Adjoint Wavefield Research Articles

Images of the Iberian Lithosphere from One Local Earthquake

Abstract On 12 August 2007, a magnitude M w 4.7 intraplate earthquake occurred near the center of the Iberian Peninsula, an area characterized by comparably simple crustal structure within the complex Iberia–Maghreb plate boundary zone, and characterized by complete azimuthal coverage with seismic broadband stations. We analyze regional intermediate-period (20- to 50-s) coda waveforms for this earthquake. They contain energy representing late-arriving surface waves that have been reflected laterally at lithospheric heterogeneities in or around Iberia, but complexity of the coda waveforms hampers a direct interpretation. We use coda recordings as the source for a back-propagating adjoint wave field and compute 3D Born sensitivity kernels for the dependence of least-squares waveform misfits of coda waves on wave speed variations. We hereby image the origin of single scattering recorded in the coda. Misfit kernels for P - and S -wave velocity show azimuth-dependent intensity variation as a result of source radiation, and an appropriate compensation significantly improves imaging quality, thereby revealing several clear lineaments. These are interpreted as surface-wave reflectors due to deep-rooted heterogeneity such as terrain boundaries or Moho topography, demonstrating the ability of the approach to unravel complex waveforms, and providing a new point of view on regional lithospheric structure.

Noise cross-correlation sensitivity kernels

SUMMARY We determine finite-frequency sensitivity kernels for seismic interferometry based upon noise cross-correlation measurements. Under the assumptions that noise is spatially uncorrelated but non-uniform, we determine ensemble-averaged cross correlations between synthetic seismograms at two geographically distinct locations. By minimizing a measure of the difference between observed and simulated ensemble cross correlations—subject to the constraint that the simulated wavefield satisfies the seismic wave equation—we obtain ensemble sensitivity kernels. These ensemble kernels reflect the sensitivity of ensemble cross-correlation measurements to variations in model parameters, for example, mass density, shear and compressional wave speeds and the spatial distribution of noise. Ensemble kernels are calculated based upon the interaction between two wavefields: an ensemble forward wavefield and an ensemble adjoint wavefield. To obtain the ensemble forward wavefield, one first calculates a generating wavefield obtained by inserting a signal determined by the characteristics of the noise at the location of the first receiver, saving the results of this calculation at locations where noise is generated, that is, typically on (a portion of) the Earth's surface. Next, one uses this generating wavefield as the source of the ensemble forward wavefield associated with the first receiver. The ensemble adjoint wavefield is obtained by using measurements between simulated and observed ensemble cross correlations as a seismic source located at the second receiver. The interaction between ensemble forward and adjoint wavefields ‘paints’ ensemble sensitivity kernels. We illustrate the construction of ensemble kernels and their nature in two and three dimensions using a spectral-element method. In addition to a ‘banana-doughnut’ feature connecting the two receivers, as in traditional finite-frequency earthquake tomography, some noise cross-correlation sensitivity kernels exhibit hyperbolic ‘jets’ protruding from each receiver in a direction away from the other receiver. Ensemble sensitivity kernels for long-period (T > ∼ 50 s) non-uniform noise in global models exhibit sensitivity along the minor and major arcs. These kernels reflect the fact that measurements typically involve long time-series that include multi-orbit surface waves. Like free oscillations, such measurements are sensitive to structure along the great circle through the two receivers. From the perspective of noise cross-correlation tomography, we discuss strategies for inversions in terrestrial and helioseismology.

Imaging lateral heterogeneity in the northern Apennines from time reversal of reflected surface waves

Prominent arrivals in the coda of seismograms from the wider Alpine area can be associated with lateral reflections of Love waves at the northern Apennines mountain chain (Italy), where structural heterogeneity causes an abrupt contrast in phase velocity. We discuss an approach to image lateral heterogeneity from reflected surface waves using intermediate-period, three-component coda waveforms as sources for an adjoint wavefield that propagates the reflections backward in time. We numerically compute three-dimensional sensitivity kernels for the dependence of coda waveforms on P velocity, S velocity and density, based upon correlations between the adjoint and the regular forward wavefields. We consider synthetic coda waveforms for a simplified model of the northern Apennines, as well as real coda observations from five moderate magnitude earthquakes (MW 4.6–5.6) in the southern Alps. Wave propagation is simulated using the spectral-element method, for which a 3-D regional earth model is used in the case of real data. Single and combined event sensitivity kernels provide clear images of the reflectivity associated with the northern Apennines in kernels for density and S-wave speed. The kernels show that surface wave reflections occur near the axial zone of the mountain chain. Apart from the Apennines, the approach is able to image other smaller reflectivity patches from the coda waveforms, like the Ivrea zone in the southern Alps. Our coda misfit kernels can be integrated in a gradient-based waveform tomography, where they could enhance the sharpness of the model at lateral discontinuities.

Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain

We propose a new approach to full seismic waveform inversion on continental and global scales. This is based on the time–frequency transform of both data and synthetic seismograms with the use of time- and frequency-dependent phase and envelope misfits. These misfits allow us to provide a complete quantification of the differences between data and synthetics while separating phase and amplitude information. The result is an efficient exploitation of waveform information that is robust and quasi-linearly related to Earth's structure. Thus, the phase and envelope misfits are usable for continental- and global-scale tomography, that is, in a scenario where the seismic wavefield is spatially undersampled and where a 3-D reference model is usually unavailable. Body waves, surface waves and interfering phases are naturally included in the analysis. We discuss and illustrate technical details of phase measurements such as the treatment of phase jumps and instability in the case of small amplitudes. The Fréchet kernels for phase and envelope misfits can be expressed in terms of their corresponding adjoint wavefields and the forward wavefield. The adjoint wavefields are uniquely determined by their respective adjoint-source time functions. We derive the adjoint-source time functions for phase and envelope misfits. The adjoint sources can be expressed as inverse time–frequency transforms of a weighted phase difference or a weighted envelope difference. In a comparative study, we establish connections between the phase and envelope misfits and the following widely used measures of seismic waveform differences: (1) cross-correlation time-shifts; (2) relative rms amplitude differences; (3) generalized seismological data functionals and (4) the L2 distance between data and synthetics used in time-domain full-waveform inversion. We illustrate the computation of Fréchet kernels for phase and envelope misfits with data from an event in the West Irian region of Indonesia, recorded on the Australian continent. The synthetic seismograms are computed for a heterogeneous 3-D velocity model of the Australian upper mantle, with a spectral-element method. The examples include P body waves, Rayleigh waves and S waves, interfering with higher-mode surface waves. All the kernels differ from the more familar kernels for cross-correlation time-shifts or relative rms amplitude differences. The differences arise from interference effects, 3-D Earth's structure and waveform dissimilarities that are due to waveform dispersion in the heterogeneous Earth.

Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods

We determine adjoint equations and Fréchet kernels for global seismic wave propagation based upon a Lagrange multiplier method. We start from the equations of motion for a rotating, self-gravitating earth model initially in hydrostatic equilibrium, and derive the corresponding adjoint equations that involve motions on an earth model that rotates in the opposite direction. Variations in the misfit function χ then may be expressed as , where δ ln m = δm/m denotes relative model perturbations in the volume V, δ ln d denotes relative topographic variations on solid–solid or fluid–solid boundaries Σ, and ∇Σδ ln d denotes surface gradients in relative topographic variations on fluid–solid boundaries ΣFS. The 3-D Fréchet kernel Km determines the sensitivity to model perturbations δ ln m, and the 2-D kernels Kd and Kd determine the sensitivity to topographic variations δ ln d. We demonstrate also how anelasticity may be incorporated within the framework of adjoint methods. Finite-frequency sensitivity kernels are calculated by simultaneously computing the adjoint wavefield forward in time and reconstructing the regular wavefield backward in time. Both the forward and adjoint simulations are based upon a spectral-element method. We apply the adjoint technique to generate finite-frequency traveltime kernels for global seismic phases (P, Pdiff, PKP, S, SKS, depth phases, surface-reflected phases, surface waves, etc.) in both 1-D and 3-D earth models. For 1-D models these adjoint-generated kernels generally agree well with results obtained from ray-based methods. However, adjoint methods do not have the same theoretical limitations as ray-based methods, and can produce sensitivity kernels for any given phase in any 3-D earth model. The Fréchet kernels presented in this paper illustrate the sensitivity of seismic observations to structural parameters and topography on internal discontinuities. These kernels form the basis of future 3-D tomographic inversions.

Full three-dimensional tomography: a comparison between the scattering-integral and adjoint-wavefield methods

SUMMARY This paper analyses the computational issues of full 3-D tomography, in which the starting model as well as the model perturbation is 3-D and the sensitivity (Frechet) kernels are cal- culated using the full physics of 3-D wave propagation. We compare two formulations of the structural inverse problem: the adjoint-wavefield (AW) method, which back-propagates the data from the receivers to image structure, and the scattering-integral (SI) method, which sets up the inverse problem by calculating and storing the Frechet kernels for each data functional. The two inverse methods are closely related, but which one is more efficient depends on the overall problem geometry, particularly on the ratio of sources to receivers, as well as trade-offs in computational resources, such as the relative costs of compute cycles to data storage. We find that the SI method is computationally more efficient than the AW method in regional waveform tomography using large sets of natural sources, although it requires more storage.

Finite-frequency tomography using adjoint methods-Methodology and examples using membrane surface waves

SUMMARY We employ adjoint methods in a series of synthetic seismic tomography experiments to recover surface wave phase-speed models of southern California. Our approach involves computing the Frechet derivative for tomographic inversions via the interaction between a forward wavefield, propagating from the source to the receivers, and an 'adjoint' wavefield, propagating from the receivers back to the source. The forward wavefield is computed using a 2-D spectral- element method (SEM) and a phase-speed model for southern California. A 'target' phase- speed model is used to generate the 'data' at the receivers. We specify an objective or misfit function that defines a measure of misfit between data and synthetics. For a given receiver, the remaining differences between data and synthetics are time-reversed and used as the source of the adjoint wavefield. For each earthquake, the interaction between the regular and adjoint wavefields is used to construct finite-frequency sensitivity kernels, which we call event kernels. An event kernel may be thought of as a weighted sum of phase-specific (e.g. P) banana- doughnut kernels, with weights determined by the measurements. The overall sensitivity is simply the sum of event kernels, which defines the misfit kernel. The misfit kernel is multiplied by convenient orthonormal basis functions that are embedded in the SEM code, resulting in the gradient of the misfit function, that is, the Frechet derivative. A non-linear conjugate gradient algorithm is used to iteratively improve the model while reducing the misfit function. We illustrate the construction of the gradient and the minimization algorithm, and consider various tomographic experiments, including source inversions, structural inversions and joint source-structure inversions. Finally, we draw connections between classical Hessian-based tomography and gradient-based adjoint tomography.

Finite-Frequency Kernels Based on Adjoint Methods

We derive the adjoint equations associated with the calculation of Frechet derivatives for tomographic inversions based upon a Lagrange multiplier method. The Frechet derivative of an objective function χ(m), where m denotes the Earth model, may be written in the generic form δχ = ∫ K_m(x) δ ln m(x) d^3x, where δ ln m = δm/m denotes the relative model perturbation and K_m the associated 3D sensitivity or Frechet kernel. Complications due to artificial absorbing boundaries for regional simulations as well as finite sources are accommodated. We construct the 3D finite-frequency “banana-doughnut” kernel K_m by simultaneously computing the so-called “adjoint” wave field forward in time and reconstructing the regular wave field backward in time. The adjoint wave field is produced by using time- reversed signals at the receivers as fictitious, simultaneous sources, while the regular wave field is reconstructed on the fly by propagating the last frame of the wave field, saved by a previous forward simulation, backward in time. The approach is based on the spectral-element method, and only two simulations are needed to produce the 3D finite-frequency sensitivity kernels. The method is applied to 1D and 3D regional models. Various 3D shear- and compressional-wave sensitivity kernels are presented for different regional body- and surface-wave arrivals in the seismograms. These kernels illustrate the sensitivity of the observations to the structural parameters and form the basis of fully 3D tomographic inversions.

The adjoint method in seismology—: II. Applications: traveltimes and sensitivity functionals

Sensitivity functionals which allow us to express the total derivative of a physical observable with respect to the model parameters, are defined on the basis of the adjoint method. The definition relies on the existence of Green’s functions for both the original and the adjoint problem. Using the acoustic wave equation in a homogeneous and unbounded medium, it is shown that the first derivative of the wavefield u with respect to the model parameter p = c − 2 ( c is the wave speed) does not contain traveltime information. This property also influences objective functions defined on u and in particular the least squares objective function. Therefore, a waveform inversion should either be complemented by a traveltime tomography or work with initially very long wavelengths that decrease in the course of the iteration. The definition of the sensitivity functionals naturally introduces waveform sensitivity kernels. Analytic examples are shown for the case of an isotropic, elastic and unbounded medium. In the case of a double couple source there are three classes of sensitivity kernels, one for each of the parameters λ , μ (Lamé parameters) and ρ (density). They decompose into P → P, P → S, S → P and S → S kernels and can be described by third-order tensors incorporating the radiation patterns of the original wavefield and the adjoint wavefield. An analysis of the sensitivity kernels for density suggests that a waveform inversion procedure should exclude either the direct waves or the source and receiver regions. S-wave sensitivity kernels, corresponding to conversions from or to S-waves, are larger than the kernels corresponding to P-waves only. This implies that S-wave residuals caused by parameter differences between the true Earth model and the numerical model will dominate a waveform inversion that does not account for that effect.

Numerical identification of linear cracks in 2D elastodynamics using the instantaneous reciprocity gap

This paper considers the identification problem of a linear crack in a body of finite extension within the framework of linear two-dimensional elastodynamics. In a series of prior papers in electricity, elasticity or acoustics, it has been proved using the reciprocity gap that three different series of adjoint wave fields determine in closed-form solution the normal of the plane of the crack, the position of the plane and finally the complete crack extension. The work developed next within the framework of linear elastodynamics defines a novel instantaneous reciprocity gap as the instantaneous work done by the adjoint tractions on the crack opening displacement. This quantity is then used to identify linear cracks in a two-dimensional problem. It is shown using a numerical example that a unique family of planar shear waves permits the identification of the normal, position and a convex hull of a linear crack through simple interpretations of the instantaneous reciprocity gap. This method is more general in the sense that it applies to three-dimensional problems as well.

Eigenmode scattering relations for plane-stratified gyrotropic media

We consider the 4×4 scattering matrixS for a plane wave incident on a plane-stratified gyrotropic multilayer slab, which is assumed to have a single symmetry axis (the magnetisation vector in a ferrite sheet, or the external magnetic field direction in a plane-stratified magnetoplasma). In the given (original) problem the plane of incidence is at an azimuthal angle ϕ with respect to the magnetic meridian plane (the plane containing the normal to the stratification, and the gyrotropic symmetry axis), and there is a corresponding “conjugate” set of wave fields, in which the plane of incidence is at an azimuthal angle (π−ϕ), with a corresponding conjugate scattering matrixS′. Adjoint wave fields, obtained by reversing the magnetic symmetry vector, are used to yield the eigenmode amplitudes required in the definition of the scattering matrices. At an interface between two adjacent layers the scattering matrices are shown to be uniquely determined by the characteristic wave polarisations, and this is used to prove that the given and conjugate scattering matrices,S andS′, for the overall multilayer system are mutually transposed, i.e.\(S = \tilde S'\).