Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

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An overview of full-waveform inversion in exploration geophysics

A least-squares point-by-point inversion of strong ground motion and teleseismic body waves is used to infer the fault rupture history of the 1979 Imperial Valley, California, earthquake. The Imperial fault is represented by a plane embedded in a half-space where the elastic properties vary with depth. The inversion yields both the spatial and temporal variations in dislocation on the fault plane for both right-lateral strike-slip and normal dip-slip components of motion. Inversions are run for different fault dips and for both constant and variable rupture velocity models. Effects of different data sets are also investigated. Inversions are compared which use the strong ground motions alone, the teleseismic body waves alone, and simultaneously the strong ground motion and teleseismic records. The inversions are stabilized by adding both smoothing and positivity constraints.
The moment is estimated to be 5.0 × 10^(25) dyne-cm and the fault dip 90° ± 5°. Dislocation in the hypocentral region south of the United States-Mexican border is relatively small and almost dies out near the border. Dislocation then increases sharply north of the border to a maximum of about 2 m under Interstate 8. Dipslip motion is minor compared to strike-slip motion and is concentrated in the sediments. The best-fitting constant rupture velocity is 80 per cent of the local shear-wave velocity. However, there is a suggestion that the rupture front accelerated from the hypocenter northward. The 1979 Imperial Valley earthquake can be characterized as a magnitude 5 earthquake at the hypocenter which then grew into or triggered a magnitude 6 earthquake north of the border.

/pdf/inversion-of-strong-ground-motion-and-teleseismic-waveform-74bcy5k05q.pdf

Inversion of strong ground motion and teleseismic waveform data for the fault rupture history of the 1979 Imperial Valley, California, earthquake

A linear model for attenuation of waves is presented, with Q, or the portion of energy lost during each cycle or wavelength, exactly independent of frequency. The wave propagation is completely specified by two parameters, e.g., Q and c0, a phase velocity at an arbitrary reference frequency ω0. A simple exact derivation leads to an expression for the phase velocity c as a function of frequency: c/c0 = (ω/ω0)γ, where γ = (1/π) tan−1 (1/Q). Scaling relationships for pulse propagation are derived and it is shown that for a material with a given value of Q, the risetime or the width of the pulse is exactly proportional to travel time. The travel time for a pulse resulting from a delta function source at x = 0 is proportional to xβ, where β = 1/(1 - γ). On the basis of this relation it is suggested that the velocity dispersion associated with anelasticity may be less ambiguously observed in the time domain than in the frequency domain. A steepest descent approximation derived by Strick gives a good time domain representation for the impulse response. The scaling relations are applied to field observations from the Pierre shale formation in Colorado, published by Ricker, who interpreted his data in terms of a Voigt solid with Q inversely proportional to frequency, and McDonal et al., who interpreted their data in terms of nonlinear friction. The constant Q theory fits both sets of data.

/pdf/constant-q-wave-propagation-and-attenuation-1jl7n33irc.pdf

Constant Q-wave propagation and attenuation

SUMMARY

In this paper we examine the connection between the westward motion of Turkey relative to Europe and the extension in and around the Aegean Sea. The principal new data available since the last attempt to synthesize the tectonics of this region by McKenzie (1978) are much improved focal mechanisms of earthquakes, constrained by P and SH body wave modelling as well as by first motions. These mechanisms show that the faulting in the western part of the Aegean region is mostly extensional in nature, on normal faults with a NW to WNW strike and with slip vectors directed NNW to NNE. There is evidence from palaeomagnetism that this western region rotates clockwise relative to stable Europe. In the central and eastern Aegean, and in NW Turkey, distributed right-lateral strike–slip is more prevalent, on faults trending NE to ENE, and with slip vectors directed NE. Palaeomagnetic data in this eastern region is more ambiguous, but consistent with very small or no rotations in the northern part and possibly anticlockwise rotations, relative to Europe, in the south. The strike–slip faulting that enters the central Aegean from the east appears to end abruptly in the SW against the NW-trending normal faults of Greece.



The kinematics of the deformation is controlled by three factors: the westward motion of Turkey relative to Europe; the continental collision between NW Greece–Albania and the Apulia–Adriatic platform in the west; and the presence of the Hellenic subduction zone to the south. As the right-lateral slip on the North Anatolian Fault enters the Aegean region it splays out, becoming distributed on several parallel faults. The continental shortening in NW Greece and Albania does not allow the rotation of the western margin of the region to be rapid enough to accommodate this distributed E–W right-lateral shear, and thus leads to E–W shortening in the northern Aegean, which is compensated by N–S extension as the southern Aegean margin can move easily over the Hellenic subduction zone. The dynamics of the system, once initiated, is self-sustaining, being driven by the high topography in eastern Turkey and by the roll-back of the subducted slab beneath the southern Aegean.



The geometry of the deformation resembles the behaviour of a system of broken slats attached to margins that rotate. In spite of its extreme simplicity, a simple model of such broken slats is able to reproduce quantitatively most of the features of the instantaneous velocity field in the Aegean region, including: the slip vectors and nature of the faulting in the eastern and western parts; the senses and approximate rates of rotation; the overall extensional velocity across the Aegean; and the distribution of strain rates, as seen in the seismicity and topography or bathymetry, and using geodetic measurements.



As part of this study, we re-examined the relation between the surface faulting and the focal parameters determined seismologically for the three large 1981 Gulf of Corinth earthquakes, and reassessed the evidence for associating particular earthquakes in the sequence with observed surface faulting. An important result is that in one of the events there is a resolvable discrepancy between the slip vector measured at the surface and that determined from the seismic body waves.

https://hal.archives-ouvertes.fr/hal-01400404/document

Active tectonics of the north and central Aegean Sea

The effective elastic moduli of a fluid-saturated solid containing thin cracks depend on the degree of interconnection between the cracks. Three separate regimes may be identified: (1) dry (drained), in which fluid in cracks can flow out of bulk regions of compression, (2) saturated isobaric, in which fluid may flow from one crack to another but no bulk flow takes place, and (3) saturated isolated, in which there is no communication of fluid between cracks. Transitions between these cases involve fluid flow, resulting in dissipation of energy. Relaxation of shear stresses in viscous fluid inclusions also results in dissipation. Viscoelastic moduli are derived, by using a self-consistent approximation, that describe the complete range of behavior. There are two characteristic frequencies near which dissipation is largest and the moduli change rapidly with frequency. The first corresponds to fluid flow between cracks, and its value can be estimated from the crack geometry or permeability. The second corresponds to the relaxation of shear stress in an isolated viscous fluid inclusion; its value may also be estimated. Variations of crack geometry result in a distribution of characteristic frequencies and cause Q to be relatively constant over many decades of frequency. Fluid flow between cracks accounts for attenuation of seismic waves in water-saturated rocks and attenuation observed in laboratory measurements on water-saturated rocks and partially molten aggregates. Attenuation in a partially molten upper mantle is probably due to fluid flow between cracks, although grain boundary relaxation in an unmelted upper mantle could also account for the seismic low-velocity zone. Grain boundary relaxation in the mantle may cause the long-term shear modulus to be around 20% less than that measured from seismic observations.

Viscoelastic properties of fluid-saturated cracked solids

Although the attenuation of body waves in an infinite medium has often been considered, little emphasis is usually given to the attendant dispersion, which is a necessary consequence of the medium absorption and is determined unambiguously by it. Dispersion equations are obtained from the application of an integral transform in the frequency domain. Such relations are of the Kramers-Kronig type, relating the dispersive part of the index of refraction of the medium to the absorptive part by means of an integral over the entire frequency range. They are a consequence of the principle of causality and follow without recourse to a specific wave equation. In particular, our results are consistent with and presume linearity for the wave motion. Three forms of the dispersion equation are exhibited. In all cases, the absorption coefficient has a linear dependence upon frequency for frequencies above a certain characteristic value—chosen low enough to be outside the range of measurement. However, it is necessary in all cases to introduce a low-frequency, nonzero cutoff for the absorption, which can in principle be taken arbitrarily small. Group and phase velocities are determined as functions of the medium Q and a characteristic velocity, which for small absorption can be related to the measured signal velocity. Seismic phenomena occur in materials where the absorption is small, Q≥30, so that the corresponding dispersion presumably will be difficult to measure in the earth. For appropriate laboratory materials the effect can be significant.

Walter I. Futterman

Papers

Dispersive body waves