On robust estimation of the location parameter

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

Geophysical, petrological, and geochemical data provide important clues about the composition of the deep continental crust. On the basis of seismic refraction data, we divide the crust into type sections associated with different tectonic provinces. Each shows a three-layer crust consisting of upper, middle, and lower crust, in which P wave velocities increase progressively with depth. There is large variation in average P wave velocity of the lower crust between different type sections, but in general, lower crustal velocities are high (>6.9 km s−1) and average middle crustal velocities range between 6.3 and 6.7 km s−1. Heat-producing elements decrease with depth in the crust owing to their depletion in felsic rocks caused by granulite facies metamorphism and an increase in the proportion of mafic rocks with depth. Studies of crustal cross sections show that in Archean regions, 50–85% of the heat flowing from the surface of the Earth is generated within the crust. Granulite terrains that experienced isobaric cooling are representative of middle or lower crust and have higher proportions of mafic rocks than do granulite terrains that experienced isothermal decompression. The latter are probably not representative of the deep crust but are merely upper crustal rocks that have been through an orogenic cycle. Granulite xenoliths provide some of the deepest samples of the continental crust and are composed largely of mafic rock types. Ultrasonic velocity measurements for a wide variety of deep crustal rocks provide a link between crustal velocity and lithology. Meta-igneous felsic, intermediate and mafic granulite, and amphibolite facies rocks are distinguishable on the basis of P and S wave velocities, but metamorphosed shales (metapelites) have velocities that overlap the complete velocity range displayed by the meta-igneous lithologies. The high heat production of metapelites, coupled with their generally limited volumetric extent in granulite terrains and xenoliths, suggests they constitute only a small proportion of the lower crust. Using average P wave velocities derived from the crustal type sections, the estimated areal extent of each type of crust, and the average compositions of different types of granulites, we estimate the average lower and middle crust composition. The lower crust is composed of rocks in the granulite facies and is lithologically heterogeneous. Its average composition is mafic, approaching that of a primitive mantle-derived basalt, but it may range to intermediate bulk compositions in some regions. The middle crust is composed of rocks in the amphibolite facies and is intermediate in bulk composition, containing significant K, Th, and U contents. Average continental crust is intermediate in composition and contains a significant proportion of the bulk silicate Earth's incompatible trace element budget (35–55% of Rb, Ba, K, Pb, Th, and U).

Nature and composition of the continental crust: A lower crustal perspective

New measurements of compressional and shear wave velocities to hydrostatic pressures of 1 GPa are summarized for 678 rocks. Emphasis was placed on obtaining high-accuracy velocity measurements, which are shown to be critical in calculating Poisson's ratios from velocities. The rocks have been divided into 29 major groups for which velocities, velocity ratios, and Poisson's ratios are presented at several pressures. Observed Poisson's ratios for the monomineralic rocks compare favorably with theoretical Poisson's ratios calculated from single-crystal elastic constants. Plagioclase feldspar composition is important in understanding rock Poisson's ratios, since Poisson's ratio of albite increases from 0.28 to a predicted value of 0.31 for anorthite. Fe substitution for Mg in pyroxene and olivine also increases Poisson's ratio. Plotting rock compressional wave velocities versus Poisson's ratios reveals a triangular distribution bounded by quartzite with low compressional wave velocity and low Poisson's ratio, dunite with high compressional wave velocity and intermediate Poisson's ratio, and serpentinite with low compressional wave velocity and high Poisson's ratio. For common plutonic igneous rocks, there is a clear trend relating Poisson's ratio to composition, in which Poisson's ratio for granitic rocks increases from 0.24 to 0.29 as composition changes to gabbro and then decreases with decreasing plagioclase and increasing olivine contents to 0.25 in dunite. Changes in Poisson's ratio with progressive metamorphism of mafic and pelitic rocks correlate reasonably well with mineral reactions. There is no simple correlation between Poisson's ratio and felsic and mafic rock compositions; however, a linear correlation of increasing Poisson's ratio with decreasing SiO2 content is observed for rocks with 55 to 75 wt % SiO2. Average Poisson's ratios for continental and oceanic crusts are estimated to be 0.265 and 0.30, respectively.

Poisson's ratio and crustal seismology

Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

SUMMARY

The localization of seismic events is of utmost importance in seismology and exploration. Current techniques rely on the fact that the recorded event is detectable at most of the stations of a seismic network. Weak events, not visible in the individual seismogram of the network, are missed out. We present an approach, where no picking of events in the seismograms of the recording network is required. The observed wavefield of the network is reversed in time and then considered as the boundary value for the reverse modelling. Assuming the correct velocity model, the reversely modelled wavefield focuses on the hypocentre of the seismic event. The origin time of the event is given by the time where maximum focussing is observed. The spatial extent of the focus resembles the resolution power of the recorded wavefield and the acquisition. This automatically provides the uncertainty in the localization with respect to the bandwidth of the recorded data. The method is particularly useful for the upcoming large passive networks since no picking is required. It has great potential for localizing very weak events, not detectable in the individual seismogram, since the reverse modelling sums the energy of all recorded traces and, therefore, enhances the signal-to-noise ratio similar to stacking in seismic exploration. The method is demonstrated by 2-D and 3-D numerical case studies, which show the potential of the technique. Events with a S/N ratio smaller than 1 where the events cannot be identified in the individual seismogram of the network are localized very well by the method.

Reverse modelling for seismic event characterization

Imaging of diffractions is a challenge in seismic processing. Standard seismic processing is tuned to enhance reflections. Separation of diffracted from reflected events is frequently used to achieve an optimized image of diffractions. We present a method to effectively separate and image diffracted events in the time domain. The method is based on the common-reflection-surface-based diffraction stacking and the application of a diffraction-filter. The diffraction-filter uses kinematic wavefield attributes determined by the common-reflection-surface approach. After the separation of seismic events, poststack time-migration velocity analysis is applied to obtain migration velocities. The velocity analysis uses a semblance based method of diffraction traveltimes. The procedure is incorporated into the conventional common-reflection-surface workflow. We apply the procedure to 2D synthetic data. The application of the method to simple and complex synthetic data shows promising results.

Common-reflection-surface-based workflow for diffraction imaging

Summary. An algorithm for the computation of travel times, ray amplitudes and ray synthetic seismograms in 3-D laterally inhomogeneous media composed of isotropic and anisotropic layers is described. All 21 independent elastic parameters may vary within the anisotropic layers. Rays and travel times are evaluated by numerical solution of the ray tracing equations. Ray amplitudes are determined by evaluating reflection/ transmission coefficients and the geometrical spreading along individual rays. The geometrical spreading is computed approximately by numerical measurement of the cross-sectional area of the ray tube formed by three neighbouring rays. A similar approximate procedure is used for the determination of the coefficients of the paraxial ray approximation. The ray paraxial approximation makes computation of synthetic seismograms on the surface of the model very efficient. Examples of ray synthetic seismograms computed with a program package based on the described algorithm are presented.

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Computation of high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

Dynamic ray tracing (DRT) is important in evaluating high-frequency seismic wave fields in complicated structures. Two formulations of the DRT equations for inhomogeneous anisotropic media are presented. One of them is represented by the classical DRT system, suggested by Cervený 15 years ago. Both systems are specified in Cartesian coordinates. The DRT equations are supplemented with formulae for the transformation of DRT across a smoothly curved interface between two inhomogeneous anisotropic media. Cervený's formulation of the DRT is applied to the computation of vertical seismic profile (VSP) synthetics. The results of DRT are used for the evaluation of the geometrical spreading and of the coefficients of the paraxial ray approximation for travel times and ray amplitudes. In addition, the DRT is also used in the interval ray tracing procedure, a procedure searching for rays starling from the source and terminating in a specified interval on a profile. Results of numerical modeling of VSP measurements in a three-dimensional laterally varying structure consisting of Isotropic and anisotropic layers separated by curved interfaces are presented. Ray diagrams of selected elementary waves, time-distance curves, and multisource three-component VSP synthetics generated for two different source locations calculated for the anisotropic model are compared with the results determined for a reference Isotropic model. The latter is obtained by averaging phase velocities of the anisotropic model. This comparison clearly shows the effects of anisotropy and the lateral variation of the model on seismic wave fields. The reliability of the ray synthetics is briefly discussed.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JB095iB07p11301%4010.1002/%28ISSN%292169-9356.SEISANIS1

Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures

Shear wave velocity, especially when used with Compressional wave velocity, is an important constraint on estimates of the composition and physical state of the continental lithosphere. We present shear velocity (νs) models derived from high-quality S wave data along three profiles in Southwest Germany, which combine with existing P wave velocity (νp) models to give Poisson's ratio (σ) in the crust and uppermost mantle. We then construct a laterally varying petrological model of the crust by comparing our νp and νs models with published laboratory-measured rock velocities. In general, Poisson's ratio is 0.25 in the uppermost crust, 0.22–0.23 in the mid-crust (but reaching a low of 0.15 in the Black Forest low-νp zone), and varies laterally from 0.24 to 0.30 in the lower crust; the average Poisson's ratio of the whole crust is everywhere about 0.25. The laminated signature of the lower crust prominent on reflection and P wave refraction data is absent on the S wave refraction data, implying a strong high/low alternation of Poisson's ratio in the lower crust. This suggests that the lower-crustal laminae represent mafic/silicic compositional layering (for example, basic sills in an originally more quartz-rich lower crust) rather than alternating zones of high and low pore pressure. There is no evidence for a shear wave low-velocity zone (LVZ) in the Black Forest mid-crust, despite the presence of a strong P wave LVZ; this suggests high quartz content and the presence of pore fluids at low (not high) pore pressure. A clear P-to-S converted Moho reflection (Pm S) is visible on several shots, indicating that the Moho is, at least locally, a first-order discontinuity. No shear wave refractions from the upper mantle are present, even where strong P wave refractions are observed, possibly indicating an increase of Poisson's ratio with depth in the upper mantle. The petrological modeling indicates that the upper and middle crust are probably of granitic composition, that the mid-crustal low-νp zones have high quartz content, and that the lower crust probably consists of granulites of laterally varying bulk composition, from granodioritic to gabbroic. The lower crust is more mafic in areas of high reflection density on seismic reflection profiles, suggesting that the laminations are caused by mafic intrusions into the lower crust, either prior to the Variscan orogeny or during (Tertiary?) volcanic activity.

Dirk Gajewski

Papers

Reverse modelling for seismic event characterization

Common-reflection-surface-based workflow for diffraction imaging

Computation of high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures

An interpretation of wide-angle compressional and shear wave data in southwest Germany: Poisson's ratio and petrological implications