# Detecting mechanical property anomalies along railway earthworks by Bayesian appraisal of MASW data

A. Burzawa, L. Bodet, A. Dhemaied, M. Dangeard, Sylvain Pasquet, Q.

Vitale, J. Boisson-Gaboriau, Y.J. Cui

► To cite this version:

A. Burzawa, L. Bodet, A. Dhemaied, M. Dangeard, Sylvain Pasquet, et al. Detecting mechanical property anomalies along railway earthworks by Bayesian appraisal of MASW data. Construction and Building Materials, 2023, 404, pp.133224. 10.1016/j.conbuildmat.2023.133224. hal-04886930

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# Detecting mechanical property anomalies along railway earthworks by Bayesian appraisal of MASW data

A. Burzawa<sup>a,b,*</sup>, L. Bodet<sup>a</sup>, A. Dhemaied<sup>b</sup>, M. Dangeard<sup>b</sup>, S. Pasquet<sup>a,c</sup>, Q. Vitale<sup>d,e</sup>, J. Boisson-Gaboriau<sup>b</sup>, Y. J. Cuif<sup>f</sup>

<sup>a</sup>Sorbonne Université, CNRS, EPHE, UMR 7619 METIS, 4 place Jussieu, 75252 Paris
05, France

<sup>b</sup>SNCF Réseau, 6 avenue François Mitterand 93210 Saint-Denis, France

<sup>c</sup>Observatoire des Sciences de l'Univers ECCE TERRA - UAR 3455, Sorbonne Université, CNRS, Paris, France

<sup>d</sup>Éveha International, 161 avenue de Verdun, 94200 Ivry-sur-Seine, France

<sup>e</sup>Université de Lyon, CNRS, Archeorient, UMR 5133, Maison de l'Orient et de la Méditerranée, France

<sup>f</sup>Ecole des Ponts, ParisTech

## Abstract

The techniques traditionally used to estimate the mechanical properties of railway earthworks (RE) are costly and of low performance. There is a great need for the development of non-destructive methods, which would allow a fast and efficient diagnosis of RE. Seismic surface-wave (SW) methods, also known as multi-channel analysis of surface waves (MASW), were rescaled here for there systematic implementation on RE. A test site was chosen along the North European High Speed Line (HSL) for its well-known correspondence between discontinuity in mechanical properties and anomalous maintenance forces. With the proposed acquisition and processing strategy, a contrast in SW propagation velocity was correlated with the change in shear modulus of the soil layers beneath the track, previously evaluated in

<footnote>*audrey.burzawa@sorbonne-universite.fr</footnote>
the laboratory. Bayesian inversion of the SW data also allowed to integrate
the strong *a priori* knowledge available on these HSL lines and to provide
quantitative results with a confidence index to help end-users in their deci-
sions.

Keywords: railway earthwork, mechanical properties, seismic methods,
surface waves, bayesian inversion
# 1. Introduction

The preservation and expansion of rail transportation networks is a ma-
jor national and international issue. Stability problems are among the pri-
orities. They depend mainly on the spatial and temporal variations of the
mechanical properties of the materials used to build and support railway
earthworks (RE). RE are dimensioned according to very strict internal con-
struction standards (SNCF Réseau IN3278, IG90260 in France for instance).
Unfortunately, during the life of the RE, and depending on its location and
the construction period, many factors such as water drainage, weight and
speed of trains, frost, etc., may cause heterogeneities in the structure of the
railway tracks. It is therefore necessary to develop tools capable of monitoring
the condition of the RE, to control the evolution of their mechanical state,
and more particularly, improve the knowledge of existing RE assets (Selig
and Waters, 1994; Trevin, 2008). Several criteria are classically used to eval-
uate the global track behaviour, such as geometry-related faults (cross-level,
alignment, longitudinal levelling, twist and gauge) or frequency and type of
maintenance operations and geotechnical surveys (Selig and Waters, 1994;
Quiroga and Schnieder, 2013; Rhayma et al., 2013; Tzanakakis, 2013).

To complement the above techniques with non-destructive characterisa-
tion and imaging of the underground RE and soil structure, geophysical
methods can be deployed with great efficiency and yield (Milsom and Erik-
sen, 2013). They are able to provide information along larger linear distances
and more densely than geotechnical methods, as well as to allow interpolation
of stratigraphy and mechanical parameters between boreholes. Such methods
are therefore of great interest because they promise rapid and efficient diag-
26 nosis of the condition of RE (see for example the review on non-destructive testing methods for the assessment and monitoring of the health of railway infrastructures by Artagan et al., 2020).

29 In the toolbox of geophysical prospecting, it is well known that seismic methods can help, by definition, image near-surface mechanical property contrasts. This is why their use, and more particularly surface-wave (SW) based approaches, has been increasingly suggested for the investigation of RE, as for example very recently described by Kyrkou et al. (2022) in 'a review of measurement practice'. These methods, also known as Mutlichannel Analysis of Surface Waves (MASW) (see e.g. Foti et al., 2018), make it possible to estimate near-surface shear-wave velocities ($V_s$), hence shear moduli ($G$).
37 A rationale for the use of seismic methods applied to RE and recent developments in SW techniques in this context, are presented in more details with associated references in a following section dedicated to the 'background methodology'. Despite the growing trend towards developments (Kyrkou et al., 2022), there seems to be a lack of specifically dedicated approaches best suited to the railway context.
42

43 The present study therefore aims at illustrating the feasibility of developing a specific SW prospecting method able to mechanically characterise RE and underlying soils along railway tracks, without perturbing the traffic. A test site along the North European High-Speed Line (HSL) has been targeted, as the RE at this location presented important stability problems, sufficiently documented to test new approach. To identify and understand the origin of the anomalies observed at the surface (mainly consisting in strong tracks motion and important loss of ballast), classical geotechnical
studies were conducted with core drilling, dynamic penetrometer and laboratory tests (Dhemaied et al., 2014a,b,c). We present how the geotechnical results (see more details in a dedicated section in the following) appeared non-conclusive except for laboratory measurements in soil samples of micro-porosimetry and $V_S$. These parameters, showing significant variations in one particular layer of the RE, justified the design of a specific seismic SW setup, dimensioned and adapted to this type of RE, in order to detect variations of $V_S$ along the line. The setup and measurement strategy (array geometry, types of sensors and source) are first presented in details. To ensure reliable SW data extraction along the considered HSL, a specific processing workflow has then been suggested and detailed as well.

This work is actually presented so as to provide practitioners and end-users with detailed descriptions and guidelines of the proposed approach, seen as a toolbox to facilitate the setup design, data processing and interpretation of results. As railway infrastructure managers need thresholds above which mechanical moduli variations can be considered significant, resulting models of the RE mechanical properties are suggested here to be given in terms of probability rather than in terms of fixed parameter values with posterior uncertainties. This can be performed by implementing a Bayesian formalism (Tarantola, 2005). It has been applied to the appraisal of surface-wave dispersion inversion results at large scale (Sambridge, 1999b,a; Bodin et al., 2012) and, for instance, to near-surface applications with the advantage of combining different types of geophysical and geotechnical data (see e.g. Killingbeck et al., 2018; Halló et al., 2021). Following the processing workflow, the probabilistic approach implemented with the Bayesian formal-
ism as mentioned above has thus been used to enable a fair quantification
of mechanical properties variations in the RE, in depth and along the line.
The study finally suggests how, in the long run, such technique could be de-
ployed with a sufficient yield for decision support, to guide choices in terms
of monitoring, diagnosis and, eventually, of appropriate maintenance.

## 2. Background methodology

### 2.1. Standard track behaviour evaluation tools

Geotechnical surveys such as boreholes, coring, in situ testing, etc., are
classically performed to test the mechanical condition of RE. Standard pa-
rameters such as the bearing capacity of base structures or the core resis-
tance of sub-ballast layers are also regularly assessed (Benz-Navarrete, 2009;
Escobar, 2015; Escobar et al., 2016; Haddani et al., 2016). But their imple-
mentation remains limited in space and extremely costly on a large scale (in
France, for instance, there are more than 28,000 km of track to be monitored),
not to mention the difficulties of accessing railway sites, which constitute an
additional constraint and reduce the cost-effectiveness of these probing tools.

To overcome these problems, Ground Penetrating Radar (GPR) is mounted
on trains and combined with high frequency levelling systems, for the imag-
ing of RE shallow layers (detailed stratigraphy, water content and presence
of mud) (see e.g. Eriksen et al., 2010; Anbazhagan et al., 2011; Hugenschmidt
et al., 2013; Khakiev et al., 2014; De Bold et al., 2015). This technique how-
ever suffers limitations in depth of investigation (DOI) in high permittivity
fine materials, such as clays which attenuate the signal and prevent access
to underlying soils. In a similar manner, geophysical approaches of possi-
ble high yields, such as electromagnetic and capacitive electrical prospecting methods (Rejkjær et al., 2021), are very difficult to apply along railways because of their sensitivity to metallic elements, underground cables, electrical wiring systems and, of course, catenaries. Electrical Resistivity Tomography (ERT) however consists in a possible tool of interest for the study of RE, as described in details by Gunn et al. (2018). Other techniques such as micro-gravimetry have been suggested by geophysicists to target cavities and/or undercompacted areas (Fauchard et al., 2004; Talfumiere and Nebieridze, 2008; Nebieridze and Leroux, 2012). Yet, none of the previously mentioned methods provide information about the RE mechanical properties in the standard framework of geotechnics.

## 2.2. Seismic method

The mechanical parameters of shallow soil layers (compression and shear moduli) can be estimated *in situ* using seismic methods. They are based on the indirect characterization of seismic-wave propagation velocities from seismograms (records of the wavefield by means of a collection of sensors implanted at the surface and detecting the particle displacements generated by a seismic source) (see Mari et al., 2004). The refraction of body waves (compressional or pressure (P) and/or shear (S) waves), for instance, makes it possible to simply define near-surface geometry, as well as its compressional and/or shear velocities (respectively $V_p$ and $V_s$). Although considered fast to implement and relatively straightforward to process, these techniques suffer from several limitations (see e.g. Mari et al., 2004; Fauchard et al., 2004). The existence of civil engineering structures with a strong 3D character and/or offering important velocity contrasts, in the vicinity of the acquisition setup
(at the surface as well as at depth), complicates the interpretation of seismograms (it becomes difficult to differentiate the various events of the wavefield which can either be due to the investigated layers at depth or to lateral '3D objects'). Conversely, refraction seismic implies the assumption of increasing velocities with depth. Lower velocity layers (LVL) will not be properly characterized and high velocity layers (HVL) will perturb the investigation of deeper structures of interest. In refraction seismic as well, accurate event detection requires a good signal-to-noise ratio (sometimes impossible to obtain in urban and suburban conditions). The study of S waves is all the more delicate, because they are more difficult to generate and identify on this type of records (Pasquet et al., 2015). It appears that the above-mentioned limitations restrict the applicability of refraction seismic (Mari et al., 2004; Fauchard et al., 2004, and moreover reflection seismic) to the characterization of RE that are typically affected by the presence of 3D structures (concrete channel for drainage and also for cables, plateforms, acces structures), HVL, LVL (high contrast between the sub-ballast layer and the capping layer), potentially significant background noise (urban context and catenary poles vibrations), etc.

From a theoretical point of view, the elastic strain generated by a mechanical dynamic source does not propagate in the Earth as body waves only. Actually, most of the energy propagates along the surface without penetrating deeply (in the upper ten meters for near-surface applications). If the mechanical properties (and more particularly the shear modulus) of the medium are vertically heterogeneous, the propagation velocity of these guided waves (the 'surface waves', also named 'PSV' or 'Rayleigh' waves in the following)
will depend on their propagation frequency. This property, called dispersion,
allows us to retrieve, *via* an inversion procedure, the V<sub>s</sub> structure of the
probed medium, below the acquisition setup. Active-source SW dispersion
measurements can be achieved using typical seismic shot gathers (Foti et al.,
2018). The inferred V<sub>s</sub> models are ‘one-dimensional’ (1D) and measurements
and interpretations are limited by several well-known theoretical and exper-
imental constraints (O’Neill, 2003; O’Neill et al., 2003; Socco and Strobbia,
2004; Bodet et al., 2005, 2009). SW methods, widely used in seismology
since the 1950s, have gone through important theoretical and experimental
developments for a wide range of near-surface applications since the 1980s
(non-destructive evaluation, civil and geotechnical engineering, environmen-
tal geophysics, natural hazards, see Socco et al., 2010). The first applications
of this technique were actually largely dedicated to the *in situ* characteriza-
tion of soils and near-surface anthropogenic structures (Heisey et al., 1982;
Nazarian and Stokoe II, 1984; Matthews et al., 1996; Hévin et al., 1998; Lai
et al., 2002; Ryden et al., 2001, 2004). The latest methodological advances
show that ‘surface-wave prospecting’ or MASW can be deployed along linear
sections or along a surface grid, in order to reconstruct a ‘two-dimensional’
(2D) or 3D model of near-surface V<sub>s</sub> distribution (Neducza, 2007; Boiero and
Socco, 2010; Ezersky et al., 2013). A detailed presentation of the method,
SW acquisition, processing and inversion workflows is for instance given by
Pasquet and Bodet (2017), along with a tutorial for most of the codes and
tools used in the following work. Further information and details can be
found in the literature (see e.g. O’Neill, 2003; O’Neill et al., 2003; Socco and
Strobbia, 2004; Bodet et al., 2005; Socco et al., 2010; Pasquet, 2014).
Given the guided nature of surface waves, these events of the wavefield
are less sensitive to the strong 3D character of the structures encountered in
civil engineering applications (Karl et al., 2011). In addition, recent studies
show a growing interest in the implementation of SW prospecting related
to geotechnical issues (Heitor et al., 2012) or in a railway context (Dono-
hue et al., 2013, 2014; Hwang and Park, 2014; Forissier, 2015; Gunn et al.,
2015; Bergamo et al., 2016a,b; Gunn et al., 2016; Sussmann Jr et al., 2017;
Kyrkou et al., 2022), in which V<sub>S</sub> is a very good criterion for determining
the mechanical state of associated materials (see e.g. Dhemaied et al., 2014a;
Byun and Tutumluer, 2017; Byun et al., 2019). However, given the spe-
cific railway environment and the geometry of the targeted structures, the
experimental conditions limit the classical implementation of seismic meth-
ods (instrumental deployments are restricted both in space and time). An
optimal setup configuration has thus to be defined by adapting the usual
experimental protocol to the scale of RE and to the specific railway environ-
ment, while ensuring sufficient resolution for the measured velocities to be
informative. Seismic methods can actually be applied to very shallow targets
(< 10 m depth) using various types of sources, sensor and acquisition setups,
depending on the objectives (see e.g. Bachrach et al., 1998; Baker et al., 1999;
Abraham et al., 2004; Schmelzbach et al., 2005), the key parameter being the
measurement frequency range. Pasquet (2014); Dangeard (2019), in the same
way as studies cited above, illustrated that a classical near-surface seismic
sources such as a 5 kg sledgehammer (or even smaller 1.25 kg hammers avail-
able) can generate a broadband signal at frequencies high enough to reach
infra-metric vertical resolutions, in media with propagation velocities lower
than 250 m.s⁻¹. In the case of RE, according to Dhemaied et al. (2014a) geotechnical results and laboratory tests, the seismic velocities can be higher than in natural unconsolidated soils (up to 600 m.s⁻¹). Care must therefore be taken to ensure that the frequencies are high enough to work at reasonable wavelengths, as for instance shown in Dangeard et al. (2021). In the following work, the strong *a priori* knowledge about RE structure along HSL and the detailed geotechnical study first presented, has made it possible to: (i) check if the setup was sufficiently simple and quick to implement (e.g. adapted to both spatial and temporal constraints along railways such as trackside occupancy limits, safety issues, operational constraints, etc.); (ii) estimate signal resolutions and anticipate for real-life data acquisitions; and (iii) verify the results provided by the inversion of SW dispersion.

## 3. Feasibility study along a high-speed line

### 3.1. Context of the test site

This study focuses on a specific location (see Figure 1) along the French north HSL (north of the town of Hattencourt) affected by an unusually frequent and local need for maintenance, more particularly observed on track 2 (V2 on Figure 1). The areas where regular and too frequent 'clogging operations' were needed (white dashed lines on Figure 1) were also strongly affected by local vertical motion of rails and sleepers at each train passage (solid white lines labelled as 'the phenomenon' on Figure 1, as observed during the geophysical survey described in the following). Since no track failures (rail, sleepers) or defects in the surface structures of the RE were observed in this area, the origins of such anomalies were assumed to be related to strong
224 variations in the nature or drainage of the underlying soils.

225 A geological and geotechnical survey has thus been suggested to better
226 characterize the RE and underlying soils in the areas identified as problematic
227 (Dhemaied et al., 2014a,c). The RE has been built on ‘natural’ loess with the
228 typical structure defined by the SNCF Réseau standards (IN3278, IG90260)
229 in terms of both geometry and mechanical properties, as shown in Figure 2
230 with (from top to bottom): the sub-ballast layer, the capping layer and the
231 subgrade (mainly consisting in *in situ* soil) overlying the lithological units
232 described below from core samples:

233 1. quaternary earthwork mainly from the RE construction and maintenance;
234

235 2. loess characterized by beige silts more or less clayey, and;

236 3. campanian chalk, whitish, poor in flint and possibly containing fine glaucous passages.
237

3.2. *Standard geotechnical tests*

239 The geotechnical study included 8 boreholes down to 3 and 12 m depths (black circles on Figure 1) as well as 5 dynamic cone penetrometer tests (DCP) 12 m deep (not presented here, see Dhemaied et al., 2014a,c; Heraibi, 2019, for more details). These *in situ* tests were complemented by laboratory measurements such as Atterberg limits, granulometry, proctor compaction tests, methylene blue values, wet bulk density and water content estimations. The results, presented by Dhemaied et al. (2014a,c), were coherent with the typical structure of HSL RE (see the Figure 2), mainly involving vertical contrasts under the ballast. These structure corresponds to: the sub-
<figure><img src="image_2.png" /></figure>

Figure 1: (a) Schematic map (modified from Bodet et al., 2017; Bodet, 2019) of the boreholes cores drillings (CD#) and seismic profiles (P1 to P7) locations. The positions of which are given according to the referenced kilometre points (KP). Local vertical motion of rails and sleepers observed at each train passage is showed by white plain lines (the phenomenon). Frequently maintained track sections are represented by the white dashed line. Photographs are given to illustrate the setups installation along the trackside: (b) preparation of the surface at each geophone location; (c) seismic source with a 1.25 kg sledgehammer vertically hitting a metallic plate; (d) setup installed along the track side.
<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic cross-section (modified from Bodet et al., 2017; Bodet, 2019) of the studied RE. The typical structure is defined by *SNCF Réseau* standards (IN3278, IG90260). The details about layers materials are based on ‘visual’ monitoring data and interpretation of core samples.</figcaption></figure>
248 layer and the capping layer (with a total thickness of 75-80 cm on average along the track), more compact than the loess layer, itself less compact than the chalk occurring between 6 and 7 m deep under the tracks. The results also showed similar water contents and densities in the loess (respectively 20 % and between 1.6 and 1.7 Mg/m³ between 1 and 3 m deep) whatever the locations of the drilling. The boreholes as well as the laboratory tests show neither significant variability of the RE structure along V2, nor from one trackside to another (see Dhemaied et al., 2014a,c). This type of geotechnical study therefore did not make it possible to identify the origins of the maintenance problem encountered along this line.

### 3.3. Shear moduli (Bender Elements) and porosimetry

To complement the classical geotechnical study summarized above, Vs measurements on samples that could be properly collected from the cores (cylindrical samples of 100 mm diameter and 50 mm height) were carried out using the 'Bender Elements' (BE) technique (Shirley and Hampton, 1978). The BE are piezoceramic plates allowing the transmission and reception of a mechanical wave through a cylindrical sample to estimate its propagation velocity (Vp or Vs according to the transmission/reception mode). This technique is widely used for the estimation of small strain shear modulus (G) in non-consolidated granular materials in general, and is the subject of active research for the study of soils in particular (see e.g. Lee and Santamarina, 2005; O'Donovan et al., 2012). The impact of water content on the propagation velocities of mechanical waves in soils was also investigated in the laboratory using this technique (Santamarina et al., 2005; Cho and Santamarina, 2001) and/or using 'acoustic' methods (see e.g. Fratta et al., 2005;
George et al., 2009). The use of BE techniques has proved interesting for the laboratory or *in situ* estimation of V<sub>S</sub> (Byun and Tutumluer, 2017; Byun et al., 2019) in RE. G, which is an important criterion for the design and monitoring of RE's condition, can then be calculated when the density (ρ) is known as well, such as

$$G = \rho V_S^2. \qquad (1)$$

For the present study, all samples were collected from the loess layer at one or several depths (depending on cores quality). The results, combined with water content and density measurements, are summarized in table 1. They show that the decrease in V<sub>S</sub> (and consequently in G) was partly correlated with the maintenance anomaly observed along V2. Additional tests (carbonate content) showed that these results are not linked to the treatment of the loess. Finally, mercury porosimetry tests were carried out on these samples. The porosity spectra presented on Figure 3 show an increase of equivalent pore diameter, thus a variability of the microstructure, also partly consistent with the maintenance variations along V2.

### 3.4. Upscaling for *in situ* characterization of RE over long distances

Such analysis unfortunately remain sparse and time consuming, both in distance along the tracks and in depths at each sample collection location. The CD1 sample collected on V1 at 2.6 m depth (area not affected by the phenomenon, see table 1 and Figure 1), was characterised by the highest G value while its porosity spectrum was spread out. The single sample collected and analysed for CD5 showed a high G value with a marked maximum porosity while located in the problematic area. To avoid such ambiguities, regular
<table><caption>Table 1: Measurements of density ($\rho_d$), water content ($w$) and orders of magnitude of velocities and shear moduli ($V_{S,min-max}$ and $G_{min-max}$, respectively) measured with BE on samples collected from boreholes at the study site. The depths correspond to the depth of the core samples centers (modified from Dhemaied et al., 2014a,b; Bodet et al., 2017; Bodet, 2019).</caption><thead><tr><th>Core<br>Drilling<br>(#)</th><th>Sample<br>depth (m)</th><th>w (%)</th><th>ρ<sub>d</sub> (Mg/m<sup>3</sup>)</th><th>V<sub>S,min-max</sub><br>(m.s<sup>-1</sup>)</th><th>G<sub>min-max</sub><br>(MPa)</th></tr></thead><tbody><tr><td>CD1</td><td>1.6</td><td>20.6</td><td>1.65</td><td>120-134</td><td>28-35</td></tr><tr><td>CD1</td><td>2.3</td><td>15.7</td><td>1.73</td><td>125-137</td><td>30-36</td></tr><tr><td>CD1</td><td>2.6</td><td>15.7</td><td>1.73</td><td>596-687</td><td>675-897</td></tr><tr><td>CD2</td><td>2.6</td><td>19.8</td><td>1.60</td><td>165-183</td><td>52-64</td></tr><tr><td>CD3</td><td>1.6</td><td>18.1</td><td>1.71</td><td>152-172</td><td>44-56</td></tr><tr><td>CD5</td><td>1.5</td><td>20.1</td><td>1.67</td><td>312-366</td><td>185-255</td></tr><tr><td>CD6</td><td>1.1</td><td>20.9</td><td>1.70</td><td>153-173</td><td>44-57</td></tr><tr><td>CD8</td><td>2.3</td><td>19.3</td><td>1.57</td><td>296-378</td><td>166-271</td></tr><tr><td>CD8</td><td>2.9</td><td>15.4</td><td>1.65</td><td>357-390</td><td>242-289</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>Figure 3: Mercury porosimetry measurements carried out on samples collected from bore-holes (see locations on Figure 1). The depths noted in the legend correspond to the depth of the core samples centers (modified from Dhemaied et al., 2014a,b; Bodet et al., 2017; Bodet, 2019).</figcaption></figure>
sampling (along the tracks as well as at depth) would be necessary. Such
studies are obviously not feasible in a systematic manner for the monitoring
of RE along great distances (typically tens or hundreds of kilometres). In
this context, it seems justify to test seismic SW prospecting methods for a
systematic non-destructive *in situ* characterisation of V<sub>S</sub> along RE. The cho-
sen test site is moreover extremely well constrained by HSL standards. Its
recent construction (with respect to the age of conventional lines) has allows
to have strong *a priori* information on the characteristics of the *in situ* soil
(geometry of the RE, nature of materials, etc.) and make it convenient to
perform such a feasibility study.

4. Surface-wave prospecting: implementation and dispersion mea-
surements

308
4.1. *Setup and measurement strategy*

With dimension and acquisition parameters based both on the *a priori*
well known structure of HSL RE and on properties provided by the geotechni-
cal results presented above, 7 seismic profiles of identical length were carried
out along V2 (see Figure 1a). The setup for each seismic profile consisted
of 72 vertical component geophones (14 Hz low cut-off frequency) spaced with
0.25 m, so as to form a 17.75 m long profile. The setups were implanted along
the trackside to ensure a good geophone coupling with near-surface materi-
als and to avoid the particular conditions of acquisition directly on ballast
(Hwang and Park, 2014; Forissier, 2015). For each seismic profile, seismic
shots were performed with a 1.25 kg hammer vertically hitting a metallic
plate (see Figure 1c) every 24 geophones (between adjacent geophones) with
320 first and last shot positions at half inter-geophone distance before and after
321 the first and last geophone, respectively. Each shot corresponded to the stack
322 of 6 hammer/plate impacts to improve the signal-to-noise ratio. The record-
323 ing length was 2 s with a sampling interval of 0.5 ms and a pre-triggering
324 delay of -0.02 s. The seismic profiles numbering, P1 to P7 (see Figure 1)
325 corresponded to the implementation strategy decided during the survey:

326 • it was initially chosen to locate the first seismic profile at the middle of
327 the area presenting the higher number of maintenance operations (P1
328 in Figure 1) and centered on CD5;

329 • then, another seismic profile was located at the middle of an area free
330 of maintenance problems (P2 in Figure 1) and centered on CD8;

331 • 5 other seismic profiles finally made it possible to observed continuous
332 variations between these two end-members profiles, with P5 slightly
333 south of the high maintenance area where highly active anomalies were
334 spotted during the survey.

335 The whole distance covered by these 7 seismic profiles thus represented a
336 120.9-m long seismic line (which is less than 7×7.75 m because of an overlap
337 between P7 and P2 due to the original choice of centering P1 on CD5 and
338 P2 on CD8). The acquisition parameters and the relative positions of the
339 seismic profiles are shown in Figure 4.

340 The first task carried out following this survey consisted in choosing the
341 appropriate geophone window size for the SW dispersion extraction from the
342 recorded sets of seismograms. This method is traditionally used to ensure
343 the best compromise between expected vertical resolution and the need to
<figure><img src="image_5.png" /></figure>

Figure 4: Schematic layout plan of the whole entire seismic line and description of a single seismic setup with 72 receivers (in blue on the left of the figure). The names of seismic profiles are written above each one. The color scale represents the position along the line with the southern seismic profiles in blue and the northern seismic profiles in orange. The position ‘0 m’ represent the beginning of P1. The boreholes cores drilling (CD#) located above the seismic profile are represented by black circles and with black arrow when they are in the same position but offset from the track. Local vertical motion of rails and sleepers observed at each train passage (the ‘phenomenon’) is showed by red plain lines. Frequently maintained track sections are represented by red dashed lines.

limit the influence of lateral variations on measurements as explained in details for instance in Pasquet and Bodet (2017) (the term ‘lateral variations’ is sometimes used by practitioners to refer to variations encountered ‘along’ the seismic profile). SW dispersion analysis is indeed limited by the classical trade-off between lateral resolution and investigation depth (Gabriels et al., 1987). On the one hand, the inverse problem formulation imposes the investigated medium to be assumed 1D under the geophone spread. Additionally, the spread itself has to be short enough to achieve lateral resolution if profiling is performed. On the other hand, long geophone spreads are required in order to: obtain high resolution dispersion spectra; record wavelengths great enough to reach expected investigation depth; mitigate near-field effects; and discriminate modes at every available frequencies (see e.g. O’Neill
et al., 2003; O'Neill, 2003; Bodet et al., 2005, 2009). Several window sizes were actually processed before eventually selecting the largest one with 72 geophones, as it offered the best spectral resolution (for more details about the process, see e.g. Pasquet and Bodet, 2017; Dangeard, 2019). The choice of this window size gives a empirical maximum investigation depth (Zmax) of 8.875 m (based on the maximum measurable wavelength (λmax), approximately equal to the length of the stacking window, divided by two, Z ≈ λ/2) (Bodet et al., 2009; Pasquet and Bodet, 2017; Foti et al., 2018). In addition, as recommended by Bodet et al. (2005) and recalled by Steinel et al. (2014), direct and reverse shots on both sides of a given spread (end-on and off-end) have to be considered to check the validity of the 1D assumption according to considered wavelengths.

### 4.2. Seismograms

For the survey presented here, the longest spreads consisted in 72 geophones. With our acquisition setup, only 2 shots were available on both side of the spread to be systematically compared for each seismic profile. This work is illustrated here for P1 and P2 which will systematically serve as examples, since they are supposed to represent two very different states of the studied RE (the disorder zone, P1, and an area with no apparent problem, P2). Figures 5a and 5b present the 72 geophones seismograms obtained with both shots for P1 and P2, respectively. The seismic traces look symmetrical as far as the guided waves are concerned, whether it is P1 or P2. The wavefield is very clearly disturbed by the catenary pole foundations at kilometre point (KP) xxx.475 for P1 (at 5 m in local coordinates). The signal recorded along P2 is perturbed by 50 Hz noise probably because of the buried electri-
<figure><img src="image_6.png" /></figure>

Figure 5: (a) Normalized seismograms (top) and spectrograms (bottom) of the direct and reverse shots for P1 (the geophone positions are given in the local coordinate system shown in Figure 4). The seismic traces are symmetrical with respect to the guided wave train. The spectrograms represents the frequency band along the seismic profile in which the wavefield is sufficiently energetic (maxima of amplitude in black) to allow dispersion interpretation. In (a), the wavefield is very clearly perturbed by the pole foundations at kilometre point (KP) xxx.475, around the geophone located at 5 m. In (b), the P2 shots show 50 Hz noise (probably due to the buried electrical networks and contact of the connectors with the wet ground after rain).

cal networks and a possible contact of the geophone connectors with the wet ground after rain during the survey. Strong ambient noise generated by the highway in the east produced low frequency wave fronts on both P1 and P2. As shown on spectrograms of Figures 5a and 5b, the frequency content of the data is very satisfactory as far as dispersion analysis is concerned, with significant energy from 20 Hz to at least 250 Hz.
387 **4.3. Dispersion extraction**

388 Each of the seismograms was then transposed, after correction for geomet-
ric attenuation, into the frequency-phase velocity domain. The result of such
389 a wavefield transformation is called a ‘dispersion image’ (see the tutorial of
390 Mokhtar et al. (1988) for more details about the slant-stack in the frequency
domain). The dispersion images obtained for each shot of seismic profiles
391 P1 and P2 are presented in Figures 6a,b and 6d,e respectively. The spectro-
grams are reproduced on these figures to show the frequency band in which
392 the wavefield is sufficiently energetic to allow dispersion interpretation. Here
393 again, whether it is P1 or P2, the direct and reverse shots produced very sim-
ilar results. In this domain, the maxima (in red) on Figure 6) correspond to
394 the dominant events of the wavefield: the PSV waves. It appears that a very
395 large number of propagation modes are available in these images. But the
396 difficulty of identifying their rank and the effective nature of the dispersion
397 (Lai and Rix, 1998; O’Neill, 2003; Lin and Ashlock, 2016) only made it pos-
398 sible to pick the first two modes (in red and white on Figures 6c and f). The
399 RE offers waveguides that are favourable to the emergence of higher modes,
400 such as guided P waves, which could be interpreted in more advanced work
401 (O’Neill et al., 2003; Maraschini et al., 2010; Boiero et al., 2013) but would
402 involve specific processing tools not adapted to the RE context yet. The
403 maxima of the dispersion image resulting from the summation of the two
404 shots are also given (Figure 6c for P1 and Figure 6f for P2) and compared
405 with the individual picks on Figure 7. It clearly shows that the obtained
406 dispersion curves are identical (taking the errors in dispersion measurements
407 into account, according to the relationship introduced by O’Neill (2003) and
408
409
410
411
essentially based on the resolution of the wavefield transformation).

These results validated the hypothesis of weak lateral variations along the 72 geophones window (compared to involved wavelengths). They validated the stacking of dispersion images from direct and reverse shots, for both the improvement of the signal-to-noise ratio and the mitigation of near-offset effects. Thanks to such stacking, it was also possible to preserve the highest frequency part of coherent wavefields present in the near-offset traces (O'Neill, 2003; Bodet et al., 2005, 2009). They also proved the weak influence on the dispersion analysis, of the noises identified on the seismograms as well as the repeatability of the measurement. As mentioned earlier, this work was systematically repeated for each seismic profile.

### 4.4. Dispersion variability along the line

As suggested by the seismograms presented previously (see Figure 5), the dispersion images and the associated picked curves obtained for P1 (Figure 6a, b and c) and P2 (Figure 6d, e and f) show a very clear difference in terms of dispersion. While the first higher mode (M1 on Figure 8) may be common to both seismic profiles for a few points, the fundamental mode (M0 on Figure 8) is clearly different from one area to another, with lower phase velocities in P1 than in P2. Since M0 is the one that reflects (very empirically) the properties of the medium, those measured in P1 and P2 suggest lower $V_s$ in the area affected by maintenance issues. The set of dispersion data obtained for the 7 seismic profiles is finally presented on Figure 9 with a color scale according to the position along the line. Again, only M0 and M1 were safely enough identifiable in dispersion images to be picked (O'Neill and Matsuoka, 2005). The observed dispersion covers a range of wavelengths
<figure><img src="image_7.png" /><figcaption>Figure 6: Normalized dispersion images and spectrograms of the direct and reverse shots (a and b for P1, d and e for P2) and stacked images for P1 (c) and P2 (f). Among the large number of propagation modes appearing on these images, only the fundamental mode (in black) and the first higher mode (in white) were picked (the errors in dispersion measurements are given empirically according to the relationship suggested by O'Neill (2003), essentially based on the resolution of the wavefield transformation). Limits of dispersion image analysis ($\lambda_{min}$ and $\lambda_{max}$) are plotted in black and white dotted lines respectively. The spectrograms represent the frequency band along the seismic profile (Gx) in which the wavefield is sufficiently energetic (maxima of amplitude in red) to allow dispersion interpretation.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Dispersion data (modified from Bodet et al., 2017; Bodet, 2019) plotted for each shot at P1 (a) and P2 (b) compared to those picked from stacked dispersion images. The fundamental mode (M0) is shown in black while the first higher mode (M1) is shown in red. The data were resampled in wavelength.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8: Comparison of the dispersion curves picked on the stacked images for P1 in blue (see Figure 6c) and P2 in orange (see Figure 6f). Only the fundamental mode (M0) and the first higher mode (M1) were considered. The data were resampled in wavelength and the error calculated according to the relation of O'Neill (2003).</figcaption></figure>
(0.3 to 18 m) considered as representative of the whole RE, including the upper part of the substrate. Particular attention has been paid to the dispersion curves picking, e.g. by taking care to respect the limits related to the resolution of the wavefield transformation and near-offsets effects, as recommended by O'Neill (2003); Bodet et al. (2005, 2009) for instance. The data show a partitioning of the dispersion that clearly corresponds to the observed differences in maintenance as shown on Figure 9, with lower phase velocities in blue in the phenomenon area, and higher phase velocities in orange in the northern part of the line, with no apparent problems.

<figure><img src="image_10.png" /><figcaption>Figure 9: Dispersion curves (resampled in wavelength) plotted for every seismic profile (P1 to P7). Only the fundamental mode (M0) and the first higher mode (M1) were considered. The colorscale represents the location of seismic profiles along the line: the orange profiles are located further south (disorder area) and the blue profiles are located further north (healthy area), see Figure 4.</figcaption></figure>

This is a first qualitative answer to the methodological questions posed
447 by the study, at least as far as this particular site is concerned. The following section will provide a more quantitative interpretation of these phase velocity measurements, by estimating 1D $V_s$ models centered on each seismic profile, solving the classical SW dispersion inverse problem.

## 5. Estimating $V_s$ along the line: the inverse problem

### 5.1. Extracted dispersion as data space

For each seismic profile, the data vector ($d^{obs}$) to be inverted consist in $N_d$ phase velocity measurements ($V\phi_i^{obs}$):

$$d^{obs} = [V\phi_1^{obs}, V\phi_2^{obs}, ..., V\phi_i^{obs}, ..., V\phi_{N_d}^{obs}]. \quad (2)$$

When 2 modes are picked (M0 and M1), $N_d$ is the total number of velocity measurements for M0 and M1 in their respective frequency range which can overlap or not, depending of the seismic profiles. Frequency measurements are independent variables of the problem. For each seismic profile, frequency vectors are simply sorted the same way as $d^{obs}$. They thus include $N_d$ values as well, possibly redundant when M0 and M1 are picked at the same frequencies. $V\phi_i^{obs}$ is expressed with an uncertainty $\sigma_i$ simulated according to the relation of O'Neill (2003). This phase velocity uncertainty is extracted from the dispersion images and actually varies non-linearly with frequency, from the order of more than 30 % of the velocity at low frequencies to less than 1 % at high frequencies, as shown by the error bars on the selected dispersion curves for P1 and P2 on Figure 8. The uncertainties actually follow a Lorentzian distribution type at low frequencies (O'Neill, 2003), usually below 25 Hz for seismic acquisition setup used here. The implementation
469 of this type of uncertainty allows accounting for the poor resolution of low frequency, possible mode mis-identifications, as well as near-offset effects.

471 **5.2. A priori information on RE and associated setup of the parameter space**

472 HSL recently built in France (since the 1990s) are structurally well con-
strained (IN3278, IG90260), as presented in section 3.1 (see Figure 2). This
473 provides very strong a priori information such as the number of near-surface
474 layers, their thicknesses (H), or the approximate depth of the bedrock when
475 geotechnical soundings are available. Based on this knowledge, the soil model
476 of the studied RE has been built with 4 layers: the sub-ballast layer (in-
477 index 1) ; the capping layer (index 2) ; a loess layer (index 3) and the half-space
478 of chalk (index 4) (see Figure 2). The actual parameters of this soil model,
479 as far as SW dispersion inversion is concerned, are then: Hj, Vsj, Vpj and ρj
480 for each soil layer j. Finally, this represents a set of 15 parameters for this
481 studied RE.
482

483 Ground surveys by SNCF Réseau confirmed the thicknesses in the study
484 area to be in agreement with the reference thicknesses from construction
485 standards for new HSL : 0.20 m and 0.55 m in the sub-ballast layer and the
486 capping layer respectively (Dhemaied et al., 2014a). ρ and Vp are considered
487 as parameters of weaker influence on SW velocity which mainly depends on
488 the V<sub>S</sub> structure (see e.g. the sensitivity study of Socco and Strobbia, 2004).
489 Density values were then set according to the construction standards as well.
490 No information was available on V<sub>p</sub> and Poisson's ratio (ν) was set to 0.33
491 (which is a classical value in soils) for all layers. Table 2 summarises the
492 fixed parameters finally chosen for this study.

493 The integration of all the *a priori* information on parameter values al-
<table><caption>Table 2: Parameters fixed according to available a priori information for the considered type of RE</caption><thead><tr><th>Description of the parameter (units)</th><th>Name</th><th>Value</th></tr></thead><tbody><tr><td>Thickness of the sub-ballast layer (m)</td><td>H<sub>1</sub></td><td>0.2</td></tr><tr><td>Thickness of the capping layer (m)</td><td>H<sub>2</sub></td><td>0.55</td></tr><tr><td>Poisson's ratio in every layer</td><td>ν</td><td>0.33</td></tr><tr><td>Density of the sub-ballast layer (kg/m<sup>3</sup>)</td><td>ρ<sub>1</sub></td><td>2210</td></tr><tr><td>Density of the capping layer (kg/m<sup>3</sup>)</td><td>ρ<sub>2</sub></td><td>2160</td></tr><tr><td>Density of the loess layer (kg/m<sup>3</sup>)</td><td>ρ<sub>3</sub></td><td>2150</td></tr><tr><td>Density of the chalk layer (kg/m<sup>3</sup>)</td><td>ρ<sub>4</sub></td><td>2150</td></tr></tbody></table>

494 lowed us to eventually constrain the model space to only 5 parameters of interest (the others being either fixed or depending on them) and thus form the following model vector **m**,

$$ \mathbf{m} = [V_{s1}, V_{s2}, V_{s3}, V_{s4}, H_3]. \qquad (3) $$

The limits of the model space were also set according to the a priori information on this RE (see table 3). The borehole data (Dhemaied et al., 2014a) indicated that the thickness of the loess layer (H₃) was between 3 and 7 m. The top of the half-space depth was consequently chosen in agreement with the classical empirical relations giving the maximum investigation depth as used by practitioners and mentioned in section 4.1 (Zmax ≈ λ/2) (see e.g. Bodet et al., 2009; Pasquet and Bodet, 2017). Preliminary inversions with greater degrees of freedom (Bodet et al., 2017; Bodet, 2019) suggested that Vs1 and Vs2 should not not exceed 300 m.s⁻¹ and that Vs3 should vary from 180 to 350 m.s⁻¹, depending on the considered zone (as confirmed by
<table><caption>Table 3: Parameter space, boundaries and sampling interval: shear-wave velocity of the sub-ballast layer ($V_{s1}$), shear-wave velocity of the capping layer ($V_{s2}$), shear-wave velocity of the loess ($V_{s3}$), shear-wave velocity of the chalk ($V_{s4}$) and thickness of the loess ($H_3$). This parameterisation creates a space composed of $N_m = 907,200$ possible models.</caption><thead><tr><th>Parameter name (unit)</th><th>Min value</th><th>Max value</th><th>Sampling interval</th><th>Number of possible values</th></tr></thead><tbody><tr><td>V<sub>s1</sub> (m.s<sup>-1</sup>)</td><td>50</td><td>270</td><td>11</td><td>21</td></tr><tr><td>V<sub>s2</sub> (m.s<sup>-1</sup>)</td><td>60</td><td>130</td><td>7</td><td>10</td></tr><tr><td>V<sub>s3</sub> (m.s<sup>-1</sup>)</td><td>180</td><td>350</td><td>10</td><td>18</td></tr><tr><td>V<sub>s4</sub> (m.s<sup>-1</sup>)</td><td>300</td><td>600</td><td>20</td><td>15</td></tr><tr><td>H<sub>3</sub> (m)</td><td>3</td><td>7</td><td>0.25</td><td>16</td></tr></tbody></table>

507 BE data). $V_{s4}$ was authorized to vary in a range of 300 m.s⁻¹ to 600 m.s⁻¹, as suggested by maximum apparent velocities observed in the data. The velocity sampling step (see table 3) was defined according to the dispersion resolution and uncertainty: with greater steps for deeper layers empirically corresponding to the dispersion response at low frequencies and; with smaller steps for shallow layers empirically corresponding to the dispersion response at high frequencies. According to empirical relationships used by the community as well (see e.g. O'Neill, 2003; Bodet et al., 2005; Dangeard et al., 2018), the best possible resolution in thickness was estimated equal to the seismograms space sampling interval (25 cm). This parameterisation finally leads to a space of $N_m = 907,200$ possible models.
5.3. Retrieving V_s from dispersion data in a Bayesian framework

From the parameter space described above, a basic grid search algorithm is used to build the following misfit function:

$$ misfit(\mathbf{m}) = \sum_{i=1}^{N_d} \frac{(V\phi_i^{calc} - V\phi_i^{obs})^2}{\sigma_i^2}, \quad (4) $$

where $V\phi_i^{calc}$ are the elements, at each measurements frequency, of the synthetic data vector ($d^{calc}$) computed for $\mathbf{m}$ using the Thomson-Haskell matrix propagator method (Thomson, 1950; Haskell, 1953), as classically used to provide theoretical SW dispersion curves in horizontally stratified media. An example of the results provided by the grid search is given in Figure 10. It shows all dispersion curves and associated V_s models represented according to their corresponding misfit value, for the data collected along P1 (Figure 10a, b) and P2 (Figure 10c, d). In theory, the solution of the inverse problem (if it exists) is the minimum of the misfit function.

Misfit minima for the parameter V_s1 are observed around 80 m.s⁻¹ for P1 and 210 m.s⁻¹ for P2. In contrast, V_s2 for P1 shows minimum misfit values around 100 m.s⁻¹ while for P2 the minima are around 75 m.s⁻¹. As expected from BE data, the minima of the misfit show higher V_s3 for P2 than for P1, with velocity of the order of 300 m.s⁻¹ and 210 m.s⁻¹ respectively. The parameters H3 and V_s4 do not present distinct misfit minima for their part, which means that it is not possible to find unique valuable values for these parameters and that the chalk layer might be deeper than the maximum investigation depth.

In order to present these grid search results with an estimation of their reliability, they have been expressed, as suggested by Tarantola and Valette
<figure><img src="image_11.png" /><figcaption>Figure 10: Grid search results: at the top for P1 (a, b) and at the bottom for P2 (c, d). (a) and (c) show dispersion curves with the fundamental mode (M0) and the first higher mode (M1). (b) and (d) represent velocity models. Each dispersion curve and each velocity model are shown with a color depending on the misfit value between the data (black dots and error bars) and the simulated dispersion. The data were resampled in wavelength and the error has been calculated according to the relation of O'Neill (2003).</figcaption></figure>
541 (1982); Tarantola (2005), in terms of the a posteriori probability density function (pdf) $\sigma(\mathbf{m}|\mathbf{d}^{\text{obs}})$, through the application of the Bayes' theorem:

$$ \sigma(\mathbf{m}|\mathbf{d}^{\text{obs}}) = \frac{p(\mathbf{d}^{\text{obs}}|\mathbf{m})\rho(\mathbf{m})}{\int_{\mathcal{M}} p(\mathbf{d}^{\text{obs}}|\mathbf{m})\rho(\mathbf{m})d\mathbf{m}}, \quad (5) $$

543 where $\sigma(\mathbf{m}|\mathbf{d}^{\text{obs}})$ estimates the probability to recover the 1D $V_S$ vertical structure of the RE below each seismic profile along the line, knowing the phase velocity data and associated uncertainties. $\rho(\mathbf{m})$ is the a priori pdf which contains all the information available on the RE structure and properties of the subsurface even before acquiring the phase velocity data. The actual a priori are the limits (minimum and maximum possible values) of the parameter space $\mathcal{M}$, as described earlier in Table 3. It thus consists in the uniform a priori and has been defined homogeneous throughout the model space with

$$ \rho(\mathbf{m}) = 1/N_m. \quad (6) $$

551 $p(\mathbf{d}^{\text{obs}}|\mathbf{m})$, on its side, is the function which sets the likelihood of every 1D $V_S$ vertical models of the RE as defined in the parameter space, when the phase velocity dispersion that could be calculated from them ($\mathbf{d}^{\text{calc}}$) is compared to actually measured data ($\mathbf{d}^{\text{obs}}$). As mentioned above, the comparison was performed (in the grid search) considering that measured phase velocity data have uncertainties and that the visited models do not reproduce them perfectly, so the likelihood depends on the misfit function in the following manner:

$$ p(\mathbf{d}^{\text{obs}}|\mathbf{m}) = e^{-\frac{1}{2}\text{misfit}(\mathbf{m})}. \quad (7) $$

559 The a posteriori pdf $\sigma(\mathbf{m}|\mathbf{d}^{\text{obs}})$ are given as example for P1 and P2 on Figure 11 and Figure 12, respectively. These figures show, in a matrix format
561 for every parameters at once, every possible 2D marginal pdfs in order to re-
veal possible covariances, as well as the 1D a priori and a posteriori marginal
562 pdfs (on the diagonal of Figure 11 and Figure 12). For both P1 and P2, the
563 2D marginal pdfs show very clearly that the systematic grid search does not
564 suggest any distinct solution for H3 nor for V54. The velocities of the 3 first
565 layers seem on the other hand well defined in the a priori limits. The 2D
566 marginal pdfs indeed show Gaussian shapes with unique maxima, in partic-
567 ular quite distinct for the (V52, V53) couple. These two parameters appear
568 to be very different depending on the measurement position along the line.
569

570 These *a posteriori* pdf σ(m|d<sup>obs</sup>) have been computed for every seismic
571 profiles and the marginal *a posteriori* pdf associated to each parameters are
572 presented in Figure 13. As clearly shown on the 2D marginals, the parameters
573 V<sub>s4</sub> and H<sub>3</sub> are not well defined (1D marginals pdfs do not present Gaussian
574 shapes neither clearly identifiable maxima). The *a posteriori* marginal pdfs
575 of V<sub>s1</sub> do not seem to be well defined, as they does not show neither clear
576 Gaussian shapes or unique maxima, except for P3. It is therefore difficult to
577 interpret the behaviour of the mechanical state in the sub-ballast layer along
578 the study zone. The marginal *a posteriori* pdfs for V<sub>s2</sub> are not very well
579 defined neither. Yet, the pdfs suggest slightly higher V<sub>s2</sub> on the southern
580 part of the line compared to the northern part. In contrast, for V<sub>s3</sub>, pdfs
581 clearly show two distinct groups with lower velocities in the disorder zone
582 compared to the ‘healthy’ area. The most important feature here is that
583 these pdfs, which do not overlap in velocity (their width are different due to
584 the contrast of velocity, hence corresponding uncertainty), do show a parti-
585 tioning in very good agreement with BE data and analyses presented earlier.
<figure><img src="image_12.png" /><figcaption>Figure 11: Bayesian appraisal of the grid search for P1, presented as a cornerplot. The diagonal presents the *a posteriori pdf* (in orange) for each parameter (label and legend given along the x-axis, the y-axis giving the probability value). The homogeneous *a priori pdf* is given in blue dashed lines. The other plots represent the *a posteriori* marginal *pdf* per pair of parameters (labels and legend presented along the x and y-axis). The colorbar present the probability value. The grey plus sign indicates the model coordinates in the parameter space.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 12: Bayesian appraisal of the grid search for P2, presented as a cornerplot. The diagonal presents the *a posteriori pdf* (in orange) for each parameter (label and legend given along the x-axis, the y-axis giving the probability value). The homogeneous *a priori pdf* is given in blue dashed lines. The other plots represent the *a posteriori marginal pdf* per pair of parameters (labels and legend presented along the x and y-axis). The colorbar present the probability value. The grey plus sign indicates the model coordinates in the parameter space.</figcaption></figure>
As shown above, the parameter that shows the best definition of *pdf* maxima and strong partitioning is $V_{s3}$.

To go further, the marginal *a posteriori pdf* on G in the loess layer (G3) was inferred in a Bayesian framework as well and presented in Figure 14. These results show the same behaviour, i.e. G3 in the northern profiles is clearly higher than in the southern profiles. The results can be compared with the laboratory tests presented in section 3.3, yet only qualitatively since these moduli are inferred from soil samples with average values per layer and under different state of stress. The comparable samples are from CD2 and CD8 (see Table 1) at 2.6 m and 2.9 m and located on P3 and P2, respectively. CD2 is representative of the medium at the southern part of the profile with low G values (of 52-64 MPa) while CD8 is typical of the good RE state with higher values (around 242-289 MPa).

## 6. Conclusions

Samples previously collected and studied in the RE along the targeted site, originally showed that laboratory $V_s$ measurements and microporosity were the only data able to identify the origin of maintenance anomalies, when classical geotechnical approaches (*in situ* or laboratory tests) would fail. $V_s$ thus appeared as a good indicator of the mechanical quality of materials in the RE. These results made it possible to propose this area of the France northern HSL as a test site for the development of non-destructive and more efficient methods to monitor RE. SW techniques with small size setups were thus suggested to estimate $V_s$ along these structures (HSL RE). SW acquisitions proved their advantage of robustness against noise and influence of
<figure><img src="image_14.png" /><figcaption>Figure 13: Marginal pdfs of each parameters of the Bayesian inversions performed for each seismic profiles. The profiles are ordered according to their position along the line (see figure 4). The blue dotted lines represent the *a priori* marginal pdfs and the black lines represent the *a posteriori* marginal pdfs.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 14: Synthetic representation of the Bayesian inversions performed along the line for the *a posteriori* marginal *pdf* of $G_3$. The seismic profiles are ordered and colored according to their position along the line: the orange profiles are located further south (disorder area) and the blue profiles are located further north (healthy area), see Figure 4. The grey values on the left are the probabilities of each maxima (the *pdfs* are plotted with equal vertical scales).</figcaption></figure>
610 3D structures in this particular context. The systematic record of both di-
rect and reverse shots seismograms allowed reliable and repeatable dispersion
611
measurements along the trackside. This approach, combined with dispersion
612
image stacking, offered sufficient spectral resolution in wide frequency and
613
velocity ranges. The dispersion images have then been meticulously analysed
614
and the choice of picking only the fundamental and first higher mode was
615
made. This was assumed to be the best compromise between the amount
616
of information accessible in the dispersion image and the safe identification
617
and inversion of propagation modes. 7 seismic profiles, intercepting the area
618
of unusual maintenance works, then produced very distinct SW propaga-
tion velocities (correlated with the previously identified phenomenon). The
619
strong a priori knowledge available for this type of RE structure, allowed
620
a very well constrained inversion of dispersion data for 1D models of V<sub>S</sub>
621
along the trackside. The inversion of each profile has been proposed in a
622
Bayesian framework, in order to estimate the a posteriori pdf on V<sub>S</sub> for each
623
layer of the RE. The presentation of these a posteriori pdf for every profiles
624
625
clearly showed a partitioning, particularly marked in the loess layer, that
626
corresponds to the observed RE stability issues. Thanks to the Bayesian
627
framework chosen for dispersion data interpretations, the suggested method
628
enabled the estimation of shear moduli in the affected layer, with levels of
629
confidence. These results are very encouraging with regards to the applicabil-
630
ity of the developed approach in an industrial manner, with the possibility to
631
consider it as a decision support tool. However, the methodology presented
632
in this article, remains to be validated in different contexts, and above all
633
to be optimised for large scale deployments. Several aspects have to thus be
634
addressed, such as the development of an operational and versatile methodology (e.g. adaptable to the variety of RE 'heritage' and the diversity of *pathologies* to be diagnosed). In order to achieve this, it will be necessary to fully extract all the available information from the dispersion images. It is therefore planned to develop tools for more automatic dispersion analysis and inversion (using machine and/or deep learning algorithms). The systematic implementation of the method eventually requires the design of deployment tools with sufficiently high yield to support decision-making and to guide choices in terms of monitoring and maintenance.

### Acknowledgements

This work has been funded by SNCF Réseau/CNRS/Université Pierre et Marie Curie-Paris6/Sorbonne Université research contracts C13/411, C15/0781 and the convention ANRT/Cifre-SNCF Réseau C21/1995. The geophysical equipment was provided by the METIS laboratory at Sorbonne Université.

The authors thank SNCF Réseau local teams for their great help and support during field works. Seismic data processing has been performed thanks to open-source software packages: SWIP (github.com/SWIPdev/SWIP/releases), Seismic Unix (github.com/JohnWStockwellJr/SeisUnix) and GEOPSY (geopsy.org).

The authors thank A-M. Tang, J.-M. Terpereau, P. Leroux, A. Robinet and S. Nebieridze for their advices and support during the study, as well as valuable discussion for the results interpretations.
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Sample CRediT author statement

**Audrey Burzawa**: Conceptualization, methodology, software, investigation, Writing - Original Draft. **Ludovic Bodet**: Conceptualization, methodology, investigation, supervision, project administration, funding acquisition, Writing - Original Draft, Writing - Review & Editing. **Amine Dhemaied**: Conceptualization, methodology, investigation, supervision, project administration. **Marine Dangeard**: Conceptualization, methodology, investigation, supervision. **Sylvain Pasquet**: Investigation, Writing - Review & Editing. **Quentin Vitale**: Investigation, Writing - Review & Editing. **Joséphine Boisson-Gaboriau**: Writing - Review & Editing, supervision. **Yu-Jun Cui**: ressource, investigation, Writing - Review & Editing.
**Declaration of interests**

[x] The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[ ] The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Highlights

Detecting mechanical property anomalies along railway earthworks
by Bayesian appraisal of MASW data

A. Burzawa, L. Bodet, A. Dhemaied, M. Dangeard, S. Pasquet, Q. Vitale, J. Boisson-Gaboriau, Y. J. Cui

* Designing non-destructive tools allowing efficient diagnosis of railway earthworks

* Geotechnical and geophysical study of a test site along an High Speed Line

* Surface-wave seismic acquisition to delineate shear-wave velocity variations

* Correlation between shear-wave velocity decrease and abnormal maintenance efforts

* Shear moduli profile presented in probability to provide a decision support tool

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