Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements
TL;DR: In this paper, the authors proposed new misfit functions for full waveform tomography based on instantaneous phase differences and envelope ratios between observed and synthetic seismograms, which can reduce non-linear behavior of waveforms due to interaction of phase and amplitude information.
Abstract: Resolution in seismic tomography intimately depends on data coverage, with different parts of seismograms sensitive to different parts of Earth's structure. In classical seismic tomography, the usable amount of data is often restricted because of approximations to the wave equation. 3-D numerical simulations of wave propagation provide new opportunities for increasing the amount of usable data in seismograms by choosing appropriate misfit functions which have direct control on Frechet derivatives. We propose new misfit functions for full waveform tomography based on instantaneous phase differences and envelope ratios between observed and synthetic seismograms. The aim is to extract as much information as possible from a single seismogram. Using the properties of the Hilbert transform, we separate phase and amplitude information in the time domain. To gain insight in the advantages and disadvantages of chosen misfit functions, we make qualitative comparisons of the corresponding finite-frequency adjoint sensitivity kernels with those from commonly used misfit functions based on cross-correlation traveltime, amplitude and waveform differences. The major advantages of our misfit functions are: (1) working in the Hilbert domain reduces non-linear behaviour of waveforms due to interaction of phase and amplitude information, and (2) we show with noise-free synthetic seismograms that it is possible to use a complete seismogram without losing information from low-amplitude phases. Complementary to instantaneous phase measurements, envelope measurements provide a way of using amplitude information of waveforms, which may also easily be extended to constrain anelastic properties. The properties of the kernels allow us to simplify the tomography problem by separating elastic and anelastic inversions. First indications are that the kernels remain well behaved in the presence of noise.
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Book•
23 Jul 2011
TL;DR: In this article, the authors proposed a numerical solution of the Elastic Wave Equation and computing sensitivity kernel for full waveform tomography for upper-mantle structure in Australasian Region.
Abstract: Introduction.- Numerical Solution of the Elastic Wave Equation.- Computing Sensitivity Kernels.- Seismological Data Functionals and their Associated Adjoint Sources.- Iterative Optimisation.- Full Waveform Tomography for Upper-mantle Structure in Australasian Region.- A Comparative Study of Local-scale full Waveform Tomographies.- Source Staking and Data Reduction in Global full Waveform Tomography.
442 citations
TL;DR: In this paper, an envelope fluctuation and decay of seismic records carries ultra low-frequency (ULF) signals that can be used to estimate the long-wavelength velocity structure.
Abstract: We recognized that the envelope fluctuation and decay of seismic records carries ultra low-frequency (ULF, i.e., the frequency below the lowest frequency in the source spectrum) signals that can be used to estimate the long-wavelength velocity structure. We then developed envelope inversion for the recovery of low-wavenumber components of media (smooth background), so that the initial model dependence of waveform inversion can be reduced. We derived the misfit function and the corresponding gradient operator for envelope inversion. To understand the long-wavelength recovery by the envelope inversion, we developed a nonlinear seismic signal model, the modulation signal model, as the basis for retrieving the ULF data and studied the nonlinear scale separation by the envelope operator. To separate the envelope data from the wavefield data (envelope extraction), a demodulation operator (envelope operator) was applied to the waveform data. Numerical tests using synthetic data for the Marmousi model pro...
358 citations
Additional excerpts
...Bozdag et al. (2011) discuss the envelope misfit functional and its use in the kernel sensitivity analysis of global seismic tomography....
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TL;DR: In this paper, the objective function consists of a data-misfit term and a penalty term, which measures how accurately the wavefields satisfy the wave-equation, and the solution is forced to solve the waveequation and fit the observed data, which leads to significant computational savings.
Abstract: Wave-equation based inversions, such as full-waveform inversion, are challenging because of their computational costs, memory requirements, and reliance on accurate initial models. To confront these issues, we propose a novel formulation of full-waveform inversion based on a penalty method. In this formulation, the objective function consists of a data-misfit term and a penalty term which measures how accurately the wavefields satisfy the wave-equation. Because we carry out the inversion over a larger search space, including both the model and synthetic wavefields, our approach suffers less from local minima. Our main contribution is the development of an efficient optimization scheme that avoids having to store and update the wavefields by explicit elimination. Compared to existing optimization strategies for full-waveform inversion, our method differers in two main aspects; i) The wavefields are solved from an augmented wave-equation, where the solution is forced to solve the wave-equation and fit the observed data, ii) no adjoint wavefields are required to update the model, which leads to significant computational savings. We demonstrate the validity of our approach by carefully selected examples and discuss possible extensions and future research.
287 citations
Cites background from "Misfit functions for full waveform ..."
...…high frequencies (Bunks 1995; Sirgue & Pratt 2004), near to far offsets (Virieux & Operto 2009) or small to large Laplace damping parameters (Shin & Cha 2009) and different misfit functionals (Cara & Lévêque 1987; Luo 1991; van Leeuwen & Mulder 2010; Bozda et al. 2011; Moghaddam & Mulder 2012)....
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TL;DR: In this study, a measure of the misfit computed with an optimal transport distance allows to account for the lateral coherency of events within the seismograms, instead of considering each seismic trace independently, as is done generally in full waveform inversion.
Abstract: Full waveform inversion using the conventional L2 distance to measure the misfit between seismograms is known to suffer from cycle skipping. An alternative strategy is proposed in this study, based on a measure of the misfit computed with an optimal transport distance. This measure allows to account for the lateral coherency of events within the seismograms, instead of considering each seismic trace independently, as is done generally in full waveform inversion. The computation of this optimal transport distance relies on a particular mathematical formulation allowing for the non-conservation of the total energy between seismograms. The numerical solution of the optimal transport problem is performed using proximal splitting techniques. Three synthetic case studies are investigated using this strategy: the Marmousi 2 model, the BP 2004 salt model, and the Chevron 2014 benchmark data. The results emphasize interesting properties of the optimal transport distance. The associated misfit function is less prone to cycle skipping. A workflow is designed to reconstruct accurately the salt structures in the BP 2004 model, starting from an initial model containing no information about these structures. A high-resolution P-wave velocity estimation is built from the Chevron 2014 benchmark data, following a frequency continuation strategy. This estimation explains accurately the data. Using the same workflow, full waveform inversion based on the L2 distance converges towards a local minimum. These results yield encouraging perspectives regarding the use of the optimal transport distance for full waveform inversion: the sensitivity to the accuracy of the initial model is reduced, the reconstruction of complex salt structure is made possible, the method is robust to noise, and the interpretation of seismic data dominated by reflections is enhanced.
264 citations
Cites methods from "Misfit functions for full waveform ..."
...A similar strategy has been proposed by Bŏzdag et al. (2011) where the ampli148 tude and travel-time information are computed following a Hilbert transform....
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...A similar strategy has been proposed by Bŏzdag et al. (2011) where the amplitude and traveltime information are computed following a Hilbert transform....
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TL;DR: In this article, the authors present a methodology to compute 3D global seismic wavefields for realistic earthquake sources in visco-elastic anisotropic media, covering applications across the observable seismic frequency band with moderate computational resources.
Abstract: . We present a methodology to compute 3-D global seismic wavefields for realistic earthquake sources in visco-elastic anisotropic media, covering applications across the observable seismic frequency band with moderate computational resources. This is accommodated by mandating axisymmetric background models that allow for a multipole expansion such that only a 2-D computational domain is needed, whereas the azimuthal third dimension is computed analytically on the fly. This dimensional collapse opens doors for storing space–time wavefields on disk that can be used to compute Frechet sensitivity kernels for waveform tomography. We use the corresponding publicly available AxiSEM ( www.axisem.info ) open-source spectral-element code, demonstrate its excellent scalability on supercomputers, a diverse range of applications ranging from normal modes to small-scale lowermost mantle structures, tomographic models, and comparison with observed data, and discuss further avenues to pursue with this methodology.
220 citations
Cites background from "Misfit functions for full waveform ..."
...Most modern measurements of “traveltimes”, such as cross-correlation (Nolet, 2008), time–frequency phase delays (Fichtner et al., 2008), or instantaneous phase (Bozdag et al., 2011) are based on waveforms, and therefore necessitate full wavefield modeling....
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..., 2008), or instantaneous phase ( Bozdag et al., 2011) are based on waveforms, and therefore necessitate full wavefield modeling....
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References
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TL;DR: In this paper, a large data set consisting of about 1000 normal mode periods, 500 summary travel time observations, 100 normal mode Q values, mass and moment of inertia have been inverted to obtain the radial distribution of elastic properties, Q values and density in the Earth's interior.
9,266 citations
"Misfit functions for full waveform ..." refers methods in this paper
...D background model, and use isotropic PREM (Dziewonski & Anderson 1981) and 3-...
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Book•
01 Jan 1980
TL;DR: This work has here attempted to give a unified treatment of those methods of seismology that are currently used in interpreting actual data and develops the theory of seismic-wave propagation in realistic Earth models.
Abstract: In the past decade, seismology has matured as a quantitative science through an extensive interplay between theoretical and experimental workers. Several specialized journals have recorded this progress in thousands of pages of research papers, yet such a forum does not bring out key concepts systematically. Because many graduate students have expressed their need for a textbook on this subject and because many methods of seismogram analysis now used almost routinely by small groups of seismologists have never been adequately explained to the wider audience of scientists and engineers who work in the peripheral areas of seismology, we have here attempted to give a unified treatment of those methods of seismology th at are currently used in interpreting actual data. We develop the theory of seismic-wave propagation in realistic Earth models. We study specialized theories of fracture and rupture propagation as models of an earthquake, and we supplement these theoretical subjects with practical descriptions of how seismographs work and how data are analyzed and inverted. Our text is arranged in two volumes. Volume I gives a systematic development of the theory of seismic-wave propagation in classical Earth models, in which material properties vary only with depth. It concludes with a chapter on seismometry. This volume is intended to be used as a textbook in basic courses for advanced students of seismology. Volume II summarizes progress made in the major frontiers of seismology during the past decade. It covers a range of special subjects, including chapters on data analysis and inversion, on successful methods for quantifying wave propagation in media varying laterally (as well as with depth), and on the kinematic and dynamic aspects of motions near a fault plane undergoing rupture. The second volume may be used as a texbook in graduate courses on tectonophysics, earthquake mechanics, inverse problems in geophysics, and geophysical data processing.n
5,291 citations
Additional excerpts
...Inserting eqs (3) into (2), the gradient becomes δχ = − N∑ r=1 ∫ T 0 ∂si g(xr , t, m) ∫ t 0 ∫ V [δρ(x′)Gi j (xr , x′; t − t ′)∂2t ′ s j (x′, t ′) + δc jklm(x′)∂ ′k Gi j (xr , x′; t − t ′)∂ ′l sm(x′, t ′)]d3x′dt ′ dt. (4) Using the reciprocity of the Green’s function (Aki & Richards 1980; Dahlen & Tromp 1998) and reversing time, it is convenient to define the adjoint wavefield s†k (x ′, t ′) = ∫ t ′ 0 ∫ V Gki (x ′, xr ; t ′ − t) f †i (x, t)d3x dt, (5) where f †i is the adjoint source given by f †i (x, t) = N∑ r=1 ∂si g(xr , T − t, m)δ(x − xr )....
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...Using asymptotic finite-frequency kernels, Li & Romanowicz (1996), Mégnin & Romanowicz (2000) and Gung & Romanowicz (2004) constructed global models based on waveforms obtained by cutting seismograms into energy wave packets where each packet was weighted appropriately by its energy, highlighting the importance of weighting to retrieve information from low-amplitude parts of data....
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...Using 3-...
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...Using relations between elastic moduli, density, P-wave (α) and S-wave (β) speeds, the gradient of the misfit function may alternatively be written as δχ = ∫ V [K ′ρ(x)δ ln ρ(x) + Kβ (x)δ ln β(x) + Kα(x)δ ln α(x)] d3x, (11) C© 2011 The Authors, GJI, 185, 845–870 Geophysical Journal International C© 2011 RAS where K ′ρ , K β and K α are given by K ′ρ = Kρ + Kκ + Kμ, (12) Kβ = 2 ( Kμ − 4 3 μ κ Kκ ) , (13) Kα = 2 ( κ + 43 μ κ Kκ ) ....
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...Using the Born approximation (Hudson 1977; Wu & Aki 1985), the ith component of δs may be expressed as δsi (x, t) = − ∫ t 0 ∫ V [δρ(x′)Gi j (x, x′; t − t ′)∂2t ′ s j (x′, t ′) + δc jklm(x′)∂ ′k Gi j (x, x′; t − t ′)∂ ′l sm(x′, t ′)]d3x′ dt ′, (3) where ρ is the density, cjklm is the fourth-order elastic tensor and δρ and δcjklm are their associated perturbations....
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TL;DR: In this paper, the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation, which is based on the generalized least squares criterion, and it can handle errors in the data set and a priori information on the model.
Abstract: The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commerc...
3,198 citations
"Misfit functions for full waveform ..." refers background in this paper
...…is based on differences between observed and synthetic waveforms combining amplitude together with phase information, and is defined as (e.g. Tarantola 1984, 1987, 1988; Nolet 1987) χ (m) = 1 2 N∑ r=1 ∫ T 0 ‖d(xr , t) − s(xr , t, m)‖2 dt, (35) C© 2011 The Authors, GJI, 185, 845–870…...
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TL;DR: In this article, a frequency-space domain approach to waveform inversion is presented, which is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data.
Abstract: Seismic waveforms contain much information that is ignored under standard processing schemes; seismic waveform inversion seeks to use the full information content of the recorded wavefield. In this paper I present, apply, and evaluate a frequency-space domain approach to waveform inversion. The method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data. The model data are computed using a full-wave equation, viscoacoustic, frequency-domain, finite-difference method. Ray asymptotics are avoided, and higher-order effects such as diffractions and multiple scattering are accounted for automatically. The theory of frequency-domain waveform/wavefield inversion can be expressed compactly using a matrix formalism that uses finite-difference/finite-element frequency-domain modeling equations. Expressions for fast, local descent inversion using back-propagation techniques then follow naturally. Implementation of these methods depends on efficient frequency-domain forward-modeling solutions; these are provided by recent developments in numerical forward modeling. The inversion approach resembles prestack, reverse-time migration but differs in that the problem is formulated in terms of velocity (not reflectivity), and the method is fully iterative. I illustrate the practical application of the frequency-domain waveform inversion approach using tomographic seismic data from a physical scale model. This allows a full evaluation and verification of the method; results with field data are presented in an accompanying paper. Several critical processes contribute to the success of the method: the estimation of a source signature, the matching of amplitudes between real and synthetic data, the selection of a time window, and the selection of suitable sequence of frequencies in the inversion. An initial model for the inversion of the scale model data is provided using standard traveltime tomographic methods, which provide a robust but low-resolution image. Twenty-five iterations of wavefield inversion are applied, using five discrete frequencies at each iteration, moving from low to high frequencies. The final results exhibit the features of the true model at subwavelength scale and account for many of the details of the observed arrivals in the data.
1,496 citations
"Misfit functions for full waveform ..." refers methods in this paper
...D numerical simulations with adjoint techniques (e.g. Tarantola 1984, 1988; Fink 1997; Talagrand & Courtier 1987; Crase et al. 1990; Pratt 1999; Akçelik et al. 2003)....
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Journal Article•
1,402 citations
Additional excerpts
...0 (Bassin et al. 2000) (S20RTS+Crust2....
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