Imaging by inversion of acoustic or electromagnetic wave fields have applications in a wide variety of areas, such as non-destructive testing, biomedical applications, and geophysical exploration. However, each modality suffers from its own application specific limitations with respect to resolution and sensitivity. To exploit the advantages of both imaging modalities, methods to combine them include image fusion, usage of spatial priors and application of joint or multi-physics inversion methods. In this work, a joint inversion algorithm based on structural similarity is presented. In particular, a joint Born inversion (BI) algorithm has been developed and tested successfully. With standard BI, an error functional based on the L2-norm of the mismatch between the measured and modeled wave field is minimized iteratively. To accomplish joint BI, we extend the standard error functional with an additional penalty term based on the L2-norm of the difference between the gradients of the acoustic and electromagnetic contrasts.

The knowledge of concrete-rock foundation interface is a key factor to evaluate the stability of gravity dams as well as understanding their mechanical behavior under water pressure. Being an inaccessible part of the structure, the exploration of this region is a complex procedure. Coring techniques can be used, but they only give limited information about a specific location and can be damaging in some situations. Hence the usefulness of non-destructive seismic waves.

We model several non-destructive seismic waves and we propose an inversion scheme for ob- taining the shape of the interface. Our approach consists in solving an inverse problem using “full-wave inversion” type techniques from wave measurements simulated by the finite element method. The inverse problem is modeled as an optimization for a least square cost functional with perimeter regularization associated with sparse data collected on the dam wall. We model different type of measurements such as elastic waves when the source is on the dam wall or acoustic waves when the source is in the water. Moreover, in order to numerically model the radiation conditions in the rock and in the water we employ PML techniques.

We present some validating results on realistic experiments. We demonstrate in particular how our proposed methodology is capable of accurately reconstructing the interface wile classical reverse time migration techniques fail. We then discuss sensitivity with respect to the position and the number of sensors, the wave number as well as to the propagation medium (for example shape of the dam) and the properties of the materials.

Data from the Kepler Space telescope have allowed stellar astrophysicists to measure the frequencies of oscillation modes in many stars. These frequencies carry information about the internal structure of the stars, providing ways to test stellar theory. One method, called structure inversions, seeks to infer differences in internal sound speed between a star and its model using the differences in oscillation frequencies. While this method was used extensively to study the structure of the Sun, the number of other stars studied with structure inversions remains low. In the case of main-sequence stars without a convective core, sound speed inversion results are currently only available for two stars other than the Sun. I will present the results of structure inversions for about 10 solar-like stars and discuss what these results imply about our current understanding of stellar structure.

The subsurface exploration, as part of the development of the human being's environment, focuses on the location of water and mineral deposits, oil, gas, geological structures, among others. Geophysical methods provide information about natural resources, besides information of structures generated by human beings, namely archaeological structures, from the analysis of their physical properties, density of a source body for instance.

The Euler’s homogeneity equation for geophysical potential data is given by:

\begin{equation*} (x-x_0)\frac{\partial T}{\partial x} + (y-y_0) \frac{\partial T}{\partial y} + (z - z_0)\frac{\partial T}{\partial z} = n(B - T), \end{equation*}

where $(x_0,y_0,z_0 )$ refers to the top of a source object, $(x,y,z)$ refers to the position of the observed potential field $T$, $n$ is the structural index, which depends on the source geometry, and $B$ is the regional value of the total field [1].

The inverse problem we are interested in, consists in locating a set of points $(x_0,y_0,z_0 )$ on the top of the source, from observed potential field data. In the classical Euler deconvolution strategy, this is achieved by solving Euler's homogeneity equation shown above. However, this strategy has an opportunity area in estimating the vertical component $z_0$ of the points composing the top of the source. This limitation is amplified when multiple source objects are present.

In the area of image processing, in particular in optical flow, the movement of pixels between two frames is analyzed. The spatial and temporal displacements are assumed to follow the Lambertian assumption: the pixels intensity remains after displacement. This assumption results in a differential equation very similar to the Euler's homogeneity equation: the Optical Flow equation:

$$\nabla T = 0, $$

where $\nabla$ indicates spatial-temporal derivatives. There exist methodological similarities between standard Euler deconvolution method, and standard Optical Flow methods such as Lucas-Kanade. Thus, our hypothesis is that the improving methods applicable to Optical Flow, such as Horn and Schunck method [2], will benefit analogously to the Euler deconvolution method. By reformulating the Euler deconvolution strategy to be similar to the Horn and Schunck method, the position of the top of the source is estimated by minimizing the energy functional:

\begin{equation*} \begin{aligned} E_{HSED}(u,v,w)=\int \int \int_{}^{}&(T_x u +T_y v +T_z w - n(B - T))^2 + \Big(\lambda_u(u_x^2+u_y^2+u_z^2)\\ & + \lambda_v(v_x^2+v_y^2+v_z^2) + \lambda_w(w_x^2+w_y^2+w_z^2)\Big) d_x d_y d_z , \end{aligned} \end{equation*}

where $u$,$v$ and $w$ are the unknowns, $\lambda_u$, $\lambda_v$ and $\lambda_w$ are regularization parameters and the sub-indices indicate partial derivation. It is noticed that the first term in the integral corresponds to Euler’s equation, meanwhile regularization terms impose smoothness on the source position reconstruction.

In this work it will be shown the results obtained after applying the methodology to synthetic 3D subsurface models. We present evidence that the horizontal location of the sources provided by Horn and Schunck based formulation is comparable to results obtained by standard Euler deconvolution strategy, with the advantage that the depth of the top of the subsurface's source is properly estimated.

[1] D.T . Thompsom., A new technique for making computer-assisted depth estimates from magnetic data.", Vol 47. 1982.

[2] Berthold K.P. Horn, Brian G. Schunck . "Determining Optical Flow", Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cam bridge, MA 02139, 1981.