# Deep learning contribution to the automatic picking of surface-wave dispersion for the characterization of railway earthworks

J. Cárdenas, A. Burzawa, N. Radic, L. Bodet, R. Vidal, K. Diop, M.

Dangeard, A. Dhemaied

► To cite this version:

J. Cárdenas, A. Burzawa, N. Radic, L. Bodet, R. Vidal, et al., Deep learning contribution to the automatic picking of surface-wave dispersion for the characterization of railway earthworks. Computers & Geosciences, 2025, 198, pp.105883. 10.1016/j.cageo.2025.105883. hal-04967067

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Research paper

# Deep learning contribution to the automatic picking of surface-wave dispersion for the characterization of railway earthworks

J. Cárdenas<sup>a,●,*,1</sup>, A. Burzawa<sup>a,b,1</sup>, N. Radic<sup>b</sup>, L. Bodet<sup>a</sup>, R. Vidal<sup>b</sup>, K. Diop<sup>a</sup>, M. Dangeard<sup>b</sup>, A. Dhemaied<sup>b</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

Keywords:

Geophysics

Surface waves

Seismic wavefield modeling

Automatic dispersion picking

Deep learning

Explainability

## ABSTRACT

Railway Trackbed (RT), which collectively describes the subgrade structures that support rail tracks, is of great importance to the effective maintenance and rehabilitation of the rail network. Therefore, a comprehensive understanding of the mechanical condition of Railway Earthwork (RE) is necessary. The development of non-destructive and efficient methods for the characterization of REs is a priority. Previous studies have investigated the potential of surface waves for the characterization of RE. Preliminary results indicate that this approach is effective, particularly when using high yield acquisition setup such as landstreamer. However, these instruments generate an amount of data that necessitates optimized and automated processing. The potential of Deep Learning (DL) to automate the processing of surface wave data is being explored. In this study, the primary objective is to identify the energy maxima and propagation modes in dispersion images. A supervised convolutional neural network (CNN), designated as ‘U-Net’, was selected to perform segmentation tasks. This model, called ‘U2-pick’, integrates two U-net architectures: one for maxima identification and another for propagation mode identification. The training dataset was constructed using synthetic data that is representative of a French High-Speed-Line (HSL) RE structure. The preliminary outcomes on the synthetic datasets are encouraging, demonstrating accurate identification of energy maxima and mode classification. However, the predictions made on field datasets revealed that while the energy peaks were identified with a high degree of accuracy, the mode assignment proved to be less satisfactory, especially in the case of higher modes. Finally, a comparison of the accuracy and picking time was performed using more standard tools like maxima search, semi-automatic, and Machine Learning (ML) tools.

## 1. Introduction

To guarantee the security and durability of the French railway network, it is essential to implement a rigorous monitoring and maintenance program. The railway superstructure (rail, sleepers, ballast) is based on a specific structure called the Railway Trackbed (RT). In the context of French high-speed line (HSL), the RT is composed of specific layers, including sub-ballast layer, capping layer, and subgrade soil, called the shallow Railway Earthwork (RE). Each layer fulfills a structural, protective, and supporting function, thereby ensuring the stability of the track. A variety of factors can lead to alterations in the substructure and subgrade, including natural disasters such as flooding, landslides, and sinkholes, as well as premature deterioration

caused by the passage of trains (Li and Selig, 1996). To evaluate the mechanical state of the RT, the methods currently employed on the French railway network are predominantly destructive, including core drilling and dynamic penetrometer tests (Benz-Navarrete, 2009; Escobar, 2015; Escobar et al., 2016). These tests provide information on the mechanical state of the RE (laboratory analysis and cone resistance respectively), but they have the significant disadvantage of being local, destructive, and particularly costly. It is therefore imperative that non-destructive, high-performance methods are employed.

The limited acquisition time and the presence of metals and electrical currents associated with railway operations represent significant challenges to the implementation of several geophysical prospecting methods. Some authors employ electrical methods to ascertain

Abbreviations: RT, Railway Trackbed; RE, Railway Earthwork; HSL, High Speed Line; GPR, Ground Penetrating Radar; LVL, Low Velocity Layer; AI, Artificial Intelligence; ML, Machine Learning; DL, Deep Learning; FD, Finite Difference; CNN, Convolutional Neural Network; OHE, One-Hot Encoding

* Corresponding author.

E-mail address: julio.cardenas@sorbonne-universite.fr (J. Cárdenas).

¹ Contributed equally - Cárdenas and Burzawa.

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

<footnote><sup>b</sup> SNCF Réseau, 6 avenue François Mitterrand 93210 Saint-Denis, France</footnote>
the petrophysical properties of the soils comprising railway earth-
works (Chambers et al., 2008; Donohue et al., 2011; Chambers et al.,
2012; Holmes et al., 2022; Ibrahim et al., 2023). Electromagnetic
methods, and in particular Ground Penetrating Radar (GPR), are used
extensively on the network (Eriksen et al., 2010; Anbazhagan et al.,
2011). GPR technique provides information on stratigraphy of the
ballast and sub-ballast layers, evaluation of the water saturation, and of
the presence of mud. Although these methods are deployed with high
efficiency over large distances for shallow RE and RT imaging, they do
not provide direct access to the mechanical properties of the medium,
in particular its shear modulus, which is important for geotechnical
studies (Artagan et al., 2020; Burzawa et al., 2023). Petrophysical
links between electrical and mechanical parameters tend to be devel-
oped (Mavko et al., 2009; Wu and Grana, 2017) but cannot yet be used
in a systematic manner.

The seismic method is the geophysical technique capable of provid-
ing a mechanical characterization of soil. Conventional seismic meth-
ods, such as refraction or reflection, are however not applicable in
the context of railway engineering due to significant three-dimensional
(3D) effects (Mari et al., 2004). However, surface waves are less af-
fected by these complex structures (Karl et al., 2011). Surface-wave
methods are then of particular interest for characterizing shallow RE
and RT (Bodet et al., 2017; Bodet, 2019; Bergamo et al., 2016a,b;
Kyrkou et al., 2022; Burzawa et al., 2023). The MASW (Multiple Analy-
sis of Surface Waves) method entails the analyses of the propagation of
surface waves (typically Rayleigh waves) within the subsurface in order
to ascertain variations in shear-wave velocity (V<sub>s</sub>) (Park et al., 1999; Xia
et al., 1999). If a variation in vertical velocity with depth is assumed,
it can be shown that each frequency component of the surface wave
propagate at a different velocity. This property is known as dispersion.
In the context of the mechanical characterization of shallow RE, Bodet
et al. (2017) demonstrated that the analysis of surface-wave dispersion
can be used to characterize variations in V<sub>s</sub> in relation to shallow RE
disorders. More recently, Burzawa et al. (2023) has demonstrated the
applicability of the method by proposing a quantification of V<sub>s</sub> profiles
and their uncertainties along track.

The processing of surface waves entails the extraction of the dis-
persion from the data, requires advanced processing techniques (Foti
et al., 2018). The dispersion is defined as energy maxima discernible
as curves in dispersion images. The identified maxima are assigned
to distinct propagation modes (for further insights into the methodol-
ogy employed in surface-wave analysis, please refer to Section 2). At
present, the majority of dispersion picking tools are either manual or
semi-automatic (e.g. Pasquet and Bodet, 2017). While these tools are
accurate, they are not sufficiently time-efficient when the number of
dispersion images is considerable. In addition, the inherent subjectivity
of the operator produces biased results. Therefore, it is important to
estimate the associated error (Khalaf, 2016; Dangeard et al., 2018;
Huynh et al., 2023).

The application of Machine Learning (ML) methods to the analysis
of dispersion images represents a topic that has been the subject of re-
cent research. For example, Wang et al. (2021) proposed an automatic
picking method of multi-mode surface-wave dispersion curves based on
unsupervised machine learning methods. Li et al. (2020) tested the use
of logistic regression, neural networks, and support vector machines to
measure their effectiveness in isolating surface waves. In particular, the
use of Deep Learning (DL) applications has been extensively tested. In
this study case, dispersion matrix can be considered as images, and Con-
volutional Neural Networks, which are a type of DL architecture, are
well suited to this type of data (LeCun et al., 2015). As demonstrated
by Dai et al. (2020) and Hu et al. (2023), a DL model can rapidly extract
dispersion curves from dispersion images and classify them by using a
clustering algorithm.

The objective of this study is to automate the dispersion picking
process, thereby facilitating the rapid processing of data and reduc-
ing the time-consuming nature of this task. The approach proposed

by Cárdenas (2022) was used as a reference to develop and validate
the methodology. In Section 3.1, a Finite Difference (FD) modeling tool
was used to generate synthetic data, taking into account RT parameters.
This step is crucial in geophysics because field datasets are often lim-
ited. To identify and characterize the modes of energy maxima in the
dispersion images, a CNN model called U-Net, specifically developed
for image segmentation, was selected and adapted. In Section 4.1,
entitled 'Data labeling', the criteria applied to the labeling of the
data and the methodologies used to ensure a balanced database are
delineated. In Section 5, the internal functioning of the DL model was
visualized through the use of a visualization tool, which demonstrated
the accuracy of the predictions and the selection of hyperparameters.
The DL model was evaluated on real data to ascertain their ability for
generalization. Furthermore, a comparison was performed between the
DL model and traditional geophysical methods for dispersion picking,
to illustrate the advantages and limitations as well as the potential
benefits of the DL model.

## 2. Background methodology

### 2.1. Surface-wave analysis for the characterization of railway earthworks

The preliminary stage of surface-wave analysis involves the de-
ployment of sensors (known as geophones) with a geometry adapted
to the depth of investigation aimed and the resolution objective for
characterizing the RT. The designed linear array of geophones records
the vertical displacements of particle velocity at the ground surface,
in response to one or more controlled seismic sources (e.g. hammer
or sledgehammer). In the context of RE investigation, a geometry was
defined with 96 geophones, spaced 0.25 meters apart (Bodet et al.,
2017; Burzawa et al., 2023). The array is placed along either side
of the track called 'cess' (Fig. 1a). The recorded seismic wavefield is
then represented in the distance-time domain (d-t) and subsequently
visualized in a seismogram (or also named shot gather) (Fig. 1b).
The propagation velocity of surface waves is subject to variation by
frequency, depending on alterations in the mechanical properties of
the subsoil at different depths (Socco and Strobbia, 2004; Foti et al.,
2018). The behavior, designated 'dispersion', is subjected to analysis
within the frequency-phase velocity domain (f - V<sub>φ</sub>). The method
used for wavefield transformation is the slant-stack (Mokhtar et al.,
1988) and the result is designated a 'dispersion image' (Fig. 1c). This
image represents the energy distribution of the seismic wavefield. If the
properties of the medium vary with depth, the energy maxima corre-
spond mainly to the propagation modes of surface waves and can be
identified as 'dispersion curves'. Empirically, the high frequencies are
associated with the near-surface, while the low frequencies represent
the medium at depth. Subsequently, these curves are extracted from
the images, with associated error bars calculated from a Lorentzian
law (O'Neill et al., 2003) (Fig. 1d). Assuming that the medium be-
neath the array under consideration is one-dimensional (1D), these
curves can be used in an inverse problem (Fig. 1e) to retrieve the
corresponding 1D V<sub>s</sub> vertical structure (Fig. 1f). This is achieved by
undertaking a comparison between the observed dispersion curves and
those calculated employing a hypothetical subsurface model (Socco and
Strobbia, 2004). Through a process of model parameter adjustment,
the simulated dispersion curves are compared to the measured ones,
thereby facilitating the reconstruction of the V<sub>s</sub> structure. It is then
possible to reconstruct a two-dimensional (2D) profile by juxtaposing
the inversion results, giving a good idea of the V<sub>s</sub> distribution along
a linear section (see Foti et al., 2018, for detailed tutorial). This V<sub>s</sub>
profile reflects the medium's properties at different depths. In theory,
low frequencies of surface waves are sensitive to deeper layers, while
high frequencies reflect the characteristics of the shallow layers. In the
railway context, this technique has been adapted and has shown that
it is capable of characterizing the RE and the shallow layers of the
supporting soil (Donohue et al., 2013, 2014; Gunn et al., 2015, 2016;
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Schematic representation of the surface-wave analysis methodology a. Photography of a typical seismic setup placed on the cess. b. Wavefield propagation represented in a seismogram. c. Associated dispersion. d. Extracted dispersion curves for the fundamental mode (M0) and the first higher mode (M1). The error bars are according to a Lorentzian law (O'Neill et al., 2003). e. 1D inversion of the extracted dispersion curves. The theoretical dispersion curves are calculated with the Thomson-Haskell matrix propagator technique (Thomson, 1950; Haskell, 1953). f. Corresponding 1D $V_s$ vertical structure. Dispersion curves and $V_s$ model are represented with a colorscale according to misfit value which corresponds to the difference between observed and simulated data.</figcaption></figure>

Source: The field data illustrated in this figure is sourced from Burzawa et al. (2023).

Sussmann et al., 2017; Gunn et al., 2018; Burzawa et al., 2023).

All these steps, acquisition, processing, and inversion necessitate a high degree of expertise and rigor. The most time-consuming step is the picking of the data when processing the dispersion images. The extraction of dispersion curves is traditionally carried out by hand or using adapted tools to search for energy maxima in a dispersion image. For example, to facilitate these maxima picking, Pasquet and Bodet (2017) proposed a semi-automatic tool that proposes the closest maxima around each manual pick. These tools are subject to the inherent subjectivity of the operator in charge of identifying the zone of maxima and associated range of propagation mode (Dangeard et al., 2018; Huynh et al., 2023). More recently, ML-based tools have been developed, notably, the work of Wang (2020), whose code was adapted for this study to compare its accuracy prediction with the DL model. A summary of the use and results of the various picking tools is given in Section 6.

## 2.2. U-Net

U-Net is a deep learning architecture that was originally proposed by Ronneberger et al. (2015) for biomedical image segmentation. It has since become well known for its effectiveness in image segmentation tasks and demonstrated strong performance in a wide range of applications that require pixel-level classification. Its architecture consists of a contracting path, named encoder, to capture context, and a symmetric

expanding path, named decoder, that enables precise localization. A key feature of U-Net is its skip connections, which directly link encoder and decoder layers to preserve spatial information, making it particularly well-suited for tasks involving the identification and segmentation of complex patterns in images (see Fig. 2).

When considering a deep learning model, the tuning of hyperparameters is important as they significantly impact the learning time and the precision of its predictions. While the architecture (layers, connections) defines the complexity of the architecture, hyperparameters (learning rate, batch size, etc.) control how effectively the model learns. These hyperparameters cannot be determined during the training phase and need to be set beforehand. Sub-optimal hyperparameter settings can lead to poor model performance known as overfitting or underfitting. In this study, the hyperparameters proposed by the original authors of the U-Net deep learning architecture were initially adopted (e.g. convolution/up-convolution kernel size, max-pooling dimension, and the activation function RELU). For the remaining hyperparameters (batch size, output activation function, learning rate, and number of convolutional layers), a range of values was defined, and a grid search was conducted to evaluate all possible combinations within this range. The accuracy of the models was compared, and the hyperparameters corresponding to the highest accuracy were selected.

A loss function is a mathematical function used to measure how well a model's predictions align with the actual target values. It quantifies the difference (or 'error') between the predicted outputs and the true
<figure><img src="image_5.png" /><figcaption>Fig. 2. Schematic representation of surface-wave dispersion extraction by using a U-Net neural network model.</figcaption></figure>

values (ground truth). In this study, a Dice Loss (Eq. (1)), as proposed by Milletari et al. (2016), was considered due to its primary application in image segmentation tasks. It is based on the Dice Similarity Coefficient (DSC), which measures the overlap between the predicted segmentation (binary mask) and the ground truth. Additionally, a Categorical Cross-Entropy (CCE) Loss (Zhang and Sabuncu, 2018), shown in Eq. (2) and usually applied for multi-class tasks, was used to compare the predicted probability assigned to the true class with 1 (as the correct label is represented as 1 in one-hot encoding) and penalize larger differences. The DICE loss is defined as follows:

$$L_{DICE} = 1 - \frac{\sum_{i=1}^{n} p_i t_i + \varepsilon}{\sum_{i=1}^{n} p_i + \sum_{i=1}^{n} t_i + \varepsilon}, \qquad (1)$$

where $p$ is the predicted value for pixel $i$, $t$ is the true value for pixel $i$ and $n$ is the total number of pixels in the image.

The CCE loss is defined as:

$$L_{CCE} = -\frac{1}{n} \sum_{i=1}^{n} \sum_{j=1}^{k} y_{i,j} \log(\hat{y}_{i,j}), \qquad (2)$$

where $y$ is the true label and $\hat{y}$ is the predicted probability for the $i$th sample and class $j$; $n$ is the number of samples, and $k$ is the number of classes.

An activation function is a mathematical function applied to the output of a neuron in a neural network to introduce non-linearity into a DL model. A variety of activation functions are applied at different layers of the network, enabling the model to discern intricate patterns and relationships within the data. The activation functions employed in this study include ReLU (Rectified Linear Unit), Sigmoid, and Softmax (Eqs. (3), (4), and (5) respectively), selected based on the specific requirements of each layer. ReLU is one of the most commonly used activation functions in deep learning due to its simplicity and efficiency. It introduces non-linearity by setting all negative values to zero, while retaining positive values. The Sigmoid function is often used in binary classification tasks, as it maps input values to a range between 0 and 1, interpreting the output as a probability. The Softmax function is used in the output layer for multi-class classification problems, where the goal is to assign probabilities to multiple classes. The Softmax function normalizes the output into a probability distribution over K classes. These three activations functions are defined as follows respectively:

$$f(x) = \max(0, x), \qquad (3)$$

$$\sigma(x) = \frac{1}{1 - e^{-x}}, \qquad (4)$$

$$Softmax(x_i) = \frac{e^{x_i}}{\sum_{j=1}^{k} e^{x_j}}, \qquad (5)$$

where $x$ is the input to the neuron and $k$ is the total number of classes.

## 3. Database generation

### 3.1. Modeling wavefield propagation

The scarcity of accessible data, compounded by difficulties in accessing sites, results in an insufficient amount of data to train DL models. Consequently, the generation of synthetic data is necessary for model training. The modeling tool used in this study is a FD code, known as ‘Denise Black Edition’, developed by Köhn et al. (2012). In the case of a RE, the ground model can be particularly complex, exhibiting the presence of Low Velocity Layer (LVL), and notable variations in velocity propagation. In this study, soil models are defined with a structure representative of the French HSL RE. The ballast layer is not considered in this study due to the inherent complexity of modeling such a heterogeneous medium. The modeling of the RE is that of a two-layer soil structure above an infinite half-space. The modeling parameters can be framed in a way that enables the study of different soils with normally and inversely dispersive media (O'Neill et al., 2003; Bodet, 2005). These structures depend on the velocity distribution between the layers. Inversely dispersive media, i.e. with a LVL at depth, are particularly representative of RE, as the capping layer is significantly less compacted than the sub-ballast layer above it.

Each layer is defined by ranges of $V_s$, compression-wave velocity $V_p$, density $\rho$, and thicknesses $H$. For reasons of simplicity, the thickness of the capping layer is initially set at 0.5 m, as it has been observed to vary to a minimal extent in practice. A total of four parameters can be obtained by considering $V_s$ and layer thicknesses. These are the $V_s$ of the sub-ballast layer which is named $V_{s1}$, and its thickness $H_1$, $V_s$ of the capping layer, which is called $V_{s2}$; and $V_s$ of the subgrade, which is named $V_{s3}$. $V_p$ of each layer is related to $V_s$ by a factor of two ($V_p = 2V_s$), representative of a Poisson ratio $\nu$ equal to 0.33 for medium soil. The range of these parameters space and their steps are detailed in Table 1. This parametrization generated a set of 1 260 synthetic seismograms. However, only 1 153 seismograms made up the database because some parameter sets do not allow the wave equation to be solved numerically and yield dispersion images that cannot be interpreted.

Once the ground model has been parameterized, the acquisition setup geometry must also be defined. In this study, a seismic acquisition geometry analogous to that employed in the field was used (see e.g. Burzawa et al., 2023). A set of 96 geophones, positioned at a distance of 0.25 m from one another, was thus defined (see Fig. 3a). The seismic source is representative of the impact of a sledgehammer on a metallic plate, with a center frequency $f_c$ of 40 Hz. As the medium modeled is laterally homogeneous, with no dip or heterogeneity, wavefield is similar on both sides of the seismic setup. So, for calculation purposes, only the direct seismic shot to the right of the setup has been considered (see Fig. 3b). The wavefield is then transposed into the $f - V_\phi$ domain, resulting in the generation of the corresponding dispersion image (see Fig. 3c).
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** a. Railway earthwork soil model in terms of $V_s$. $V_p$ is defined as $V_p = 2 * V_s$ for a Poisson ratio of 0.33 and $\rho$ is fixed at 2 000 kg/m³ for each layer. The acquisition setup is made of 96 geophones spaced with 0.25 m. b. Synthetic seismogram of the defined RE model. c. Associated dispersion image with the theoretical dispersion curves in a white dotted line.</figcaption></figure>

<caption>**Table 1**</caption>

Table of possible values for $V_{s1}$, $V_{s2}$, $V_{s3}$ and the thickness of the sub-ballast layer $H_3$ for considered modeling purposes. The set of parameter space is defined according to previous works of Burzawa et al. (2023).

<table><thead><tr><th>Description of parameters (units)</th><th>Name</th><th>Min value</th><th>Max value</th><th>Step</th></tr></thead><tbody><tr><td>Thickness of the sub-ballast layer (m)</td><td>H<sub>1</sub></td><td>0.3</td><td>0.9</td><td>0.3</td></tr><tr><td>Shear-wave velocity of the sub-ballast layer (m/s)</td><td>V<sub>s1</sub></td><td>50</td><td>260</td><td>15</td></tr><tr><td>Shear-wave velocity of the capping layer (m/s)</td><td>V<sub>s2</sub></td><td>60</td><td>120</td><td>20</td></tr><tr><td>Shear-wave velocity of the subgrade (m/s)</td><td>V<sub>s3</sub></td><td>180</td><td>330</td><td>25</td></tr></tbody></table>

## 3.2. Synthetic dispersion images generation

The dispersion images are defined in $V_\phi$ with a range from 1 m/s to 599 m/s (with a step of 1 m/s) on the vertical axis. The frequency range was defined from 0.5 Hz to 100 Hz (with a step of 0.5 Hz) in the horizontal axis depending on the maximum frequency at which the field data can be analyzed. These images are, in fact, matrices of dimension 599 × 200, where each element of the matrix corresponds to the amplitude of the energy. It should be noted that this amplitude is normalized at each frequency between the values: [0.0, 1.0]. In this study, it was assumed that these matrices represent images with a single channel, corresponding to the amplitude (in contrast to the three RGB channels typically used in color images).

As mentioned in Section 3.1, the database is composed of normally dispersive and inversely dispersive models. The dispersion images associated with the normally dispersive and inversely dispersive models, respectively, are shown in Figs. 4a, b. The database is made up of dispersion images showing up to four distinct propagation modes, from the fundamental mode (M0) to the third higher mode (M3). To enhance the complexity of the database, a subset of noisy seismograms has been added to the database. The noise is generated from sources with equivalent frequency positioned randomly in the time-distance domain. The distribution of energy in the propagation modes represented in synthetic dispersion images is therefore noisier. The dispersion image associated with the noisy seismogram is shown in Fig. 4c. The database

is then composed of 1 153 synthetic and 118 noisy synthetic dispersion images.

# 4. Dataset engineering

## 4.1. Datasets labeling

The data labeling stage is of significant importance, as it serves to define the objective of the training output by establishing the requisite masks. All the dispersion images in the training database were manually picked and uniformly labeled. A set of rules specific to the picking process has been established. Firstly, no picking is permitted within the spectral aliasing zone. This phenomenon occurs when the spatial resolution is too low (due to an inappropriate measurement scale or sampling frequency) and aliasing artifacts can then appear (Foti, 2000). These artifacts are often the cause of high-frequency propagation mode confusion during the picking step (O'Neill et al., 2006). This very distinct area in the dispersion image is illustrated by purple dotted lines in Fig. 5. Thereby ensuring that the algorithm is aware that this particular area is to be excluded from the picking process. A second rule applied during the labeling of the training data is the modification of the propagation mode in response to each perceptible perturbation of the dispersion curve (Fig. 5a). In theory, M0 is always the lowest in phase velocity and mode crossing does not occur. If a high-frequency mode exhibits a phase velocity that exceeds that of its corresponding mode (which may be indicative of a jump), the modes are nevertheless considered to follow one another. It is also imperative to identify as many maxima as possible, even in instances where they do not align with the maximum indicated at a given frequency (Fig. 5b). In cases where a clear distinction in dispersion curves cannot be made, it is essential to define a criterion for differentiating between propagation mode ranges. Accordingly, the mode change was defined at the trough of the 'L' visible on the dispersion curve, as shown in Fig. 5c. These rules are based on empirical observations and the expertise of the geophysicist. It is also important to note that, the extraction of the dispersion of surface waves is, in the majority of cases, reliant upon the identification of the effective dispersion (see e.g. O'Neill et al., 2003; Forbriger, 2003a,b; Foti et al., 2014).
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Dispersion images obtained with different soil model velocity distributions. a. Normally dispersive medium with $V_{S1} < V_{S2} < V_{S3}$. b. Inversely dispersive medium with $V_{S2} < V_{S1} < V_{S3}$. c. Normal dispersion medium with noise generated from sources with equivalent frequency positioned randomly in the time-distance domain.</figcaption></figure>

## 4.2. Balanced database

The final step in dataset engineering is to assign an appropriate level of importance to each labeling class. This involves giving greater importance to dispersion curves than to the image background. Picks mode coordinates are notably infrequent in all matrices, with approximately one hundred maxima coordinates for a total of approximately 120 000 points per matrix (given a matrix size of 599 × 200). The database is thus imbalanced and the Categorical Cross-Entropy loss function is not robust to this kind of problem. In the absence of weight addition, there is a significant risk of decreasing the model's accuracy in correctly classifying all dispersion modes. The addition of weights to the loss function entails the introduction of a factor for each class. A weight less than one serves to reduce the importance of a class, while a weight greater than one has the opposite effect. Fig. 6a shows dispersion picking with no specified weights. Few picks of M0 are located and no higher modes are identified. The model determines that treating the entire matrix as background is the optimal solution. In the second case represented in Fig. 6b, the weights are not optimally distributed. Consequently, the model principally identifies the M0, with the majority of maxima being allocated to it. The distribution of weights is of equal importance. The optimal weight of each class for a database can be calculated using the appropriate functions. The code utilized to evaluate the weight calculation function is derived from the scikit-learn library (Pedregosa et al., 2011). Despite a few errors in mode assignment (especially at low frequencies), this weight distribution enables the three modes visible on the dispersion matrix to be accurately identified and classified. Fig. 6c shows the results of the optimal case of weight distribution.

# 5. U2-pick and data explainability

## 5.1. Deep learning architecture

Initially, a single DL architecture was employed to perform both maxima picking and mode identification simultaneously. Despite various efforts to optimize the model, this approach failed to deliver satisfactory results, with the architecture struggling to accurately perform both tasks concurrently. It is hypothesized that the complexity and differing requirements of each task led to this underperformance. To address this issue, a restructuring of the approach was implemented, whereby the tasks were separated and two U-Net architectures were employed, each with the same backbone but differing outputs to align with the specific requirements of each task. To ascertain the efficacy of combining the outputs of the two models, a first DL model was trained to confirm its ability to identify multiple propagation modes (called ‘Multimode’ in this study). Once this step was validated, a second DL model was trained for the maxima picking (called ‘Monomode’ in this study).

In the case of the Multimode model, the probability of the presence of each class is obtained using the ‘softmax’ function at the output of U-Net. Then, as illustrated in Fig. 7, a threshold filter is employed to eliminate uncertain predictions while accepting the greatest number of accurate data points. In practice, this confidence threshold has been set to 0.8. Consequently, if the probability of the presence of all modes other than the background is less than 80%, it is classified as the background. Finally, the ‘argmax’ function is employed to transform the five images, each delineating the coordinates of their respective
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Dispersion images representative of different picking rules for the labeling. **a.** Representation of the rule: every break is a mode change. **b.** Picked dispersion curves must be continuous, particularly for the fundamental mode (M0). **c.** M0 is the lowest velocity mode. It is not possible to have a higher mode below M0. This method has been designated the ‘L’ rule.</figcaption></figure>

modes, into a single image. The function compares the probability of presence of each class, retaining the class with the highest probability and assigning it a number (0 for background, 1 for M0, 2 for M1, 3 for M2, and 4 for M3).

The attribution of propagation modes is then effectively accomplished, although the maxima picking is approximate. The Monomode model was developed to address the aforementioned shortcomings in the maxima search task. The objective of this second learning is to identify only energy maxima in dispersion images. The issue can thus be conceptualized as a binary problem similar to the use of masks. The structure is identical to that of the initial CNN presented above, except for the final activation function, which is now the ‘sigmoid’ function. Subsequently, the probability of presence is filtered to 10% (also determined empirically), and the loss function is set to the Dice loss function. This loss function is robust to the unbalanced database problem, the addition of weights is no longer necessary. Hyperparameters of each learning are represented in the Table 3.

Once the U-Net architectures have been trained, the next step is to merge the outputs of the two models. To accomplish this, the mode of the nearest point in the Multimode training is assigned to each point in the Monomode training using the Euclidean distance. The complete structure of the proposed architecture, named ‘U2-pick’ for ‘Double U-Net picking’, is illustrated in Fig. 8. The results, in a synthetic case, of the Monomode and Multimode training are shown in Figs. 9a, b, respectively. Multimode training results in M0 reaching its picks at frequencies between 20 Hz and 30 Hz, a phenomenon not observed in binary training. This phenomenon also occurs concerning M1 within

this frequency range. The combination of the two training results is shown in Fig. 9c.

It is necessary to determine the loss curve of each learning process (Monomode and Multimode). The model corresponding to the iteration with the smallest loss will be selected. A summary table of the CCE loss and Dice coefficient values obtained for train, validation, and test set can be found in Table 2. These losses and accuracies are satisfactory, given that the objective is to develop a decision-making tool that suggests an accurate pick to the user, who can then change it manually if necessary. The learning process of both models can be evaluated by examining the shape of the curves. In both cases (Figs. 10a, b), the validation curve decreases over iterations and follows the training curve. In addition, a confusion matrix was calculated from test data to assess the model’s classification precision. As illustrated in Fig. 11, the model demonstrates successful classification of the background and dispersion modes from 1 to 3, though it lacks precision in the last mode. Finally, the evolution of these curves, the CCE loss and Dice coefficient values obtained on test data, and the confusion matrix suggest that the current scenario does not involve overfitting or underfitting.

## 5.2. Data explainability

Understanding the intricate operations of a DL model is a challenge. The data is transmitted as the input, and a result is subsequently presented without any accompanying explanation. These intricate structures may appear opaque, analogous to ‘black boxes’, in the absence of external analytical tools. Nevertheless, it is crucial to comprehend the
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Example of a dispersion image showing the picking dispersion curves results with different class weights. **a.** No weight is specified. **b.** Only the weight of the background is reduced. **c.** The weight is adjusted for each class (background and modes).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Diagram of the processing of two pixels of the matrix, which passes through three stages (softmax, filtering, and argmax) between the training output and the final output.</figcaption></figure>

<table><caption>**Table 2**

Summary table of the Dice coefficient and the CCE loss obtained for the train, validation, and test set.</caption><thead><tr><th></th><th>Monomode (Dice coefficient)</th><th>Multimode (CCE loss)</th></tr></thead><tbody><tr><td>Train</td><td>0.75</td><td>0.27</td></tr><tr><td>Valid</td><td>0.72</td><td>0.28</td></tr><tr><td>Test</td><td>0.71</td><td>0.30</td></tr></tbody></table>

operational principles and the underlying foundations of these models when making decisions so that they can be examined, evaluated, and enhanced. This mechanism is referred to as the explicability or interpretability of models. As the name implies, it allows for the explanation

of the data obtained, as well as the quantification of the learning state and the indirect optimization of the hyperparameters of the U-Net structures.

The tool, named ‘Grad-CAM’ (Selvaraju et al., 2017) was used to provide a visual representation of the image activity zones in each convolution. Once the structure has been translated into latent space, its activity zones serve as the basis for decision-making, specifically in the selection or mode assignment process. To accomplish this, the gradient of each image pixel is calculated during the convolution process. The gradient value correlates with the pixel’s significance in the decision-making process. A pixel with a stronger gradient value will have a greater influence on the final decision. It is thus essential to guarantee that the U-Net model considers the most pertinent areas
<figure><img src="image_11.png" /><figcaption>Fig. 8. Representation of the two U-Nets architectures with the same structure (the monomode and multimode model, respectively). Their outputs are merged into a single result. The name of this architecture is ‘U2-pick’ for ‘Double U-Net picking’.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Dispersion images with the result of two U-Net architectures: a. Picking energy maxima (Monomode learning) and b. Mode attribution (Multimode learning). c. The two predictions are associated to obtain the final picking.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Learning curves obtained during the training phase of the monomode and multimode models. On the left (a.), the CCE loss evolution of the multimode model and, on the right (b.), the Dice coefficient evolution of the monomode model.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Confusion matrix calculated from the multimode model when considering test data.</figcaption></figure>

<table><caption>**Table 3**
Table of hyperparameters for the U-Net structure represent Fig. 8.</caption><thead><tr><th>Learning</th><th>Monomode</th><th>Multimode</th></tr></thead><tbody><tr><td>Activation function</td><td>ReLU and Sigmoid</td><td>ReLU and Softmax</td></tr><tr><td>Number of layers (without latent space)</td><td>4</td><td>4</td></tr><tr><td>Size of convolution filters</td><td>3 x 3 with stride = 1</td><td>3 x 3 with stride = 1</td></tr><tr><td>Number of filters per layer</td><td>16,32,64,128 and 256 for latent space</td><td>16,32,64,128 and 256 for latent space</td></tr><tr><td>Maxpooling dimension</td><td>2 x 2</td><td>2 x 2</td></tr><tr><td>Kernel size for transpose</td><td>2 x 2 with stride = 2</td><td>2 x 2 with stride = 2</td></tr><tr><td>Number of output matrix(es)</td><td>1</td><td>5</td></tr><tr><td>Loss function</td><td>Dice loss</td><td>Categorical cross-entropy</td></tr><tr><td>Learning rate</td><td>1 x 10<sup>-5</sup></td><td>3 x 10<sup>-5</sup></td></tr></tbody></table>

before the latent space. It is therefore crucial for the learning process to consider the entirety of the dispersion curve when selecting and assigning modes. This study focuses on the convolutions of the ‘Encoder’ component, situated prior to the latent space of the Multimode learning

model (Fig. 12). In the initial stage, the entire matrix is subjected to analysis of convolution number 1. Subsequently, convolution number 2 is applied, whereby only the background is activated. The initial stage of the structure allows to differentiate the background from the dispersion curve. During the max pooling operation, the model will retain the information represented by dispersion curves, rather than the background. In convolutions 3 and 4, the model is tasked with studying the dispersion curve exclusively. This indicates that the number of convolutions and filters per convolution is adequate. The removal of filters would result in a reduction in precision, while the addition of further filters would render the architectural design more complex without resulting in an improvement in precision. However, exceeding convolution number 5 prevents the learning process from accounting for the entire dispersion curve, limiting its scope to a partial representation. As demonstrated in Fig. 12, the final convolutions (6, 7, and 8) appear to concentrate on the higher frequencies of the curve. It is necessary to consider whether focusing on this part of the curve with so many filters is genuinely beneficial for learning purposes. Further investigation is required at higher frequencies, given that the dispersion curve typically exhibits a greater number of modes. This may explain the necessity of employing a multitude of filters. The number of filters is a hyperparameter that has been evaluated and adjusted by comparing
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Heat map, obtained with Grad-CAM tool, representing the importance of the pixels in each convolution present in the 'Encoder' part of the U-Net architecture (see Fig. 8). The greener the pixel, the more important it is for decision-making. The first convolutions are used to separate the background and the dispersion curve. Convolutions 3 to 5 cover the whole dispersion curve and the last convolutions are used to study the high-frequency curve data.</figcaption></figure>

loss, predictions on synthetic and field data, and results obtained when applying the Grad-CAM tool to several models.

# 6. Results and evaluation of the performance

## 6.1. Dispersion curves extraction on synthetic data

The application of the dispersion picking model was carried out on three different synthetic $V_s$ models with normal $V_s$ distribution, inversely $V_s$ distribution, and noisy synthetic case. The results are respectively shown in Figs. 13a, b, c. Firstly, the picks of the maxima are centered on the energy lobes, thereby enabling a precise identification of the dispersion curves. It is important to note that there is a slight phase velocity shift for picks located at high frequencies (>60 Hz). This difference may be attributed to a discrepancy in the model. In accordance with the picking rules defined in Section 4.1, no picks are present within the aliasing zone. The maxima picking is then highly satisfactory for every $V_s$ distribution. With regard to mode attribution, the results are in accordance with the expected outcomes (see the associated semi-automatic picking carried out by a user in Fig. 14b). Fig. 13a shows that the mode attribution adheres to the established criteria with the change mode following a break at approximately 20 Hz and 30 Hz. Fig. 13b shows a single mode correctly defined as M0 over the entire frequency range. The presence of noise in the dispersion images represented in Fig. 13c, does not appear to alter the attribution of modes, in this case M0 and M1. These results are encouraging, as they demonstrate that U2-pick is well-suited to the tasks of this study, even when dealing with noisy data. Furthermore, the number of examples chosen was representative enough for the synthetic case.

Once the results of U2-pick have been evaluated, it is interesting to compare this model with other dispersion picking tools. In this study, the results of a simple maxima search tool in a matrix (Fig. 14a), a semi-automatic picking tool from Pasquet and Bodet (2017) (Fig. 14b), a ML-based tool re-implemented from the work of Wang (2020) (Fig. 14c), and finally the DL picking tool (Fig. 14a) developed were compared. These tools all correctly identify the energy maxima in the synthetic dispersion image. On the other hand, there are significant differences in mode attribution. In fact, there are no

differences in modes between the maxima search tool and the ML-based tool. They are all identified as M0. The results are similar for semi-automatic picking and DL. However, it should be noted that semi-automatic picking has to be performed by an operator, unlike DL, which is applied automatically.

## 6.2. Dispersion curves extraction on field data

The field data presented in this section has been acquired along French HSL. The site 1 is located on the North European HSL at two distinct positions (see more details in Burzawa et al., 2023). The site 2 is located on the South East European HSL (see more details in Burzawa et al., 2024). The acquisition setup for each of the two sites consists of 96 geophones (with a lower cutoff frequency of 14 Hz), 0.25 m spaced, for a total length of 23.75 m. The geophones were placed on the cess to assure good coupling between geophones and the ground. The seismic signal is generated as the impact of a sledgehammer on a metal plate, with a delay of 0.02 s before impact and a recording duration of 2 s at a sampling frequency of 0.5 ms. Ultimately, to enhance the signal-to-noise ratio, each dispersion image is derived from the stack of the direct and reverse seismic shots.

The results of the U2-pick architecture on the field data are illustrated in Fig. 15a and b for the site 1 and on Fig. 15c for site 2. The selection of maxima is, on the whole, satisfactory. The model does not pick any noise in dispersion images. No pick are assigned in the aliasing zone or in a heavily disturbed area. With regard to mode assignment, the results are unsatisfactory, particularly in the case of higher propagation mode. In fact, the propagation modes are mixed, with, for the case shown in Fig. 15a, M1 picked first and then M0. For the case represented in Fig. 15b, M0 is truncated to reveal M1 while the theoretical M1 is pointed at M0. Finally, for the last field case shown in Fig. 15c, M0 is also mixed with M1 and does not appear as the lowest velocity mode.

In the same way as for the synthetic data presented above, a comparison of results between different picking tools is carried out in this study. Fig. 16 shows the results for respectively the simple maxima search tool in a matrix (Fig. 16a), a semi-automatic pointing tool from Pasquet and Bodet (2017) (Fig. 16b), a ML-based tool re-implemented from the work of Wang (2020) (Fig. 16c), and finally
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Prediction of picked dispersion curves on synthetic dispersion images, from the test group. Each image represents a different velocity distribution (see Fig. 4) with a. $V_{S1} < V_{S2} < V_{S3}$; b. $V_{S2} < V_{S3} < V_{S1}$ and c. normal dispersive media with noise.</figcaption></figure>

the DL picking tool developed (Fig. 16a). Firstly, maxima search picks out all the energy maxima in the image, particularly in the low-frequency aliasing zone. Semi-automatic picking shows good maxima identification and mode assignment. The ML-based picking tool also shows good mode attribution but poor mode identification at low frequencies (aliasing zone). Finally, the results of the DL model show good identification of maxima but poor attribution of two low-frequency picks identified as M1 instead of M0.

A detailed comparison of the picking tools is provided in Table 4. The time required to perform the picking operation is estimated for the same synthetic image. The tools employed to automatically pick the images are the maxima search, the ML, and the DL. However, the semi-automatic tool requires the action of an operator to identify maxima. The ML and DL tools assign modes automatically. The maximum search tool does not allow the modes to be differentiated and the semi-automatic picking tool must be guided by an operator. Ultimately, the only tool that can both pick and assign modes automatically in a reasonable calculation time is the DL tool.

## 7. Conclusions and future work

The U2-pick model is a DL algorithm that automates the process of extracting dispersion curves when processing surface waves in the context of RE characterization. The model has two U-net type architectures. The first allows for the accurate attribution of propagation modes. The second allows for the precise picking of energy maxima. The outputs of these two learning processes are merged into a single result. The data used for training and testing is based on soil models derived

<caption>**Table 4**</caption>

Comparison of time calculation for one dispersion image picking and mode attribution by using a desktop computer (Macbook Pro 2 019, Intel Core i9, 8 cores, and 16.00 GB 2 666 MHz DDR4 RAM).

<table><thead><tr><td>Parameter</td><td>Time (s)</td><td>Automatic picking</td><td>Attribution mode</td><td>User adjustment</td></tr></thead><tbody><tr><td>Maximum search</td><td>5</td><td>Yes</td><td>No</td><td>Yes</td></tr><tr><td>Semi-automatic picking</td><td>120</td><td>No</td><td>No</td><td>Yes</td></tr><tr><td>Machine learning</td><td>4</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Deep learning</td><td>5</td><td>Yes</td><td>Yes</td><td>No</td></tr></tbody></table>

from RT and shallow RE standards in the context of French HSL. The synthetic seismograms were calculated using a finite difference modeling tool. The application of U2-pick to synthetic data yielded optimal results, exhibiting satisfactory accuracy. In contrast, when the model is applied to real data, the resulting accuracy is lower, yet still encouraging. The results and associated loss curves indicate that the database is not sufficiently adapted to the problem posed in this study. The database must therefore be improved with a larger number of data and particularly more complex data (noisy data, other velocity distribution, variable geometry with dip, etc.). The results on field data also show that there is probably a bias in the picking induced by the rules defined when labeling the data. Once the database has been expanded, it would be interesting to examine the influence of labeling on the picking results applied to real data.

Subsequently, U2-pick will be incorporated into a graphical user interface. This will facilitate the user's ability to perform quality control
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Comparison of different dispersion picking tool on synthetic images. The soil model associated with the image is shown as an inset in the top right-hand corner of each dispersion image. The dispersion picking tools compared are: a. Maximum search; b. semi-automatic picking (Pasquet and Bodet, 2017); c. Machine Learning; d. U2-pick based on Deep Learning.</figcaption></figure>

and, for instance, modify mode values in the event of confusion. Furthermore, this will facilitate the comprehension and utilization of the processing tools on an industrial scale. This functional toolbox will facilitate the automation of seismic data processing, which can then be employed for a comprehensive diagnosis of the RE via the surface wave seismic method. Future works will consist of using a different CNN for dispersion inversion. The objective is to develop a DL model for the estimation of soil model parameters representing the mechanical state of shallow RE. The specific model architectures that will be employed for this task are currently under study.

## CRediT authorship contribution statement

**J. Cárdenas:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

**A. Burzawa:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

**N. Radic:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation. **L. Bodet:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition.

**R. Vidal:** Writing – review & editing, Visualization, Methodology, Formal analysis, Data curation, Conceptualization. **K. Diop:** Investigation, Data curation. **M. Dangeard:** Validation, Project administration, Formal analysis. **A. Dhemaied:** Validation, Supervision, Resources, Funding acquisition.

## Code availability section

Name of the code/library : U2-pick

Contact: julio.cardenas_chapellin@sorbonne-universite.fr, nicolas.radic@etu.sorbonne-universite.fr

Hardware requirements: Google Collab

Program language: Python

Program size: 26.88 kB (learning.ipynb); 5.60 kB (show-predictions.ipynb); 11.58 kB (Show-Grad-CAM.ipynb)

The source codes are available for downloading at the link:

* U2-pick: https://gitlab.in2p3.fr/nicolas.radic/U2-Pick

* Machine learning: https://gitlab.in2p3.fr/julio.cardenas_chapelli/n/semi_automatic_picking

* Semi-automatic (SWIP): https://github.com/spasquet/SWIP

## Declaration of competing interest

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.

## Acknowledgments

This work has been funded by SNCF Réseau/CNRS/Université Pierre et Marie Curie - Paris 6/Sorbonne Université research contracts C13/411, C15/0781 and the convention ANRT/Cifre-SNCF Réseau C21/1 995. This work has been funded by the European Union. Views and opinion expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the granting authority. Neither the European Union nor the granting authority can be held responsible for them. The project is supported by the Europe's Rail Joint Undertaking and its members. This project has received funding for them. The project is supported by the Europe's Rail Joint Undertaking
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Prediction of picked dispersion curves on dispersion images from field data. **a.** and **b.** The data was acquired on the cess along the French North HSL (Burzawa et al., 2023). **c.** The data was acquired on the cess along the French South-East HSL (Burzawa et al., 2024).</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Comparison of different dispersion picking tool on field data (from Burzawa et al., 2023). The soil model associated with the image is shown as an inset in the top right-hand corner of each dispersion image. The dispersion picking tools compared are: a. Maximum search; b. semi-automatic picking (Pasquet and Bodet, 2017); c. Machine Learning; d. U2-pick based on Deep Learning.</figcaption></figure>

and its members. This project has received funding from the European Union's Horizon Europe research and innovation programme under Grant Agreement No 101101966. The geophysical equipment was provided by the METIS laboratory at Sorbonne Université. The authors would like to thank the SNCF Réseau and METIS laboratory team for their help in the field and during methodology development.

## Data availability

Data will be made available on request.

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