In this study, we investigate the use of ground-penetrating radar (GPR) time-lapse monitoring of artificial soil infiltration experiments. The aim is to evaluate this protocol in the context of estimating the hydrodynamic unsaturated soil parameter values and their associated uncertainties. The originality of this work is to suggest a statistical parameter estimation approach using Markov chain Monte Carlo (MCMC) methods to have direct estimates of the parameter uncertainties. Using the GPR time data from the moving wetting front only does not provide reliable results. Thus, we propose to use additional information from other types of reflectors to optimize the quality of the parameter estimation. Water movement and electromagnetic wave propagation in the unsaturated zone are modeled using a one-dimensional hydrogeophysical model. The GPR travel time data are analyzed for different reflectors: a moving reflector (the infiltration wetting front) and three fixed reflectors located at different depths in the soil. Global sensitivity analysis (GSA) is employed to assess the influence of the saturated hydraulic conductivity

The vadose zone is defined by the region between the ground surface and the groundwater table. Because of its location, it is at the center of the interactive atmospheric–surface–underground water system. Hence, understanding water flow in the vadose zone is crucial for hydrological modeling and forecasting, which can be useful for water resources management, agricultural practices optimization, or geotechnical studies. The porous medium in the vadose zone is filled by both water and air phases. The air phase is considered infinitely mobile and remains at atmospheric pressure. The movement of water has a non-linear behavior and is characterized by two fundamental hydraulic relationships, namely, the water retention and the hydraulic conductivity functions. Various mathematical expressions can describe these functions in terms of dependent variables and fitting parameters. In this work, we use the Mualem–van Genuchten (

Different approaches can be applied to estimate the unsaturated soil parameters. In soil physics, the reference method relies on laboratory measurements conducted on soil core samples. Such experiments can use various techniques such as thermogravimetry or tensiometry, but common practices rely on measurements of hydraulic fluxes

At the field scale, the soil hydraulic properties and state variables can be estimated using numerous approaches. Measurements of the soil water content, water pressure, and hydraulic conductivity can show significant variations because of their sensitivity to different hydrological processes. As a consequence, such measurements are convenient for the estimation of soil parameters of the subsurface at the field scale by inverse modeling. Measuring techniques for soil hydraulic properties and state variables can be classified into two categories based on whether the measuring devices have to be in direct contact with the soil or not. In the former, when the measuring devices must be in direct contact with the soil, measurements can present a spatial support around the micro (mm–cm) and local scale (cm–m) with water content sensing techniques (using thermal or electromagnetic sensors, e.g., capacitance or time domain reflectometry,

Other techniques use non-invasive devices that do not have to be in direct contact with the soil, such as remote sensing and hydrogeophysical methods. Remote sensing techniques use devices that operate remotely and relatively far from the ground, such as unmanned aerial vehicle-based thermal infrared imagery

Common hydrogeophysical methods include electromagnetic induction

Today, GPR is widely used in the field of hydrogeophysics. Different techniques have been reviewed and discussed by

Various studies have investigated the monitoring of different types of flow processes with time-lapse GPR in the context of evaluating the soil hydraulic states, hydraulic properties, or unsaturated soil parameters (e.g.,

At the laboratory scale,

Test case and experimental device illustration at an advanced time step

In the present study, we are interested in using a quick, easy-to-apply, and cheap field-scale method to characterize the unsaturated soil parameters. To this end, time-lapse GPR monitoring of artificial infiltration is a well-suited protocol. It is similar to ring infiltrometry methods but with additional information from GPR measurements. In the literature, the work of

considering different reflectors at different depths: a moving reflector which corresponds to the infiltration dynamic wetting front and two fixed reflectors located at different depths in the soil;

investigating the influence of all soil parameters (the saturated hydraulic conductivity, the saturated and residual water contents, and the Mualem–van Genuchten shape parameters

performing statistical calibration of soil parameters using the Markov chain Monte Carlo (MCMC) method and evaluating the reliability of the estimated parameters by analyzing not only the calibrated model parameters but also their associated uncertainty;

evaluating the impact of the type of reflector (moving or fixed) by analyzing the calibrated model parameters and their confidence intervals for different scenarios.

The plan of the paper is as follows: Sect. 2 describes the test case as well as the mathematical and numerical hydrogeophysical models. Section 3 reports on the GSA results of the different TWT signals. Then, Sect. 4 discusses the results of the soil parameter estimation with MCMC for different scenarios including varying soil types, water table depths, and surface boundary conditions.

In this work, we conduct a synthetic study on the time-lapse GPR monitoring of artificial infiltration protocol, prior to applying it in real conditions. The idea is to perform synthetic experiments under the same conditions as real experiments so as to better understand the pertinence of the investigated protocol when used for estimating the unsaturated soil parameters. The test case considered is a hypothetical one-dimensional experiment of water infiltration in a homogeneous sandy soil of 150

Water infiltration in unsaturated or saturated soils is governed by the one-dimensional Richards equation

The interdependencies of the pressure head, conductivity, and water content are described using the standard models of

In GPR sounding, pulses of radiofrequency (

To describe the dependency of the dielectric permittivity on the water content, we use the complex refractive index model

In this work, the soil is considered as a linear and isotropic non-magnetic medium. When working with frequencies below 1

Equations (4) and (5) evidence that GPR waves propagate at a much lower speed in wet conditions. Any source of reflection in the sounded soil produces a reflected wave that is recorded at a time corresponding to the duration of its propagation, from the transmitting antenna, down to the source of reflection, then back up to the receiving antenna, i.e., the TWT of the reflected wave.

We consider a one-dimensional scenario (the offset between the antennas is null) and discretize the domain into

Prior intervals of the unsaturated soil parameters for both GSA and Bayesian estimation.

A reflection occurs at the interface between two successive elements if the reflection coefficient is not zero. The reflection coefficient expresses the contrast of dielectric constant (due to the contrast of water content) at the interface between the two elements

For an 800

Note that one could easily consider a non-perpendicular incidence of the GPR wave at the interface, introducing the incidence angle in Eq. (

The variation of the water content in the soil during the infiltration is computed using the WAMOS-1D code

Hydrogeophysical model responses for

The time-lapse signal

Summary of the working process of the forward hydrogeophysical model and how it is used to build the PCE surrogate model.

Note that

The GSA method evaluates how the outputs of a model are influenced by the variation of the input parameters

Given a model with a set of

the first-order sensitivity index:

the total sensitivity index:

To perform a variance-based GSA, a practical approach (to save computational time) is to use polynomial chaos expansion (PCE;

In this work, Legendre polynomials are used since uniform distributions are assumed for all uncertain parameters. Uniform distributions express the absence of prior information. This makes all parameter values in the given prior intervals equally likely. Large prior distribution intervals are considered for all unsaturated soil parameters (Table

The number of coefficients for a full PCE representation is

A PCE is constructed at each time step for all model responses (

Variance of

Time distribution of the variance of

The temporal distribution of the output variance of the three TWT signals (

Difference between the total (

Marginal effects of the unsaturated soil parameters

To evaluate further the effect of the unsaturated soil parameters on the

The sensitivity of

The van Genuchten parameter

The sensitivity of

The variance of the

As for the

The saturated water content

As for the

The van Genuchten parameter

Surprisingly, and contrarily to the

In this section, we estimate the unsaturated soil parameters in a Bayesian framework using the Markov chain Monte Carlo (MCMC) sampler (

Can we obtain an appropriate estimation of all unsaturated soil parameters from TWT data?

What is the impact of the kind of TWT data (moving/fixed reflectors) and of the number of reflectors on the calibrated model parameters and their confidence intervals?

What is the optimal set of TWT measurements that yields good reliability of all unsaturated soil parameters?

The MCMC method has been successfully employed in various inverse hydrological problems (e.g.,

Reference values used to build the synthetic calibration data for the different scenarios (estimated mean values in bold, size of the posterior confidence intervals between parentheses, and ratio of prior to posterior intervals in italics)

In the sequel, the MCMC method is performed with the

The

The reliability of the unsaturated soil parameters is assessed for five different scenarios of measurement sets. In the first scenario, only data of the wetting-front

In the five scenarios, the MCMC sampler uses three parallel chains and a total number of 50 000 runs. The last 25 % of the runs that adequately fit the model onto observations are used to estimate the joint posterior distribution.

The MCMC results of the five studied scenarios are given in Table

The results in Table 3 for scenario 1 using only data of the

An accurate estimation of

A fair estimate of the parameters

The parameter

There is a poor estimation of

The

In summary, using only data of the

The estimation of the unsaturated soil parameters for scenarios 2 and 3, using only data of the

The parameters

The soil parameters

The results of scenario 4, which combines data of

Parameters

Parameters

MCMC solutions using scenarios 1 (a1–a5), 3 (b1–b5), and 4 (c1–c5) for calibrating the hydrogeophysical model. The histograms are built from the posterior distributions. The estimated mean values are represented as a dashed black line and compared to the exact target value (solid red line). The displayed parameter intervals correspond to the prior upper and lower limits of Table

Figure

Finally, the results of the last scenario, which combines data of

MCMC solutions using

The results of MCMC for this last scenario are shown in Fig.

Note that the parameter

To complete the numerical study, the protocol was tested varying the boundary conditions. One may wonder how much the thickness of the vadose zone would impact the calibration of the hydraulic parameters. For that purpose, three scenarios were considered, varying the water table depth from 50

MCMC solutions considering a water table at 50

Last, the efficiency of the protocol was numerically tested on three types of soils used in the experimental platform SCERES in Strasbourg (Bohy et al., 2006). A coarse, a medium, and a fine sand are considered (see Table

Hydraulic parameters for three types of sand.

These results evidence that the GPR signal data of both the wetting front and a fixed reflector can provide very different but complementary information for the identification of the unsaturated soil parameters. They also point to the significant benefit of combining the GPR signal data of a fixed reflector, preferably located sufficiently deep in the soil, with the TWT signal of the moving infiltration wetting front. This combination leads to good reliability of almost all soil parameters with very narrow posterior intervals in comparison with the prior ones. In particular, the van Genuchten parameter

The aim of the present study was to optimize a cheap method used at the field scale to characterize the hydraulic parameters of a porous medium. To this end, we investigated a particular protocol: time-lapse GPR monitoring of artificial infiltration experiments. Water infiltration into an initially unsaturated sandy soil was simulated using a one-dimensional hydrogeophysical model. GPR time signals were analyzed from the reflection of the electromagnetic wave on the moving wetting front and on two fixed reflectors located at different depths. GSA, based on PCE decomposition, was used to assess the effect of the unsaturated soil parameters (saturated hydraulic conductivity, saturated and residual water contents, and Mualem–van Genuchten shape parameters

The results of GSA showed that the

The reliability of the unsaturated soil parameters was assessed for five different scenarios of TWT measurement sets. When only data of the

The results of this study highlight the significant benefit of combining TWT signals of fixed and moving (infiltration wetting front) reflectors for very good identification of all the unsaturated soil parameters. It also points out the role of GSA in assessing the influence of the parameters on the output signals and the necessity of performing statistical calibration to assess the reliability of model parameters by evaluating not only estimated parameter values but also their associated uncertainties.

The hydrological code WAMOS-1D, the coupled hydrogeophysical model, and the global sensitivity analysis tools can be provided by the authors upon request.

All data presented in this paper are available from the authors upon request.

JFG and NL formulated the aims of the research. BB, FL, and AY provided the hydrological code. AY provided the resources for sensitivity analysis and Bayesian estimation studies. RM established the hydrogeophysical coupled model and ran all experiments under the supervision of JFG and NL, and validation of all co-authors. RM prepared the manuscript with reviewing and editing contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors would like to acknowledge the High-Performance Computing Center of the University of Strasbourg for supporting this work by providing scientific support and access to computing resources. Part of the computing resources was funded by the Equipex Equip@Meso project (Programme Investissements d'Avenir) and the CPER Alsacalcul/Big Data. Computing time was provided by the HPC-UdS.

The doctoral position of the first author is co-funded by the Grand-Est Region and the University of Strasbourg. This research work is partly funded by the French National Research Agency through the Exciting project (grant no. ANR-17-CE06-0012-01).

This paper was edited by Marnik Vanclooster and reviewed by two anonymous referees.