Waves in complex media
From coda waves interferometry to active ultrasounds experiments
For the last decade, a new paradigm arised in seismology called the ambiant noise tomography [Boué2014, Poli2012, Larose2006, Campillo2003]. This allows the sismologists to produce images of the earth (see the image above) without the need of a big seismic event.
This technique is based on the fact that the movements detectable at the surface of the earth are caused by microsources emitting their noises in every directions (the ambiant noise). By correlating signals measured simultaneaously by seismometers at different locations, it is possible to retrieve the different times of flight (Green's functions) between the sensors.
As illustrated on the figure on the side, the correlation between two signals measured simultaneaously at locations A and B allows (see a) or not (see b) the retrieval of the time of flight. This is completely dependent on the position of the initial pulsed source (broadband). In addition, in this elementary configuration we understand that the lateral position (west or east) of the source in regard of the receivers will be coded in positive (causal - see c) or negative (anticausal - see d) times of the correlation. Then, in a diffusive configuration where the noise comes from everywhere (see e) the derivative of the correlation function gives the anti-causal and causal part of the Green's function. This is the principle of Coda Waves Interferometry used to produce the map of the first figure. So, coda waves interferometry is a broadband technique that takes advantage of the complexity of a wavefield to extract spatio-temporal information.
It is strongly related to another technique that I have studied: the Time Reversal. If we consider the configuration of the previous figure e, this technique corresponds to the situation where:
- first, a short pulse is emitted from A and recorded at every "sources" locations (in this step, the sources are actually receivers),
- second, after having flipped all the received signals in time, we re-emit them from the sources (now actives).
What we would obtain in B is the correlation function equivalently to the one obtain with passive Coda Waves Interferometry method previously described. While in A, the energy from all the sources would synchronise to give birth to a spatiotemporal collapse (or focusing) of the waves. This technique is widely used with ultrasounds for medical applications.
An alternative to Time Reversal: the Inverse Filter techniques
This section is a complement to the resume of the article published in Appl. Phys. Lett. in 2014 given in an other post of this blog.
In a closed cavity with highly reflective boundaries, the emission of a short pulse gives birth to a special case of diffuse wavefield where the lost of the initial spatiotemporal coherence is due to the numerous reflections. Using ultrasounds, we have performed focusing experiments using a single point of emission that take advantage of the strong reverberation inside a closed cavity. Thus the only degrees of freedom available were the temporal ones. But the cavity we studied (a regular cylinder) presented a very low level of degeneracy for the trajectories of the waves. When emitting a short pulse at the surface and in the direction of the axis of symmetry, most of the energy is trapped in the axial modes which are coherent over the whole surface. This produces echoes that prevent the time reversal to focus optimally the waves. Instead we demonstrate that it necessary to use the information surrounding the focusing point to synthesized a single emission signal that make an optimal use of all the modes of the cavity by reducing the axial ones. This is based on the single channel focusing technique that we called: the One Channel Inverse Filter. Of course, a similar technique was successfully applied to the passive configuration of Coda Waves Interferometry [Gallot2012]. This inverse filter approach is very interesting as it allows the use of weak information the optimally reconstruct the Green's functions in a passive imaging purpose or to optimally re-focus the waves in a time reversal experiment. I am know studying the possibility offers by the approach to manipulate both temporal and spatial degrees of freedom in a closed cavity with using multi-emitters arrays.
Wavelength tomography: a smart way to produce images not limited by the Shannon-Nyquist sampling criteria
This section is a complement to the resume of the article published in Appl. Phys. Lett. in 2014 given in an other post of this blog.
Another configuration that I studied is the shear waves propagating in soft solids. They are commonly studied in medical imaging for their similarities to the elastic waves propagating in the human body. Once again, we used reverberant cavities, but we added inclusions of different elasticities (hard or soft). From diffuse field approximation, we show that the determination of the local wavelength at a point of the cavity is proportional to the amplitude of the autocorrelation (self-focusing in a time reversal experiment) of the displacement field and of the strain field (both evaluated at this point and at the only time t=0). But, as a difference with the velocity field, the latter does not involve the time variable and it is shown that in the presence of the diffuse field, one can produce images of the elasticity of a medium from greatly undersampled signals. The experiments we performed were realized using the elastography technique which involves the multi-wave technique of Ultrasonic Speckle Interferometry. Recently [Zorgani2015], this work was the basis of the first realization of the in-vivo images of the elasticity of the brain based on passive elastography from very low frame rate MRI images (1 image acquired every 1.5 s).
