This is the reason why the long-term precision and accuracy of dynamic ratios is, in general, much better than that of static ratios.

Results and Discussion

Precision and accuracy of standard measurements

The long-term static average of 84 Sr/ 86 Sr for the NIST SRM 987 Sr standard is 0.056495 ± 0.000003, (2SD; n=52). The data are in good agreement with those reported by other groups (Table 3; [START_REF] Andreasen | Mixing and homogenisation in the early solar system: clues from Sr, Ba, Nd, and Sm isotopes in meteorites[END_REF][START_REF] Moynier | Planetary-scale strontium isotopic heterogeneity and the age of volatile depletion of early solar system materials[END_REF]Hans et al., 2013a;[START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF][START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF]. The slight differences in absolute ratios for standards between this study and others do not affect the data comparison between the studies because epsilon values are calculated relative to the standards for each analytical session. Furthermore, our long-term dynamic average of 84 Sr/ 86 Sr for the NIST SRM 987

Sr standard is 0.056497 ± 0.000003 is more accurate as it is not dependent on Faraday cup deterioration. Repeat measurements of the NIST SRM 987 Sr standard that was processed through ion chromatography define a dynamic value of ε 84 Sr -0.08 ± 0.19 (2SE; n= 5, Table 4) and is thus zero within uncertainty. This demonstrates that the chemical separation procedure does not affect the Sr isotopic composition of samples.

Precision and accuracy of sample measurements

For data acquired in dynamic mode, the terrestrial rocks define an average ε 84 Sr value of -0.04 ± 0.09 (2SE), which is within uncertainty of 0, as defined by the NIST SRM 987 Sr standard (Table 4; supplementary material Table A1). In contrast, for static acquisition, the mean of all terrestrial rocks is resolved from 0, an ε 84 Sr value of -0.15 ± 0.09 (2SE).

These averages are based on 27 separate samples measured over two years. The data broadly agree with those in the literature within uncertainty and reproduce the negative offset observed in terrestrial material for ε 84 Sr when measured in static mode (Table 4).

This offset is not observed in dynamic mode (Fig. 1 and2).

For BCR-2 and BHVO-2 in particular, our static measurements also confirm the previously reported negative offsets in ε 84 Sr (Table 4, [START_REF] Andreasen | Mixing and homogenisation in the early solar system: clues from Sr, Ba, Nd, and Sm isotopes in meteorites[END_REF][START_REF] Paton | Identification of an 84 Sr-depleted carrier in primitive meteorites and implications for thermal processing in the solar protoplanetary disk[END_REF][START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF]. However, the average ε 84 Sr values for dynamic measurements of BCR-2 (-0.12 ± 0.11) and BHVO-2 (0.04 ± 0.24) do not display a resolved negative offset (Fig. 1 and2). Terrestrial Sr is not expected to show non-mass dependent variations in 84 Sr/ 86 Sr ratios. Thus, the absence of ε 84 Sr deficits for dynamic measurements is consistent with expectations. This indicates that, for 84 Sr, dynamic measurements are more accurate than static ones, consistent with dynamic data for ε 84 Sr reported by Hans et al. (2013a) and [START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF]. The external reproducibility (2SD) on ε 84 Sr, based on 10 replicate analyses of BCR-2, is 0.36 and is similar to the longterm reproducibility of SRM-987 (± 0.51). The values above demonstrate the high degree of accuracy and precision of the measurements.

Dynamic vs. static data acquisition

Static measurements have the advantage of time efficiency and good internal precision.

However, a static routine fails to account for changes in individual cup efficiencies over time. Dynamic acquisition mode overcomes this problem (e.g., Hans et al., 2013a;[START_REF] Roth | Neodymium isotopes and evolution of early terrestrial crust[END_REF]. The discrepancy between dynamic and static measurements, with dynamic analyses yielding values closer to zero (Hans et al., 2013a,b;[START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF] is reproduced in this study (Fig. 2). For example, the static line 1 results for BCR-2 are characterised by a negative average ε 84 Sr value of -0.20 ± 0.15 (2SE), which is within uncertainty of the range of -0.10 to -0.18 reported by other studies using only a static routine [START_REF] Andreasen | Mixing and homogenisation in the early solar system: clues from Sr, Ba, Nd, and Sm isotopes in meteorites[END_REF][START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF][START_REF] Paton | Identification of an 84 Sr-depleted carrier in primitive meteorites and implications for thermal processing in the solar protoplanetary disk[END_REF]. Static line 2 results also display a negative offset of slightly smaller magnitude than line 1. This unusual signature in terrestrial material has been attributed to possible contamination of the NIST SRM Sr standard SRM 987 during production or nuclear field shift effects [START_REF] Moynier | Planetary-scale strontium isotopic heterogeneity and the age of volatile depletion of early solar system materials[END_REF][START_REF] Fujii | The nuclear field shift effect in chemical exchange reactions[END_REF].

Dynamic analyses, however, yield ε 84 Sr data that do not show resolved negative shifts, which indicates that the shift may be of analytical origin. Possible explanations include:

(i) varying faraday cup efficiencies that are cancelled out by the dynamic routine, (ii) rapid changes in mass fractionation between magnet settings that lead to inaccurate mass bias correction for the dynamic data, (iii) insufficient settling of the amplifiers between magnet jumps from one line to the next (signal decay), (iv) matrix effects that lead to erroneous mass bias correction, (v) interferences, or (vi) ion beam truncation.

Faraday cup efficiency

Changes in faraday collector efficiencies depend on instrument use and the amount of ions collected in the cups through their life. Because previous studies reported similar negative ε 84 Sr values for static data but likely did not use their instrument the same way, the static-dynamic offset is unlikely to be due to problems with varying cup efficiencies through time. Moreover, there is no obvious reason for the collector efficiencies to be different between standards and samples, as required to explain the static-dynamic offset between terrestrial samples and the NIST SRM Sr standard.

Rapid changes in instrumental mass bias

Roth et al. ( 2014) reported that fast rates of isotopic fractionation during dynamic measurements of Nd isotopic ratios by TIMS can introduce an analytical bias. If the rate of instrumental mass-dependent isotopic fractionation is fast relative to the time between measurement lines, the isotopic ratio used for internal normalisation can be unsuitable. Consequently, Nd isotopic anomalies relative to terrestrial values reported by [START_REF] Upadhyay | 142 Nd evidence for an enriched Hadean reservoir in cratonic roots[END_REF] were suggested to be analytical artefacts [START_REF] Roth | Revisiting the 142 Nd deficits in the 1.48 Ga Khariar alkaline rocks, India[END_REF]. This effect might also be significant for dynamic measurements of 84 Sr/ 86 Sr ratios by TIMS.

Fractionation of 88 Sr/ 86 Sr ratios between lines one and two could cause line two 88 Sr/ 86 Sr ratios (measured 8 seconds after line 1) to be unsuitable for the mass bias correction of line one 84 Sr/ 86 Sr ratios. Given the collector configuration (Table 2), it is expected that the normal fractionation of Sr through the run (i.e. 88 Sr/ 86 Sr increase through time) would bias the dynamic 84 Sr/ 86 Sr ratio towards higher than true value for high fractionation rate. In their study, [START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF] argued that the average fractionation rate of their runs was too low to account for the observed dynamic-static offset, therefore ruling out the possibility of significantly shifting the dynamic 84 Sr/ 86 Sr ratios with this process.

Our runs have, on average, higher fractionation rates than [START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF], but like them, we do not believe that the instrumental fractionation of Sr in between acquisition lines is responsible for the measured static-dynamic offset in ε 84 Sr values. The detailed analyses of several of our runs revealed no obvious correlation between the mean fractionation rate of the runs and the resulting dynamic 84 Sr/ 86 Sr ratios. Moreover, one of our BCR-2 run displays a reverse fractionation trend (run with a negative mean fractionation rate), which should theoretically results in a dynamic ε 84 Sr value lower than the true value. Nevertheless, this particular run still returns a less negative dynamic ε 84 Sr value (dynamic = -0.13 ± 0.29 (2SE)) than the static value from line one (ε 84 Srstatic line 1 = -0.29 ± 0.25 (2SE)). Therefore, instrumental fractionation of 88 Sr/ 86 Sr ratios between line 1 and line 2 likely does not cause the observed offset between static and dynamic ε 84 Sr values.

Settling time of the amplifiers

This issue has been tested by Hans et al. (2013a,b) on the same Triton multicollector thermal ionisation mass spectrometer (TIMS) at ETH Zürich that was used for our study.

These authors showed that remnant signal is not an issue because it decays to background levels within the delay times used during the measurement routine (i.e. 4 and 8 seconds after lines one and two, respectively).

Matrix effects

The offsets may be caused because the exponential law used for mass bias correction may not accurately describe Sr isotopic fractionation in detail. Since the offset is observed for rock samples and not for the processed NIST SRM 987 Sr standard (Fig. 1, Table 4), the problem may be due to matrix element(s) from the rock samples that were left behind after ion exchange chemistry. We did not observe significant levels of Ca, Mg, or Al in the sample solution after the separation of Sr. Thus, the static offset would require an as yet unrecognised matrix effect and for the dynamic routine to be more robust against such matrix effects.

Interferences

Matrix elements present in the analysed solution could also introduce isobaric interferences on Sr isotopes (e.g., 68 Zn 16 O on 84 Sr). This scenario would require that the same amount of interfering element is always left in the sample solution and that the dynamic routine fortuitously corrects for this offset. This is rather unlikely because different studies [START_REF] Andreasen | Mixing and homogenisation in the early solar system: clues from Sr, Ba, Nd, and Sm isotopes in meteorites[END_REF][START_REF] Moynier | Planetary-scale strontium isotopic heterogeneity and the age of volatile depletion of early solar system materials[END_REF]Hans et al., 2013a;[START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF][START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF] observed similar negative offsets despite using different Sr purification protocols. In general, the amount and composition of any remaining matrix depends on the ion exchange chemistry protocol and would not be constant across the above studies.

Ion beam truncation

Yobregat et al. (2017) proposed that truncation of the 88 Sr ion beam was responsible for the difference in e 84 Sr values measured in static mode versus those measured in dynamic mode and that these effects depend on the location of the 88 Sr beam relative to the axial cup (i.e., H3 vs. H1 on lines 1 and 2, respectively); the farther away the 88 Sr beam is from the axial cup, the more prone it is to truncation. [START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF] also observed a constant, and large difference in the mean mass fractionation factors (f) between rock samples and SRM 987 and showed that such a large difference, combined with beam truncation, could lead to an apparent offset of 25 ppm in e 84 Sr values. However, this hypothesis is not able to explain why their static line 2 values show larger e 84 Sr effects than those from static line 1 because the 88 Sr beam is collected closer to the axial cup on line 2. Thus, truncation of the 88 Sr beam does not provide an internally consistent explanation for the difference between static and dynamic results in their dataset.

In addition, beam truncation can explain the difference between static and dynamic results only if the mean mass fractionation factors observed for analyses of rock samples compared to those for SRM 987 are distinct. [START_REF] Yobregat | A new method for TIMS high precision analysis of Ba and Sr isotopes for cosmochemical studies[END_REF] observed a significant difference (i.e., 0.3), whereas we observed a difference of only 0.004. Moreover, t-tests performed on our data confirm that there is no significant difference between mass fractionation factors obtained from terrestrial rock analyses compared to those from SRM 987 for either static line 1 (t = 0.21) or static line 2 (t = 0.24). Nevertheless, our analyses show a systematic offset in e 84 Sr static values for terrestrial samples. Therefore, we conclude that truncation of the 88 Sr ion beam is an unlikely explanation for the difference between e 84 Sr values measured in static mode versus those measured in dynamic mode observed in our study.

In summary, it cannot be excluded that dynamic acquisition may produce erroneous Sr compositions that cause the analysed samples to overlap with the NIST SRM 987 Sr isotopic standard by coincidence. However, we consider this scenario unlikely. Our data confirm that dynamic analyses consistently produce ε 84 Sr values for terrestrial rock and mineral samples that are indistinguishable within uncertainty from SRM 987, whereas static analyses show reproducible negative offsets. While the origin of the offsets observed using a static routine needs further investigation, the consistency of the dynamic data strongly suggests that the acquisition of high precision ε 84 Sr data requires a dynamic routine to obtain accurate results.

Conclusions

An extensive dataset collected over more than two years confirms that our analytical 2017) is confirmed, and the dynamic mode is found to be more accurate. Previously reported deficits for ε 84 Sr in terrestrial material [START_REF] Andreasen | Mixing and homogenisation in the early solar system: clues from Sr, Ba, Nd, and Sm isotopes in meteorites[END_REF][START_REF] Moynier | Planetary-scale strontium isotopic heterogeneity and the age of volatile depletion of early solar system materials[END_REF][START_REF] Brennecka | Evidence for supernova injection into the solar nebula and the decoupling of r-process nucleosynthesis[END_REF][START_REF] Paton | Identification of an 84 Sr-depleted carrier in primitive meteorites and implications for thermal processing in the solar protoplanetary disk[END_REF] were reproduced in this study and are best understood as analytical artefacts caused by data acquisition in static mode. Looking to the future, Sr isotopic measurements for a similar range of terrestrial rocks by inductively coupled plasma mass spectrometry (ICP-MS) would provide a means of testing if the deficit is unique to TIMS measurements carried out in static mode, which then may provide insight into the origin of the artefacts. *Both studies used the same Triton TIMS instrument at ETH Zürich.

  Where m84, m86, and m88 are the atomic masses of isotopes 84 Sr, 86 Sr, and 88 Sr, the ratios measured on acquisition line 1. Since the measured Sr ion-beam intensities are function of faraday collector efficiencies and amplifier gains, we can write: " true " intensities of the different Sr isotope beams on acquisition lines 1; C S0 , C U0 , and C UX are the efficiencies of the faraday collectors L1, H1, and H3 that change with time depending on instrument use; G S0 , G U0 , and G UX are the gains of the amplifiers to which the faraday collectors L1, H1, and H3 are attached.Using this notation, (Eq.1) can be re-written as follows:

  procedure provides precise and accurate Sr isotopic measurements. Separation of Sr is achieved by ion exchange chromatography and isotopic compositions are measured by TIMS. Data for terrestrial rocks acquired in dynamic mode display ε 84 Sr values within uncertainty of 0. The discrepancy between static and dynamic acquisition schemes reported byHans et al. (2013a) andYobregat et al. (

Fig. 1 .

 1 Fig. 1. Strontium isotopic data for terrestrial samples acquired in dynamic and static (line 1) modes. The ε 84 Sr values were calculated relative to the NIST SRM 987 Sr isotopic standard. The grey lines are ± 49 ppm (dynamic) and ± 45 ppm (static) (2SD for all rocks). The dashed lines represent the average ε 84 Sr values of all rocks: -0.04 (dynamic) and -0.15 (static).

Fig. 2 .

 2 Fig. 2. Mean e 84 Sr values for BCR-2 (n=10), BHVO-2 (n=6), and all anorthosites (AN-G, JL-27, AN-1, AN-2; n=10) for dynamic (filled symbols) and static line 1 (open symbols) data acquisition modes. Error bars are 2SE. The black dashed line (dynamic) and gray dashed line (static) are the averages of all terrestrial samples analyzed (n=27).

  Table 2). Dynamic ratios are calculated assuming that Sr mass fractionation follows an exponential law during the run. The dynamic 84 Sr/ 86 Sr

	relative difference of efficiencies and amplifier gains between Faraday collectors L1 and
	H1. This gives:
	1	Sr Sr 46 7 45 mno 0	= 1	Sr Sr 46 7 45 9*-: 0	× <	m 45 m 46	>	?	with f =	ln	⎝ ⎛	< < 44 44 46 > Sr Sr (#)* Sr Sr 46 > 9*-: p ⎠ ⎞ ln ! m 44 m 46 '
	⟺ 1	Sr Sr 46 7 45 mno 0	= 1	Sr Sr 46 7 45 9*-: 0	× 1	Sr Sr 46 7 44 (#)* `×	1	44 46 7 b` Sr 9*-: p Sr	(Eq. 2)
	With ion-beam intensities, this gives:
	1	Sr Sr 46 7 45 mno 0	=	I Sr 45 I Sr 46	0 0	× 1	Sr Sr 46 7 44 (#)* `× 1 I Sr I Sr 46 44	p p	7 b`
	ratio, !	%+ %& ' "# "# mno 0	, is obtained by combining the 84 Sr/ 86 Sr ratio measured on line 1,
	!	"# "# %& ' %+	9*-: 0	, with the 88 Sr/ 86 Sr ratio measured on line 2, !	%% %& ' "# "#	9*-: p	, to minimize the

Table 3

 3 The Sr isotopic composition for NIST SRM-987.

		84 Sr/ 86 Sr	2SD	Mode	n
	This study*	0.056497	0.000003	Dynamic 52
	Hans et al. (2013a)*	0.056495	0.000003	Dynamic 30
	Yobregat et al. (2017)	0.056492	0.000001	Dynamic 35
	Brennecka et al. (2013)	0.056493	0.000003	Static	15
	Moynier et al. (2012) (Tokyo Tech)	0.056492	0.000002	Static	53
	Andreasen and Sharma (2007)	0.056494	0.000002	Static	12

Table 4

 4 The e 84 Sr values for BCR-2, BHVO-2, and BSE acquired in static (line 1) and dynamic modes.

		Static1 ε 84 Sr	2SD	2SE	n
	SRM 987 (column processed, this study)	-0.06	0.39	0.17
	BCR-2
	This study	-0.20	0.46	0.15	10
	Hans et al. (2013b)	-0.32	0.26	0.11
	Brennecka et al. (2013)	-0.10	0.50	0.25
	Andreasen and Sharma (2007)	-0.16	0.45	0.17
	Paton et al. (2013)	-0.18	0.30	0.10	10
	BHVO-2
	This study	-0.13	0.35	0.14
	Hans et al. (2013b)	-0.16	0.30	0.17
	Yobregat et al. (2017)	-0.07	0.31	0.10
	Moynier et al. (2012)	-0.18	0.31	0.22
	Paton et al. (2013)	-0.25	0.22	0.06	16
	BSE
	This study	-0.15	0.45	0.09	27
	Hans et al. (2013b)	-0.25	0.26	0.07	14
	Yobregat et al. (2017)	-0.14	0.41	0.10	17
	Moynier et al. (2012) (TT)	-0.29	0.20	0.08
	Paton et al. (2013)	-0.22	0.26	0.05	26
		Dynamic ε 84 Sr	2SD	2SE	n
	SRM 987 (column processed, this study)	-0.08	0.43	0.19
	BCR-2
	This study	-0.12	0.36	0.11	10
	Hans et al. (2013a)	0.04	0.09	0.04
	BHVO-2
	This study	0.04	0.60	0.24
	Hans et al. (2013a)	0.04	0.31	0.18
	Yobregat et al. (2017)	-0.07	0.31	0.10
	BSE
	This study	-0.04	0.49	0.09	28
	Hans et al. (2013a)	0.02	0.24	0.07	14
	Yobregat et al. (2017)	-0.03	0.29	0.07	17

Acknowledgements

This work has been carried out within the frame of the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation. MG received funds from the Ambizione program of the Swiss National Science Foundation under the grant N o PZ00P2_161218. We thank Julien Leuthold for providing anorthosite sample material and Irene Ivanov-Bucher for her help in sample preparation and mineral separation. We also thank Colin Maden and Donat Niederer for helping to maintain the Triton TIMS instrument at ETH Zürich.

565

Table 2 Faraday Cup configuration for TIMS measurements. The boxes indicate the overlaps exploited for the dynamic analyses. The 88 Sr/ 86 Sr ratio from line 2 is used for the mass bias correction of the 84 Sr/ 86 Sr ratio from line 1.


Mathis Boukhellouf

Nicolas Chiabaut

Christine Buisson

High-occupancy vehicle lanes: modeling the upstream interface capacity

Keywords: capacity drop, high-occupancy vehicle lanes, microscopic traffic simulation, shared mobility, traffic flow theory, weaving section

This work contributes to the understanding of the behavior of the upstream end of a high-occupancy vehicle (HOV) lane. Indeed, just upstream of this lane, vehicles may change lanes to be sure they are allowed to use the downstream lane (either HOV or general-purpose lane(s)). This upstream end is therefore the place of lane changes that may reduce the capacity. To estimate this drop, we adapt an existing weaving section analytical model to this upstream interface. We assess this model by comparison with the maximal theoretical supply and with simulation results. Our model adaptation indeed reproduces a capacity drop. Although with a less important drop, simulation results are coherent, except for cases with 0 or 100% of the vehicles being highly occupied. Further work should focus on treating the successive lane changes when considering more than two upstream lanes, and on comparing the results with the ones of other simulators.

INTRODUCTION

Since the end of the 2010s, European public authorities, and particularly in France, have been moving more and more towards the deployment of high-occupancy vehicle (HOV) lanes (HLs) to regulate highway traffic. Opened in the United States since the 1970s on networks with bottlenecks, these lanes are only accessible to vehicles carrying people who share their mobility whether through carpooling or using public transport, and in some cases to low-emission vehicles. HLs being designed to be uncongested, one can reward people adopting a virtuous behavior towards car use with shorter travel times. Meanwhile, self-drivers remain in the congested general-purpose lanes (GLs). This type of planning is particularly deployed on roads leading to a central business district (CBD), as they are heavily traveled during workdays' peak periods. In the long term, the strategic objective is therefore to foster a modal shift towards cleaner mobility. However, it should first be ensured that in the short term, traffic conditions induced by the HL activation avoid the propagation of queues to upstream strategic nodes, as well as guaranteeing a convincing travel time differential between carpoolers and self-drivers.

In Europe, rather than building a new lane or marking permanently the road surface as is mostly the case in the United States, the leftmost lane on the highway is temporarily allocated to HOVs. A typical case is shown in figure 1, with an HL-section of length ℓ H leading to a downstream bottleneck whose supply is Ω. In this situation, [START_REF] Menendez | Effects of HOV lanes on freeway bottlenecks[END_REF] and [START_REF] Daganzo | Effects of high occupancy vehicle lanes on freeway congestion[END_REF] give conditions on the flows of each type of lane so that the HL activation is not

q x ,    ∀x ∈ [0 ; ℓ H ], q H (x) ≤ q x ∀x ∈ [0 ; ℓ H ], q G (x) ≤ (n -1)q x q H (x = ℓ H ) ≤ Ω/m (1)
The first two conditions reflect physical constraints, as the flows cannot overpass the capacity. It means that traffic conditions should be checked at any potential bottleneck (HL (de)activation, onor off-ramp, lane drop, ...) so that if not met at any position in the section, congestion starts from this position. The last condition applies at the downstream end of the HL, assuming there are m downstream lanes. It is stronger than the first one because if not met, the HL is saturating from downstream, meaning that the deployment fails, as HOVs cannot gain travel time.

By adapting conditions (1), [START_REF] Hans | Évaluation a priori des voies réservées au covoiturage sur voies structurantes d'agglomération[END_REF] and [START_REF] Rousic | Voies structurantes d'agglomération. Aménagement des voies réservées au covoiturage et à certaines catégories de véhicules[END_REF] model the use domain of an HL-section. It ensures fluidity in the HL and congestion from downstream in the GLs so that the HL acts like a bottleneck for self-drivers. They assume that either there is no ramp or the inflows equal the outflows so that the flows arriving at the end of the section shall roughly be the same as the ones at x = 0. Thus, the conditions of this use domain focus on the upstream demands:

   ∆ ≥ Ω (1 -α)∆ ≤ (n -1)q x α∆ ≤ Ω/m (2)
α being the HOVs proportion. The first condition states that the section is congested, and the second one ensures the GLs should not saturate from upstream. The last condition is the same as (1), ensuring the HL does not saturate from downstream.

The main shortcoming of this framework is that it does not consider the phenomena occurring at the upstream and downstream interfaces, i.e. the way vehicles join their assigned lane and how they rearrange once the HL is deactivated, generating lane changes (LCs). The aggregation of these individual behaviors may induce a capacity drop [START_REF] Leclercq | Capacity drops at merges: New analytical investigations[END_REF], so that the two last conditions of (2), regarding the inflows at x = 0, should be modified due to the exogenous limitations of the capacities. Indeed, underestimating the possible drop upstream of the HL entrance leads to underestimating the increase in travel times and in queues length, which may then propagate up to sensitive traffic nodes. Thus, to make the use domain more reliable, this work investigates analytically and in simulation the HL upstream interface capacity.

METHODOLOGY

Our aim is to provide tools to better estimate the upstream capacity of sections where one of the downstream lanes is dedicated to HOVs. Table 1 presents all variables and parameters describing C u Priority ratio in upstream general saturation δ the upstream limit of an HL-section. The α ratio, corresponding to the HOVs proportion, as well as n, the number of upstream lanes, are the key factors for the operation of the upstream limit of those zones.

To reach our aim, for now, we rely on the tools available without experiment, because we want to make the estimation before the experimental deployment of HLs. Varying α from 0 to 1 and n from 2 to 3, we use three methods:

• first, a lower-bound capacity is derived, based on the endogenous capacity computed with a model inspired from the weaving sections,

• secondly, an upper-bound capacity (or theoretical supply) is defined, according to equation (1),

• thirdly, we use a microscopic tool to explore the simulation results for this configuration.

Lower-bound capacity estimation

As the movements upstream an HL-section are similar to the ones all along a weaving section (see figure 2), we consider the upstream interface to be a specific type of weaving section. Thus, to derive the capacity, we chose to adapt the findings of [START_REF] Marczak | A Macroscopic Model for Freeway Weaving Sections[END_REF]'s work. It explains how traffic operates on the highway discontinuity induced by the proximity between an on-ramp and an off-ramp and by the individual differences in the origin and the destination choice. A proportion β i of the demand λ i coming from lane i takes the off-ramp, while a proportion β j of the demand λ j coming from the on-ramp (lane j) joins lane i, see figure 2(a). Macroscopically, this weaving section is modeled as the superposition of two merges and two diverges, since on each lane vehicles are inserting (merging operation) and others are deserting (diverge operation). For each operation, individual behaviors are considered, such as acceleration after changing lane to reach the free-flow speed, or that LCs occur along a zone of a certain length and create voids, so that the capacity drop caused by weaving flows is endogenously derived. This formulation relies on an allocation scheme of the flows, following the Newell-Daganzo model [START_REF] Newell | Applications of Queueing Theory[END_REF][START_REF] Daganzo | Effects of high occupancy vehicle lanes on freeway congestion[END_REF], which returns three possible ways of allocating the capacity C on the two upstream lanes i and j, depending on the demands' level and distribution: (i) both lanes free flow (q i = λ i , q j = λ j );

(ii) one lane saturates and the other free flows (q i = λ i , q j = Cλ i ; or both lanes saturate, the total effective flow being shared between both lanes with respect to a priority coefficient δ (q j /q i = δ ). Thus, effective flows are derived when at least one lane is congested. Assuming this lane is lane i, there are two possibilities to its macroscopic operation:

q i = C -λ j , q j = λ j ); (iii) (a) (b) C u (α) ∆ 1 -α HL GLs α C(λ i , λ j , β i , β j ) λ j β j Li L j λ i β i 1 lane n - 1 lanes
• merge operation: vehicles inserting from lane j worsen the traffic conditions on lane i. This operation lies on [START_REF] Leclercq | Capacity drops at merges: New analytical investigations[END_REF])'s merge model and derives effective flow q m i ,

• diverge operation: vehicles deserting from lane i worsen the traffic conditions on this lane by anticipating their change to lane j. This operation lies on [START_REF] Laval | Stochastic Processes of Moving Bottlenecks: Approximate Formulas for Highway Capacity[END_REF][START_REF] Laval | Effects of geometric design on freeway capacity: Impacts of truck lane restrictions[END_REF]'s work and derives effective flow q d i .

The actual operation mode on lane i minimizes the effective flow, i.e. q i = min(q m i ; q d i ). The weaving section capacity is then numerically derived -all calculation details are in [START_REF] Marczak | A Macroscopic Model for Freeway Weaving Sections[END_REF] -as C(λ i , λ j , β i , β j ) = q i + q j . However, some adaptations in Marczak's model should be done to take into account the specificities of HLs weave, assuming there is only one HL:

• we consider the two upstream lanes as two pipes of 1 lane (upstream of the HL) and n -1 lanes (upstream of the GLs), and assume the effective flows in general congestion are equally shared, i.e. δ = 1/(n -1),

• in Marczak's model, the LCs zone is strictly limited by the weaving section length. In its application to HLs, as there is no geometrical constraint, we make the strong assumption that LCs are possible all along a zone of length L upstream of the activation,

• as it is an exogenous parameter in the original model, macroscopic traffic relaxation is not considered,

• interestingly, the number of input parameters is reduced from four to one, as pictured in figure 2:

as vehicles come from the same origin, we can legitimately assume they are evenly distributed on the upstream lanes, i.e. λ H = ∆/n and λ G = (n -1)∆/n. Then, follow-ing Newell-Daganzo's projection rule, only two traffic states are possible: fluidity or saturation on both upstream pipes. Thus, ∆ is not a parameter anymore, as a consequence, HOVs are also uniformly distributed upstream. Since weaving flows depend on the HOVs proportion, we have β H = 1α and β G = α.

The adapted model then derives lower-bound capacity C u (α), which matches with solid lines in figure 3.

Upper-bound capacity derivation

The upper-bound capacity coincides with the theoretical case without a capacity drop. It is derived depending on which type of lane is limiting due to a too important demand of selfdrivers for the GLs ((1α)∆ ≥ (n -1)q x ), or of HOVs for the HL (α∆ ≥ q x ). It thus equals min (n -1)q x 1α ; q x /α and matches with dashed lines in figure 3.

Simulation framework

Simulations were performed with the microscopic traffic tool Aimsun [START_REF] Aimsun | Aimsun next 20 user's manual[END_REF], with calibrated fundamental diagram parameters matching with the values of (u, w, κ) in table 1. Varying α from 0 to 1 and inflowing an upstream demand of nq x , effective flows are collected 250 meters downstream of the HL activation, to be sure vehicles reached the free-flow speed. To be statistically relevant due to the simulations stochasticity, 15 one-hour-simulations are considered, for each value of α. To elapse a potential warm-up phase, the flows collection holds for the 15 last simulated minutes. The simulated datasets match with the boxplots in figure 3, white dots representing the mean flow.

It is important to note that 11 replications, representing 3.33% of the total, lead to a complete gridlock before the end of the simulated hour. This is because of a vehicle arriving at the interface upstream of the wrong lane, and unable to change lanes due to the presence of another one arriving upstream of the wrong lane too, at exactly the same time instant. Those cases were discarded and replaced by new ones that do not present this problem.

RESULTS AND DISCUSSION

Figure 3 first compares the lower-and upper-bounds. It shows that our analytical formulation indeed integrates a capacity drop. Moreover, it exhibits the same variations as the theoretical supply, even if in the case the GLs are limiting, the capacity increase is less significant. It is also worth noting that for n = 3, the proportion maximizing the capacity is less than 1/3, the theoretical one. It means that weaving flows start worsening the traffic conditions for lower HOVs proportions than when they are not considered. As a consequence, as shown in figure 4, the capacity drop, defined as the complement of the ratio between the lower-and the upper-bound, is more significant for n = 3. Indeed, an additional lane is affected by the weaving phenomena, generating more traffic instabilities induced by insertions or desertions.

When it comes to simulation results, we notice that for both numbers of lanes, simulated flows fall between the lower-and the upper-bounds, corroborating our model is realistic. Flows simulated for proportions near to the one maximizing the capacity are even close to the lower-bound ones. However, the closer we get to the borderline cases α ∈ {0, 1}, the less capacity drop there is. When α = 0 (resp. α = 1), i.e. there is no carpooler (resp. no self-driver), the interface is a 1 + (n -1)to-n -1 lanes (resp. 1 + (n -1)-to-1 lane) merge. Nonetheless, for n = 2, the drop has been reported when using the same simulation configuration for an extended merge [START_REF] Boukhellouf | Merging and diverging operations: benchmark of three European microscopic simulation tools and comparison with analytical formulations[END_REF], making the upstream interface simulation results surprising. Another surprising result is the simulated borderline cases of n = 3, as we observe a drop for α = 1 but not for α = 0. This means that the 1 + 2-to-1 lane merge induces a capacity drop but not the lane drop 1 + 2-to-2 lanes one, because of a more important volume getting through a smaller bottleneck.

CONCLUSIONS

To fill the gap in the HLs upstream interface modeling, we proposed here to adapt an analytical model based on a weaving section operation. This adaptation is of flexible use, as the HOVs proportion is the only parameter. It integrates a capacity drop and gives consistent results so that the upstream interface is indeed a type of weaving section. However, when it comes to validation, Aimsun's results do not present a capacity drop when reaching high or low values of HOVs proportions. Thus, additional parameters of this tool, as LCs related ones, should be changed, or another simulator should be tested to validate the model. Finally, to improve the model, multilane merges [START_REF] Marczak | Capacity drops at merges: Analytical expressions for multilane freeways[END_REF] should be considered to take into account more precisely the impact of HOVs successive LCs upstream of the GLs on the capacity drop.

Figure 1 :

 1 Figure 1: Depiction of a HL deployment project, showcasing the potential issues due to the network geometry

Figure 2 :

 2 Figure 2: From the classic weaving section model (a) to the HL upstream interface weaving model (b)

Figure 3 :Figure 4 :

 34 Figure 3: Upstream interface capacity derivation: comparing the theoretical supply with our model and with simulation results for two upstream lanes (a) and three upstream lanes (b)

Table 1 :

 1 Variables and parameters used in the modeling

	Variable/parameter	Notation	Value
	HOVs proportion	α
	Upstream demand (all lanes)	∆
	Number of upstream lanes	n
	Free-flow speed	u	19.44 m/s
	Congestion wave maximum speed	w	5.4 m/s
	Jam density (one lane)	κ	0.15 veh/m
	Theoretical capacity (one lane)	q x	2282 veh/h
	Acceleration after changing lanes	a	3 m/s 2
	Weaving vehicles proportion from lane i	β i
	LCs zone length	L	1000 m
	Upstream demand (lane i)	λ i
	Effective flow (lane i)	q i
	Upstream interface capacity (all lanes)


Aymene Mohammed Bouayed

Samuel Deslauriers-Gauthier

Rachid Deriche

Rotation invariant features for Alzheimer's disease identification using convolutional neural networks

R 22 ⊕ R 22224 R 22 R 22 ⊕ R 22224 ⊕ R 2224 ⊕ R 0 R 22 ⊕ R 22224 ⊕ R 2224

Class Activation Map

Class Activation Map can be extracted from our architecture and provides the location of salient voxels used for the classification of AD and CN subjects.

Subject CNN model

The proposed Subject CNN model that uses the encoding of each subject's slice to take the classification decision.

Data set

The data set split over the ADNI-SIEMENS 1 data set.

Single RIF performance

Identification of the most useful RIFs to the task of AD patient classification.

Multi-RIF performance

Combining the best RIFs as to improve performance.

Class Activation Maps on the used data set

Class Activation Maps extracted from our best performing model that uses and has a B-score of 82.67%. The colours red and bleu are used for CAMs for AD and NC classes respectively.

R 22 ⊕ R 22224

Conclusion

This work allows us to conclude that the 12 algebraically independent 4th degree RIFs contain information that is very useful to AD identification and we can achieve a high performance on the ADNI-SIEMENS data set. It can also be noted that not all the RIFs are equivalently informative to the task at hand and careful RIF selection is important.

Problem statement : How useful can the 4th order Rotation Invariant Features (RIFs) be in the context of Alzheimer's disease identification?

And are all these RIFs useful to the task?

Abstract

Rotation Invariant Features (RIFs) [START_REF] Zucchelli | A computational framework for generating rotation invariant features and its application in diffusion MRI[END_REF] extracted from dMRI scans represent a generalisation of the usually used 2nd order invariants such as Fractional Anisotropy (FA) and Mean Diffusivity (MD). This work studies the usefulness all of the 12 algebraically independent RIFs extracted from Spherical Harmonics up to degree 4 in the context of Alzheimer Disease (AD) identification. To do so, we introduce a fair metric (B-score) that we use to evaluate the proposed deep Convolutional Neural Network (Subject CNN) which operates on subject slices to classify the whole subject while avoiding over-fitting. On the ADNI-SIEMENS 1 data set that contains 46 AD and 352 Normal Connectivity (NC) subjects respectively, we observe that the 12 algebraically independent 4th degree RIFs are not equivalently useful to the classification task. A particular combination of a low degree RIF ( ) with a high degree one ( ) achieves the best performance of 82.67% B-score and 84.88% accuracy on this data set. Also, the generated 3D Class Activation Maps (CAMs) [START_REF] Bolei | Learning deep features for discriminative localization[END_REF] show that to classify a subject as AD or NC the model focuses on the value of the RIFs in the white matter around the ventricules. Formula for extracting Rotation Invariant Features (RIFs) [START_REF] Zucchelli | A computational framework for generating rotation invariant features and its application in diffusion MRI[END_REF] f(θ, ϕ)

≈ D ∑ l=0 | l%2=0 l ∑ m=-l c m l Y m l (θ, ϕ) R l 1 ,⋯,l d = l 1 ∑ m 1 =-l 1 ⋯ l d ∑ m d =-l d c l 1 m 1 ⋯c l d m d G(l 1 , m 1 | ⋯ | l d , m d ) a)

  Rotation Invariant FeaturesRepresenting the diffusion signal as a weighted sum of spherical harmonics.

  Sagittal view. b) Coronal view. c) Axial view.MetricA fair metric that gives equal weight to both classes regardless of the number of data points is crucial in the used imbalanced data set.

edu/ This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.