Quantitative knowledge of the subsurface material distribution and its effective soil hydraulic material properties is essential to predict soil water movement. Ground-penetrating radar (GPR) is a noninvasive and nondestructive geophysical measurement method that is suitable to monitor hydraulic processes. Previous studies showed that the GPR signal from a fluctuating groundwater table is sensitive to the soil water characteristic and the hydraulic conductivity function. In this work, we show that the GPR signal originating from both the subsurface architecture and the fluctuating groundwater table is suitable to estimate the position of layers within the subsurface architecture together with the associated effective soil hydraulic material properties with inversion methods. To that end, we parameterize the subsurface architecture, solve the Richards equation, convert the resulting water content to relative permittivity with the complex refractive index model (CRIM), and solve Maxwell's equations numerically. In order to analyze the GPR signal, we implemented a new heuristic algorithm that detects relevant signals in the radargram (events) and extracts the corresponding signal travel time and amplitude. This algorithm is applied to simulated as well as measured radargrams and the detected events are associated automatically. Using events instead of the full wave regularizes the inversion focussing on the relevant measurement signal. For optimization, we use a global–local approach with preconditioning. Starting from an ensemble of initial parameter sets drawn with a Latin hypercube algorithm, we sequentially couple a simulated annealing algorithm with a Levenberg–Marquardt algorithm. The method is applied to synthetic as well as measured data from the ASSESS test site. We show that the method yields reasonable estimates for the position of the layers as well as for the soil hydraulic material properties by comparing the results to references derived from ground truth data as well as from time domain reflectometry (TDR).

Quantitative understanding of soil water movement is in particular
based on accurate knowledge of the subsurface architecture and the
hydraulic material properties. As direct measurements are
time-consuming and near to impossible at larger scales, soil hydraulic
material properties are typically determined with indirect
identification methods, such as inversion

Available research studies regarding the estimation of hydraulic properties from GPR measurements may be categorized according to the applied methods for the different components of the research study, such as the (i) GPR measurement procedure, (ii) experiment type, (iii) GPR simulation method, (iv) optimization method, and (v) evaluation method of the GPR signal.

Most of these studies either use on-ground, off-ground, or borehole
GPR measurements. On-ground measurements

The applied experiment types range from infiltration, fluctuating
groundwater table, to evaporation. Infiltration experiments

The applied models to simulate the GPR signal are faced by an inherent
tradeoff between performance and accuracy. Ray tracing

Due to the inherent oscillating nature of the electromagnetic signal,
inversion of GPR data generally demands globally convergent and robust
optimization techniques. Sequentially coupling a globally convergent
search algorithm, e.g., the global multilevel coordinate search
algorithm

The GPR signal has to be processed automatically for parameter
estimation. Many full waveform inversion approaches directly use the
resulting Green's function

In homogeneous materials, the transition zone above the groundwater
table exhibits a smooth variation of the relative permittivity. Since
the resulting GPR reflection is a superposition of a series of
infinitesimal contributions along the transition zone, the detailed
form of this reflection is sensitive to the variation of the relative
permittivity. For simplicity, we refer to this reflection as
transition zone reflection.

ASSESS provides an effectively 2-D subsurface architecture
consisting of three kinds of sand (A, B, and C). During the experiment that
is evaluated in this work, the groundwater table was manipulated via
a groundwater well (white square at

This work builds upon previously published methods for simultaneous
estimation of the subsurface architecture and the effective water
content based on on-ground multi-offset GPR measurement data

The measurement data for this work are acquired at an approximately

Quantitative understanding of a system of interest requires its
mathematical representation. Based on

The standard model to describe the volumetric water content

We choose the Brooks–Corey parameterization

The hydraulic conductivity function

In order to provide the measurement data for this study, ASSESS was
forced with a fluctuating groundwater table leading to two
characteristic phases comprising an initial drainage phase and
a multistep imbibition phase (Fig.

During the experiment with two distinct phases (initial drainage and
multistep imbibition – separated by the vertical line), the position of the
groundwater table was measured manually in the groundwater well and
automatically with the tensiometer (Fig.

During the experiment, the groundwater table was measured (i) automatically
with a tensiometer and (ii) manually via the groundwater well
(Fig.

The observation operator required to compare the simulated hydraulic
state with the GPR measurement data involves the solution of the
time-dependent Maxwell equations in linear macroscopic isotropic
media. These equations quantify the propagation of the electric field

The relative permittivity of the subsurface

To analyze the GPR data, we follow

The GPR data evaluation method presented in this section consists of
four main steps. In the first step, the signal is processed
(Sect.

The processing of the GPR signal includes the following step: (i)
time-zero correction, (ii) dewow filter, (iii) 2-D to 3-D conversion,
(iv) removal of the direct signal (particularly including the direct
wave and the ground wave) and the trailing signal (signal at the end
of the trace which is disturbed by the dewow filter), and (v)
normalization (Fig.

Since the time-zero of the GPR antennas changes over time, we pick the
direct signal and subtract the evaluated travel time from each trace
of the radargram for time-zero correction. Subsequently, a dewow
filter is applied to subtract inherent low-frequency wow noise of the
GPR signal. Because the observation is in 3-D and the simulation in
2-D, we convert the simulated signal to 2.5-D, meaning to 3-D with
translational symmetry perpendicular to the survey line and parallel
to the ground surface

In this step, events are detected in each trace separately (Fig.

After the event detection, the measured signal and the detected events
(Sect.

The detected events of the first trace of the synthetic radargram
analyzed in Sect.

Example of the association of simulated (s) and measured (m) events
with indices 1–6 and 1–7, respectively. The color of the dots indicates the
sign of amplitude of the events.

The selected events extracted from the measured data have to be
associated with the detected events extracted from the simulated data for the
parameter estimation. To this end,

In order to exclude combinations a priori, the detected events are
aggregated in clusters (Fig.

After the association of the events, outliers are detected by
calculating the mean and standard deviation of the travel time
differences. All associations that exhibit an absolute travel time
difference larger than

Inversion of GPR data typically requires globally convergent parameter
estimation algorithms which are computationally expensive. In order to
keep the parameter estimation procedure efficient, we use an iterative
strategy (Fig.

We choose a sequentially coupled parameter estimation procedure
which (i) allows minimizing the computational cost and (ii) facilitates the
implementation of tagging (Sect.

The most expensive operation of the forward simulation is the
calculation of the observation operator, which includes the solution
of Maxwell's equations (Sect.

Assuming

Due to the oscillating nature of the GPR signal and due to the applied
GPR data evaluation (Sect.

We choose the simulated annealing algorithm

If the parameter update is drawn from the whole parameter space, the
simulated annealing algorithm is globally convergent. However, this
approach is typically inefficient. Hence, we search the neighborhood
for better parameters starting from the Latin-hypercube-sampled initial
parameters

The Levenberg–Marquardt algorithm is implemented as described by

We choose

In this section, the methods presented in the previous section are
applied to GPR data. First, the setup of the case study, its
implementation and the detailed setup of the parameter estimation
procedure are explained (Sect.

In this work, the subsurface architecture of ASSESS is represented
with layers. The position of these layers is parameterized and can be
estimated. For illustration, the setup is shown in
Fig.

For the simulation of the GPR signal, we assume a layered subsurface
architecture (Fig.

To simulate the temporal propagation of the electromagnetic signal, we
solve Maxwell's equations (Sect.

The fit range limits the parameter space available for parameter
estimation and is in particular used by the simulated annealing algorithm to
draw parameter updates (Sect.

We use one-tenth of the minimal wavelength

To avoid multiple reflections at the air–soil boundary, we set the
relative permittivity above the soil to

The GPR data are evaluated according to Sect.

The general setup of the optimization is sketched with this figure.
The available hydraulic potential

The general setup of the optimization is explained with
Fig.

In order to evaluate the performance of the ensemble members, the mean
absolute error in normalized travel time

The standard deviations of the measured data,

In this section, the synthetic data generated along the lines given in
Sect.

The phenomenology of the transition zone reflection for characteristic
times during imbibition, equilibration, and drainage was discussed by

The mean and the standard deviation are calculated using the
resulting parameters from the 10 best ensemble members
(Sect.

The true synthetic data are simulated with hydraulic parameters that
represent coarse-textured sandy soils (Table

Initialized with static hydraulic equilibrium, the simulation starts
with an initial drainage step where the groundwater table is
lowered. Hence, the material at the upper end of the capillary fringe
with high initial water content is desaturated. After the subsequent
equilibration step, the groundwater table is raised during the
subsequent imbibition step. The Brooks–Corey parameterization
(Eq.

During the equilibration step after the first imbibition, the additional kink smoothes. Thus, the water content increases in the material with low water content (3) and decreases in the material at the upper end of the capillary fringe (4). This smoothing depends on both the soil water characteristic and the hydraulic conductivity function. Sharpening and smoothing of the transition zone are repeated consistently for the other subsequent imbibition and equilibration phases (5–8).

According to the CRIM (Sect.

In summary, this numerical simulation confirms qualitatively (i) that the dynamics of the fluctuating groundwater table are sensitive to both the soil water characteristic and the hydraulic conductivity function and (ii) that the transition zone reflection leads to tractable reflections during the imbibition step.

The resulting material parameters estimated from synthetic data are
shown for the

After the inversion of the synthetic data, we find that the resulting
soil water characteristics for material
A (Fig.

Since the architecture is a layered structure where material C is
located above material A (Fig.

The correlation coefficients for the mean parameter set show in
particular that the porosity of the gravel layer
(

This figure shows

The evaluation of the synthetic GPR data is separated into three
parts

The saturated hydraulic conductivity of material
A (Fig.

The uncertainty of the soil water characteristic of
material A (Fig.

The uncertainty of the saturated hydraulic conductivity of material
A (Fig.

The correlation coefficients
(Fig.

In order to further investigate the quality of the mean parameter set,
we simulated the resulting water content distribution
(Fig.

The remaining deviations in soil water content after the parameter
estimation cause residuals in the GPR signal
(Fig.

Similar to the analysis of the deviation in water content
(Fig.

In this section, the measured data (Sect.

Before starting the experiment, a single-offset measurement was
acquired to analyze the initial state of ASSESS revealing material
interfaces as well as compaction interfaces
(Fig.

ASSESS is confined by walls at all four sides. Reflections from
confining walls are most visible around

As an aside, closer scrutiny of the radargrams reveals that the single-offset and the time-lapse data were measured with different but structurally identical antennas. Thus, in particular the measured GPR signals of the direct wave and the ground wave are slightly different.

The time-lapse GPR measurement was recorded at

Corresponding to the analysis of the synthetic data
(Sect.

Together with the water content distribution, the time-lapse GPR data also contain information about the subsurface architecture. However, separating signal contribution from the subsurface architecture and the hydraulic dynamics is not always possible. Here, this is most prominent for the reflection of the material interface (V). Initially, the amplitude of this reflection is large, because the water content in material C is near the residual water content, whereas the water content in material A is significantly higher at the material interface. As soon as both materials are water saturated, the amplitude of the material interface reflection (V) is low since the effective porosities of the two materials are similar. Thus, the amplitude of the reflected signal originating from the material interfaces may change depending on the hydraulic state. Additional information about the subsurface architecture can be inferred from the reflection at the material interface between material A and the gravel layer (VI) and from the reflection at the material interface of the gravel layer and the concrete basement (VII). These reflections are in particular suitable to analyze the total change of water content over time.

In summary, we note that the characteristic properties of the
transition zone reflection during the imbibition and equilibration
steps that were identified in the simulation
(Fig.

Since the GPR measurements cover only a small portion of the
subsurface architecture of ASSESS, the hydraulic representation is
restricted to 1-D (Sect.

Investigating the resulting material properties of the inversion
(Fig.

The resulting material parameters estimated from measured data are
shown for the

The mean and the standard deviation are calculated using the
resulting parameters from the 10 best ensemble members
(Sect.

Compared to the uncertainties based on synthetic data
(Table

Analogous to Fig.

The parameter

The resulting value for parameter

Similar to parameter

The resulting value for parameter

Concerning the position of the material interfaces, we find that the
estimated interface position of material A and C
(

An analysis of the remaining residuals in travel time after the
optimization (Fig.

The measured reflection (6) interferes with the reflection of the compaction interface, (i) leading to a compressed reflected wavelet in the measurement. Similarly, reflections (3) and (5) also interfere with the compaction interface (i). Since interferences cannot be correctly evaluated if not all contributions are represented, this analysis shows that representing compaction interfaces is relevant in ASSESS.

As a side remark, note that the error originating from assuming a constant soil temperature for the calculation of the relative permittivity of water is relatively small regarding the total residuum. However, it is worth noting that the corresponding residuals easily exceed 1 standard deviation in signal travel time.

The distribution and the support of the measurement data (i) differs
between the TDR sensors and GPR measurements
(Fig.

TDR measurements are a standard method that provides measurement data for the estimation of soil hydraulic material properties. However, this invasive measurement method yields point-scale measurements and typically requires a local measurement station. Hence, it is difficult to apply at larger scales or to transfer the sensors to another field site. In contrast, GPR is a noninvasive measurement method that is traditionally used for subsurface characterization including subsurface architecture and effective water content. The analysis of GPR measurements is much more challenging than that of TDR and there is still a need for efficient quantitative evaluation methods.

In this study, we propose a new heuristic semiautomatic evaluation approach to identify, extract, and associate relevant information from GPR data. Focussing the optimization on this relevant information regularizes the parameter estimation. The suitability of the proposed methods to accurately identify the subsurface architecture and the soil hydraulic material properties was analyzed for synthetic and measured time-lapse GPR data.

The developed GPR data evaluation method first detects the most important extrema of the signal (events) in the measurement and in the simulation. Subsequently, the detected measured events are associated with the detected simulated events. All plausible combinations of simulated and measured events are analyzed to identify the optimal pair association of these events. To decrease the computational effort, the detected events are grouped in clusters. First, the clusters are associated. Then follows the association of the events contained in these clusters. In order to estimate the subsurface architecture and the corresponding soil hydraulic material properties, the difference in the signal travel time and amplitude of the associated events is minimized with inversion methods. Using events instead of the full GPR signal regularizes the optimization.

Synthetic and measured single-offset time-lapse GPR data are first analyzed qualitatively. It was confirmed that a fluctuating groundwater table experiment introduces characteristic transition zone reflections that are likely to provide valuable information for the parameter estimation. Subsequently, the subsurface architecture and soil hydraulic material properties are estimated based on the GPR data using a global–local parameter estimation approach with preconditioning. The preconditioning step starts from an ensemble of 60 Latin-hypercube-sampled initial parameters sets. They are used to initialize a preconditioning step in which a simulated annealing algorithm and a Levenberg–Marquardt algorithm are sequentially coupled. In this step, these algorithms optimize parameters based on a subsampled data set with a limited number of iterations. The resulting parameter sets are then used to initialize the Levenberg–Marquardt algorithm that operates on the full data set.

Employing the presented approaches on synthetic data shows that the
true parameters are within 1 standard deviation of the resulting
mean parameter set based on the 10 best ensemble members. This mean
parameter set describes the hydraulic dynamics with a mean absolute
error in volumetric water content of

The resulting parameters for the measured data are mostly consistent with results from reference TDR measurement data. We discussed the deviations of the parameters and basically associated them with representation errors and the lack of available measurement data. Relevant representation errors in the GPR data analysis comprise in particular the neglected (i) compaction interfaces and (ii) roughness of the material interfaces.

The three major drawbacks of the presented approach comprise (i) the computational effort which is required to solve Richards' and Maxwell's equations, (ii) the limited number of events that can be analyzed due to the pairwise event association which investigates all plausible combinations of simulated and measured events, and (iii) the fact that the hyperparameters for the GPR evaluation algorithm have to be determined a priori. The latter is difficult and requires expert knowledge, especially as the shape of the radargram is likely to change considerably during the optimization procedure.

Although, the proposed methods have been shown in 1-D, going to 2-D and even to 3-D is first and foremost a matter of computational effort with 2-D already demanding significant time on a large compute cluster. No concepts or methods beyond what we demonstrated in this paper are required, however.

Previous work showed that the location of moderately complicated layer
interfaces and of the mean water content between them can be obtained from multi-offset
measurements

The underlying measurement data are available at

SJ designed and conducted the experiment, developed the main ideas, implemented the algorithms, and analyzed the measurement data. KR contributed with guiding discussions. SJ prepared the manuscript with contributions of both authors.

We thank Jens S. Buchner for software to process (i) the raw data of the ASSESS architecture and (ii) GPR data according to the constructive inversion approach. We are grateful to Angelika Gassama for technical assistance with respect to ASSESS. We especially thank Patrick Klenk and Elwira Zur for assistance during the experiment. The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grants INST 35/1134-1 FUGG and RO 1080/12-1. We are also grateful to the editor Insa Neuweiler and to two anonymous referees, who helped to improve the manuscript significantly. Edited by: Insa Neuweiler Reviewed by: two anonymous referees