In this study, we present an inversion approach to detect and localize inclusions in thick walls under natural solicitations. The approach is based on a preliminary analysis of surface temperature field evolution with time (for instance acquired by infrared thermography); subsequently, this analysis is improved by taking advantage of a priori information provided by ground-penetrating radar reconstruction of the structure under investigation. In this way, it is possible to improve the accuracy of the images achievable with the stand-alone thermal reconstruction method in the case of quasi-periodic natural excitation.

The integration of geophysical and non-invasive diagnostic methods is a topic of timely interest, and several fields can benefit of this strategy, such as monitoring of the environment, infrastructure protection and cultural heritage management.

In this scientific frame, several efforts have been made and interesting
results and strategies are drawn in

Here, we propose to combine two complementary approaches by coupling thermal and electromagnetic reconstructions both based on non-invasive sensing techniques. The non-destructive diagnostic approach proposed here aims at detecting and localizing inclusions in thick walls by exploiting surface thermal and ground-penetrating radar (GPR) surveys. Here, we consider inclusions in the structure, which may represent for example delaminations or inner cavities, whose prompt identification can be crucial in structural health monitoring applications.

The identification of thermal sources (heat flux) or thermal properties
(thermal conductivity and/or capacity) of a material has numerous
applications. Indeed, a variation of the thermal parameters may be the
signature of an inclusion, and the identification process is typically
performed by solving an inverse thermal problem. Several reconstruction
methods have been proposed to solve the inverse thermal problem, either for
source reconstruction

In this frame, a method based on the adjoint-state and finite-element method has
been developed in recent years at IFSTTAR, with the aim to reconstruct the
thermal field over an investigated domain

In order to mitigate these spurious effects, in this work we propose a strategy where the reconstruction performance of the thermal method is improved by resorting to the information provided by the electromagnetic approach based on ground-penetrating radar. To this end, the data are collected by long-term thermal monitoring with an adapted infrared thermography system in combination with a GPR.

As is well known, GPR imaging has proved to be useful in several application
fields, e.g., the detection of buried pipes, land mines, defects in structures
like bridges or roads and the investigation of archaeological sites

The working principle of the GPR is the same as the traditional radar save
for the fact that wave propagation occurs in a lossy dielectric medium
instead of free space. A wideband electromagnetic pulse is radiated in the
investigated medium; owing to the presence of subsurface anomalies, a
part of this wave is scattered/reflected, and this backscattered signal is
collected by a receiving antenna

The diagnostic approach proposed in this work couples both thermal and electromagnetic reconstruction methods and consists of three stages. First a preliminary thermal inversion is carried out to identify possible anomalies in the structure. Then, a GPR reconstruction is performed to refine the preliminary and rough localization of potential inclusions achieved by the thermal method. Finally, the information about the scenario provided by the GPR reconstructed image is exploited in the thermal inverse modeling to improve the accuracy of this method.

The paper is organized as follows. The thermal inverse model is detailed in
Sect.

Geometry of the problem.

The 2-D geometry relevant to the thermal reconstruction problem is sketched
in Fig.

A direct thermal model is first established to get numerical temperature data at the surface of the investigated wall. Thereafter, a thermal inversion is performed with the adjoint-state method in order to get the reconstruction of the shallower part of the investigated wall. The direct and inverse thermal models are described below.

As shown in Fig.

The heat modeling for the investigated wall

We denote with

The variational and Dirichlet formulations

For a numerical solution of the thermal problem, we consider a mesh of
elements

Eq. (

From Eq. (

This equation is temporally solved with an implicit Euler algorithm, so the temperature is obtained at each node of the mesh and for each time step. Finally, in order to provide a more realistic modeling, white Gaussian noise is added to the computed temperature.

The aim of the inverse thermal problem is to achieve a reconstruction of the
thermal capacity

In the above formula,

The inverse problem consists in finding

To solve this non-quadratic optimization problem, we apply the
Levenberg–Marquardt algorithm, which is frequently used to solve nonlinear
inverse problems, as thermal reconstruction

At each iteration, the functional

The minimum of

To minimize

The adjoint-state method initially developed in the control theory
(

Then, we introduce

The adjoint model (Eq.

The differential of

The conjugate gradient algorithm is applied at each iteration of the
Levenberg–Marquardt algorithm, with

The GPR imaging of an unknown wall requires recording electric field measurements at the air–wall surface. The GPR is composed of a transmitting and a receiving antenna having a spatial common offset, which are moved simultaneously to gather data along a scanning line. The transmitting antenna radiates an electromagnetic signal in the wall, and a part of this signal is reflected by buried anomalies and detected by the receiving GPR antenna.

For GPR signal modeling, we consider a 2-D geometry consisting in a
three-layered medium where the upper medium is free space, the central medium
represents the lossy dielectric wall and the lower medium is free space (see
Fig.

In order to simulate the electric field at the measurement points along the
air–wall surface, the forward problem has to be solved based on the known
geometrical and electromagnetic properties of the scenario. This problem is
herein solved numerically by using the popular gprMax tool developed by

The goal of GPR imaging is to reconstruct the dielectric properties of the
investigated domain

The unknown of the inverse scattering problem is described in terms of the
electric contrast function

The scattered electric field is related to the unknown contrast function

The electromagnetic inverse scattering problem is nonlinear and ill-posed. This fact implies the necessity to apply a regularization scheme in order to obtain a stable solution. Furthermore, the nonlinear nature of the problem brings additional difficulties related to the presence of false solutions (local minima).

In order to avoid the nonlinearity problem and reduce the computation
complexity of the related data processing algorithms, the Born approximation
is applied to linearize the problem in Eq. (

In order to retrieve the unknown contrast function

The GPR imaging method presented in Sect. 3 has several drawbacks. First of
all, the retrieved contrast function depends on the electrical properties of
the background

The thermal reconstruction method also exhibits several drawbacks. The
effusivity ratio has an important effect on the quality of the
reconstructions

In this work, we propose to reduce the area of the thermal investigation domain in order to mitigate the side effects and noise sensitivity if there is no inclusion close to the measurement points. This refinement of the investigation domain extent is performed when the GPR image indicates that there is no inclusion near the measurement points.

Accordingly, we first perform a reconstruction of the wall with the GPR. The GPR reconstruction allows defining a smaller investigation region where an inclusion is likely to be located. Thereafter, a thermal reconstruction is performed over this smaller domain.

To get a first location of the inclusions with the GPR, the spatial map
defined by the normalized amplitude of the contrast function

The normalized contrast function is null where the dielectric permittivity is
equal to that of the background and different from 0 over target regions.
As introduced and discussed in

Based on the information retrieved from GPR images, we define three useful
subdomains for the thermal reconstruction (see Fig.

The first subdomain

The second subdomain

Thermal subdomains.

To reduce the discontinuity between the two below subdomains, we consider a
third subdomain

Let us define

The subdomain

The subdomain

Joint thermal and electromagnetic imaging approach.

Geometry of the simulated scenario.

The lower part of the inclusion might be badly located

The subdomain

The location of these subdomains is shown in Fig.

In the case

We propose to take into account the a priori information coming from the
electromagnetic characteristic function

The subdomain

To take into account the fact that the subdomain

Reconstruction of inclusion with the thermal method without a priori information.

We define

The optimization problem becomes the following: find

To
minimize

Thermal and electric properties of the materials.

Lengths used to generate the three subdomains.

Thermal and electric properties of the materials.

This new formulation of the functional

The numerical examples reported in this study are concerned with a limestone
wall containing a wood inclusion (see Fig.

Synthetic GPR data are computed using the gprMax code, by considering a
Ricker pulse radiated at the central frequency of 1300 MHz. The data are
gathered along the observation line ranging from

The investigation domain for GPR reconstruction has the same size as the wall, and it is discretized with a step of 1.4 cm in both directions. For the GPR reconstruction, we exploit a frequency band [100, 2600] MHz, which is discretized with a frequency step equal to 30 MHz.

The TSVD algorithm is applied with a threshold on the singular values equal
to

Then, to get the three investigation subdomains, we use the parameters
summarized in Table

Reconstruction of inclusions with the thermal method without a priori information.

Reconstruction of inclusions with the thermal method with a priori information.

Properties of the reconstructed inclusions.

With regard to the thermal direct model, we define the following synthetic
periodic thermal loading at the monitored surface

The reconstructed fields for thermal parameters without a priori information
(not geometry information inferred by GPR inversion) are shown in
Fig.

In order to analyze these images in a simple but quantitative way, we
implement the following approach to identify the contour shape of the
reconstructed anomaly. To this end, we analyze both the values of the
reconstructed thermal parameters and their gradients. An element of the mesh
is part of a contour if both reconstructed values,

We suppose that the thermal parameters have strong variability along the
contour of the inclusion and assume that, at the boundaries of the inclusion,
the norm of the gradient of both reconstructed parameters is higher than
elsewhere in the investigated domain. On the basis of this consideration, we
assume that a point

After processing, closed contours that verified the above condition
Eq. (

In order to have a first estimation of the contours, we start with values of
the threshold

With both criteria, in the case of no a priori information, we get the
reconstruction of the inclusion shown in Fig.

Let us turn to consider the case of the thermal reconstruction when the
information about the geometry is gained by GPR imaging. The reconstructed
(normalized) contrast function is shown in Fig.

The inferred thermal subdomains are shown in Fig.

The reconstruction of the inclusion is shown in Fig.

To complete our study, we focus on the case in which the inclusion is comprised of
air or steel instead of wood. The air is characterized by a relative
dielectric permittivity equal to 1, whereas the steel can be assumed as a
perfectly electrical-conducting object. The thermal properties of these
materials are described in Table

As can be observed in Figs.

In practice, we focus the minimization of the functional

However, we can notice that the reconstructed thermal properties' (thermal
conductivity and heat capacity) values encompass not only the location of the
inclusion but also connected areas in the non-affected part of the material
matrix. The relative differences between the reconstructions and the targets
are reported in Table

In this numerical study, we have proposed a diagnostic approach, for thick wall structures, combining thermal and GPR imaging approaches in order to improve the fidelity of the reconstruction results obtained with only one method.

The thermal method is based on the minimization of a functional, which is achieved with the Levenberg–Marquardt algorithm by computing a gradient with the adjoint-state method.

The GPR imaging problem has been formulated by resorting to a linearized inverse scattering model based on the Born approximation. The resulting linear problem has been inverted via the truncated singular value decomposition scheme.

The combination of both reconstruction methods has then been presented. Based on GPR reconstruction, three subdomains have been identified. The first one is the most likely to contain inclusions; the second one is far from potential inclusions revealed by the GPR. The third one is located among the previous ones. A constraint term has been added to the functional to minimize it in order to take into account the a priori information provided by GPR images.

Numerical examples have been reported and have confirmed improved performance in terms of location of the inclusion, shape of the inclusion, area and thermal parameters. However, the thermal conductivity is still far from the true inclusion conductivity.

Future research work will focus on the validation of the proposed approach against experimental data at a dedicated test site in outdoor conditions.

The authors declare that they have no conflict of interest. Edited by: L. Eppelbaum Reviewed by: one anonymous referee