Nonlinear Waveform Inversion Research Articles

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Reduced Order Model Based Nonlinear Waveform Inversion for the 1D Helmholtz Equation

Automated salt model building: From compaction trend to final velocity model using waveform inversion

Conventional seismic velocity model building in complicated salt-affected areas requires the explicit identification of salt boundaries in migrated images and typically involves testing of possible subsurface scenarios through multiple generations. The resulting velocity models are slow to generate and may contain interpreter-driven features that are difficult to verify. We show that it is possible to build a full final velocity model using advanced forms of full-waveform inversion applied directly to raw field data, starting from a model that contains only a simple 1D compaction trend. This approach rapidly generates the final velocity model and migrates processed reflection data at least as accurately as conventionally generated models. We demonstrate this methodology using an ocean-bottom-node data set acquired in deep water over Walker Ridge in the Gulf of Mexico. Our approach does not require exceptionally long offsets or the deployment of special low-frequency sources. We restrict the inversion so it does not use significant energy below 3 Hz or offsets longer than 14 km. We use three advanced forms of waveform inversion to recover the final model. The first is adaptive waveform inversion to proceed from models that begin far from the true model. The second is nonlinear reflection waveform inversion to recover subsalt velocity structure from reflections and their long-period multiples. The third is constrained waveform inversion to produce salt- and sediment-like velocity floods without explicitly identifying salt boundaries or velocities. In combination, these three algorithms successively improve the velocity model so it fully predicts the raw field data and accurately migrates primary reflections, though explicit migration forms no part of the workflow. Thus, model building via waveform inversion is able to proceed from field data to the final model in just a few weeks. It entirely avoids the many cycles of model rebuilding that may otherwise be required.

Recovery of the Alberta Basin Sedimentary Structures in Southern‐Central Alberta Using Nonlinear Inversions of Receiver Functions

AbstractSecondary converted waves from receiver functions are highly sensitive to physical properties below the Earth's surface. When modeled properly, the waveforms of converted waves offer direct constraints on the impedance contrast, depth, and P‐to‐S velocity ratio pertaining to sedimentary, crustal and mantle interfaces. In this study we introduce a nonlinear waveform inversion algorithm that matches the first 5 s of receiver functions recorded in the Alberta Basin within the Western Canada Sedimentary Basin (WCSB). Our algorithm searches for the optimal thickness of the sedimentary cover and shear velocities of appropriately selected layers within and below it. Combining inversions with forward simulations, we determine the supracrustal stratigraphy from 80 regional broadband seismic stations in the WCSB. The inverted models show east tapering sedimentary layers with their thicknesses ranging from ∼6 km beneath the foothills of the Canadian Rocky Mountains to 3–4 km beneath the Alberta Basin. This finding is consistent with the sedimentary strata determined from regional well‐logging data. The sedimentary layer contains low velocity zones of variable thicknesses and amplitudes that, depending on the locations, may be caused by mechanisms involving deposition, composition or deformation history. Our shear velocity models near the top of the basement complement the existing sonic‐logs or single component seismic data and offer new constraints on the subsidence history of the WCSB. The resolved range of depths (0–14 km) effectively bridges the gap between the vertical scales of well logging (0–6 km) and those of traditional broadband analysis (>10 km) involving receiver functions and surface waves.

Nonlinear Waveform Inversion for Quantitative Ultrasound

Due to its non-invasive and non-radiating nature, along with its low cost, ultrasound (US) imaging is widely used in medical applications. Typical B-mode US images have limited resolution and contrast and weak physical interpretation. Inverse US methods were developed to reconstruct the media's speed-of-sound (SoS) based on a linear acoustic model. However, the wave propagation in medical US is governed by nonlinear acoustics, which introduces more complex behaviors neglected in the linear model. In this work we propose a nonlinear waveform inversion (NWI) approach for quantitative US, that considers a nonlinear acoustics model to simultaneously reconstruct multiple material properties, including the medium's SoS, density, attenuation, and nonlinearity parameter. We thus broaden current inverse US approaches, such as the full waveform inversion (FWI) algorithm, by considering nonlinear media, and additional physical parameters. We represent the nonlinear acoustic model by means of a recurrent neural network, which enables us to apply advanced optimization algorithms borrowed from the deep learning toolbox and achieve more efficient reconstructions compared to the FWI method. We evaluate the performance of our approach on in-silico data and show that neglecting nonlinear effects may result in substantial degradation in the reconstruction, paving the way of NWI into clinical applications.

Data-driven prestack-waveform inversion using genetic algorithm: Methodology and examples

The choice of an initial model for seismic waveform inversion is important. In matured exploration areas with adequate well control, we can generate a suitable initial model using well information. However, in new areas where well control is sparse or unavailable, such an initial model is compromised and/or biased by the regions with more well controls. Even in matured exploration areas, if we use time-lapse seismic data to predict dynamic reservoir properties, an initial model that we obtain from the existing preproduction wells could be incorrect. In this work, we outline a new methodology and workflow for a nonlinear prestack isotropic elastic waveform inversion. We call this method a data-driven inversion, meaning that we derive the initial model entirely from the seismic data without using any well information. By assuming a locally horizontal stratification for every common midpoint and starting from the interval P-wave velocity, estimated entirely from seismic data, our method generates pseudowells by running a two-pass 1D isotropic elastic prestack-waveform inversion that uses the reflectivity method for forward modeling and the genetic algorithm for optimization. We then use the estimated pseudowells to build the initial model for seismic inversion. By applying this methodology to real seismic data from two different geologic settings, we determine the usefulness of our method. We believe that our new method is potentially applicable for subsurface characterization in areas where well information is sparse or unavailable. Additional research is, however, necessary to improve the computational efficiency of the methodology.

Hessian-based reflection waveform inversion: Theory and applications to synthetic and field data

Wave-equation reflection waveform inversion (RWI) is a promising method to reconstruct the background velocity model with reflection data. But it is difficult to precondition this highly nonlinear inverse problem for efficient convergence and reliable model updating. In the context of full-waveform inversion (FWI), the second-order derivative of the objective function with respect to the model parameters (i.e., the Hessian operator) can be used with the second-order optimization to overcome similar difficulties. To our knowledge, however, there are few such studies for RWI. This motivates us to derive and investigate the Hessian operator using the Born approximation in the context of RWI. For the same experimental setup with a toy model, we observe that the Hessian matrix of RWI exhibits a more compact structure than its counterpart in the context of FWI because they are constructed with the sensitivity kernels of very different spatial distribution features. Accordingly, we have developed a truncated Gauss-Newton RWI method consisting of two nested loops: An outer loop updates the background model in the manner of nonlinear waveform inversion, and an inner loop aims to iteratively estimate the background model update in a matrix-free manner by solving the Gauss-Newton equation. The workflow consists of alternating estimation of the perturbation and the background velocity model components, of which the former is done with a linear least-squares reverse time migration using the conjugate-gradient method. A synthetic example on the overthrust model and a field data example demonstrate that the Hessian-based RWI can significantly improve the estimation of intermediate-wavenumber components and promote velocity model building and seismic imaging with precritical reflection data.

Nonlinear Moment-Tensor Inversion of Repetitive Long-Periods Events Recorded at Pacaya Volcano, Guatemala

Detailed models of low-frequency seismicity at volcanoes provide insights into conduit structure and dynamics of magmatic systems. We examine explosion-related long-period (LP) events from Pacaya volcano, Guatemala, that were recorded during a temporary installation of four broadband seismic stations from October 2013 to November 2013. The repetitive LP events are identified with the aid of infrasound measurements using a matched filter due to the high level of background tremor and the small magnitude of the recorded events. We derive a representative seismic signal from the phase-weighted stack of 8,587 of these similar events, and invert for a source moment tensor. To address the limitations posed by the limited number of stations of the local network, we employ a non-linear waveform inversion that uses a grid search for source type to obtain a quantitative measure of the source mechanism reliability. With only four stations, Pacaya represents a case of limited observational data, where a quantitative description of moment tensor uncertainty is needed before any interpretation is to be attempted. Results point to a shallow source mechanism somewhat like a tension crack, dipping ~ 40° to the east, consistent with the dominant E-W motion in the seismic records from stations west, north, and east of the source. The uncertainties determined from the non-linear inversion are not insignificant, but clearly constrain the mechanism to be a source dominated by isotropic components

Characterization of a viscoelastic heterogeneous object with an effective model by nonlinear full waveform inversion

The determination of equivalent viscoelastic properties of heterogeneous objects remains challenging in various scientific fields such as (geo)mechanics, geophysics or biomechanics. The present investigation addresses the issue of the identification of effective constitutive properties of a binary object by using a nonlinear and full waveform inversion scheme. The inversion process, without any regularization technique or a priori information, aims at minimizing directly the discrepancy between the full waveform responses of a bi-material viscoelastic cylindrical object and its corresponding effective homogeneous object. It involves the retrieval of five constitutive equivalent parameters. Numerical simulations are performed in a laboratory-scale two-dimensional configuration: a transient acoustic plane wave impacts the object and the diffracted fluid pressure, solid stress or velocity component fields are determined using a semi-analytical approach. Results show that the retrieval of the density and of the real parts of both the compressional and the shear wave velocities have been carried out successfully regarding the number and location of sensors, the type of sensors, the size of the searching space, the frequency range of the incident plane pressure wave, and the change in the geometric or mechanical constitution of the bi-material object. The retrieval of the imaginary parts of the wave velocities can reveal in some cases the limitations of the proposed approach.

Waveform inversion of SS precursors: An investigation of the northwestern Pacific subduction zones and intraplate volcanoes in China

The arrival time and amplitude of underside reflections from mantle seismic discontinuities (SS precursors) have made major contributions to the understanding of mantle composition and dynamics. In this study, we introduce a nonlinear waveform inversion technique to simultaneously constrain shear velocities and discontinuity depths beneath the northwestern Pacific subduction system. Based exclusively on a large SS precursor waveform dataset, we are able to clearly delineate the morphology of the descending Pacific plate, which flattens at the base of the upper mantle and extends westward by ~1500km toward northern-central China. Our grid search yields the maximum correlation between shear velocity and transition zone thickness at an angle of ~30°, consistent with the reported average slab dip beneath the study region. The strongly positive correlation suggests predominantly thermal, rather than compositional, variations along the descending Pacific plate. The joint depth-velocity solution also shows a 5–10km depression of the 410km discontinuity and an average decrease of 1.2% in upper mantle shear velocity beneath the intraplate volcanic fields in northeastern China. This anomaly, which reaches the middle of the upper mantle transition zone beneath the Changbai hotspot, initiates at a significantly shallower (~320km) depth beneath the Wudalianchi region. High amplitude reflections at depths greater than 410km suggest a water-poor melt layer in possible association with 1) decompression melting from passive upwelling and/or 2) active upwelling through a slab window.

Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T-matrix formalism for wave-equation tomography

We derived the renormalized nonlinear sensitivity operator and the related inverse thin-slab propagator (ITSP) for nonlinear tomographic waveform inversion based on the theory of nonlinear partial derivative operator and its De Wolf approximation. The inverse propagator is based on a renormalization procedure to the forward and inverse transition matrix scattering series. The ITSP eliminates the divergence of the inverse Born series for strong perturbations by stepwise partial summation (renormalization). Numerical tests showed that the inverse Born T-series starts to diverge at moderate perturbation (20% for the given model of Gaussian ball with a radius of 5 wavelength), while the ITSP has no divergence problem for any strong perturbations (up to 100% perturbation for test model). In addition, the ITSP is a non-iterative, marching algorithm with only one sweep, and therefore very efficient in comparison with the iterative inversion based on the inverse-Born scattering series. This convergence and efficiency improvement has potential applications to the iterative procedure of waveform inversion.

Full waveform inversion in the frequency domain using direct iterative T-matrix methods

We present two direct iterative solutions to the nonlinear seismic waveform inversion problem that are based on volume integral equation methods for seismic forward modelling in the acoustic approximation. The solutions are presented in the frequency domain, where accurate inversion results can often be obtained using a relatively low number of frequency components. Our inverse scattering approach effectively replaces an ill-posed nonlinear inverse problem with a series of linear ill-posed inverse problems, for which there already exist efficient (regularized) solution methods. Both these solutions update the wavefield within the scattering domain after each iteration. The main difference is that the background medium Green functions are kept fixed in the first solution, but updated after each iteration in the second solution. This means that our solutions are very similar to the Born iterative (BI) and the distorted Born iterative (DBI) methods that are commonly used in acoustic and electromagnetic inverse scattering. However, we have eliminated the need to perform a full forward simulation (or to invert a huge matrix) at each iteration via the use of an iterative T-matrix method for fixed background media for the BI method and a variational T-matrix method for dynamic background media for the DBI method. The T-matrix (variation) is linearly related with the seismic wavefield data (residuals), but related with the unknown scattering potential model parameter (updates) in a non-linear manner, which is independent of the source-receiver configuration. This mathematical structure, which allows one to peel off the effects of the source-receiver configuration, is very attractive when dealing with multiple (simultaneous) sources, and is also compatible with the (future) use of renormalization methods for dealing with local minima problems. To illustrate the performance and potential of the two direct iterative methods for FWI, we performed a series of numerical experiments on synthetic seismic waveform data associated with a simple 2D model and the more complicated Marmousi model. The results of these numerical experiments suggest that the use of a fixed (e.g. smooth and ray-tracing friendly) background medium may be adequate for some applications with moderately large velocity contrasts, but the solution based on a dynamic (non-smooth and constantly updated) background medium will normally provide superiour inversion results; also in the case of low signal-to-noise ratios.

Intricate heterogeneous structures of the top 300 km of the Earth's inner core inferred from global array data: II. Frequency dependence of inner core attenuation and its implication

We apply a nonlinear waveform inversion method that we developed previously (Iritani et al., 2010, 2014) to broadband (0.02–2 Hz) core phase data recorded by global seismic arrays in order to investigate the frequency dependence of inner core attenuation. The results show a prominent difference of the frequency dependence of attenuation between the eastern hemisphere and the western hemisphere: while attenuation is frequency dependent in the western hemisphere, not so in the eastern hemisphere. Moreover, in the western hemisphere, a stronger frequency dependence of the attenuation is observed beneath Africa compared to beneath North America. This intricate quasi-hemispherical property of the frequency dependence of the inner core attenuation, together with the quasi-hemispherical variability of seismic velocity and attenuation strength at around 1 Hz that we reported earlier, can be consistently understood in terms of the relative location of the seismic observation frequency band and the absorption band peaks that are different for regions. The variability of the absorption band peak may reflect the variability of the iron grain size of the inner core: the grain size is larger in the eastern hemisphere than the western hemisphere, and in the western hemisphere beneath North America, it gradually increases with depth. However, caution should be taken in this inference as the physical mechanism of attenuation in the inner core is still poorly constrained both theoretically and observationally.

Cauchy prior distribution-based AVO elastic parameter estimation via weakly nonlinear waveform inversion

Cauchy priori distribution-based Bayesian AVO reflectivity inversion may lead to sparse estimates that are sensitive to large reflectivities. For the inversion, the computation of the covariance matrix and regularized terms requires prior estimation of model parameters, which makes the iterative inversion weakly nonlinear. At the same time, the relations among the model parameters are assumed linear. Furthermore, the reflectivities, the results of the inversion, or the elastic parameters with cumulative error recovered by integrating reflectivities are not well suited for detecting hydrocarbons and fluids. In contrast, in Bayesian linear AVO inversion, the elastic parameters can be directly extracted from prestack seismic data without linear assumptions for the model parameters. Considering the advantages of the abovementioned methods, the Bayesian AVO reflectivity inversion process is modified and Cauchy distribution is explored as a prior probability distribution and the time-variant covariance is also considered. Finally, we propose a new method for the weakly nonlinear AVO waveform inversion. Furthermore, the linear assumptions are abandoned and elastic parameters, such as P-wave velocity, S-wave velocity, and density, can be directly recovered from seismic data especially for interfaces with large reflectivities. Numerical analysis demonstrates that all the elastic parameters can be estimated from prestack seismic data even when the signal-to-noise ratio of the seismic data is low.

Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation

Multiparameter full waveform inversion (FWI) is a challenging quantitative seismic imaging method for lithological characterization and reservoir monitoring. The difficulties in multiparameter FWI arise from the variable influence of the different parameter classes on the phase and amplitude of the data, and the trade-off between these. In this framework, choosing a suitable parametrization of the subsurface and designing the suitable FWI workflow are two key methodological issues in non-linear waveform inversion. We assess frequency-domain visco-acoustic FWI to reconstruct the compressive velocity (VP), the density (ρ) or the impedance (IP) and the quality factor (QP), from the hydrophone component, using a synthetic data set that is representative of the Valhall oil field in the North Sea. We first assess which of the (VP, ρ) and (VP, IP) parametrizations provides the most reliable FWI results when dealing with wide-aperture data. Contrary to widely accepted ideas, we show that the (VP, ρ) parametrization allows a better reconstruction of both the VP, ρ and IP parameters, first because it favours the broad-band reconstruction of the dominant VP parameter, and secondly because the trade-off effects between velocity and density at short-to-intermediate scattering angles can be removed by multiplication, to build an impedance model. This allows for the matching of the reflection amplitudes, while the broad-band velocity model accurately describes the kinematic attributes of both the diving waves and reflections. Then, we assess different inversion strategies to recover the quality factor QP, in addition to parameters VP and ρ. A difficulty related to attenuation estimation arises because, on the one hand the values of QP are on average one order of magnitude smaller than those of VP and ρ, and on the other hands model perturbations relative to the starting models can be much higher for QP than for VP and ρ during FWI. In this framework, we show that an empirical tuning of the FWI regularization, which is adapted to each parameter class, is a key issue to correctly account for the attenuation in the inversion. We promote a hierarchical approach where the dominant parameter VP is reconstructed first from the full data set (i.e. without any data preconditioning) to build a velocity model as kinematically accurate as possible before performing the joint update of the three parameter classes during a second step. This hierarchical imaging of compressive wave speed, density and attenuation is applied to a real wide-aperture ocean-bottom-cable data set from the Valhall oil field. Several geological features, such as accumulation of gas below barriers of claystone and soft quaternary sediment are interpreted in the FWI models of density and attenuation. The models of VP, ρ and QP that have been developed by visco-acoustic FWI of the hydrophone data can be used as initial models to perform visco-elastic FWI of the geophone data for the joint update of the compressive and shear wave speeds.

A Nonlinear Method to Estimate Source Parameters, Amplitude, and Travel Times of Teleseismic Body Waves

Abstract We present a nonlinear waveform inversion method to determine source parameters under point‐source approximation (depth, source time function, centroid moment tensor, and seismic moment) and also travel times and amplitudes of body waves from teleseismic records. We demonstrate that our source time function estimates significantly improve correlation between observed and synthetics broadband seismograms. The scaling of source dimensions with seismic moment is similar for deep and shallow events in the Japanese subduction area for moment magnitudes between 6.1 and 7.5. We compare event depths and moment tensors obtained with our method to those reported by different institutions for a set of Japanese earthquakes. This comparison reveals that our method gives hypocentral depths closer to those obtained with regional arrays than other teleseismic methods. The seismic moment is properly recovered, but data weighting strongly influences the moment tensor solution. Focal mechanisms are biased by high‐frequency body‐wave amplitude variations due to crustal structure below the stations and by errors on the instrument responses. Body‐wave differential travel times are improved compared with phase‐picking measurements. The high‐quality travel times will be exploited to obtain more precise event locations and higher resolution tomographic models. Online Material: Coordinates of Japanese earthquakes processed in this study.

Seismic attenuation imaging with causality

Seismic data enable imaging of the Earth, not only of velocity and density but also of attenuation contrasts. Unfortunately, the Born approximation of the constant-density visco-acoustic wave equation, which can serve as a forward modelling operator related to seismic migration, exhibits an ambiguity when attenuation is included. Different scattering models involving velocity and attenuation perturbations may provide nearly identical data. This result was obtained earlier for scatterers that did not contain a correction term for causality. Such a term leads to dispersion when considering a range of frequencies. We demonstrate that with this term, linearized inversion or iterative migration will almost, but not fully, remove the ambiguity. We also investigate if attenuation imaging suffers from the same ambiguity when using non-linear or full waveform inversion. A numerical experiment shows that non-linear inversion with causality convergences to the true model, whereas without causality, a substantial difference with the true model remains even after a very large number of iterations. For both linearized and non-linear inversion, the initial update in a gradient-based optimization scheme that minimizes the difference between modelled and observed data is still affected by the ambiguity and does not provide a good result. This first update corresponds to a classic migration operation. In our numerical experiments, the reconstructed model started to approximate the true model only after a large number of iterations.

Automated multimode phase speed measurements for high-resolution regional-scale tomography: application to North America

SUMMARY A fully automated method for obtaining multimode phase speed measurements from a single seismogram has been developed and applied to a large data set of three-component longperiod seismograms in North America, constructing high-resolution phase speed maps on a continental scale. The method of our phase speed estimation is based on a fully non-linear waveform inversion by Yoshizawa & Kennett working with a global search method (the Neighbourhood Algorithm). The entire process of waveform fitting and the evaluation of the estimated phase speed have been fully automated employing several empirical quantitative measures, assessing the quality of waveform fit and the relative contributions of each mode in a chosen time window. The measured phase speed data undergo automatic screening for quality control, comprising the threshold evaluation of their reliability and outlier detection and removal. This new automated method has been applied to a large data set recorded at North American stations, including the latest transportable stations of USArray. Using long-period three-component seismograms recorded during the past eight years, we have successfully retrieved large numbers of regional surface wave paths, including over 20000 paths for the fundamental-mode Rayleigh waves over a wide range of frequencies, and over 10000 paths for the higher mode Rayleigh as well as the fundamental-mode Love waves. The consistentresultsoftheautomatedmeasurementproceduresuggestthatthemethodworkswell at regional distances, allowing us to perform a high-resolution mapping of multimode phase speeds in North America. The results of the automated waveform analysis also indicate some intrinsic limitations in the higher mode phase speed measurements from a single seismogram particularly in the short period range, mainly due to the overlapping of higher mode arrivals as well as coupling between mode branches. Despite such an innate difficulty in the higher mode dispersion measurements, the automated method allows us to construct reliable multimode phase speed maps. The current data set of ray paths is significantly biased towards the western half of North America, resulting in non-uniform horizontal resolution across the continent. This issue will be resolved by the future migration of the USArray stations to cover the central and eastern United States. The new automated method can be a useful tool for high-resolution mapping of regional 3-D shear wave structure including possible anisotropy.

Migration velocity analysis and waveform inversion

ABSTRACTLeast‐squares inversion of seismic reflection waveform data can reconstruct remarkably detailed models of subsurface structure and take into account essentially any physics of seismic wave propagation that can be modelled. However, the waveform inversion objective has many spurious local minima, hence convergence of descent methods (mandatory because of problem size) to useful Earth models requires accurate initial estimates of long‐scale velocity structure. Migration velocity analysis, on the other hand, is capable of correcting substantially erroneous initial estimates of velocity at long scales. Migration velocity analysis is based on prestack depth migration, which is in turn based on linearized acoustic modelling (Born or single‐scattering approximation). Two major variants of prestack depth migration, using binning of surface data and Claerbout's survey‐sinking concept respectively, are in widespread use. Each type of prestack migration produces an image volume depending on redundant parameters and supplies a condition on the image volume, which expresses consistency between data and velocity model and is hence a basis for velocity analysis. The survey‐sinking (depth‐oriented) approach to prestack migration is less subject to kinematic artefacts than is the binning‐based (surface‐oriented) approach. Because kinematic artefacts strongly violate the consistency or semblance conditions, this observation suggests that velocity analysis based on depth‐oriented prestack migration may be more appropriate in kinematically complex areas. Appropriate choice of objective (differential semblance) turns either form of migration velocity analysis into an optimization problem, for which Newton‐like methods exhibit little tendency to stagnate at nonglobal minima. The extended modelling concept links migration velocity analysis to the apparently unrelated waveform inversion approach to estimation of Earth structure: from this point of view, migration velocity analysis is a solution method for the linearized waveform inversion problem. Extended modelling also provides a basis for a nonlinear generalization of migration velocity analysis. Preliminary numerical evidence suggests a new approach to nonlinear waveform inversion, which may combine the global convergence of velocity analysis with the physical fidelity of model‐based data fitting.

Surface wave tomography of North America and the Caribbean using global and regional broad-band networks: phase velocity maps and limitations of ray theory

SUMMARY We present phase velocity maps of fundamental mode Rayleigh waves across the North American and Caribbean plates. Our data set consists of 1846 waveforms from 172 events recorded at 91 broad-band stations operating in North America. We compute phase velocity maps in four narrow period bands between 50 and 150 s using a non-linear waveform inversion method that solves for phase velocity perturbations relative to a reference Earth model (PREM). Our results show a strong velocity contrast between high velocities beneath the stable North American craton, and lower velocities in the tectonically active western margin, in agreement with other regional and global surface wave tomography studies. We perform detailed comparisons with global model results, which display good agreement between phase velocity maps in the location and amplitude of the anomalies. However, forward modelling shows that regional maps are more accurate for predicting waveforms. In addition, at long periods, the amplitude of the velocity anomalies imaged in our regional phase velocity maps is three time larger than in global phase velocity models. This amplitude factor is necessary to explain the data accurately, showing that regional models provide a better image of velocity structures. Synthetic tests show that the raypath coverage used in this study enables one to resolve velocity features of the order of 800–1000 km. However, only larger length-scale features are observed in the phase velocity maps. The limitation in resolution of our maps can be attributed to the wave propagation theory used in the inversion. Ray theory does not account for off-great-circle ray propagation effects, such as ray bending or scattering. For wavelengths less than 1000 km, scattering effects are significant and may need to be considered.

Non-linear waveform inversion for surface waves with a neighbourhood algorithm-application to multimode dispersion measurements

SUMMARY A new technique for multimode dispersion measurement has been developed by employing fully non-linear waveform inversion for a path-specific 1-D profile using a neighbourhood algorithm (NA). One-dimensional models derived from waveform inversion are quite sensitive to the model parametrization and the reference model used to start the inversion. With different approaches to the parametrization of the shear wave speed profile, we can find models with significant differences in velocity variation with depth, which provide similar levels of fit to the observed waveforms. Although the models differ, the calculated phase dispersion for the first few modes of the surface waves are very close indeed. We therefore regard the 1-D models derived from the multimode waveform inversion as an implicit description of the path-specific dispersion for each of the modes. The inversion procedure using NA samples a substantial number of models and we select the 1-D wave speed model that achieves minimum misfit. Phase speeds for each mode branch are then calculated from the model parameters with the minimum misfit. Because we use the 1-D models as a summary of the average multimode dispersion along the path, we do not need to make assumptions concerning the nature of anisotropy. We can employ simple isotropic models and the method can be applied to both Rayleigh and Love waves independently. The new method is applied to sets of paths in the Australian region and provides stable measurements of multimode dispersion from a single seismogram. The multimode phase speeds measured using the NA inversion can be used to retrieve phase velocity maps as a function of frequency for higher-mode branches as well as the fundamental mode, which will be crucial constraints on 3-D Earth models.