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## Wave propagation through random media

**Transverse review**

The study of WPRM is shared by many fields since media from very small scales (rocks, or-ganic layers) to extremely large ones (interstellar medium) exhibit inhomogeneities that induce scattering of the propagating waves (Ishimaru 1978), due to local changes in index of refrac-tion. The case of the ocean is treated in section 1.2.2 in terms of sources of fluctuations and description of the physical impact on the traveling of acoustic waves. We focus here on the analogous phenomena existing in the nature.

The propagation of ultrasonic waves as a diagnostic tool in order to image organic tissues to prevent or monitor the evolution of diseases is a relevant example. In fact, sound speed inho-mogeneities are found in large organs such as the liver, or breast, inducing some distortions of the propagated wavefronts (Zhu and Steinberg 1992). The resolution of the images is hence limited by the capability to correctly describe the observed fluctuations. On the other hand, the measurement of a scattered wave may represent a way to reveal the presence of an anomaly. Similarly, ultrasonic wavefront distortions can be observed in non-destructive testing (NDT) of materials. However, the presence of a scatterer in the medium of propagation in this context is often the reason of the testing in the first place. In this case, the perturbation of the propagated wave is sought out, since it may hint an unwilling intrinsic characteristic of the material (e.g. anisotropy, cracks).

The earth interior is also an important source of inhomogeneities. Understanding the effect of the multi-scale heterogeneities is essential to interpret the behavior of seismic waves. As an example, the multiple scattering was recognized to be the source of the so-called coda (i.e. late arrivals of signals) (Aki 1980). Probing geological media therefore requires some statistical knowledge of the characteristics of the medium.

Optical scintillation due to atmospheric turbulence has been recognized as a limitating factor for the size and the resolution of telescopes (Newton 1704). Indeed, the index of refraction variations due to temperature fluctuations induce what is known as the “‘twinkling” of stars. The Kolmogorov spectrum allowing to statistically describe turbulence was developed in this context (Kolmogorov 1941). The topic of optical scintillation due to atmospheric turbulence was theorized by Tatarskii (Tatarskii 1971). Figure 1.2 shows an example of the star twinkling:

Vincent Van Gogh’s painting Starry Night is perhaps the most famous representation of the phenomenon. In fact, the study of the statistics of the luminance of the painting showed that it described quite accurately the Kolmogorov spectrum of turbulence (Aragon´ et al. 2008).

FIGURE 1.2: Starry Night by Vincent Van Gogh. From vangoghgallery.com.

Other examples of WPRM can be found in engineering applications such as radio commu-nications. It was indeed demonstrated that fluctuations in the ionosphere in terms of electron density were responsible for the scintillation of radio sources (Briggs and Parkin 1963, Buckley 1975). Rickett (Rickett 1977) also gathered information about the fluctuations in the interstellar plasma which cause distortions of radio waves from far radio sources, such as quasars and pulsars. The interstellar scattering is described in more details in (Rickett 1990), highlighting the limitations caused by this phenomenon. The analogy between this phenomenon and the topic of this dissertation is shown in figure 1.3:

At interplanetary scale, solar wind is also considered to generate strong fluctuations leading to perturbations in the electromagnetic wave transmission. Magnetohydrodynamic (MHD) turbulence is, for example, caused by solar winds (Matthaeus and Goldstein 1982) which have also been proven to interact with interplanetary magnetic field (IMF) (Gosling et al. 1987, Zank 1999). Solar winds are also responsible for a well-known phenomenon, the appearance of aurora borealis (Dessler 1966).

All these examples emphasize the fluctuating aspect of any propagation medium. At all space and time scales, perturbations in the measurement environments are observed. The case of sound propagation through the ocean medium is not different, as presented in section 1.2.2.

**Wave propagation in a random ocean**

**Sources of fluctuation**

The ocean is in constant motion. Multiple physical phenomena contribute to the spatio-temporal variation of the world’s oceans, from very large basin-scale heterogeinites, such as gyres, to small, meter-scale, turbulence. Driven by wind (Ekman transport) and the Coriolis effect, ocean gyres present a scale of the order of magnitude of the size of the ocean. Smaller events, such as eddies, are characterized by a diameter of 10 to 500 km and typical periods of days to months. Some examples of temperature and salinity heterogeneities induced by eddies are given in Tychensky and Carton (1998). The phenomenon of up-welling, of space and time scales up to respectively a few hundreds km and a few tens of days, is also well known to produce ocean temperature fluctuations. Tides are an example of non-wind driven event that cause ocean motion. In fact, the relationship between tides and the moon was raised by Pyth-eas as early as in the IVth century BC. A graphical representation of the space-time scale of the phenomena is proposed in Graham (1993)

However, these large-scale phenomena all present a quite long time period, which leads ocean engineering scientists to focus mainly on smaller-scale events such as those located in the bottom left corner of figure 1.4. For instance, the interactions of the propagated wave with the interfaces of the medium were proven to induce severe degradation in the signal coher-ence (Kuperman and Ingenito 1977, Kuperman and Schmidt 1989) when the interface is rough.

There is a tremendous amount of literature about the issue of scattering from rough surfaces, especially in the case of high-frequency acoustic waves interacting with an agitated sea surface. The studies conducted during World War II were extended by Eckart (Eckart 1953), where a theoretical analysis of the problem is provided. A comprehensive study of the ocean surface roughness and the associated model for sound propagation is presented in Marsh et al. (1961) and in Urick (1973). McDaniel (McDaniel 1993) also reviewed the topic of scattering from the sea surface, addressing it as a twofold problem: the scattering from the roughness of the sur-face and the volume attenuation due to the presence of bubbles or bubble layers and plumes. Statistical parameters such as the root-mean-square (rms) surface wave-height are classically used to describe the sea surface (Zhou et al. 2007). Statistical models for the sea surface can also be found, such as in Elfouhaily et al. (1997). The evaluation of the backscattering strength depends on the frequency of the signal since it involves the ratio between the acoustic wave-length and the surface rugosity. Several models are available in the literature, including the Chapman-Harris model for mid-frequencies (Chapman and Harris 1962), the Ogden-Erskine model for low frequencies (Ogden and Erskine 1992) and the Crowther model for high fre-quencies (Crowther 1980). The problem, overall, lies in the spatio-temporal dependence of the ocean channel impulse response (CIR), when the surface is agitated. The coherence time of such CIR, defined as the time during which the channel remains constant, can be as small as a few seconds (Li and Preisig 2007) which leads to a fading of the underwater acoustic channel response.

The roughness of the seabed is also a source of loss of coherence of acoustic signals propa-gated in the sea. Due to its extraordinary variability both in sediment nature and roughness, it is excruciatingly difficult to provide a model for the seabed. High-frequency signals show a high sensitivity to the grain size and Rayleigh scattering can be observed (Jackson et al. 1996). On the other hand, lower frequency signals penetrate inside the sea bottom and are therefore impacted by the internal burrowed geological structure. Similarly to what was presented for the sea surface agitation case, several models are used to evaluate the scattering strength of the rough seabed: for example, the formula proposed by DelBalzo (Leclere et al. 1997) is accurate for the 300 Hz − 1:5 kHz frequency band, whereas the Jackson model can be applied for waves around the 30 kHz center frequency (Jackson et al. 1986).

Besides the effects of ocean interfaces, an increasing interest is found in volume fluctuations. Especially, internal waves (IW) have been proven to induce spatial and temporal fluctuations in the sound speed distribution. In the early to mid 1970s, observations and analytical de-scriptions of the IW spectrum have been the subject of numerous studies (Boyce 1975, Garrett and Munk 1972; 1975, Munk and Zachariasen 1976, Desaubies 1976). The resulting model of these contributions is known as the GM model (for Garret and Munk model). The idea is to synthesize the available measurements of internal wave energy and to propose a model spec-trum describing the variation of energy in terms of wavenumber (vertical and horizontal) and frequency. The internal wave energy per unit mass can therefore be expressed as a function of frequency ! and mode number j:

cal ocean (Munk profile (Garrett and Munk 1972)) E0 = 6:10−5 J:m−2, the buoyancy frequency N (z) = N0e(z−H)~H , where N0 is the Brunt-Vais¨al¨a¨ frequency (or buoyancy frequency), the SO-FAR (SOund Fixing and Ranging) depth is H = 1:3 km, fC is the Coriolis parameter and j∗ = 3.

Besides the GM model, qualitative descriptions of the sound field fluctuations induced by IW were provided in Munk and Zachariasen (1976). Waveguide invariant studies also allowed to relate the medium fluctuations to the phase and group velocities of the propagated wave (Ku-perman et al. 2012, Roux et al. 2013). Ocean acoustic tomography was proven to provide spatial and temporal measurements on the temperature fluctuations due to IW at ultrasonic scale as well (Roux et al. 2011).

The ratio between vertical and horizontal fluctuations due to internal waves is found to be ap-proximately 10, which imparts to the sound speed fluctuations field an anisotropic behavior. Internal waves are also found to be the main source at the origin of volume inhomogeneities, over cycles as long as hours or days (Kuperman and Lynch 2004). They are predominant with respect to meter-scale turbulence (Levine and Irish 1981). Various at-sea measurements demonstrate the influence of internal waves on acoustic wave propagation. Section 1.2.2.2 shows how such fluctuations impact the formulation of the wave equation and the way some classical theories can tackle this issue. The range dependency of the sound speed is illustrated by the measurements presented in Rouseff et al.

Other examples of field measurements and the corresponding data processing are given in the following sections

**Effects on wave propagation**

The Helmholtz equation is classically solved using either ray theory (high frequency hypoth-esis), or normal modes theory (low-frequency hypothesis), when the configuration studied is “range-independent” (Jensen et al. 2011). In the case of an ocean medium perturbed by internal waves, the dependence in range appears in the term c(x; t). Therefore, the range-dependent Helmholtz equation is solved using either, in the high-frequency case, ray theory taking into account the effects of the sound speed fluctuations (Esswein and Flatte´ 1980; 1981), or, in the low-frequency case, coupled mode theory (Evans 1983). Nevertheless, strong lim-itations of these techniques are observed: in fact, ray theory fails near caustics and shadow zones induced by sound speed fluctuations (Flatte´ and Rovner 2000), and mode theory is too expensive in terms of numerical calculations at high frequencies, since the number of modes becomes very important.

An efficient way to tackle the middle-frequency band (100 Hz − 10 kHz) is the parabolic equa-tion (PE) (Jensen et al. 2011). Based on the two main hypotheses of weak fluctuations of the medium and narrow-angle propagation, the parabolic equation was applied to optical wave propagation through weak turbulence (Tatarskii 1971) and allows to perform a step-by-step solving with a given initial condition (Flatte´ and Tappert 1975, Dashen et al. 2010). This pro-cedure is valid because of the first order derivative in distance of propagation presented by the standard PE. It was shown otherwise in Flatte´ and Vera (2003) that full wave equations are not necessary to accurately describe the influence of IW on underwater acoustic propa-gation. The radiation transport equation was also used in order to extend PE methods to higher frequency cases (Wilson and Tappert 1979). Analytical solutions for the parabolic equa-tion in a randomly fluctuating ocean have also been proposed, using Rytov’s method partic-ularly (Munk and Zachariasen 1976). Path-integral techniques can also be used to solve the standard parabolic equation (Dashen et al. 1985).

An a priori qualitative characterization of the acoustic field using dimensional parameters is classically used in WPRM (Wolf 1975) and its most spread version in underwater acoustic was developed by Flatte´ (Dashen et al. 2010). In this case, the dimensional parameters are

• the strength parameter, , which characterizes the amplitude of the acoustic field distor-tions. In the geometrical limit, it is defined as the standard deviation of the random phase fluctuations of the signal.

• the diffraction parameter, , which characterizes the qualitative nature of the distortions.

Regimes of fluctuations are then defined, depending on the values of and , as depicted by figure 1.6, in the case of a single-scale medium:

If >> 1 and < 1, the Rytov approximation can be applied, which means that the pres-sure field may be approached using a perturbation expansion. When < 1 and ≈ 1, a single eigenray occurs, exhibiting a small displacement in vertical correlation length: this is the unsaturated regime. The configuration were > 1 and 2 > 1 is called partial saturation. The eigenray splits into multiple well-correlated eigenrays. Finally, if > 1 and > 1, the eigenray splits into uncorrelated eigenpaths. The appearance of caustics and shadow zones is characteristic of the saturation. Physically, the unsaturated case corresponds to configurations where weak fluctuations occur at short ranges of propagation, and the saturated regime cor-respond to cases where strong fluctuations or long range propagation occurs. The boundaries between the various regimes of fluctuations should not be considered as strict delimiters, since their domain of validity may overlap. They are used in the present manuscript in order to provide qualitative information about the signals propagated through fluctuating media and they should not be taken as absolute predictions. Examples of the interpretation of the images resulting from atmospheric turbulence in terms of regimes of fluctuation can also be found, such as shown in figure 1.7 (Texereau 1948):

FIGURE 1.7: Evolution of an image at the output of a telescope in presence of very calm at-mosphere (V), calm atmosphere (IV), agitated atmosphere -or unsaturation- (III), strongly ag-itated atmosphere -or partial saturation- (II), and very strongly agitated atmosphere – or full saturation- (I).

Most of the published materials focus on the calculation of the moments of the acoustic field, since derivations and predictions of the detailed realization of the pressure field in a complex random environment itself seem unreasonable and deprived of interest. Statistics of the pressure field propagated through IW have been provided, using the various theories described earlier. For example, path-integral resolution of the parabolic equation was used to derive expressions for the mutual coherence function (MCF), second-order moment of the sound pressure, noted . The spatial MCF was therefore approximated as follows in Esswein and Flatte´ (1980).

where s denotes the spacing between two sensors and D ( s) is the phase-structure func-tion, defined in (Dashen et al. 2010). It was shown in Flatte´ (2002) that the second-order moment for changes in depth could be expressed as a Gaussian function, i.e. a quadratic form for the phase-structure function:

The measurement and the modeling of the loss of spatial (horizontal and vertical) and tempo-ral coherence were extensively investigated using path-integral methods (Flatte´ and Stoughton 1988, Flatte´ and Vera 2003, Yang 2008), coupled and adiabatic modes (Voronovich and Osta-shev 2006), transport theory (Colosi et al. 2013, Chandrayadula et al. 2013) and numerical PE codes (Tielburger¨ et al. 1997, Oba and Finette 2002, Flatte´ 2002, Vera 2007). Alternative meth-ods, such as polynomial chaos, can also be found (Finette 2006, Creamer 2006). Combinations of horizontal ray theory and vertical mode theory is also used (Badiey et al. 2005) in order to characterize the variation of acoustic intensity. This last method is used to cope with the strong anisotropy of the sound speed fluctuations induced by internal waves. The statistical distribu-tion of the acoustic pressure field (Dashen et al. 2010) and intensity (Flatte´ et al. 1987, Colosi et al. 2001) are also of great interest, since it was shown in these papers that a discrimination between the regimes of fluctuations was possible from the analysis of these quantities.

The direct link between WPRM and the limitation and degradation of the array gain was stud-ied in Laval and Labasque (1981), Carey (1998). This means that the fluctuations of the prop-agating medium have to be considered in the design of sonar arrays, especially in the case of large arrays. In an ocean perturbed by internal waves, horizontal coherence lengths of 10 to 100 wavelengths and vertical coherence lengths less than 10 wavelengths are found (Gorodet-skaya et al. 1999). It is nonetheless difficult to anticipate for the degradation solely caused by the effect of internal waves, since at-sea measurements involved numerous phenomena con-currently, including surface and bottom scattering, distortions of the array and dispersion of sensors properties. The development of signal processing techniques in order to mitigate the detection gain degradation is hence limited to numerical configurations. Could scaled experi-ment performed in a controlled and reproducible manner emulate the array gain degradation due to fluctuations in the ocean ? We will try to answer this question throughout the present document. Section 1.3 investigate the emulators of WPRM in various domains, mainly in op-tics.

**Emulators of WPRM**

Due to the tremendous complexity of the phenomena described in section 1.2.1, researchers found an interest in trying to reproduce the physical phenomena, or their impact on wave propagation, in controlled environments. We present here the main motivations behind these developments of scaled experimental protocols, and we provide examples in various domains of physics.

**Motivations**

The numerous phenomena concurrently involved in measurements of WPRM induces uncer-tainties on the quantification of the influence of the phenomenon of interest. For example, in the case of fluctuations of acoustic signals propagated in an ocean perturbed by IW, it is diffi-cult to evaluate the influence of the process studied with respect to other sources of fluctuations (listed in section 1.2.2). Moreover, models describing extremely complicated phenomena such as IW are based on at-sea measurements involving high costs. A (cheap) way to isolate the involved phenomena and quantify its influence on wave propagation and signal processing represents therefore a strong interest. This can be performed in controlled environments, such as water tanks, where the development of a reproducible protocol allowing to acquire acoustic data perturbed in a similar fashion to what can be observed in the ocean is possible. The ques-tion is now to give a more accurate meaning to the word “similar”.

The ability to work with acoustic data acquired in controlled environments represents a way to benchmark signal processing techniques developed in order to mitigate the loss of coherence of the signals. This procedure can also be performed using numerical models as well, but we see a vivid interest in being able to compare the results with experimental data.

Several fields related to WPRM were investigated using a comparable approach. A non-exhaustive review is given in the following section.

**Existing protocols**

In the 1960s, the fluctuations of high frequency sound waves traveling through temperature microstructure were investigated in water tanks under laboratory conditions. The range-dependence of the coefficient of variation was studied in (Stone and Mintzer 1962; 1965), where sound pulses were propagated in a water tank heated from below. The time dependence of ultrasonic waves amplitude was measured using a similar protocol (Campanella and Favret 1969). The frequency-dependence of this phenomenon is presented in LaCasce Jr et al. (1962). Transverse spatial correlations close to the definition of the coherence function described ear-lier were provided in Sederowitz and Favret (1969). In 1979, Chotiros and Smith (Chotiros and Smith 1979) compared measurements under similar conditions to theoretical description of tur-bulence (Tatarskii 1971). In his Ph.D. dissertation, Dobbins summarizes the results presented in these papers (Dobbins 1989). The measurement of the effect of turbulence in the propaga-tion medium of ultrasonic waves was enhanced in Blanc-Benon and Juve´ (1993), where the experiment was conducted in air instead of water tanks, avoiding the appearance of air bub-bles, possibly responsible for a bias in the analysis of the measurements. A classification of the experimental configuration in terms of regimes of fluctuations is proposed, as well as an analysis of the intensity distribution, whose shape is characteristic of the associated regime of fluctuations.

**Table of contents :**

**1 Introduction and State of the Art **

1.1 General objective

1.2 Wave propagation through random media

1.2.1 Transverse review

1.2.2 Wave propagation in a random ocean

1.2.2.1 Sources of fluctuation

1.2.2.2 Effects on wave propagation

1.3 Emulators of WPRM

1.3.1 Motivations

1.3.2 Existing protocols

1.4 Corrective Signal Processing

1.5 Thesis Content

**2 Dimensional Analysis **

2.1 Introduction

2.2 Characteristic parameters in the “natural” oceanic case

2.2.1 Flatt´e’s dimensional analysis

2.2.2 Acoustic field correlation length

2.3 Sound field calculation in the lens case

2.3.1 The Small Slope Approximation in the case of two semi-infinite media

2.3.2 Application to the lens case

2.4 Sound field statistics

2.4.1 First-order statistics

2.4.2 Second-order statistics

2.5 The Fresnel radius and the diffraction parameter

2.6 Summary and conclusion

**3 Development of a Scaled Experiment **

3.1 Introduction

3.2 Experimental protocol

3.3 RAndom Faced Acoustic Lens manufacturing

3.4 Laboratory equipment

3.5 Numerical tools

3.5.1 Simulation tools in the scaled experiment configuration

3.5.2 Simulation tool in the corresponding oceanic configuration

3.6 Conclusion

**4 Experimental results**

4.1 Investigated configurations

4.2 Complex Pressure Distribution

4.2.1 Unsaturated regime

4.2.2 Partially saturated regime

4.2.3 Fully saturated regime

4.2.4 Discussion

4.3 Second-order moment analysis

4.3.1 Unsaturated regime

4.3.2 Partially saturated regime

4.3.3 Fully saturated regime

4.3.4 Discussion

4.4 Fourth-order moment analysis: intensity fluctuations

4.4.1 Unsaturated regime

4.4.2 Partially saturated regime

4.4.3 Fully saturated regime

4.5 Conclusion

**5 Influence of Medium Fluctuations on Detection Performance **

5.1 Introduction

5.2 Array gain degradation

5.3 Performance of classical techniques

5.3.1 Matched-Field Beamforming

5.3.2 High-resolution MUSIC algorithm

5.4 Corrective signal processing techniques

5.5 Discussion

**6 At-sea Measurements: the ALMA Experiment**

6.1 Introduction

6.2 Experimental configuration

6.2.1 Source sub-system

6.2.2 Receiver sub-system

6.3 First data gathering

6.4 A glimpse of the experimental results

6.5 Conclusion

**7 Conclusion and FutureWork **

7.1 Concluding remarks

7.2 Future work