Ludovic Margerin

Lopsided Growth of Earth's Inner Core

Clearing Up the Inner Core The behavior of Earths core controls the planets heat budget and magnetic field, yet its structure remains enigmatic. For instance, the seismic properties of the solid inner core suggest hemispherical structural asymmetry, but questions remain as to how these variations arose (see the Perspective by Buffett). Monnereau et al. (p. 1014, published online 15 April) modeled grain sizes of crystalline iron—the predicted dominant mineral phase in the core—and found that a slow translational motion eastward may trigger melting in the Eastern Hemisphere and solidification in the Western Hemisphere, creating a lopsided core. Deuss et al. (p. 1018, published online 15 April) examined the normal-mode seismic structure of the inner core, collected from 90 large earthquakes, which reveal not just simple hemispherical variations, but more nuanced regional structures. The overlap of the seismic data with Earths magnetic field suggests that directionally dependent crystal alignment in the inner core formed during the solidification of the core or as a consequence of strong forces exerted by magnetism. The asymmetry of the inner core is explained by iron crystallization on one side and melting on the other. Hemispherical asymmetry is a prominent feature of Earth’s inner core, but how this asymmetry relates to core growth is unknown. Based on multiple-scattering modeling of seismic velocity and attenuation measurements sampling the whole uppermost inner core, we propose that the growth of the solid core implies an eastward drift of the material, driven by crystallization in the Western Hemisphere and melting in the Eastern Hemisphere. This self-sustained translational motion generates an asymmetric distribution of sizes of iron crystals, which grow during their translation. The invoked dynamical process is still active today, which supports the idea of a young inner core.

Monte Carlo simulation of multiple scattering of elastic waves

We study multiple scattering of elastic waves with a Monte Carlo method. We take into account the mode conversions and the polarization of the S waves. Some important physical parameters relevant to the description of the polarization are recalled, such as the definition and properties of the elastic Stokes vector. We briefly derive the scattering and Mueller matrices, as well as the differential and total scattering cross sections for one spherical inclusion embedded in a homogeneous matrix. The results of the single-scattering problem are used as a building block for multiple scattering. A Monte Carlo method to simulate the propagation of full elastic waves is presented. We pay a special attention to the convergence toward the diffusive regime which exhibits the equilibration of the P and S energy densities. Our simulations show the shear energy to become very rapidly dominant in the coda and the S to P energy density ratio to tend to 10.4 for a Poisson solid, as predicted by the equipartition theorem. However, the typical timescale and length scale to reach equipartition heavily depend on the scattering parameter kpa, where kp is the P wave number and a is the sphere radius. For Rayleigh scattering (kpa ≪ 1) we find a smooth evolution of energy density with time and a slow convergence toward the equilibration, mainly because of the large difference between the P and S scattering mean free paths in this case. On the other hand, for Rayleigh-Gans scattering (kpa ∼ 1.2, 1.6) a peak of energy associated with the forward scattered waves is observed, followed by a slow decay according to the diffusion approximation. We find that after only a few mean free times, equipartition is reached in spite of the strong anisotropy of the scattering in this regime. As the scattering parameter kpa increases, we find that equipartition is again delayed because the transport mean free paths become quite large. We find that a large source-station distance favors a rapid equilibration. This effect is seen to be very pronounced for Rayleigh scatterers. When a source of P waves is considered, the equipartition time can be twice as long as compared with a shear source. The time evolution of the Ep/Es ratio could be used as a marker for the different scattering mechanisms.

Locating a small change in a multiple scattering environment

This article presents an imaging technique to locate a weak perturbation in a multiple scattering environment. We derive a formula to predict the spatiotemporal decorrelation of diffuse coda waves induced by an extra scatterer. Locating this new defect is formulated as an inverse problem which is solved by a maximum likelihood approach. Using elastic waves in the 50–400 kHz frequency band, we recover the position of a millimetric hole drilled in a concrete sample with a precision of a few centimeter. Note that the size of the defect is comparable to the size of the myriads of heterogeneities constituting the sample.

Locating a weak change using diffuse waves: Theoretical approach and inversion procedure

We describe a time-resolved monitoring technique for heterogeneous media. Our approach is based on the spatial variations of the cross-coherence of diffuse waves acquired at fixed positions but at different dates. The technique applies to all kind of waves, provided that waveforms can be acquired with a sampling frequency much larger than the wave frequency. To locate and characterize a weak change that occurred between successive acquisitions, we use a maximum likelihood approach combined with a diffusive propagation model. We characterize this technique, locating a weak change using diffuse waves, called LOCADIFF, with the aid of numerical simulations. In several illustrative examples, we show that the change can be located with a precision of a few wavelengths and that its effective scattering cross-section can be retrieved. We investigate how the accuracy and precision of the method depends on the number of source-receiver pairs, on the time window used to compute the cross-correlation and on the errors in the propagation model. Applications can be found in nondestructive testing, seismology, radar, and sonar location.

Velocity and attenuation of scalar and elastic waves in random media: A spectral function approach

This paper investigates the scattering of scalar and elastic waves in two-phase materials and single-mineral-cubic, hexagonal, orthorhombic-polycrystalline aggregates with randomly oriented grains. Based on the Dyson equation for the mean field, explicit expressions for the imaginary part of Greens function in the frequency-wavenumber domain (ω, p), also known as the spectral function, are derived. This approach allows the identification of propagating modes with their relative contribution, and the computation of both attenuation and phase velocity for each mode. The results should be valid from the Rayleigh (low-frequency) to the geometrical optics (high-frequency) regime. Comparisons with other approaches are presented for both scalar and elastic waves.

Generalized eigenfunctions of layered elastic media and application to diffuse fields.

The spectral decomposition of the elastic wave operator in a layered isotropic half-space is derived by means of standard functional analysis methods. Particular attention is paid to the coupled P-SV waves. The problem is formulated directly in terms of displacements which leads to a 2 x 2 Sturm-Liouville system. The resolvent kernel (Greens function) is expressed in terms of simple plane-wave solutions. Application of Stones formula leads naturally to eigenfunction expansions in terms of generalized eigenvectors with oscillatory behavior at infinity. The generalized eigenfunction expansion is employed to define a diffuse field as a white noise process in modal space. By means of a Wigner transform, we calculate vertical to horizontal kinetic energy ratios in layered media, as a function of depth and frequency. Several illustrative examples are considered including energy ratios near a free surface, in the presence of a soft layer. Numerical comparisons between the generalized eigenfunction summation and a classical locked-mode approximation demonstrate the validity of the approach. The impact of the local velocity structure on the energy partitioning of a diffuse field is illustrated.

Imaging multiple local changes in heterogeneous media with diffuse waves

This study focuses on imaging local changes in heterogeneous media. The method employed is demonstrated and validated using numerical experiments of acoustic wave propagation in a multiple scattering medium. Changes are simulated by adding new scatterers of different sizes at various positions in the medium, and the induced decorrelation of the diffuse (coda) waveforms is measured for different pairs of sensors. The spatial and temporal dependences of the decorrelation are modeled through a diffuse sensitivity kernel, based on the intensity transport in the medium. The inverse problem is then solved with a linear least square algorithm, which leads to a map of scattering cross section density of the changes.

Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium.

This paper investigates the reconstruction of elastic Greens function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Greens function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Greens function reconstruction in a scattering medium is discussed.

Multimethod Characterization of the French-Pyrenean Valley of Bagnères-de-Bigorre for Seismic-Hazard Evaluation: Observations and Models

A narrow rectilinear valley in the French Pyrenees, affected in the past by damaging earthquakes, has been chosen as a test site for soil response characteriza- tion. The main purpose of this initiative was to compare experimental and numerical approaches. A temporary network of 10 stations has been deployed along and across the valley during two years; parallel various experiments have been conducted, in particular ambient noise recording, and seismic profiles with active sources for struc- ture determination at the 10 sites. Classical observables have been measured for site amplification evaluation, such as spectral ratios of horizontal or vertical motions between site and reference stations using direct S waves and S coda, and spectral ratios between horizontal and vertical (H/V) motions at single stations using noise and S-coda records. Vertical shear-velocity profiles at the stations have first been obtained from a joint inversion of Rayleigh wave dispersion curves and ellipticity. They have subsequently been used to model the H/V spectral ratios of noise data from synthetic seismograms, the H/V ratio of S-coda waves based on equipartition theory, and the 3D seismic response of the basin using the spectral element method. General good agreement is found between simulations and observations. The 3D simulation reveals that topography has a much lower contribution to site effects than sedimentary filling, except at the narrow ridge crests. We find clear evidence of a basin edge effect, with an increase of the amplitude of ground motion at some distance from the edge inside the basin and a decrease immediately at the slope foot.

Nonparametric estimation of the heterogeneity of a random medium using compound Poisson process modeling of wave multiple scattering.

In this paper, we present a nonparametric method to estimate the heterogeneity of a random medium from the angular distribution of intensity of waves transmitted through a slab of random material. Our approach is based on the modeling of forward multiple scattering using compound Poisson processes on compact Lie groups. The estimation technique is validated through numerical simulations based on radiative transfer theory.