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# Neural network inversion of seismic wave velocities for vadose zone water content profile

Quentin Didier, Victor Sauvage, Léna Pellorce, Rémi Valois, Slimane Arhab,

Arnaud Mesgouez

► To cite this version:

Quentin Didier, Victor Sauvage, Léna Pellorce, Rémi Valois, Slimane Arhab, et al. Neural network inversion of seismic wave velocities for vadose zone water content profile. Applied Computing and Geosciences, 2025, 27, pp.100285. 10.1016/j.acags.2025.100285. hal-05296723

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Submitted on 3 Oct 2025

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# Neural network inversion of seismic wave velocities for vadose zone water content profile

Quentin Didier, Victor Sauvage, Léna Pellorce, Rémi Valois, Slimane Arhab, Arnaud Mesgouez*

Avignon Université, INRAE, UMR 1114 EMMAH, F-84000 Avignon, France

## ARTICLE INFO

**Keywords:**

Water saturation estimation

Vadose zone

Seismic inversion

Neural networks

Geophysics

## ABSTRACT

Accurate estimation of water saturation in the vadose zone is crucial for hydrological and agricultural applications. However, traditional seismic inversion methods often struggle with non-linearity and sensitivity to noise, limiting their generalisation capabilities. To address this challenge, we propose a neural network-based inversion approach that allows the assessment of the vertical distribution of water saturation from compressional ($v_p$) and shear ($v_s$) wave velocities. Specifically, our architecture is based on a regression model and incorporates an autoencoder layer to improve robustness against noise. As a result, this enhances its ability to invert complete saturation profiles and increases its adaptability to complex hydrological conditions. Furthermore, the model demonstrates strong performance across varying water table depths, with low error metrics and high resilience to input noise with a RMSE of $3.34 \times 10^{-2}$ and a $R^2$ of 0.978 for 5% noise. Our current approach has been trained exclusively on noisy synthetic data. We plan to validate it in the near future against experimental field data we have recorded for an agricultural soil. Overall, this study establishes a foundation for future applications of deep learning in hydrogeophysical inversion and underscores the need for validation with real-world data.

## 1. Introduction

As the subsurface plays a crucial role in sustaining both our ecosystems and food supply, characterising and modelling subsurface processes has been a growing subject in the scientific community (Singha et al., 2024). In particular, the effects of climate change, especially in the agricultural sector, have highlighted the vulnerability of subsurface hydrosystems, with the impact on the water supply being the most significant challenge. Understanding the distribution of water content in the vadose zone is therefore critical for agricultural soils, as it directly determines the amount of water available to crops. It is also important in hydrogeology, as aquifer recharge generally occurs through the vadose zone. An accurate assessment of spatial and temporal variations in water content in this zone is crucial for understanding critical zone processes, particularly those involving subsurface water storage and movement. In recent years, artificial intelligence (AI) has become an increasingly important tool in the field of hydrogeophysics, which combines geophysical methods with hydrological processes to better understand the structure and behaviour of vadose zones and aquifers. AI provides powerful capabilities for processing complex data obtained from electrical resistivity tomography (ERT), induced polarisation, seismic imaging, and hydrological sensors. It enables better

interpretation, prediction of subsurface flow dynamics, and mapping of aquifer systems. To highlight recent developments and applications of AI in hydrogeophysics, Table 1 presents a review of applied AI literature in hydrogeophysics from the last fifteen years.

Advances in data analysis and computational methods have greatly enhanced the ability to extract quantitative information from geo-physical observations. For example, artificial neural networks have been successfully applied to predict water saturation and fluid distribution in complex reservoir settings (Al-Bulushi et al., 2009; Amiri et al., 2015). At the same time, the study of soil properties highlights the broader environmental significance of near-surface geophysics in linking hydrological, biological, and geochemical processes (Banwart et al., 2019). More recently, the integration of advanced machine learning approaches, including Bayesian deep learning and specialised predictive models, has shown promising results in reservoir characterisation and drilling operations (Beheshtian et al., 2024b,a). In the context of seismic exploration, recent studies have critically assessed the potential and current limitations of machine learning approaches, emphasising both their opportunities and challenges for future geophysical applications (Khosro et al., 2024). These developments underscore

<footnote>* Corresponding author.
E-mail address: arnaud.mesgouez@univ-avignon.fr (A. Mesgouez).</footnote>
<table><caption>**Table 1**

Chronological summary of 13 AI-based papers in hydrogeophysics (2008–2025).</caption><thead><tr><td>References</td><td>Main objective</td><td>AI methodology</td><td>Key findings</td></tr></thead><tbody><tr><td>Mutlu et al. (2008)</td><td>Streamflow prediction with two different NN</td><td>MLP and radial basis function neural network (RBFNN)</td><td>Best performance from MLP model with precipitation and flow history</td></tr><tr><td>Edincliler et al. (2013)</td><td>Prediction of damping ratio and shear modulus for sand–waste tyres mixtures</td><td>NN and Neuro–Fuzzy (NF) models</td><td>Better performance of NN over NF</td></tr><tr><td>Travassos et al. (2021)</td><td>ANN & GPR data interpretation review</td><td>Survey of ANN/ML techniques</td><td>Improved GPR from detection to imaging</td></tr><tr><td>Giannakis et al. (2019)</td><td>Speed up GPR forward modelling</td><td>Convolutional NN-based forward solver</td><td>Real-time forward models for full-waveform inversion</td></tr><tr><td>Xie et al. (2020)</td><td>Map dielectric permittivity from GPR</td><td>U-Net + instance normalisation</td><td>Achieved low MSE and better instance-norm</td></tr><tr><td>Chen and Saygin (2021)</td><td>Recover velocity from poor initial models</td><td>Autoencoder + latent-space inversion</td><td>Robust multiscale velocity recovery</td></tr><tr><td>Aleardi et al. (2022)</td><td>Expedite probabilistic ERT inversion</td><td>ResNet-based forward surrogate + MCMC</td><td>Fast probabilistic inversion preserving uncertainty</td></tr><tr><td>Roncoroni et al. (2022)</td><td>Fast and accurate DL methodology in synthetic seismic reflection 1D data</td><td>NN and Long Short Term Memory architecture</td><td>Fast Methodology, 1D noisy synthetic data, for seismic and GPR data</td></tr><tr><td>Wang et al. (2022a)</td><td>Update lithofacies via local statistics</td><td>Multi-point geostatistical inversion + ML probability updating</td><td>Enhanced facies delineation in complex media</td></tr><tr><td>Junkai et al. (2024)</td><td>Reconstruct permittivity from GPR</td><td>TransUNet with self-attention</td><td>Superior reconstruction on synthetic or field data</td></tr><tr><td>Li et al. (2024)</td><td>Invert velocity from microseismic data</td><td>U-Net + data augmentation</td><td>High-resolution inversion from sparse data</td></tr><tr><td>Xue et al. (2024)</td><td>Joint permittivity and resistivity inversion</td><td>DeepLabV3+ -style CNN</td><td>Real-time, accurate dual-parameter inversion</td></tr><tr><td>Lu et al. (2025)</td><td>Prediction of Ground Water Level (GWL) from past data</td><td>Machine learning and CNN</td><td>Identification of key parameters to evaluate GWL</td></tr></tbody></table>

the growing importance of combining geophysical insights with modern computational tools to address both scientific and engineering challenges.

When focusing only on seismic studies to characterise unconsolidated and unsaturated environments in the vadose zone, estimates of water content can be derived from compressional (P) and shear (S) waves velocities (de Barros, 2007; Bodet, 2019; Solazzi et al., 2021). As seismic waves are highly sensitive to pore volume and saturation, they can detect the extent to which pore spaces are filled with water, referred to as water saturation. This sensitivity makes seismic methods particularly valuable for mapping the water table and determining moisture distribution with depth. Integrating both P-wave and S-wave velocity measurements (hereafter referred to as $v_p$ and $v_s$) provides a more robust framework for estimating porosity and fluid properties (Mavko et al., 2009). This approach has demonstrated effectiveness in identifying and characterising vadose zones containing coexisting gas and liquid phases (Gu et al., 2020; Liu et al., 2022; Pasquet et al., 2022).

The relationship between seismic wave velocities and water saturation is particularly challenging in unsaturated and unconsolidated subsurface environments, such as the vadose zone. Conventional mechanical models frequently misrepresent seismic wave velocities in these conditions, often overestimating them to a considerable degree. To address this issue, we adopt an approach similar to that developed by Solazzi et al. (2021). This approach employs the following models and principles: (i) the Biot–Gassmann framework, which is used to describe the elastic properties of unsaturated media (Gassmann, 1951; Biot, 1956); (ii) the Hertz–Mindlin model, which characterises the solid phase skeleton and its response under varying stress and saturation conditions (Hertz, 1882; Mindlin, 1949); (iii) the Bishop’s effective stress framework is employed (Bishop, 1959; Aitchison and Bishop, 1960), incorporating additional parameters such as the depth of the water table. Finally, (iv) the water retention curve (in this case, the van Genuchten–Mualem one van Genuchten, 1980) is used, which explicitly

links water content with soil water potential or capillary pressure in soils.

The model developed above is highly complex, and retrieving water content from shear and compressional waves requires inversion. Given its significant non-linearity and the absence of an analytical inverse solution, only a few approaches are feasible. While some studies rely on *in situ* measurements to calibrate their models (Liu et al., 2022), our objective is to develop a self-sufficient inversion strategy – conceptually aligned with the grid search approach employed by Pasquet et al. (2022) – but leveraging artificial intelligence (AI) to reduce computational cost and enhance the inversion performance. This kind of approach involves optimisation algorithms, which can be categorised as local or global (Lagarias et al., 1998; Nocedal and Wright, 2006; Nefeslioglu et al., 2008). These algorithms aim to minimise a function representing the misfit between observed and calculated shear and compressional wave velocities for a given set of parameters. However, local optimisation methods often suffer from convergence to local minima, whereas global optimisation methods are computationally intensive owing to the large number of parameters involved. In this context, neural networks offer a promising alternative – serving as metamodels that mimic the behaviour of the sought inverse model – and as valuable tools in geophysical data processing, particularly for tasks such as de-noising and inversion. De-noising is crucial in geophysical data analysis owing to the prevalence of noise from human activity and natural sources, which can obscure data interpretation. A well-known neural network architecture, the autoencoder, is frequently employed for image de-noising. An autoencoder consists of two main components: an encoder which reduces the input size through successive convolutional layers and projects it into a latent space representation; and a decoder that reconstructs the input by progressively increasing the layer sizes. For inversion tasks, the objective is to determine the parameters of interest from known fields, such as wave velocities. In this scenario, the neural network’s primary function is to
establish a non-linear mapping between inputs (wave velocity profiles) and outputs (water saturation profile). The Multi-Layer Perceptron (MLP) (Gardner and Dorling, 1998; Ziaee et al., 2014; Abdel-aziem and Soliman, 2023), a fully connected neural network (NN), is commonly used to construct neural network-based inversion models.

By integrating neural networks into geophysical data analysis, we address the limitations of traditional inversion methods and enhance the accuracy of subsurface property estimations. This study presents an inversion approach that allows water saturation vertical distribution estimates using both compressional and shear wave velocities. Unlike methods that rely on sparse features, our approach reconstructs the full saturation curve, providing a richer training signal that enables the neural network to learn underlying hydrological patterns more effectively. The use of a synthetic dataset generated from a petrophysical model representative of a range of agricultural soils – as employed in this study – enables the training of neural networks with broader applicability, effectively serving as a metamodel. This broader representation supports enhanced generalisation, even to scenarios involving dynamic saturation profiles, despite the network being trained under the assumption of hydric equilibrium.

In the first part of Section 2, we outline the theoretical framework governing seismic wave propagation in the vadose zone, including the Biot-Gassmann approach and the Hertz-Mindlin model. We also detail the water retention curve parametrisation and the mechanical assumptions underlying our dataset generation. The second part of Section 2 describes the database construction process, the selection of key geophysical and hydrological parameters, and our strategy for ensuring a well-distributed representation of water saturation curves. Additionally, we discuss the impact of noise in geophysical inversion and how our method addresses this issue. In the last part of Section 2, we introduce the architecture of our neural network, designed to maintain inversion accuracy despite noisy geophysical inputs. We highlight the incorporation of an autoencoder layer to enhance noise robustness, and we describe the training and validation procedures. Section 3 provides a comprehensive evaluation of our model’s performance under different conditions. We first assess its ability to retrieve water saturation with a fixed water table before extending the analysis to varying water table depths, demonstrating its capacity to generalise across different hydrological settings. Finally, Section 4 summarises our findings and explores the broader implications of applying deep learning to geophysical inversion. We also discuss potential future directions, including extensions to 3D datasets and real-world applications.

## 2. Models and methods

### 2.1. Mechanical choices for describing seismic propagation in the near field vadose zone

Natural or agricultural soils are multiphase composites of minerals and fluids. Such porous media exhibit varying elastic properties depending on their structural constitution and water content. One of the most common approaches to obtain seismic velocities of such materials lies in the Biot-Gassmann theory. Considering the vadose zone as a two-dimensional homogeneous and isotropic medium from a macroscopic perspective gives the following compressional and shear velocity expressions (Gassmann, 1951; Biot, 1956; Bourbié et al., 1987):

$$ egin{cases} v_p = \sqrt{\frac{K + \frac{4}{3}\mu}{\rho_{pm}}} \\ v_s = \sqrt{\frac{\mu}{\rho_{pm}}} \end{cases} \qquad (1) $$

with

$$ egin{cases} K = \frac{K_o + \left(1 - \frac{K_o}{K_s}\right)^2}{\frac{\phi}{K_f} + \frac{1-\phi}{K_s} - \frac{K_o}{K_s^2}} \\ \mu = \mu_o \\ \rho_{pm} = (1-\phi)\rho_s + \phi(S_w\rho_w + (1-S_w)\rho_a) \end{cases} \qquad (2) $$

where the following parameters are considered: porosity ($\phi$), water saturation ($S_w$), density ($\rho$), bulk modulus ($K$) and shear modulus ($\mu$). Subscripts are used to distinguish components: pm for the porous medium, o for an open or drained porous medium, s for the solid skeleton, a for the air phase, and w for the water phase.

One of the main assumptions of the Biot-Gassmann approach is that the porous medium has to be fully saturated by one fluid. In our case, since we deal with the vadose zone, the mixture of air and water can be modelled as an equivalent or effective fluid. A mixture law is used, assuming no interaction between the different fluids. This approach is also valid when several minerals constitute the solid skeleton are considered. For the fluid part, the arithmetic mean of the inverse individual bulk moduli may be used as follows (Domenico, 1976; Berryman et al., 2000):

$$ K_f = \left[ \frac{S_w}{K_w} + \frac{1 - S_w}{K_a} \right]^{-1} \qquad (3) $$

where the subscript of the bulk modulus $K$ is $f$ for the effective fluid. If several minerals constitute the solid skeleton, a Hill-type law of mixtures is applied to obtain the effective solid skeleton, as follows (Hill, 1952; Chung and Buessem, 1967):

$$ K_s = \frac{1}{2} \left[ \sum_{i=1}^{n} \omega_i K_s^i + \left( \sum_{i=1}^{n} \frac{\omega_i}{K_s^i} \right)^{-1} \right] \quad \text{and} \quad \mu_s = \frac{1}{2} \left[ \sum_{i=1}^{n} \omega_i \mu_s^i + \left( \sum_{i=1}^{n} \frac{\omega_i}{\mu_s^i} \right)^{-1} \right] \qquad (4) $$

and

$$ \rho_s = \sum_{i=1}^{n} \omega_i \rho_s^i \qquad (5) $$

with $n$ as the number of individual constituents, $\omega_i$ the volumetric fraction of individual constituents, and $K_s^i$, $\mu_s^i$, and $\rho_s^i$ are the bulk modulus, the shear modulus, and density of the individual components, respectively.

The assumption of consolidated or unconsolidated medium is crucial with respect to the seismic waves responses and the spatial scale under investigation. In this paper, since we deal with the vadose zone and particularly agricultural soils with a decimetre spatial scale, consolidated approaches tend to strongly overestimate seismic velocities. Moreover, the constitution of agricultural soils is inherently unconsolidated, at least for the first metres below the surface.

Although Pride's theory (Pride, 2005) can be useful as a first approach to describe unconsolidated soils, a better method is to use the Hertz-Mindlin model (Mavko et al., 2009). The Hertz-Mindlin approach models the solid part of the soil as an arrangement of spherical particles with stiffness behaviour strongly dependent on the presence of water in the porous medium. Among others, Bodet (2019), Solazzi et al. (2021) and Mavko et al. (2009) propose the following expressions for the open bulk and shear moduli:

$$ K_o = \left[ \frac{N^2(1-\phi)^2 \mu_s^2}{18\pi^2(1-\nu_s)^2} \times \sigma_e \right]^{1/3} \qquad (6) $$

and

$$ \mu_o = \frac{2 + 3f - (1 + 3f)\nu_s}{5(2 - \nu_s)} \left[ \frac{3N^2(1 - \phi)^2 \mu_s^2}{2\pi^2(1 - \nu_s)^2} \times \sigma_e \right]^{1/3} \qquad (7) $$
<figure><img src="image_5.png" /><figcaption>Fig. 1. Water saturation curve for different soil compositions and a water table at a depth of 5 m.</figcaption></figure>

where $v_s$ is the Poisson coefficient of the solid skeleton, $N$ is the average number of contacts per solid particle, $f$ is the fraction of non-slipping particles, and $\sigma_e$ is the effective stress. In addition to $N$ and $f$, which are key parameters to describe the unconsolidated aspect of the material, the key point is the introduction of the effective stress, which is going to strongly influences the seismic velocities due to the partial saturation of the investigated vadose zone.

The water content $\theta$ (with $\theta = \phi S_w$) and the effective saturation $\chi$ are defined as follows:

$$ \chi = \frac{\theta - \theta_r}{\phi - \theta_r} = \frac{S_w - S_{w,r}}{1 - S_{w,r}} \qquad (8) $$

where $\theta_r$ is the residual water content and $\theta_s$ is the saturated water content. Note that $\theta_s$ is often taken to be equal to $\phi$ and $\theta_r$ is not equal to zero.

Here, we use the well-known van Genuchten model (van Genuchten, 1980) to obtain the water retention curve linking volumetric water content and matrix water potential:

$$ \chi = [1 + (\alpha |h|)^n]^{-m} \qquad (9) $$

where $h$ is the pressure head, $1/\alpha = h_e$ is the air-entry pressure head, $n$ and $m$ are the shape parameters with $m = 1 - 1/n$. Fig. 1 shows some retention curves for different materials. Our goal is to estimate the entire vertical water saturation distribution rather than derive hydrological parameters from the van Genuchten model. This approach is particularly advantageous in applications where the hydrostatic assumption may not hold, as it only requires adjustment of the pressure head estimate.

The effective stress is defined as follows (Terzaghi, 1936; Guerriero, 2022):

$$ \sigma_e = \sigma - p_a + (p_a - p_w) \chi \qquad (10) $$

where $\sigma = \rho_{pm} g z$ is called the overburden stress depending on the gravitational acceleration $g$ and the depth $z$. $p_a = \rho_a g z$ is the air pressure which often can be neglected and $p_w$ is the water pressure. In multiphase systems, it is important to consider the capillary forces resulting from the pressure difference of immiscible fluids within a porous medium (Solazzi et al., 2021). This pressure difference is caused by surface tension and the curvature of the fluid interface at the pore scale. The capillary pressure is given by $p_c = p_a - p_w$ and $\sigma_s = p_c \chi$ represents the capillary suction stress. We then introduce the water pressure head (also called the matrix water potential) as:

$$ h = \frac{p_w}{\rho_w g} = - \frac{p_c}{\rho_w g} \qquad (11) $$

Under a hydrostatic assumption, the water potential is given by $|h| = z - z_{wt}$, where $z$ is the depth and $z_{wt}$ corresponds to the water table level. By introducing the previous terms into the Hertz-Mindlin model and the Biot-Gassmann expression for seismic velocities, we obtain the final expressions for $v_p$ and $v_s$. Thus, these expressions provide a complete description of the vadose zone as an unconsolidated and partially saturated medium, such as soil.

## 2.2. Database creation workflow and noise in geophysics

### 2.2.1. Database creation workflow

The most important step in designing a Neural Network (NN) is the creation of a labelled learning database that is as representative as possible to encompass most plausible scenarios. The direct modelling used to calculate compressional and shear wave velocities defined above enables us to generate a large quantity of data efficiently. The number of unknowns in this model is very large and includes: $K_s$, $\mu_s$, $\phi$, $N$, $f$, $h$, $n$ and $\alpha$. There are too many variables in this model to feasibly create a database that spans all possible configurations. Since the goal of the present work is to focus on the retrieval of hydrological behaviour of soils, and specifically on the vertical water saturation distribution, we choose to fix solid parameters $K_s$, $\mu_s$, $\phi$, $N$, $f$, which relate to the skeleton part of the medium.

The selection of these parameters is based on a well-documented site, called “Lysimeter”, located in Avignon on the INRAE campus near the Durance River, southern France as illustrated by Fig. 2. This site is of significant interest owing to its extensive instrumentation, including piezometric measurements and numerous geotechnical studies. The lysimeter experiment is specifically designed to deepen our understanding of soil water infiltration processes and serves as a benchmark for advancing non-invasive geophysical techniques. The geological environment is composed of alluvial deposits, with a shallow water table that can vary from 3 to 6 metres below ground level. The medium’s composition is well characterised, allowing us to fix the composition at 43% clay and 57% quartz. Additionally, we have insights into other physical parameters, such as $\phi$ rounded to 0.38, while we fixed $N$ and $f$ at 8 and 0.3, respectively. In order to create the database, we compute water saturation curves (i.e. vertical distribution) under the hydrostatic assumption and then generate $v_p$ and $v_s$ curves. Therefore, it is necessary to consider variations in the soil water retention parameters $\alpha$ and $n$, and the water table depth. We used the following intervals: $0.01 \le \alpha \le 15$ and $1.02 \le n \le 6.66$ as reported in Ghanbarian-Alavijeh et al. (2010), and $2 \le z_{wt} \le 7$ based on piezometric measurements.

In our initial approach, we linearly discretised the values of $\alpha$, $n$, and $z_{wt}$ to generate all possible combinations. This method resulted in an excessively large dataset which created challenges in optimising the neural network’s learning efficiency. For a fixed water table level, the direct model computed extreme water saturation curves that did not adequately cover the entire parameter space, as shown in Fig. 3(a).

We address this issue by developing a workflow that transforms unevenly distributed curves into a uniformly distributed set. To ensure a good coverage of the parameter space, the workflow generates a large number of $\alpha$ and $n$ configurations for each water table depth—yielding a dataset 100 times larger than the target size. Curves are thus selected based on their near surface water distribution, which improved the sampling of the water saturation curves. The result of such optimised database creation is shown in Fig. 3(b). The workflow used can be summed up thanks to Fig. 4.

In Fig. 4, $N_{wt}$ and $N_{curve}$ represent the number of water table depths and the total number of curves to be generated. After creating the database, we randomly selected around 15% as validation set, on which the NN was not trained. This dataset allows us to measure the generalisation of our NN during training and helps to prevent overfitting.
<figure><img src="image_6.png" /><figcaption>Fig. 2. Experimental site which inspired the synthetic medium with (a) regional view, (b) local view and (c) field view.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Water saturation curve distribution for (a) linearly distributed van Genuchten parameter workflow and (b) the optimised workflow for 100 curves.</figcaption></figure>

### 2.2.2. Noisy input

Since field data are inherently noisy, the inversion of $v_p$ and $v_s$ introduces uncertainties that can also be regarded as noise. Extensive research has been conducted to quantify seismic noise (Fleming, 2010; Zhong et al., 2015), which may be entirely random or caused by human activity. Some techniques exist to remove noise under specific conditions, as described in Li et al. (2017) and Schmidt et al. (2020). Nonetheless, noise cannot generally be completely eliminated and must be taken into account during the inversion process.

While noise in displacement field data is often assumed to be purely random, either uniformly or Gaussian distributed, this assumption does not hold for velocity profiles distribution obtained after the inversion step. In fact, the noise in the input displacement field data is incoherent and has a magnitude of approximately 3%–5%. However, after inversion, it becomes coherent, partially filtered, and distorted, fundamentally different in nature. Handling seismic noise in the inversion process is crucial for quantifying reconstruction uncertainty, as demonstrated by Zelt and Smith (1992), Dettmer et al. (2007) and
<figure><img src="image_8.png" /><figcaption>Fig. 4. Database creation workflow.</figcaption></figure>

Owoc et al. (2018). These uncertainties are unavoidable, making it essential to train a neural network that remains stable despite noisy inputs.

Since our goal is to apply the neural network to real data, ensuring robustness to noise is crucial. To achieve this, we incorporate noisy data into the training process. Specifically, to introduce noise that is both coherent and within the 3%–5% amplitude range, we add an additional step during database creation. In step 5 of the database generation, we randomly sample values between 1 – 0.05 and 1 + 0.05, smooth these values, and then multiply the computed $v_p$ and $v_s$ by these factors to introduce noise as shown in Fig. 5. Note that the labels, which represent the saturation curves, remain unchanged. The smoothing step ensures that the noise is correlated rather than uncorrelated. The added noise follows both Gaussian and uniform distributions, as the network is trained on both.

## 2.3. Neural network architecture, training phase and validation criterion

We developed a comprehensive database that accurately represents the diversity of water saturation curves varying according to hydrodynamic properties. Our primary objective is to design a Neural Network (NN) (Schmidhuber, 2015; Sengupta et al., 2020; Dong et al., 2021; Antamis et al., 2024) capable of solving the inverse problem of determining water saturation using $v_p$ and $v_s$ curves. A key technical decision is to directly estimate the saturation curve rather than derive hydrological parameters from the van Genuchten model. This approach is particularly advantageous in applications where the hydrostatic assumption may not hold. Consequently, we selected a Multi-Layer Perceptron (MLP) architecture (Gardner and Dorling, 1998; Ziaee et al., 2014; Abdel-aziem and Soliman, 2023), renowned for its efficacy in regression tasks. However, traditional MLPs often exhibit sensitivity to noise, which can adversely affect their performance. To enhance noise robustness, we integrated an autoencoder layer into our NN architecture, serving as a denoising filter. This strategy aligns with methodologies proposed in recent studies, where autoencoders have

been effectively employed to improve the resilience of neural networks to noisy data (Roncoroni et al., 2022; Gao et al., 2022; Berahmand et al., 2024).

Fig. 6(a) illustrates the architecture employed to retrieve water saturation curves. We discretised our input curves into 100 points spanning depths from 0 to 10 m, resulting in an NN input size of 200 (100 points on the $v_p$ curve and 100 on the $v_s$ curve) and an output size of 100. The encoder's primary function is to transform the input into a compact latent space, thereby reducing dimensionality and mitigating noise. Conversely, the decoder reconstructs the initial input from this latent representation, aiming to restore the original data with reduced noise levels. This autoencoder operates similarly to a nonlinear Principal Component Analysis (PCA) (Bailey, 2012; Jolliffe and Cadima, 2016), enabling effective noise reduction. However, it is important to note that autoencoders and PCA are not identical, as autoencoders can capture more complex nonlinear relationships beyond the linear transformations of PCA.

Specifically, the encoder comprises three hidden layers with 100, 50, and 10 neurons, while the decoder mirrors this structure with layers of 10, 50, and 100 neurons as illustrated in Fig. 6(b). We utilise the Sigmoid Linear Unit (SiLU) activation function (Szandala, 2021; Dubey et al., 2022; Zhang et al., 2024) between each linear transformation, as it has been shown to enhance the network's ability to handle noise more effectively than the Rectified Linear Unit (ReLU) and to mitigate the gradient vanishing problem associated with ReLU (Elfwing et al., 2018). The Multi-Layer Perceptron (MLP) component consists of three hidden layers, with 500 neurons each and an output layer with 100 neurons. Here, too, the SiLU activation function is applied between layers. Given that water saturation levels are bounded between 0 and 1, we incorporate a sigmoid activation function after the output layer to ensure the neural network predicts physically plausible values.

The neural network described above needs to be trained on a dataset using an optimisation algorithm. At this stage, we aim to enhance the robustness of our neural network against noise by selecting both a loss function minimally sensitive to noise and an appropriate optimiser. The classical loss function used in regression tasks is the L2 norm, or in some cases the L1 norm (Wang et al., 2022b; Terven et al., 2025). However, the L2 loss is highly sensitive to noise, whereas the L1 loss is not differentiable near the origin. To address these issues, we use the Huber loss, which combines the advantages of both and produces a loss function that is less sensitive to noise, as demonstrated in Ghafari et al. (2021). The most commonly used optimiser for training neural networks is Stochastic Gradient Descent (SGD), but it is highly sensitive to gradient oscillations caused by noise and converges slowly. The Adam optimiser has been shown to be less sensitive to noise and provides faster convergence (Sun et al., 2024).

One of the strengths of this approach is the optimised network architecture, which integrates a compact autoencoder performing a nonlinear PCA-like analysis to denoise input data, followed by a regression layer. This design allows for efficient training, taking roughly one hour for data with a fixed water table and around one day for variable water table depths on a standard computer equipped with an NVIDIA 3060 Ti GPU using CUDA.

## 3. Numerical results

In this section, we discuss the results that assess the quality of our neural network. To analyse the increasing complexity of this regression task, we first focus on a single water table level, both with and without noise. Then, in a second phase, we train our neural network on a dataset containing multiple water table depths. In summary, we present four sets of results derived from training our neural network on different datasets.

It is important to note that neural networks trained using clean data are generally evaluated only on clean input samples, whereas networks trained on noisy inputs are evaluated on both clean and noisy inputs. It is also crucial to note that all results originate from the same network architecture—the only variable is the training dataset, which may contain multiple water table depths and be either noisy or clean.
<figure><img src="image_9.png" /><figcaption>Fig. 5. Illustration of noise added to wave velocity curve for $v_p$ (a) and $v_s$ (b).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Neural network architecture: (a) conceptual scheme; (b) detailed diagram showing neurons and layer specifications.</figcaption></figure>

### 3.1. Performance criteria

To quantify the performance of our neural network, we use several commonly employed regression metrics following Hyndman and Koehler (2006) and Jadon et al. (2004). With $S_{pred}$ as the predicted saturation and $S_{true}$ as the true saturation, these metrics include:

* the Root Mean Square Error (RMSE):

$$RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (S_{true}^{i} - S_{pred}^{i})^2}$$

* the Mean Correlation Coefficient (MCC):

$$MCC = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{S_{pred}^{i} - \bar{S}_{pred}}{\sigma_{S_{pred}}} \right) \left( \frac{S_{true}^{i} - \bar{S}_{true}}{\sigma_{S_{true}}} \right)$$

* the Mean Residual Error (MRE):

$$MRE = \frac{1}{N} \sum_{i=1}^{N} \frac{|S_{true}^{i} - S_{pred}^{i}|}{|S_{true}^{i}|}$$

* the Mean Bias Error (MBE):
<figure><img src="image_11.png" /><figcaption>Fig. 7. Saturation prediction with clean data and fixed water table depth on the left panel and RMSE distribution over 10,000 randomly drawn curve on the right panel.</figcaption></figure>

$$MBE = \frac{1}{N} \sum_{i=1}^{N} (S_{true}^{i} - S_{pred}^{i})$$

• the Mean Absolute Error (MAE):

$$MAE = \frac{1}{N} \sum_{i=1}^{N} |S_{true}^{i} - S_{pred}^{i}|$$

• the Regression coefficient (R):

$$R = 1 - \frac{\sum_{i=1}^{N} (S_{true}^{i} - S_{pred}^{i})^2}{\sum_{i=1}^{N} (S_{true}^{i} - \bar{S}_{pred})^2}$$

• the Regression coefficient squared (R²)

## 3.2. Fixed water table

In a first attempt to invert wave velocities to retrieve the water saturation curve, we chose to fix the water table depth to ensure that the NN performed well in the regression task and remained stable to noise. The metrics are evaluated on 10,000 curves drawn randomly from the van Genuchten coefficient range provided in Ghanbarian-Alavijeh et al. (2010). We choose to evaluate the NN only on water table depths identical to those used for training. The first way to quantify the performance of the NN is to compare true saturation curves with predicted ones.

Fig. 7 illustrates the strong agreement between several true saturation curves and their corresponding predicted ones. This plot is generated using the neural network trained exclusively on clean data with a fixed water table. The results are highly satisfactory, as both the unsaturated and saturated regions are accurately retrieved. Furthermore, even extreme cases, such as those with abrupt transitions between unsaturated and saturated zones, are well reconstructed. The main challenge here is to accurately recover both extreme and smooth cases. In fact, the relationship between wave velocities and the saturation curve is highly non-linear, leading to very similar wave velocity curves for significantly different saturation profiles. Thus, we conclude that our approach effectively addresses the ill-posed nature of the problem, particularly the nonlinearity of the inversion and the weak sensitivity to certain parameters. Moreover, the left panel, which shows the RMSE distribution over 10,000 randomly drawn curves, demonstrates that most predicted curves have an RMSE value less than $10^{-3}$, which is very good.

The next step is to test our approach on noisy data to evaluate its robustness. Fig. 8 presents the synthetic and inverted vertical water saturation profiles, but for noisy data. Here, we consider noise amplitude

<caption>Table 2</caption>

Mean metrics after training on a fixed water table for 10,000 randomly drawn curve.

<table><thead><tr><th></th><th>Clean data</th><th>3% noisy</th><th>5% noisy</th></tr></thead><tbody><tr><td>RMSE</td><td>7.8 × 10<sup>-4</sup></td><td>1.48 × 10<sup>-2</sup></td><td>3.48 × 10<sup>-2</sup></td></tr><tr><td>MCC</td><td>0.999</td><td>0.997</td><td>0.994</td></tr><tr><td>MRE</td><td>6.35 × 10<sup>-4</sup></td><td>8.27 × 10<sup>-3</sup></td><td>1.99 × 10<sup>-2</sup></td></tr><tr><td>MBE</td><td>-1.7 × 10<sup>-5</sup></td><td>-5.93 × 10<sup>-4</sup></td><td>-1.13 × 10<sup>-3</sup></td></tr><tr><td>MAE</td><td>2.23 × 10<sup>-4</sup></td><td>3.91 × 10<sup>-3</sup></td><td>9.41 × 10<sup>-3</sup></td></tr><tr><td>R</td><td>0.999</td><td>0.998</td><td>0.989</td></tr><tr><td>R<sup>2</sup></td><td>0.998</td><td>0.996</td><td>0.978</td></tr></tbody></table>

of 3% and 5%. Along with the histogram of RMSE values, we selected three saturation curves with three distinct RMSE values. The blue and red curves are relative to two saturation profiles with an RMSE lower than $2 \times 10^{-2}$ and the black curve correspond to one saturation profile with an RMSE lower than $10^{-1}$, but higher to $2 \times 10^{-2}$. Moreover, the histograms distinguish two zones, coloured in green and red, which allow separating the data with a very good fit between synthetics and inversed water profiles from those with a lower fit. It is important to note that, in this case, the neural network was trained on both noisy and clean data to enhance robustness. We analyse Fig. 8 in three stages. First, we examine the histograms of RMSE values computed for 10,000 randomly sampled curves. Next, we compare three of these curves with their corresponding true saturation curves. Finally, we conduct a cross-analysis of the results. In the first stage, the histograms for noise levels of 3% and 5% indicate that most randomly sampled curves are accurately reconstructed, exhibiting an RMSE below $2 \times 10^{-2}$. This threshold is arbitrarily chosen but distinguishes curves with excessively high RMSE values, which result in the poorest reconstruction of the unsaturated zone. Applying this criterion, nearly 90% of the curves are well reconstructed with 3% noise, while more than 75% are well reconstructed with 5% noise. Focusing on the three plotted curves below, we note that extreme saturation curves are accurately reconstructed, even with 5%. However, this observation is less valid for moderate cases, where deviations from the true curves become more pronounced depending on the applied noise level. These findings can be correlated with the histogram analysis: well-reconstructed curves, particularly the extreme ones, correspond to the lower RMSE region of the histogram, whereas less extreme curves tend to fall within the region associated with higher RMSE values. Nevertheless, it is important to highlight that, in real field conditions, the observed saturation curves are rarely extreme. As a result, they predominantly fall within the region where our neural network performs well. We could also note that even for strong noisy cases, the saturated part is always well reconstructed, which is important when the water table depth will be variable.

After this initial qualitative analysis, we turn to a quantitative evaluation of our results, summarised in Table 2. The most commonly
<figure><img src="image_12.png" /><figcaption>Fig. 8. Saturation prediction with noisy data and fixed water table depth with 3% noise (a) and 5% noise (b). Above, the histograms show the distribution of curves by RMSE (Root Mean Square Error) values over a set of 10,000 randomly drawn curves. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

used metric for assessing neural network performance in regression
tasks is the Root Mean Square Error (RMSE), which quantifies the
Euclidean distance between predicted and true values. In our case,
the mean RMSE remains very low for noise-free data and increases
slightly when noise is introduced. However, in absolute terms, the
RMSE values remain low, demonstrating the robustness of our neural
network. Consequently, poorly reconstructed curves represent only
a small fraction of the parameter space. Another key metric is the
regression coefficient R, which ranges from -1 to 1 and quantifies
the correlation between predicted and true values. In both clean and
noisy cases, the R coefficient remains close to 1, confirming a strong
agreement between predictions and ground truth, even at high noise
levels (5%). Additionally, other error metrics, including the Mean
Residual Error (MRE), Mean Bias Error (MBE) and Mean Absolute Error
(MAE), remain consistently low across both noise levels. These find-
ings validate the effectiveness of our approach in solving the inverse
problem of retrieving water saturation distribution from wave velocity
data under the assumption of a fixed water table. To further assess
the generalisation capabilities of our model, we introduce variability
in the water table depth to evaluate whether our neural network can
effectively handle more complex and diverse datasets.

3.3. *Varying water table*

In this section, the assumption of a fixed water table no longer
holds, and we test our neural network with various water table depths
randomly drawn within the previously defined range. The main chal-
lenge here is to accurately capture the transition between unsaturated
and saturated zones, which varies with water table depth, while also
assessing the network's robustness to noise. It is important to emphasise

that the neural network architecture remains unchanged from the
previous subsection. The only difference lies in the training dataset,
which now includes a broader range of water table depths.

On clean data, Fig. 9 demonstrates prediction results, nearly as accurate as those in Fig. 7. This figure highlights the ability of our NN, when trained on multiple water table depths, to effectively capture the transition from the saturated to the unsaturated zone. Additionally, it successfully reproduces behaviours in the unsaturated region, including the most extreme cases. A key takeaway is that the same NN can handle highly complex behaviours and perform reliable inversion even as the forward model complexity increases. Furthermore, we tested the network on data with water table depths outside of the training range (<2 m or >7 m for the water table depth), and it was still able to provide accurate predictions, demonstrating strong generalisation capabilities. However, given the sensitivity of results to noise, it is necessary to evaluate the performance of our NN under noisy conditions.

It is important to note that the neural network's training approach
remains unchanged—noise is introduced during the training phase to
enhance its ability to handle it effectively. First on Fig. 10, the RMSE
histograms indicate that a large majority of the reconstructed curves
exhibit an RMSE below 2 × 10⁻², confirming that most predictions
are accurate. Naturally, since this task is more complex due to the
variability in water table depth, the overall performance is slightly
lower than in the fixed water table case. However, when focusing on
the curve reconstructions, we observe that extreme saturation curves
are well reproduced, along with some less extreme ones, particularly
at a noise level of 3%. Once again, we can associate specific curves
with regions of the histogram based on their RMSE values. Even in
cases where RMSE is slightly higher, the reconstructed curves generally
remain close to the ground truth. If we now examine the saturated zone
<figure><img src="image_13.png" /><figcaption>Fig. 9. Saturation prediction with clean data and variable water table depth on the left panel and RMSE distribution over 10,000 randomly drawn curve on the right panel.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 10. Saturation prediction with noisy data and variable water table depth with 3% noise (a) and 5% noise (b). Above, the histograms show the distribution of curves by RMSE (Root Mean Square Error) values over a set of 10,000 curves.</figcaption></figure>

and the neural network's ability to predict the water table depth, we observe a consistently high level of accuracy. Regardless of noise level or curve type, the saturated region is consistently well reconstructed, allowing for an easy identification of the water table depth. Thus, from a qualitative perspective, our NN demonstrates strong generalisation capabilities and robustness to noise.

All these observations are confirmed when analysing the quantitative metrics previously used for the fixed water table case. As expected, the results in Table 3 are slightly worse than those in Table 2. This can be explained by the fact that the neural network architecture remains unchanged, with only the training dataset differing. The NN still delivers strong performance on both clean and noisy data, as indicated by the RMSE. Furthermore, RMSE histograms in Fig. 10 show degradation

<table><caption>**Table 3**
Metrics after training on a variable water table.</caption><thead><tr><th></th><th>Clean data</th><th>3% noisy</th><th>5% noisy</th></tr></thead><tbody><tr><td>RMSE</td><td>4.59 × 10<sup>-3</sup></td><td>3.2 × 10<sup>-2</sup></td><td>3.34 × 10<sup>-2</sup></td></tr><tr><td>MCC</td><td>0.999</td><td>0.995</td><td>0.993</td></tr><tr><td>MRE</td><td>1.80 × 10<sup>-4</sup></td><td>2.05 × 10<sup>-2</sup></td><td>2.1 × 10<sup>-2</sup></td></tr><tr><td>MBE</td><td>-1.24 × 10<sup>-5</sup></td><td>2.40 × 10<sup>-3</sup></td><td>2.65 × 10<sup>-3</sup></td></tr><tr><td>MAE</td><td>8.40 × 10<sup>-4</sup></td><td>9.70 × 10<sup>-3</sup></td><td>1.01 × 10<sup>-2</sup></td></tr><tr><td>R</td><td>0.999</td><td>0.991</td><td>0.989</td></tr><tr><td>R<sup>2</sup></td><td>0.998</td><td>0.982</td><td>0.978</td></tr></tbody></table>
<table><caption>**Table 4**
Summary of metrics reported in deep learning papers relative to Table 1 in hydrogeophysics.</caption><thead><tr><th>References</th><th>RMSE</th><th>MAE</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>Mutlu et al. (2008)</td><td></td><td></td><td>0.79 to 0.86</td></tr><tr><td>Edincliler et al. (2013)</td><td></td><td></td><td>0.98 to 0.99</td></tr><tr><td>Xie et al. (2020)</td><td>0.29 (permittivity)</td><td></td><td></td></tr><tr><td>Aleardi et al. (2022)</td><td>6.23 (resistivity)</td><td></td><td></td></tr><tr><td>Li et al. (2024)</td><td></td><td>93.143 (v<sub>p</sub>)/48.308 (v<sub>s</sub>)</td><td></td></tr><tr><td>Lu et al. (2025)</td><td>0.1832 (ground water level)</td><td></td><td>0.9924</td></tr></tbody></table>

in the NN's ability to retrieve non-extreme saturation curves. The main coefficient reflecting reduced ability to reconstruct the saturation curves is the R coefficient, which drops to approximately 0.991 for noisy data. This explains the deviations from the true saturation curves observed in Fig. 10. However, other quality metrics such as MRE, MBE and MAE remain low, demonstrating the ability of our relatively small network to handle the task effectively.

To conclude, we propose to compare all our previous metrics to the ones (when available) coming from Table 1 proposed in the introduction and summarised in Table 4, which presents a review of the literature on AI applied to hydrogeophysics. As the reader can see, even in our worse-case scenario corresponding to the variable water table with noisy data, our metrics demonstrate the ability of our approach to handle the task very efficiently, with high accuracy and reliability.

## 4. Conclusion

In this study, we developed a neural network-based inversion approach to retrieve water saturation vertical profiles from compressional ($v_p$) and shear ($v_s$) wave velocities. A key strength of our methodology lies in the neural network architecture, which integrates an autoencoder layer to enhance noise robustness, followed by a regression model capable of predicting the entire saturation curve rather than isolated parameters. The performance evaluation demonstrates that our neural network effectively reconstructs water saturation profiles with high accuracy, even in the presence of noise. It successfully retrieves both extreme and smooth saturation curves and remains stable under varying water table conditions. The quantitative metrics confirm the robustness of our model, with low error values across different testing scenarios, highlighting its potential as a reliable tool for geophysical inversion.

Despite these promising results, our study is based entirely on synthetic data derived from numerical modelling. While this ensures a well-controlled training environment, real-world applications require validation against experimental data. Future work should focus on testing the neural network with field measurements and, addressing potential discrepancies between simulated and observed seismic responses. Further developments may also explore the integration of time varying saturation curves when the hydrostatic assumption may not hold. More complex geological media with layers or inclusions may also be explored. By demonstrating the feasibility of using neural networks for retrieving water saturation profiles, this study lays the foundation for future advancements in hydrogeophysical inversion.

### CRediT authorship contribution statement

**Quentin Didier:** Writing – original draft, Software, Methodology, Conceptualization.
**Victor Sauvage:** Software, Conceptualization.
**Léna Pellorce:** Writing – original draft, Visualization, Methodology, Conceptualization.
**Rémi Valois:** Writing – review & editing.
**Slimane Arhab:** Writing – review & editing.
**Arnaud Mesgouez:** Writing – review & editing.
**Writing – original draft, Project administration, Methodology, Conceptualization.**

### 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.

### Appendix. Abbreviations

<table><tr><td>AI</td><td>Artificial Intelligence</td></tr><tr><td>ANN</td><td>Artificial Neural Network</td></tr><tr><td>CNN</td><td>Convolutional Neural Network</td></tr><tr><td>DL</td><td>Deep Learning</td></tr><tr><td>ERT</td><td>Electrical Resistivity Tomography</td></tr><tr><td>GPR</td><td>Ground Penetrating Radar</td></tr><tr><td>NN</td><td>Neural Network</td></tr><tr><td>MAE</td><td>Mean Absolute Error</td></tr><tr><td>MBE</td><td>Mean Bias Error</td></tr><tr><td>MCC</td><td>Mean Correlation Coefficient</td></tr><tr><td>MCMC</td><td>Markov Chain Monte-Carlo</td></tr><tr><td>ML</td><td>Machine Learning</td></tr><tr><td>MLP</td><td>Multi-Layer Perceptron</td></tr><tr><td>MRE</td><td>Mean Residual Error</td></tr><tr><td>MSE</td><td>Mean Squared Error</td></tr><tr><td>PCA</td><td>Principal Component Analysis</td></tr><tr><td>R</td><td>Regression coefficient</td></tr><tr><td>ReLU</td><td>Rectified Linear Unit</td></tr><tr><td>RMSE</td><td>Root Mean Square Error</td></tr><tr><td>SGD</td><td>Stochastic Gradient Descent</td></tr><tr><td>SiLU</td><td>Sigmoid Linear Unit</td></tr><tr><td>v<sub>p</sub></td><td>P-wave velocity</td></tr><tr><td>v<sub>s</sub></td><td>S-wave velocity</td></tr></table>

### Data availability

Data will be made available on request.

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