Liouville partial-differential-equation methods for computing 2-D complex-valued eikonals in the multivalued sense in attenuating media

We present Liouville partial-differential-equation (PDE) based methods for computing complex-valued eikonals in the multivalued (or multiple arrival) sense in attenuating media. Since the earth is comprised of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is a crucial ingredient for describing wave propagation in attenuating media, since it is a unique quantity which summarizes two wave properties into one function: its real and imaginary parts are able to capture the effects of phase dispersions and amplitude attenuations, respectively. Because the usual ordinary-differential-equation (ODE) based ray-tracing methods for computing complex-valued eikonals distribute the eikonal solution irregularly in real space, we are motivated to develop PDE based Eulerian methods for computing complex-valued eikonals on regular meshes. Therefore, we propose to solve novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We dub the resulting method the Liouville PDE method for complex multivalued eikonals in attenuating media. We also provide Liouville PDE formulations for computing multi-valued amplitudes. Numerical examples, including a synthetic gas-cloud model, demonstrate that the proposed methods yield highly accurate complex-valued eikonals in the multivalued sense.

Gravitational wave propagation beyond general relativity: waveform distortions and echoes

We study the cosmological propagation of gravitational waves (GWs) beyond general relativity (GR) across homogeneous and isotropic backgrounds. We consider scenarios in which GWs interact with an additional tensor field and use a parametrized phenomenological approach that generically describes their coupled equations of motion. We analyze four distinct classes of derivative and non-derivative interactions: mass, friction, velocity, and chiral. We apply the WKB formalism to account for the cosmological evolution and obtain analytical solutions to these equations. We corroborate these results by analyzing numerically the propagation of a toy GW signal. We then proceed to use the analytical results to study the modified propagation of realistic GWs from merging compact binaries, assuming that the GW signal emitted is the same as in GR. We generically find that tensor interactions lead to copies of the originally emitted GW signal, each one with its own possibly modified dispersion relation. These copies can travel coherently and interfere with each other leading to a scrambled GW signal, or propagate decoherently and lead to echoes arriving at different times at the observer that could be misidentified as independent GW events. Depending on the type of tensor interaction, the detected GW signal may exhibit amplitude and phase distortions with respect to a GW waveform in GR, as well as birefringence effects. We discuss observational probes of these tensor interactions with both individual GW events, as well as population studies for both ground- and space-based detectors.

Determination of the Probability Density Function of the Mixture Signal and Additive Noise under Influence of Multiplicative Noise

Issues related to effect of multiplicative (modulating) (otherwise amplitude distortions) and additive noises on processed signal are considered. Analysis of statistical characteristics of probability density function (PDF) of instantaneous signal values against background of modulating interference is carried out. Expressions of the joint PDF envelope, phase and instantaneous signal values are obtained, for the case of only phase distortions, as well as the case when fluctuations in amplitude and phase are independent of each other. It is noted that in the second case, with the independence of phase and amplitude distortions, the PDF of the signal, first of all, is determined by the PDF of its instantaneous values and practically does not depend on the PDF of the signal phase. Expressions are obtained for the PDF useful signal and the most common PDF envelope when affecting the signal with modulating noise for deep phase distortions and for its uniform distribution over the interval (0, 2π). In addition, for the case of some functional relationship of phase and amplitude fluctuations, the expression for PDF instantaneous signal values is defined. The presence of such a functional connection allows calculating the PDF of the signal through the statistical characteristics of its envelope. It is obtained that PDF of mixture of signal and additive noise contains arbitrary distribution of phase and amplitude, as well as arbitrary law of distribution of envelope of processed signal. It is obtained that weight coefficients are determined by derivative of characteristic function of amplitude distortion in case when PDFs of said mixture are distributed according to normal law and functional connection exists between amplitude and phase distortions.

A Cross-Correlation-Based Approach to Pattern Distortion and Mutual Coupling for Shared-Aperture Antennas

This paper proposes a pattern distortion coefficient as a new figure of merit to quantitatively evaluate both mutual coupling and pattern distortions in multi-antenna systems. The proposed coefficient is defined as a cross correlation between unaffected and affected far-field patterns of antennas under test, and the input patterns are weighted using a Gaussian function to consider the target operation angle. The feasibility of the proposed approach is validated using a two-antenna system composed of an inverted-F antenna and a microstrip patch antenna, and the amount of mutual coupling is adjusted by changing the distance between the two antennas. The evaluation is further extended to a single-antenna system with a conducting wall that produces strong platform effects with serious pattern distortions. The results demonstrate that the proposed figure of merit provides quantitative insight into the amplitude and phase distortions of far-field patterns that can be caused by both mutual coupling and platform effects.

Ultrafast laser cutting of transparent materials: the trend towards tailored edges and curved surfaces

We report on ultrashort pulsed laser fabrication strategies for glass articles with customized edges and curved surfaces. To achieve single-pass, full-thickness modifications along the entire substrate, processing optics are presented that allow for beam shaping of non-diffracting beams and, additionally, for aberration compensation of phase distortions occurring at the tilted or curved interfaces. The efficacy of our concepts is presented by evaluating the surface and edge qualities of separated glass tubes with complex inner and outer contours as well as glass chamfer structures.

Passive seismic matrix imaging of La Soufrière of Guadeloupe volcano

<p>Volcanoes are among the most challenging media for seismic imaging given their highly localized and abrupt variations in physical parameters, extreme landforms, fractures, and the presence of magma and other fluids. Because of this high level of heterogeneity and the resulting difficulty to access the wave velocity distribution in the medium, reflection seismic imaging of volcanoes usually suffers from a loss of resolution and contrast. Here, we present a passive seismic imaging technique applied to the case of La Soufrière of Guadeloupe volcano. Inspired by previous works in optics (Badon <em>et al</em>., 2020), in acoustics (Lambert <em>et al</em>., 2020), and recently introduced in seismology (Touma <em>et al</em>., 2020), this technique relies on a matrix approach of passive reflection imaging, which requires only a rough approximation about the medium background velocity. This makes it robust even applied to extreme environments as volcanoes or fault zones. In this approach, the Green’s functions between an array of 76 geophones placed at the surface of the volcano are retrieved by cross-correlation of ambient seismic noise. This set of 2850 inter-element impulse responses forms a reflection matrix. Focusing operations are applied to this reflection matrix at emission and reception to project it in–depth. The focusing process allows to extract body wave components from seismic noise and thus, to retrieve information about reflectivity of in-depth structures. However, at this point, reflectivity images of the subsurface still suffer from phase distortions induced by long-range variations of the seismic velocity. This results in blurred images and hinders appropriate imaging. To overcome these issues, a novel operator is introduced: the distortion matrix. This operator is derived from the focused reflection matrix and connects any point in the medium with the distortion that a wavefront emitted from that point would experience due to heterogeneity. A time-reversal analysis of the distortion matrix allows to retrieve aberrations phase laws and hence to compensate for phase distortions. This correction enables to recover 3D-images of the volcano’s subsurface for the first 10km below the summit with optimized contrast and with an increased resolution. Interestingly, the restored resolution is even at least one half below the diffraction limit imposed by the geophone array angular aperture at the surface. The obtained gain in resolution and contrast allows to unveil internal structures of La Soufrière as hypothetical volcanic vents, magma reservoirs and lateral drainage conduits.</p><p><strong>References</strong></p><p>[Badon <em>et al</em>., 2020] Badon, A., Barolle, V., Irsch, K., Boccara, A. C., Fink, M., and Aubry, A. (2020). Distortion matrix concept for deep optical imaging in scattering media. <em>Science Advances,</em> 6(30).</p><p>[Lambert <em>et al.</em>, 2020] Lambert, W., Cobus, L. A., Frappart, T., Fink, M., and Aubry, A. (2020). Distortion matrix approach for ultrasound imaging of random scattering media. <em>Proceedings of the National Academy of Sciences,</em> 117(26):14645-14656.</p><p>[Touma<em> et al.</em>, 2020] Touma, R., Blondel, T., Derode, A., Campillo, M., & Aubry, A. (2020). A Distortion Matrix Framework for High-Resolution Passive Seismic 3D Imaging: Application to the San Jacinto Fault Zone, California.<em> arXiv preprint arXiv</em>:2008.01608.</p>

Analysis of the effect of fast and slow multiplicative noise on the distortion of the radiation pattern of the «array-receiver» system.

The effect of fast and slow multiplicative noise on the distortion of the beam pattern of the «array-receiver» system in the conditions of the smoothing effect of the receiver and taking into account the time of relative lag from the grating elements was analyzed. Expressions are obtained for instantaneous beam patterns of the «array-receiver» system in the absence and presence of multiplicative noise. It is shown that in the case of action of multiplicative noise the beam pattern of the «array-receiver» system is obtained by averaging the instantaneous pattern on the grating correctness at a time interval equal to the signal duration. Influence of periodic multiplicative noise on distortion of beam pattern of system «array-receiver» is investigted. It is shown that for all signals except frequency-modulated, the beam pattern expression is maximized at the time t0=0. It is also shown that the beam pattern distortion of the «array-receiver» system when exposed to periodic multiplicative noise is less than for the separately considered phased antenna array. The degree of reduction of distortion from smoothing action of matched receiver filter is determined; and the distortion is less, the greater the ratio of the spectrum width of the noise modulation function to the signal spectrum width. Analysis of the influence of fluctuation multiplicative noise on the distortion of beam pattern of «array-receiver» system for the case of matching with signal of receiver filter is performed. The power-average beam pattern of the «array-receiver» system is determined through the energy spectrum of the noise modulation function. It is noted that multiplicative noise, all other things being equal, has less effect on the beam-receiver pattern of the system when using broadband pulse signals without intra-pulse modulation and at the same time resolution in range. It is noted that in case of deep phase distortions for gratings with reversible phases, the distortion of the beam pattern of the «array-receiver» system caused by multiplicative noise is significantly more than those distortions associated with the extremity of the signal spectrum width. The newly smoothing effect of the receiver reduces the expansion of the beam lobe caused by multiplicative noise. An expression is obtained for the average power of the beam pattern of the «array-receiver» system with normally distributed phase distortions of the signal. The effect of the signal duration on beam distortions caused by multiplicative noise was quantified.