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

SUMMARY By specifying a discrete matrix formulation for the frequency^space modelling problem for linear partial diierential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non-linear inverse methods, such as the gradient (steepest descent) method, the Gauss^Newton method and the full Newton method We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency-domain inversion (FDI)The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general The FDI methods only require that the original partial diierential equations can be expressed as a discrete boundary-value problem (that is as a matrix problem) Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the mis¢t function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a ¢nal computation to compute a step length) This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss^Newton method, and the full Hessian matrix used in the full Newton method In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss^Newton method, can be e⁄ciently computed using a backpropagation approach similar to that used to compute the gradient vector The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of ¢rst-order multiples (such as free-surface multiples in seismic data) Another interpretation is that this term predicts changes in the gradient vector due to second-order non-linear eiects In a numerical test, the Gauss^Newton and full Newton methods prove eiective in helping to solve the di⁄cult non-linear problem of extracting a smooth background velocity model from surface seismic-re£ection data

Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion

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

Phanerozoic tectonics of the South China Block: Key observations and controversies

The golden transformation of the Cretaceous plate subduction in the West Pacific

Stability and efficiency are two issues of general concern in inverse Q filtering. This paper presents a stable, efficient approach to inverse Q filtering, based on the theory of wavefield downward continuation. It is implemented in a layered manner, assuming a depth-dependent, layered-earth Q model. For each individual constant Q layer, the seismic wavefield recorded at the surface is first extrapolated down to the top of the current layer and a constant Q inverse filter is then applied to the current layer. When extrapolating within the overburden, instead of applying wavefield downward continuation directly, a reversed, upward continuation system is solved to obtain a stabilized solution. Within the current constant Q layer, the amplitude compensation operator, which is a 2-D function of traveltime and frequency, is approximated optimally as the product of two 1-D functions depending, respectively, on time and frequency. The constant Q inverse filter that compensates simultaneously for phase and amplitude effects is then implemented efficiently in the Fourier domain.

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A stable and efficient approach of inverse Q filtering

A principal limitation on seismic resolution is the earth attenuation, or Q -effect, including the energy dissipation of high-frequency wave components and the velocity dispersion that distorts seismic wavelets. An inverse Q -filtering procedure attempts to remove the Q -effect to produce high-resolution seismic data, but some existing methods either reduce the S/N ratio, which limits spatial resolution, or generate an illusory high-resolution wavelet that contains no more subsurface information than the original low-resolution data. In this paper, seismic inverse Q -filtering is implemented in a stabilized manner to produce high-quality data in terms of resolution and S/N ratio. Stabilization is applied to only the amplitude compensation operator of a full inverse Q -filter because its phase operator is unconditionally stable, but the scheme neither amplifies nor suppresses high frequencies at late times where the data contain mostly ambient noise. The latter property makes the process invertible, differ...

/pdf/inverse-q-filter-for-seismic-resolution-enhancement-4vitt2j9mf.pdf

Inverse Q-filter for seismic resolution enhancement

A seismic trace may be decomposed into a series of wavelets that match their time-frequency signature by using a matching pursuit algorithm, an iterative procedure of wavelet selection among a large and redundant dictionary. For reflection seismic signals, the Morlet wavelet may be employed, because it can represent quantitatively the energy attenuation and velocity dispersion of acoustic waves propagating through porous media. The efficiency of an adaptive wavelet selection is improved by making first a preliminary estimate and then a localized refining search, whereas complex-trace attributes and derived analytical expressions are also used in various stages. For a constituent wavelet, the scale is an important adaptive parameter that controls the width of wavelet in time and the bandwidth of the frequency spectrum. After matching pursuit decomposition, deleting wavelets with either very small or very large scale values can suppress spikes and sinusoid functions effectively from the time-frequency spect...

/pdf/seismic-time-frequency-spectral-decomposition-by-matching-34epdyftfv.pdf

Seismic time-frequency spectral decomposition by matching pursuit

[1] We analyse receiver functions from 29 broad-band seismographs along a 380-km profile across the Longmenshan (LMS) fault belt to determine crustal structure beneath the east Tibetan margin and Sichuan basin. The Moho deepens from about 50 km under Songpan-Ganzi in east Tibet to about 60 km beneath the LMS and then shallows to about 35 km under the western Sichuan basin. The average crustal Vp/Vs ratios vary in the range 1.75-1.88 under Songpan-Ganzi in east Tibet, 1.8-2.0 under the LMS, and decrease systematically across the NW part of the Sichuan basin to less than 1.70. A negative phase arrival above the Moho under Songpan-Ganzi and Sichuan basin is interpreted as a PS conversion from the top of a low-velocity layer in the lower crust. The very high crustal Vp/Vs ratio and negative polarity PS conversion at the top of lower crust in east Tibet are inferred to be seismic signatures of a low-viscosity channel in the eastern margin of the Tibetan plateau. The lateral variation of Moho topography, crustal Vp/Vs ratio and negative polarity PS conversion at the top of the lower crust along the profile seem consistent with a model of lower crust flow or tectonic escape.

https://www.cas.cn/ky/kyjz/200912/W020091207404481906959.pdf

Crustal structure across Longmenshan fault belt from passive source seismic profiling

The Ricker wavelet is theoretically a solution of the Stokes differential equation, which takes into account the effect of Newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. In this paper, we defined the time-domain breadth and the frequency-domain bandwidth of the Ricker wavelet and developed quantities analytically in terms of the Lambert W function. We determined that the central frequency, the geometric center of the frequency band, is close to the mean frequency statistically evaluated using the power spectrum, rather than the amplitude spectrum used in some of the published literature. We also proved that the standard deviation from the mean frequency is not, as suggested by the literature, the half-bandwidth of the frequency spectrum of the Ricker wavelet. Moreover, we established mathematically the relationships between the theoretical frequencies (the central frequency and the half-bandwidth) and the numerical measurements (the mean frequency and its standard deviation) and produced each of these frequency quantities analytically in terms of the peak frequency of the Ricker wavelet.

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Yanghua Wang

Papers

A stable and efficient approach of inverse Q filtering

Inverse Q-filter for seismic resolution enhancement

Seismic time-frequency spectral decomposition by matching pursuit

Crustal structure across Longmenshan fault belt from passive source seismic profiling

Frequencies of the Ricker wavelet