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SUMMARYWe present a theory and conceptual examples for fibre-optic deformation sensing based on phase changes of transmitted light. As a first result, we establish an exact relation between observable phase changes and the deformation tensor along the fibre. This relation is nonlinear and includes effects related to both local changes in fibre length and deformation-induced changes of the local refractive index. In cases where the norm of the deformation tensor is much smaller than 1, a useful first-order relation can be derived. It connects phase changes to an integral over in-line strain along the fibre times the local refractive index. When spatial variations of the refractive index are negligible, this permits the calculation of phase change measurements from distributed strain measurements, for instance, from distributed acoustic sensing (DAS). An alternative form of the first-order relation reveals that a directional sensitivity determines the ability of a point along the fibre to measure deformation. This directional sensitivity is proportional to fibre curvature and spatial variability of the refractive index. In a series of simple conceptual examples, we illustrate how a seismic wavefield is represented in a phase change time-series and what the role of higher-order effects may be. Specifically, we demonstrate that variable curvature along the fibre may lead to a multiplication of seismic waves, meaning that a single seismic wave appears multiple times in a recording of optical phase changes. Furthermore, we show that higher-order effects may be observable in specific scenarios, including deformation exactly perpendicular to the fibre orientation. Though higher-order effects may be realized in controlled laboratory settings, they are unlikely to occur in seismic experiments where fibre geometries are irregular and waves asymptotically propagate in all directions with all possible polarizations as a consequence of 3-D heterogeneity. Our results provide the mathematical foundation for the analysis of emerging transmission-based fibre-optic sensing data, and their later use in seismic event characterization and studies of Earth structure.

Ambient noise cross-correlations can be used as self-consistent observables, opening novel possibilities for investigating ambient noise sources. To optimise the forward-modelling of global ambient noise cross-correlations for any given distribution of noise sources in the microseismic frequency range up to 0.2 Hz, we implement (i) pre-computed wavefields and (ii) spatially variable grids. This enables rapid inversions for microseismic noise sources based on finite-frequency source sensitivity kernels. We use this advancement to perform regional and global gradient-based iterative inversions of the logarithmic energy ratio in the causal and acausal branches of micro-seismic noise cross-correlations. Synthetic inversions show promising results, with good recovery of the main dominant noise sources of the target model. Data inversions for several consecutive days at the beginning of October 2019 demonstrate the capability of inverting for the spatio-temporal variations of the sources of secondary microseisms in the ocean. This paves the way for daily ambient noise source inversions which could help improve full-waveform ambient noise tomography and subsurface monitoring methods., Geophysical Journal International, 227 (1), ISSN:0956-540X, ISSN:1365-246X

We present a new approach to full-waveform inversion (FWI) that enables the assimilation of data sets that expand over time without the need to reinvert all data. This evolutionary inversion rests on a reinterpretation of stochastic Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS), which randomly exploits redundancies to achieve convergence without ever considering the data set as a whole. Specifically for seismological applications, we consider a dynamic mini-batch stochastic L-BFGS, where the size of mini-batches adapts to the number of sources needed to approximate the complete gradient. As an illustration we present an evolutionary FWI for upper-mantle structure beneath Africa. Starting from a 1-D model and data recorded until 1995, we sequentially add contemporary data into an ongoing inversion, showing how (i) new events can be added without compromising convergence, (ii) a consistent measure of misfit can be maintained and (iii) the model evolves over times as a function of data coverage. Though applied retrospectively in this example, our method constitutes a possible approach to the continuous assimilation of seismic data volumes that often tend to grow exponentially., Geophysical Journal International, 224 (1), ISSN:0956-540X, ISSN:1365-246X

We present a computational framework for the assimilation of local to global seismic data into a consistent model describing Earth structure on all seismically accessible scales. This Collaborative Seismic Earth Model (CSEM) is designed to meet the following requirements: (i) Flexible geometric parametrization, capable of capturing topography and bathymetry, as well as all aspects of potentially resolvable structure, including small-scale heterogeneities and deformations of internal discontinuities. (ii) Independence of any particular wave equation solver, in order to enable the combination of inversion techniques suitable for different types of seismic data. (iii) Physical parametrization that allows for full anisotropy and for variations in attenuation and density. While not all of these parameters are always resolvable, the assimilation of data that constrain any parameter subset should be possible. (iv) Ability to accommodate successive refinements through the incorporation of updates on any scale as new data or inversion techniques become available. (v) Enable collaborative Earth model construction. The structure of the initial CSEM is represented on a variable-resolution tetrahedral mesh. It is assembled from a long-wavelength 3-D global model into which several regional-scale tomographies are embedded. We illustrate the CSEM workflow of successive updating with two examples from Japan and the Western Mediterranean, where we constrain smaller scale structure using full-waveform inversion. Furthermore, we demonstrate the ability of the CSEM to act as a vehicle for the combination of different tomographic techniques with a joint full-waveform and traveltime ray tomography of Europe. This combination broadens the exploitable frequency range of the individual techniques, thereby improving resolution. We perform two iterations of a whole-Earth full-waveform inversion using a long-period reference data set from 225 globally recorded earthquakes. At this early stage of the CSEM development, the broad global updates mostly act to remove artefacts from the assembly of the initial CSEM. During the future evolution of the CSEM, the reference data set will be used to account for the influence of small-scale refinements on large-scale global structure. The CSEM as a computational framework is intended to help bridging the gap between local, regional and global tomography, and to contribute to the development of a global multiscale Earth model. While the current construction serves as a first proof of concept, future refinements and additions will require community involvement, which is welcome at this stage already.

The mantle component of the Australian Seismological Reference Model (AuSREM) has been constructed from Australian-specific sources, primarily exploiting the wealth of seismic sources at regional distances around Australia recorded at portable and permanent stations on the continent. AuSREM is designed to bring together the existing information on Australia, from both body wave and surface wave studies and provide a synthesis in the form of a 3-D model that can provide the basis for future refinement. The model is grid based with a 0.5° sampling in latitude and longitude, and is designed to be fully interpolable, so that properties can be extracted at any point. For the upper mantle the primary source of information comes from seismic surface wave tomography, supplemented by analysis of body wave arrivals and regional tomography which provide useful constraints on the relation between P- and S-wave speeds in the mantle lithosphere. A representative model has been developed to capture the features of mantle structure drawing on a range of studies. The mantle structure is represented by grid values at 25 km intervals in depth from 75 to 300 km. Shallower structure is linked to the AuSREM crust through the recent Moho depth model of Kennett et al., which exploits all available sources of seismological information. Below 300 km depth and in the surrounding area AuSREM is linked to the S40RTS model of Ritsema et al.

SummaryThe comparison of seismograms plays a central role in seismology in diverse ways such as relative time-shifts, propagation effects between stations for a common source, and inversion for source or structural studies. Different measures for comparison have been used in the various situations, but all can be linked by the use of the concept of a transfer operator between a reference seismogram and a comparator trace. Transfer operators are implicit in various methods of phase velocity estimation, receiver functions and anisotropy studies, and measures for estimating arrival times and amplitude variations.Such transfer operators have a number of important roles; first they allow a visual assessment of the similarities of seismograms, secondly they provide a useful description of propagation effects for a common source in terms of the evolution from a reference station, and thirdly the transfer operator provides a means of representing seismogram differences in inversion without dominance by the largest amplitude arrivals.Whereas many time-domain measures of the degree of fit between an observed seismogram and the corresponding synthetic seismogram depend on the difference between the traces, which can be readily disturbed by minor misalignment, the transfer operator can readily represent a time offset while retaining a suitable measure of the similarities between the traces.The transfer operator concept can be applied with weighting or windowing of seismograms, and can be expressed in the time and frequency domains, or even in frequency time. This approach provides a means of representing and quantifying differences in the character of two seismograms that are visually apparent, in the time or frequency domain, but which get suppressed in any single measure of fit.We show how transfer operators can be usefully employed in many aspects of seismology with emphasis on frequency-domain representations at low frequency, and the time domain for higher frequency applications. We can express the general goal of inversion as the reduction of the transfer operator between observed and synthetic seismograms to the identity, thereby avoiding dominance by the largest arrivals and enhancing the influence of the full range of propagation processes. Broad classes of measures for comparison of times of arrival and amplitudes with quasi-linear properties can be constructed from the transfer operators through the use of a simple weighting function. This versatility highlights the unifying character of the transfer operator; and greatly simplifies the design of measurements targeted at specific aspects of the Earth's structure.

In a synthetic tomographic experiment, we succeeded to recover an idealized narrow mantle plume reaching deep into the lower mantle by using a misfit based on the instantaneous phase difference. A misfit based on simple cross-correlation traveltime shifts leaves the lower mantle part of the plume largely unresolved, despite the use of finite-frequency sensitivity kernels. The time-continuous and amplitude-independent instantaneous phase misfit allows us to measure the interaction between direct and diffracted waves as a function of time, which is difficult to capture by simple cross-correlation traveltime measurements. The diffracted waves arriving later than the main phase are essential to improve the tomographic result. The measurement of diffracted waves yields the necessary information to recover the plume correctly even in the lower mantle. The instantaneous phase measurement is ideal to capture this interaction, but other time- or frequency-dependent measurements may give similar results. We also investigated the effect of wavefront healing on cross-correlation traveltime shifts for a range of differently sized idealized mantle plumes.We confirm that wavefront healing severely reduces traveltime shifts when the plume conduit is considerably thinner than the width of the first Fresnel zone. For plume conduits with a diameter on the order of 100 km, even traveltime shifts measured at periods as short as T = 1 s are affected.

We propose a new method for the quantitative resolution analysis in full seismic waveform inversion that overcomes the limitations of classical synthetic inversions while being computationally more efficient and applicable to any misfit measure. The method rests on (1) the local quadratic approximation of the misfit functional in the vicinity of an optimal earth model, (2) the parametrization of the Hessian in terms of a parent function and its successive derivatives and (3) the computation of the space-dependent parameters via Fourier transforms of the Hessian, calculated with the help of adjoint techniques. In the simplest case of a Gaussian approximation, we can infer rigorously defined 3-D distributions of direction-dependent resolution lengths and the image distortion introduced by the tomographic method. We illustrate these concepts with a realistic full waveform inversion for upper-mantle structure beneath Europe. As a corollary to the method for resolution analysis, we propose several improvements to full waveform inversion techniques. These include a pre-conditioner for optimization schemes of the conjugate-gradient type, a new family of Newton-like methods, an approach to adaptive parametrization independent from ray theory and a strategy for objective functional design that aims at maximizing resolution. The computational requirements of our approach are less than for a typical synthetic inversion, but yield a much more complete picture of resolution and trade-offs. While the examples presented in this paper are rather specific, the underlying idea is very general. It allows for problem-dependent variations of the theme and for adaptations to exploration scenarios and other wave-equation-based tomography techniques that employ, for instance, georadar or microwave data

SUMMARY We present an extension of the adjoint method that allows us to compute the second derivatives of seismic data functionals with respect to Earth model parameters. This work is intended to serve as a technical prelude to the implementation of Newton-like optimization schemes and the development of quantitative resolution analyses in time-domain full seismic waveform inversion. The Hessian operator H applied to a model perturbation δm can be expressed in terms of four different wavefields. The forward field u excited by the regular source, the adjoint field u † excited by the adjoint source located at the receiver position, the scattered forward field δu and the scattered adjoint field δu † . The formalism naturally leads to the notion of Hessian kernels, which are the volumetric densities of H δm. The Hessian kernels appear as the superposition of (1) a first-order influence zone that represents the approximate Hessian, and (2) second-order influence zones that represent second-order scattering. To aid in the development of physical intuition we provide several examples of Hessian kernels for finite-frequency traveltime measurements on both surface and body waves. As expected, second-order scattering is efficient only when at least one of the model perturbations is located within the first Fresnel zone of the Fr´ echet kernel. Second-order effects from density heterogeneities are generally negligible in transmission tomography, provided that the Earth model is parameterized in terms of density and seismic wave speeds. With a realistic full waveform inversion for European upper-mantle structure, we demonstrate that significant differences can exist between the approximate Hessian and the full Hessian—despite the near-optimality of the tomographic model. These differences are largest for the off-diagonal elements, meaning that the approximate Hessian can lead to erroneous inferencesconcerningparametertrade-offs.ThefullHessian,incontrast,allowsustocorrectly account for the effect of non-linearity on model resolution.

SUMMARY We present a full seismic waveform tomography for upper-mantle structure in the Australasian region. Our method is based on spectral-element simulations of seismic wave propagation in 3-D heterogeneous earth models. The accurate solution of the forward problem ensures that waveform misfits are solely due to as yet undiscovered Earth structure and imprecise source descriptions, thus leading to more realistic tomographic images and source parameter estimates. To reduce the computational costs, we implement a long-wavelength equivalent crustal model. We quantify differences between the observed and the synthetic waveforms using time–frequency (TF) misfits. Their principal advantages are the separation of phase and amplitude misfits, the exploitation of complete waveform information and a quasi-linear relation to 3-D Earth structure. Fr´ echet kernels for the TF misfits are computed via the adjoint method. We propose a simple data compression scheme and an accuracy-adaptive time integration of the wavefields that allows us to reduce the storage requirements of the adjoint method by almost two orders of magnitude. To minimize the waveform phase misfit, we implement a pre-conditioned conjugate gradient algorithm. Amplitude information is incorporated indirectly by a restricted line search. This ensures that the cumulative envelope misfit does not increase during the inversion. An efficient pre-conditioner is found empirically through numerical experiments. It prevents the concentration of structural heterogeneity near the sources and receivers. We apply our waveform tomographic method to ≈1000 high-quality vertical-component seismograms, recorded in the Australasian region between 1993 and 2008. The waveforms comprise fundamental- and higher-mode surface and long-period S body waves in the period range from 50 to 200 s. To improve the convergence of the algorithm, we implement a 3-D initial model that contains the long-wavelength features of the Australasian region. Resolution tests indicate that our algorithm converges after around 10 iterations and that both long- and short-wavelength features in the uppermost mantle are well resolved. There is evidence for effects related to the non-linearity in the inversion procedure. After 11 iterations we fit the data waveforms acceptably well; with no significant further improvements to be expected. During the inversion the total fitted seismogram length increases by 46 per cent, providing a clear indication of the efficiency and consistency of the iterative optimization algorithm. The resulting SV -wave velocity model reveals structural features of the Australasian upper mantle with great detail. We confirm the existence of a pronounced low-velocity band along the eastern margin of the continent that can be clearly distinguished against Precambrian Australia and the microcontinental Lord Howe Rise. The transition from Precambrian to Phanerozoic Australia (the Tasman Line) appears to be sharp down to at least 200 km depth. It mostly occurs further east of where it is inferred from gravity and magnetic anomalies. Also clearly visible are the Archean and Proterozoic cratons, the northward continuation of the continent and anomalously low S-wave velocities in the upper mantle in central Australia.

SUMMARY 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 Frechet 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 Frechet 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.

SUMMARY We present a method for reducing the computational costs of numerical surface wave modelling. It is based on the smoothing of thin near-surface layers and discontinuities. Optimal smooth models are found via a constrained non-linear matching of dispersion curves in the period range of interest. The major advantages of our method are that it is independent of the numerical techniques employed and that it does not require modifications of pre-existing codes or meshes. It is, moreover, applicable in cases where the layer thickness is of the order of one wavelength and automatically yields estimates of the appropriateness of the smoothed model. Even though our analysis is based on 1-D media, we demonstrate with a numerical example that the dispersion curve matching can yield satisfactory results when it is applied regionally to laterally heterogeneous models. Also in that case it can lead to considerable reductions of the computational requirements.