Research topics

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Research focus– 2017-2022

–  Ocean acoustic tomography from vertical arrays of sources and receivers and experimental investigations at the laboratory scale : the sensitivity kernel approach.
–  Experimental, numerical and theoretical study of spatio-temporal coherence of seismic ambient noise, with applications to high-resolution surface-wave imaging and monitoring.
–  Advanced array processing on dense arrays to detect/localize microseismic events buried in noise (https://resolve.osug.fr/).
–  Ultrafast imaging of transient deformation in 3D gels, with applications to high-resolution imaging of rupture dynamics of the friction surface.
–  Metamaterial physics at the mesoscopic scale with laboratory & geophysics experiments (https://metaforet.osug.fr/).
–  Ultrasonic experiments in fish schools in the multiple scattering regime : evaluation and monitoring of the biomass in lakes and aquaculture cages (https://echofish.osug.fr/).

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Summary of Research Activity – 2017-2022

I like to describe my research career as a series of fundamental encounters that have strongly influenced my thematic inflections since my doctorate. Three high level researchers were key persons in my career : Mathias Fink in Paris, W.A. Kuperman in San Diego and Michel Campillo in Grenoble. These three internationally renowned researchers are the best in the field of ultrasound for Mr. Fink, underwater acoustics for W.A. Kuperman and seismology for Mr. Campillo. My scientific career is situated at the interface between these three fields for which the propagation of acoustic and/or elastic waves is a privileged means of study. I owe a lot to these three people, at the scientific and human level, and I hope to have contributed to amplify their work by opening roads for future young researchers interested in wave physics.

If I had to characterize in two words these different roads that I have taken and sometimes cleared, I would say that they are related to (1) the study of the spatio-temporal coherence of ambient noise in acoustics and seismology, (2) the use of dense networks of sensors in geophysics and (3) the development of laboratory experiments to apprehend the complex physics of the waves by going back and forth between the small and the large scale. It is sometimes by mixing these three aspects of my research that I obtained the most convincing scientific results...

Wave physics is at the interface of several disciplines that are particularly active in the Rhône-Alpes region and more particularly in Grenoble : physics, mechanics, geophysics, signal processing, medicine and biology. Thus, several laboratories in Grenoble have included wave propagation, in solid or fluid media, among their research topics. These include ISTerre for geophysics, LIPhy in the field of life physics, LPM2C in the fundamental study of multiple scattering, LEGI in the field of sound-vorticity interaction, and GIPSA-Lab in signal processing. For more than twenty years, the researchers involved have been able to compare their approaches and results through the succession of interdisciplinary research groups POAN, PRIMA, ONDES, IMCODE, MESOIMAGE, META and COMPLEXE. I consider myself as a child of this multidisciplinary school "à la française" and I consider it my duty to give back to the younger ones by organizing regularly (every odd year) summer schools of the same type since 2011.

Since my return to France (2005), one of the main axes of my work has been to develop the Experimental Acoustics Team within ISTerre. The specificity and originality of the laboratory is to work with a multi-scale experimental platform with a multidisciplinary vocation on which several laboratories in the field of wave physics can rely.

This research work is based on my scientific background in ultrasonic acoustics and on several years of experience in a large American university (UCSD, California). Indeed, during my second stay in San Diego (Jan. 2002-July 2005), I had created and developed an ultrasonic laboratory in which we experimented on a reduced scale the acoustic propagation phenomena observed in the ocean. By controlling all the parameters of the environment (wave, number and depth of sources/receivers, temperature or density fluctuations, depth of the waveguide), this tool allowed us to make methodological progress in the field of ocean tomography, target detection in shallow water, underwater communication,... I reproduced this ultrasound platform in Grenoble as soon as I arrived and my goal was to extend its field of investigation to geophysics and more generally to the propagation of acousto-elastic waves in complex media. Thus, a model of the earth’s crust can be reproduced at the laboratory scale to study the conversion of energy generated by oceanic microseisms into Rayleigh waves on the continent. Similarly, acoustic propagation in the multiple scattering regime in bubble clouds provides a mesoscopic (and therefore easy to manipulate) account of wave phenomena observed at the crystalline scale.

Based on this experimental platform, two keywords are at the basis of my research work : (1) multi-scale approaches and (2) dense multi-sensor networks.

– (1) A complex medium for waves is any medium for which the propagation is dominated by refraction, dispersion, scattering and/or reverberation phenomena. To approach a complex wave phenomenon simultaneously at the real scale and at the laboratory scale allows to simplify or even to control this complexity by playing separately on different parameters. Thus, using the example of ocean tomography, an underwater acoustic channel is transformed into a waveguide at the ultrasonic scale, a waveguide in which one can control the height of the waves which play an important role in the randomness of the acoustic propagation.

– (2) At the real scale as well as at the laboratory scale, there is no longer any technological limitation to the use of source/receiver arrays in the study of wave phenomena in acoustics and elasticity. For example, the LAUM (Le Mans) is now equipped with a 3D laser vibrometer combined with a mechanized robot that can record the three components of the elastic field at any point on the surface of mechanical parts of any shape and volume. Similarly, geophysicists are no longer afraid to install several thousand seismic sensors on particular geological objects such as faults or volcanoes to better understand the limits of spatial resolution of inversion processes.

However, the use of multi-element arrays is a double-edged sword : on the one hand, it multiplies the angles of view on wave propagation and thus improves the resolution, as has been demonstrated regularly since Shapiro et al (2005) in passive seismic tomography ; on the other hand, it enormously increases the amount of information to be recorded, processed and exploited, at the risk of losing sight of the different physical phenomena that affect wave propagation. For waves in a complex environment (which can be observed in the magma chamber of a volcano or in the heart of a dense school of fish in the sea), I believe we have entered an era where the data as a whole is of better quality than the physical or numerical understanding we have of the mechanisms at play. Even if the emergence of artificial intelligence in the field of waves seems attractive and in any case inevitable, I am not sure that supervised learning techniques (Deep Learning, Machine Learning or others) have the answer to everything...

In practice, by combining multi-scale approaches and dense sensor networks, one way forward is to simplify the complexity of natural environments (land, sea) while maintaining the spatial resolution necessary for imaging and/or monitoring complex wave phenomena. The ultimate goal is to use waves to better characterize a natural environment, to appreciate its complexity and model it, to describe it for imaging or monitoring purposes in geophysics, acoustics or fundamental physics.

To carry out multi-scale and multi-element experiments in the field and in the laboratory, I have equipped the Experimental Acoustics Team at ISTerre with several acquisition electronics and associated source/receiver sensors. These tools allow the study of wave physics phenomena over six frequency decades ranging from Hz to MHz. For example, the latest acquisition (Sept. 2014) is an ultrafast ultrasound scanner that performs real-time channel formation and dynamic acquisition for the study of transient deformations of a gel under frictional stress.

Using these systems, (for more description, see the movies at the bottom of the web page https://www.isterre.fr/annuaire/member-web-pages/philippe-roux/article/description-of-acoustic-experimental-facilities.html), my group multiplies multi-scale approaches and the use of transducer arrays to understand acoustic and elastic wave propagation in geophysics, underwater acoustics (Fig. 1) and in ultrasonic regimes mixing multiple scattering, sound-vorticity interaction or transient deformations.

Fig. 1 : Ultrasonic scale experimental setup to reproduce the physics of a complex ocean waveguide (left). The scaling ratios between the wavelength, the propagation distance R and the depth of the waveguide D respect the full-scale experimental conditions.

In addition, ISTerre has structured during the last five years a part of its engineers/technicians in service and one of them, the Service for Geophysical Instrumentation (SIG), gathers 7 ITA who organize and manage with the researchers the large scale field experiments. For example, I benefited from this service to organize a geophysical experiment in Oct. 2016 with the use of more than 1000 geophones and 150 seismic sources in the framework of the ANR METAFORET. Similarly, the RESOLVE experiment allowed me to install 100 geophones on the Argentiere glacier at 2500 m altitude in April 2018, with a whole set of independent measurements obtained by GPS, subglacial pressure sensor or electromagnetic radar. In the next 5 years, the geophysical equipment deployed by the laboratory (and shared at the Grenoble Observatory level) will be considerably expanded with the addition of 400 geophones funded by an ERC Consolidator (PI Florent Brenguier, 2018).

ISTerre’s asset as an internationally recognized laboratory in earth sciences (Grenoble Alpes University is ranked 4th in Geochemistry/Geophysics in the latest CWUR 2018-1019 ranking) and my specific scientific contribution to this renown thus reside in these multiscale approaches that allow to isolate and study in the laboratory particular physical mechanisms and to verify in the field by measurement and observation the combination of all physical processes involved.

In the rest of this activity report, I will describe some applications of wave physics in complex environments through small and large scale experiments and the use of dense sensor networks. My research project will allow to complete the gaps of this report, which does not pretend to be exhaustive, by insisting on the last experimental achievements and the ongoing projects.

1- Correlation of ambient seismic noise : a new method for imaging and monitoring the earth.

The best example of the interdisciplinarity at the heart of my research topics is the current dynamism around the theme of passive imaging in geophysics, underwater acoustics and ultrasound. The craze around the use of ambient noise in wave physics has led Michel Campillo and myself to co-lead four one-week workshops in Cargese in 2011, 2013, 2015 and 2017 : "Passive Imaging and Monitoring in Wave Physics : from Seismology to Ultrasound". A fifth edition will take place in September 2019, this time organized by the new wave of young geophysicists/seismologists of the laboratory.

Broadly speaking, passive tomography boils down to the following dual problem :
1) Can we extract from the ambient noise measured at two points the Green’s function between these two points, this Green’s function including both surface and volume waves encountered in seismology, for example ?
2) Is the quality of the Green’s function extracted from the ambient acoustic or seismic noise sufficient to reconstruct an image of the medium ?

In recent years, experiments have validated the first point in all fields of wave physics. The Green’s function emerges from the correlation of a diffuse field measured at two points over long periods of time. The diffuse field is derived from ambient noise as in geophysics [Campillo and Roux, 2014] and underwater acoustics [Fried et al, 2008 ; Leroy et al, 2012 ; Lani et al, 2013]) or from codas of active sources in the multiple scattering regime as in geophysics [Campillo and Paul, 2003 ; Froment et al, 2010] and ultrasonic acoustics [Lobkis and Weaver, 2001 ; Derode et al, 2003 ; Larose et al, 2008].

However, in most cases, only an estimate of the Green’s function is obtained, an estimate that depends mainly on the spatial and temporal distribution of the noise sources used. For example, in geophysics, the Rayleigh wave (surface wave) is easily obtained because this wave is largely excited by ambient "ocean" type noise sources in the frequency range [0.1-0.5 Hz]. In underwater acoustics, the Green’s function is amplitude-weighted by the fact that the noise sources are mainly present at the ocean surface (wave-related bubble clouds and ship noise). Many theoretical works have studied the problem in free space, in a waveguide or in a cavity [Snieder, 2004 ; Roux et al, 2005 ; Colombi et al, 2014]. The variance of the correlation function and its convergence to the Green’s function have been studied theoretically and experimentally. Finally, since ambient noise is often difficult to control, the problem has also been approached using a distribution of incoherent and uncontrolled sources, such as the noise of a ship along its trajectory at sea or the coda of a collection of earthquakes in geophysics [Roux et al, 2004 ; Chaput et al, 2016].

Regarding the second point mentioned above, there is of course still much to do and understand in the field of passive tomography. In particular, reconstructing an image of the medium from the simple ambient noise is an exciting but still open problem. The first surface wave velocity map was obtained in Southern California in early 2005 from Rayleigh waves [Shapiro et al, 2005]. More recently (2009), the San Andreas Fault was mapped in 3D in the Parkfield area (California) via Love wave extraction from ambient seismic noise (Fig. 2a). Even more recent images of the San Jacinto fault (2019) provide even more detail on the "yarrow" structure of the damaged zone from a dense network of about 1100 surface sensors (Fig. 2b).

Fig. 2 : (Left) San Andreas Fault (SAF, California) : 3D image by ambient seismic noise correlation (Roux, 2009). (Right) San Jacinto Fault (California) : Iso-velocity representation (Vs=850 m/s) showing compaction of the fractured zone into several vertical sheets over the first few hundred meters of depth. The fault trace is shown on the surface by the purple solid line (Mordret et al, 2019).

These very encouraging results show that we can achieve a super-resolution goal in geophysical imaging at local (<20 km, Roux et al, 2011) and global ( 1000 km, Boue et al, 2014) scales via surface waves (Rayleigh or Love). In both cases, the tomographic inversion is done from measured travel times between seismometers using a ray tracing based inversion kernel. Contrary to a classical tomography where the time measurement comes from controlled active sources, the passive tomography is the result of the correlation of ambient noise which is expected to satisfy conditions of stationarity and isotropy in the considered spectral band. The quality or bias observed in the tomographic inversion of "ambient noise" is thus directly related to the spatial and temporal properties of the seismic noise and the surface coverage of the sensor network.

In recent years, two major advances have been made in the still very active field of passive seismic tomography. On the one hand, the development of dense sensor networks has become the norm in earth science, both at small and large scales (Roux et al, 2016). For example, we exploited the core part of the US USArray network (400 seismic sensors) and an antenna processing adapted to the ambient seismic noise (Fig. 3) to produce a surface wave phase velocity map with a spatial resolution never before achieved (Boue et al, 2014). Thus, it is indeed via dense arrays and modern beamforming methods that we will continue to improve spatial resolution in passive seismic tomography.

Fig. 3 : (Left) Selection of seismic stations from the "Transportable Array (USArray)" over the period Nov. 2009 - Jan. 2010. Two sub-arrays (blue) organized around each station (red) are used to extract surface waves (Love + Rayleigh) from antenna processing. The white rectangle corresponds to the area over which the high resolution seismic inversion is produced. (Center) Illustration of the Dual Channel Formation (DBF) process between two sub-arrays of 9 stations. Slowness vectors (UA and UB) are associated with azimuthal rotation angles (A and B) to optimize the extraction of surface waves from the ambient seismic noise. (Right) Phase velocity map obtained by inversion of propagation times after DBF. Low velocity areas correspond to sedimentary basins. Those with higher velocities to mountainous massifs (Boue et al, 2014).

On the other hand, seismic tomography can also be enriched by the use of sensitivity kernels based on a finite frequency diffraction approach that goes beyond, in terms of spatial resolution, the classical use of ray tracings. Applied on dense arrays at smaller scales ( some km), sensitivity kernels for surface waves (Fig. 4) allow us to revisit seismic tomography for major applications in oil exploration (Chmiel et al, 2018).

Another potential application of ambient noise via acquisition on dense seismic arrays is the dynamic localization of ambient noise sources. Take for example the noise generated by a geyser a few meters below the ground surface : Old Faithfull, in the heart of Yellowstone National Park, so named for the regularity of its eruptions (with a period of about 40 minutes).

Fig. 4 : Experimental diffraction kernels for surface waves obtained by correlation at F=4.2 Hz for two receivers (stars) separated by (a) 870 m, (b) 1443 m, (c) 2100 m and (d) 2737 m. The diffraction kernels reveal the smallest variations of the wave field velocity. Acquisition here is performed by the oil company CGG from an ultra-dense seismic network ( 50000 geophones) with a measurement point every x=y=30 m (Chmiel et al, 2018).

In 1992, a seismic network of 96 stations had been deployed around the geyser (Fig. 5) to measure its seismic activity. The recorded signals showed a permanent tremor activity, i.e. a continuous intense noise whose amplitude modulation corresponded to the eruption periods of the geyser. By revisiting these ambient seismic noise data, twenty years later, via the use of antenna processing algorithms (or Matched Field Processing, which is similar to a correlation process over short time windows), we were able to isolate and relocate the main sources of seismic noise and their temporal dynamics during the eruption cycle (Cros et al, 2012 ; Vandemeulebrouck et al, 2013). It appears (1) that the dominant source of noise comes from the geyser conduit with a progressive rise of this boiling noise during the cycle (Figs. 5c and d), and (2) that a recharge zone is present next to the geyser as is sometimes described in the literature (Fig. 5b) for this type of hydrothermal phenomena.

Fig. 5 : Temporal monitoring of ambient seismic noise sources at Old Faithfull Geyser, Yellowstone National Park, USA. (a) Geographical distribution of the 96 seismic stations around the main geyser conduit over a 40 m x 40 m area. (b) Model of the structure of a bubble trap geyser. (c) Spatial distribution of seismic noise sources during two cycles of the geyser. Sources in the main conduit are in blue, sources in the recharge zone are in red. An angle of 20 degrees is observed with respect to the vertical for the main conduit. (d) Dynamic monitoring (at depth on this plot) of ambient seismic noise sources. Note : (1) the progressive rise at the beginning of the cycle to a depth of about 10 m and (2) the activation of the recharge zone at the end of the eruption cycle (Vandemeulebrouck et al, 2013).

2- Ultrasonic acoustics : waveguide tomography via dual antenna processing.

What is it like to use dense arrays at the laboratory scale ? What scientific progress can be made in the field of wave propagation ?

The combined use of a transmitter antenna and an acoustic receiver antenna on either side of a waveguide allows the separation of the different arrivals specific to propagation in a reverberant environment via the Double Beamforming (DBF) antenna processing described above. By transforming the received data from position space to angle space (as shown in Fig. 6 for vertical linear antennas of piezoelectric elements that cover the entire water column in an ultrasonic waveguide of 5 cm depth and 1. 2 m in length), each of the wavefield intensity maxima after DBF is allowed to be identified with acoustic paths (or acoustic ray) propagating between the two antennas (Roux et al, 2008 ; Le Touze et al, 2010).

Fig. 6 : (Left) Schematic representation of the ultrasonic waveguide materialized by the two red interfaces and the acoustic rays connecting the center of the transmitting array (left) to the center of the receiving array (right). (Right) Representation of the intensity maxima (experimental data at 1 MHz) in 3D space [transmit angle, receive angle, time] of each acoustic beam after DBF. Each "beam" is numbered according to the total number of reflections on the interfaces, with the + or - sign corresponding to a first reflection on the bottom or surface of the waveguide (Le Touze et al., 2010).

The advantage of these multiple acoustic rays traced between each transmitter-receiver sub-antenna (formed of about ten piezoelectric elements) is that they cross the waveguide along the x and y axes (length and depth), providing for each of them information in time, amplitude and angle on each of the acoustic "pixels" crossed. In practice, more than 2000 rays are identifiable in the experimental configuration of Figure 6 and one can imagine performing a tomographic inversion in the waveguide associated with a local density or velocity perturbation based on all these acoustic rays. In practice, we have sought to combine DBF with the physics of sensitivity kernels to relate the variations in time, amplitude or angles of each "beam" (or acoustic beam) to the physical parameters of the fluctuation : local change in density (a target in the water), local perturbation in velocity (a temperature plume rising from the bottom of the waveguide) or local change in the water-air interface (a wave at the surface).

Unlike acoustic beams (geometric or high frequency approximation), sensitivity kernels applied to acoustic beams allow to integrate diffraction effects related to the limited bandwidth of piezoelectric transducers (Fig. 7). This physics is not new and was first developed in the context of near-surface geophysical imaging (Dalhen et al, 2000), but its generalization to amplitude/time/angle observables specific to acoustics in reverberant media (waveguide) is a novelty (Marandet et al, 2011 ; Aulanier et al, 2013).

Fig. 7 : Representation of the sensitivity kernel of the amplitude of an acoustic beam at 3 MHz for a path with one reflection at the surface and one at the bottom of the waveguide. Longitudinal (a) and transverse (b) sections in the propagation plane for transceiver antennas consisting of 3 elements. (c) and (d) Same for two transceiver antennas with 21 elements (Marandet et al, 2011)

This work led us to perform three types of inversion in ultrasonic waveguides :

(1) detection/localization of a wavelength-sized target in a harbor-like environment (Marandet et al, 2011), as depicted in Fig. 8,
(2) the imaging of a heat plume (Roux et al, 2011), observed during the rise of a convective plume from the bottom of the waveguide (Fig. 9).
(3) the inversion of a local waveguide surface disturbance (Roux and Barbara, 2014) due to the passage of a wave on the waveguide surface (Fig. 10).

Fig. 8 : (Left) Schematic representation of the waveguide with characteristic lengths. (Right) Detection and localization of a spherical lead target of diameter a for a product ka 20 (wavelength 0.5 mm) in an ultrasonic waveguide of length 1100 mm and depth 52 mm, bounded by two transceiver transducer antennas that span the entire height of the guide. The color scale represents the probability of target presence (Marandet et al, 2011) when the ball is in the center of the guide.

All this work was carried out in close collaboration with the GIPSA-Lab (INP, Grenoble) in the framework of the ANR Jeune Chercheur TOTS (2010-2013), led by Barbara Nicolas. Two thesis grants financed by the Direction Général de l’Armement (DGA) allowed us to train students on these physical problems linking ultrasonic acoustics and signal processing.

Fig. 9 : (Left) Schematic representation of the waveguide with characteristic lengths. (Right) Tomographic inversion from acoustic beam times for a thermal convection plume initiated at the base of the ultrasonic waveguide (red dot). At 8.5s after convection initiation, the rise of the plume from the bottom of the waveguide to the surface is clearly observed (Roux et al, 2011).

Fig. 10 : (Left) Schematic representation of the waveguide with the characteristic lengths. Note that the surface wave velocity (V 0.4 m/s) is very small compared to the ultrasonic wave velocity in water (c 1500 m/s). (Right) Spatio-temporal representation of the surface deformation during the propagation of a wave through the transceiver plane. The x-axis corresponds to the distance between the transceiver antennas. Each panel corresponds to the waveguide surface inversion produced from one of the 500 acoustic acquisitions made in the waveguide. The inversions are produced from the amplitude variations of the acoustic "beams" under the influence of the deformation of the waveguide surface (Roux and Barbara, JASA, 2014).

For the DGA, the purpose of this research work is to develop a methodological approach based on transceiver antennas that allows the inversion of a disturbance of any size and characteristic in underwater acoustic channels. In practice, the military issue is in the field of port protection with the detection/location of divers or mini submarines (Automated Underwater Vehicle, AUV).

For national security, the question is : how to protect the harbours of Brest or Toulon against terrorist attacks when all the classical sonar systems see their efficiency strongly decreased in reverberating environments for acoustic waves ?

The latest developments on this research theme relate to the nature of the observables used for the inversion of a disturbance in the waveguide. Indeed, if the time and amplitude variations of the acoustic beams have been used until now thanks to the sensitivity kernels (Figs. 8-10), the variation of the emission and reception angles of each of these beams had not been exploited yet. This is now done for a disturbance generated on the surface of the waveguide by a laser shock and the inversion result is very spectacular (Fig. 11).

Fig. 11 : (Left) Schematic representation of the waveguide with the laser shot impact at the center of it (red arrow). (Right) Result of the waveguide surface inversion using both transmit and receive angles of a collection of 2285 acoustic beams. The x-axis is the length of the waveguide ; the y-axis is the time relative to the laser shot generating the gravitocapillary surface wave. The color scale corresponds to the deformation of the surface in meters.

Surprisingly, we find that the inversion of the surface disturbance is much cleaner (less ghosting, better spatial resolution) with the use of angle variables instead of time or amplitude variables. This observation leads to interesting and fundamental conclusions in the field of acoustic tomography. Indeed, tomography has always been interested in the travel time (or the variation of the travel time) of a wave to obtain the velocity map (or its fluctuations) of the propagation medium. To obtain a good time measurement, a perfect synchronization between the source and the receiver is required. What is easy at the ultrasonic scale in the laboratory becomes very complex and expensive to implement in the ocean where several thousands of kilometers separate transmitters and receivers in the rare attempts at acoustic tomography at this scale (see more general info on the wikipedia site https://en.wikipedia.org/wiki/Ocean_acoustic_tomography). To show, as in Figure 11, that the same tomography can be obtained by varying the angle of the acoustic rays rather than their travel time means that only the local synchronization of each element of the transmitter/receiver antennas is required and not the complete synchronization between the two antennas.

Undoubtedly, this major paradigm shift will lead in the future to new large-scale tomographic experiments with the objective of accurately measuring spatial and temporal variations in the temperature of the surface layers of the ocean, which are indicators of ongoing climate change. It should be noted that the transposition to the real scale of an oceanic waveguide of these experimental results performed at the ultrasonic scale was already the subject of a publication (Roux et al, 2013).

My work in underwater acoustics was awarded in 2013 with the "Medwin Prize in Acoustical Oceanography" by the Acoustical Society of America.

3- Sub-wavelength resonant metamaterials : an original experimental approach conducted at the laboratory and geophysical scales.

This work concerns the experimental and numerical study of the effect of sub-wavelength uniaxial resonators on the propagation of elastic bending waves in a thin aluminum plate (Lamb A0 mode). The resonators consist of simple aluminum rods glued to the plate (see Fig. 12 for the description and evolution of the experimental setup). They derive their sub-wavelength character from the important ratio between their dimensions (length/diameter) reaching almost two decades. When we arrange them (periodically or not) on a sub-wavelength scale, we obtain a locally resonant medium which behaves like a metamaterial. In the two experimental configurations of Fig. 12, the degrees of freedom that this metamaterial offers on the control of the wave field, are large. On the one hand, the propagation medium is a plate which, in the frequency range studied, has two components : longitudinal (Lamb mode S0) and transverse (Lamb mode A0). On the other hand, the resonators present both bending resonances (related to the S0 component in the plate) and compression resonances (excited by the Lamb A0 mode). The metamaterial consists of 100 to 400 rods that can be spatially organized in an ordered or disordered manner.

Fig. 12 : (Left) Experimental setup for the first metamaterial experiments in the laboratory from 2013. A vibrometer (1) generates a Lamb A0 wave in the aluminum plate. The wave field is measured by a Doppler velocimeter (2) whose measurement point is moved on the top side of the plate to each point of the rectangular surface (5) and controlled by a PC and a set of two motorized mirrors. The metamaterial (6) is attached to the bottom side of it (Rupin et al, 2014 & 2015). (Right) Evolution of the experimental setup from 2018. The source is a piezoelectric element (a) glued to the plate. The Doppler velocimeter (c) is now connected to a motorized robot arm (e) controlled by PC (d), allowing for a larger analysis area (Lott and Roux, 2019a). In both experimental setups, the metamaterial (a) consists of 100-400 vertical aluminum rods that are glued to the underside of the plate (g). The recorded signal (b) is highly dispersed due to the low intrinsic attenuation of the plate. The temporal dispersion in response to a short pulse exceeds 0.2s, which corresponds to more than 20 round trips in the plate.

The originality of the experiment, in addition to its mesoscopic dimension which is quite unusual in the field of metamaterials, lies in the mapping of the wave field over a large area including the metamaterial, thanks to the sequential acquisition of all the pulse responses, using a laser velocimeter (Fig. 12). Analysis of the data reveals the presence of three broad frequency band gaps, which begin at the location of the compressional resonances of each rod (Fig. 13b & Fig. 14). We tested an arrangement, both periodic and random, of the resonators and the results are quite identical. The band gaps are thus related to the resonant nature of the elementary cell of the metamaterial, and not to its periodic character (Bragg diffraction). On the other hand, we have also highlighted apparent velocities lower or higher than those measured in the bare plate for the frequencies located at the edge of the band gap (Fig. 14).

Fig. 13 : Spatial representation of the normal field velocity measured experimentally on the plate (rectangular area delimited on Fig. 9). The area covered by the metamaterial (formed by 10 x 10 vertical rods) is placed on the left between 0 and 0.2 m. The wavefield for the A0 mode is presented at three frequencies : (top) just before a band gap, (middle) within a band gap, (bottom) just after a band gap. Before and after the band gap, the apparent velocities (measured from the wavelengths extracted from the speckle pattern within the metamaterial) are slower or faster than in the bare plate (Williams et al, 2015).

A global view of all these interference phenomena is obtained via the determination of the dispersion relation within the metamaterial (Fig. 14). We then find a deep modification of the dispersion curve of the A0 mode, with band gaps (or propagative bands) that are the result of repulsion effects (called "hybridizations") linked essentially to compressional resonances. We also show that Bloch’s theorem allows for excellent modeling of the dispersion relation in all its complexity (Williams et al, 2015).

Fig. 14 : Comparison between the experimental and theoretical dispersion relations within the metamaterial. (a) The wave number corresponding to the propagative part of the field is in black, with the dispersion relation resulting from a numerical calculation (red). Two hybridizations (with the binding and the anti-binding branches on both sides of the resonance) are clearly distinguished, each opening a band gap. (b) The measured attenuation within the metamaterial in the band gaps is shown in black, to be compared with the numerical prediction (red). The agreement between experimental and numerical results is excellent (Williams et al, 2015).

In another aspect of this work, we used numerical finite element modeling, via the numerical code SpecFem3D, to better understand the complexity of this elastic metamaterial. We first showed that wave propagation in this type of metamaterial can be approximated by a 1D system (support beam + resonators), which is less expensive in terms of computational resources. We were then able to look at the hybridizations induced by each of the 2 types of rod resonances, bending and compression, independently of each other (Colquitt et al, 2017). This allowed us to establish the singular character of elastic metamaterials. Indeed, the bending resonances of the rods result in an energy transfer between the transverse component of the wave field in the plate (A0) (which is the only one initially excited) and the longitudinal component (S0).

From the point of view of the dispersion relation, this results in a particular hybridization, which makes a third propagation mode appear. This hybridization accounts for the coupling between the A0 and S0 modes in the plate due to the bending resonances. It is all the more apparent as the plate is thin (and therefore flexible), and is manifested on Figure 15 by the appearance of transmission bands or very narrow band gaps associated with the bending resonances (Lott & Roux, 2019b).

Fig. 15 : Influence of the plate stiffness on the coupling between the rods and the plate at bending resonances. (a) Dispersion curve obtained experimentally with h = 6 mm wide plate. (b) As for (a), with a plate of h = 2 mm width on a restricted part of the frequency spectrum (red dashed square in (a)). For the thinner plate, the plate plus rod system shows stronger interaction with the rod bending resonances (blue dots), both inside and outside the band gap.

In conclusion, the dispersion relation obtained in these locally elastic resonant media is dominated by hybridization effects due to the rod compressional resonances within the metamaterial. However, hybridization on the longitudinal component of the field is also at work, due to the bending resonances. Finally, we were able to test a number of 1D configurations (support beam + sub-wavelength resonators), which showed the great richness of possible configurations with this beam + rods assembly. In particular, we have observed that a stiffening of the plate (Fig. 15), obtained by increasing its thickness, leads to less and less marked hybridization effects for the bending resonances. The influence of the inter-resonator spacing shows the possibility of obtaining negative group velocities, linked to hybridization by the bending resonances. This confirms the interest of this type of metamaterials for applications such as invisibility cloaks.

Currently, we are looking to obtain this type of cloaking for Lamb waves. We have initiated the development of an algorithm based on the minimization (in the sense of least squares) of the difference between the wave field observed in the bare plate (without obstacle or metamaterial) and the one observed when an obstacle surrounded by the metamaterial is added. The parameters used for the minimization are the length of the rods and their spacing. The general arrangement of the resonators follows that used recently with this same type of waves by Farhat et al. (2009). An example of a configuration with resonators of different lengths, is given in Figure 16. It shows that it is possible to obtain a slowing down or an acceleration of the Lamb A0 waves according to the bandwidth that we select. We notice that the forward diffraction pattern is less marked in the case where the waves travel faster in the metamaterial. Here, obtaining a true "cloaking" effect requires obtaining an effective anisotropy on the propagation velocity of the Lamb A0 wave (Colombi et al, 2015, 2016a).

Fig. 16 : Illustration of the work in progress for the development of an invisibility cloak for Lamb A0 waves. (a) Example of the configuration studied : a set of rods of different lengths are arranged in a star shape. (b) and (c) Allure of the wave field (vertical component) within the metamaterial (materialized by the yellow circles) in two different frequency ranges. The slowing down (left) or the acceleration (right) of the waves is obvious, the latter giving rise to a better reconstruction of the diffracted field towards the front.

The generalization of these physical phenomena to the geophysical scale, where the plate then becomes a ground that can be modeled as a semi-infinite medium, would then show that Rayleigh waves (surface waves) undergo the same effect when interacting with a forest consisting of trees 20 m tall and spaced every 3 or 4 m (Colombi et al, 2016a), opening new possibilities in the field of seismic protection (Fig. 17).

This is the goal of the METAFORET project (https://metaforet.osug.fr/), which was funded by ANR "Défi de tous les savoirs" in 2016. Indeed, the observation that was at the origin of the METAFORET project is the following : why does wave physics present few large-scale complex physics experiments ? Since waves generally obey the same propagation equation, why do we rarely observe wave physics phenomena on the geophysical scale, for example ? Of course, what seems easy in a controlled environment at the laboratory scale can be very difficult to implement at large scales where it is sometimes impossible to deploy a large number of autonomous sensors.

Fig. 17 : Examples of locally resonant metamaterials at different scales for seismo-elastic waves. (a) Seismic deployment of 1000 geophones (yellow dot) on the side of Mimizan (Landes) at the interface between an open field and a dense pine forest. (b) Laboratory-scale surface covered by a random arrangement of vertical metal rods glued to a thin aluminum plate. (c) Mechanical similarities of the unit resonant cell for both systems, with their respective frequency bands (Lott et al, 2019).

In recent years, however, the earth sciences, and geophysics in particular, have been undergoing a technological revolution with the multiplication of acquisitions on very dense seismometer networks, sometimes including more than ten thousand sensors. Until recently, these seismic campaigns were the prerogative of rich oil companies. But things are changing and the financial cost of these acquisitions is now becoming affordable for academic research. Looking ahead, we are even close to a critical point where the excellence of the geophysical data will exceed our understanding of the underlying physical phenomena.

Figure 18 : Top : META-FORET experiment setup (Oct. 2016). The objective was to install 961 three-component geophones on a 120 m × 120 m grid with 4 m spacing between elements. The seismic array (red) is placed at the interface of an open field and a dense pine forest (80 trees per 400 m²). Bottom left : Continuous recording of ambient noise was made over 12 days with FairFieldNodal wireless seismic sensors. Bottom right : In addition to this ambient noise, active source signals were recorded with a vibrometer placed at different locations (top : blue ellipses) in the open field (inside and outside the array) and in the forest.

The METAFORET project aims to fill this gap with a multidisciplinary approach proposed by a team of physicists, geophysicists and engineers who share a common interest in wave propagation in complex media. The goal of the project is to reconcile complex wave physics with large-scale observations.

Specifically, we aim to perform experiments on metamaterial physics in two geophysical-specific configurations (Fig. 18). In the first, we have shown that a natural forest behaves as a metamaterial for seismic waves (Roux et al, 2017). The idea is that each tree in the forest acts as a resonator that traps a small portion of the seismic surface waves. The collective behavior of the trees would then correspond to that observed at the very small scale (millimeter) in optical metamaterials. In the second experimental configuration (to come in 2020), we will show that a particular spatial distribution of buried concrete columns, classically used in civil engineering for soil compaction, can also behave as a seismic lens for surface waves, with the effect of refracting the waves around the center of the lens leaving this area free of any seismic vibration.

In parallel with 3D numerical simulations (Colombi et al, 2016b, 2016c, 2017) and a theoretical approach based on conformational geometry (Farhat et al, 2009), the primary goal of the META-FORET project is thus to carry out two ambitious and innovative experiments where 1000 seismic sensors deployed over an area of about one hectare will have the purpose of measuring the seismic wave field in the two geophysical metamaterials proposed above. This high spatial density of sensors is mandatory to accurately measure the dispersion curves (and thus the surface wave velocity) inside and outside the metamaterial.

Fig. 19 : METAFORET 2016 experiment. (a)-(d) Spatial representation (x-y) of the seismic wavefield measured on the seismic array (vertical component) for a source inside the forest in position (x=60 m, y=30 m) and displayed at four different times from time t=0 of the seismic shot (from left to right) (a) t=0.09 s ; (b) t=0.12 s ; (c) t=0.15 s ; (a) t=0.18 s. The seismic wave field was filtered in the 20 Hz-50 Hz band. In each image, the horizontal red line represents the forest-field boundary. (e)-(f) Same representation as above for the seismic field filtered in the 50 Hz-80 Hz frequency band. The low frequency part of the wavefield (<50 Hz) shows a spatially coherent surface wave of large amplitude while the high frequency part (>50 Hz) has a much smaller amplitude (see the different color bars in the upper and lower panels) and shows a very reduced spatial coherence.

In practice, two types of source were used during the 2016 experiment : (1) ambient seismic noise, probably of anthropogenic origin at the frequencies considered (>10 Hz) and (2) a source signal controlled and transmitted to the ground by a vibrating pot (Fig. 18). The analysis of the wave dispersion within the two metamaterials (a natural forest or a soil compaction site with buried columns) will allow us to consider potential applications to high frequency seismic cloaking in civil engineering (Fig. 19).

I am convinced that this project has important applications in geophysics and civil engineering in the longer term. For example, the cloaking frequency bands could be exploited to reduce ambient seismic noise at locations where ground vibrations may be a problem for the quality of high-precision scientific measurements (local vibrations of large astronomical antennas).

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