Regional tomographic inversion of the amplitude and phase of Rayleigh waves with 2-D sensitivity kernels (original) (raw)
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Non‐linear inversion of scattered seismic surface waves
Geophysical Journal …, 2007
Seismic surface wave analysis has provided important insight in the Earth's crustal and upper mantle structure and has recently become a standard tool in geotechnical engineering. Most current surface wave inversion methods are aimed at recovering near-surface (shear) velocity profiles from dispersion curves, assuming a (smoothly varying) horizontally layered Earth. In some cases, however, one is interested in the location, strength or shape of local heterogeneities in the shallow subsurface. In this paper we focus on estimating the strength of near-surface heterogeneity from scattered surface waves. This is a non-linear inversion problem as the wavefield in the scatterer also depends on the contrast. For this reason the inversion is cast as an optimization problem in which we minimize the difference between the observed data and the modelled scattered field. The minimization problem is solved using a conjugate gradient algorithm. To accurately estimate the contrast we account for non-linear interactions such as multiple scattering within the scattering domain in the forward problem. To do so, we use a domain-type integral representation to express the near-surface scattered wavefield, which is solved using the method of moments. The entire inversion is carried out in the frequency domain. This way, we may take advantage of the fact that the decay of surface waves with depth depends on frequency. We study 3-D sensitivity kernels for the given inversion problem and observe that sensitivity of the wavefield with respect to heterogeneity in depth depends on frequency in the same way. We therefore, expect to be able to constrain heterogeneity in depth. A numerical example illustrates this and we conclude that it is in principle possible to reliably resolve heterogeneities and estimate their strengths. We compare the results from our algorithm to a similar but more efficient inversion scheme based on the Born approximation and show that, for the shallowest heterogeneities, this inversion can also recover the contrast well.
Alternative waveform inversion for surface wave analysis in 2-D media
Geophysical Journal International, 2014
In the context of near surface seismic imaging (a few hundreds of metres), we propose an alternative approach for inversion of surface waves in 2-D media with laterally varying velocities. It is based on Full Waveform Inversion (FWI) but using an alternative objective function formulated in the frequency-wavenumber f − k domain. The classical FWI objective function suffers from severe local minima problems in the presence of surface waves. It thus requires a very accurate initial model. The proposed objective function is similar to the one used in classical surface wave analysis. In this approach, the data are first split using sliding windows in the time-space t − x domain. For each window, the amplitude of the f − k spectrum is computed. The objective function measures the least-squares misfit between the amplitude of observed and modelled 2-D Fourier transformed data sets. We call this formulation the windowed-amplitude waveform inversion (w-AWI). The w-AWI objective function reduces some local minima problems as shown here through numerical examples. The global minimum basin is wider in the w-AWI approach than in FWI. Synthetic examples show that w-AWI may achieve convergence if the lowest data frequency content is twice higher than the one needed by FWI. For elastic inversion, w-AWI can be used to reconstruct a velocity model explaining surface waves. This surface wave inversion procedure can be used to retrieve near-surface model parameters in lateral-varying media.
Diffraction tomography using multimode surface waves
Journal of Geophysical Research, 1997
A new method is described that makes it feasible to include scattered and converted surface waves into waveform inversions for the three-dimensional (3-D) structure of the Earth. The single scattering (Born) approximation forms the basis of the method. In order to minimize the amplitude of the scattered wave field, the background model is first adapted to correct for nonconverted, forward-scattered wave energy. We then perform Born inversion of the difference between the measured and synthetic waveforms, including a suite of Love and l•ayleigh modes. The Born approximation yields linear equations of the form A•?: •u Børn, which allow the determination of the three-dimensional perturbations ? to the background model from the scattered wave field •u Bøm. This procedure is followed separately for each source-receiver pair to allow for optimized background models for each signal, as well as to minimize the computational burden. We winnow the data vector for each path by performing singular value decomposition using a diagonalization of AA T.
From surface wave inversion to seismic site response prediction: Beyond the 1D approach
Soil Dynamics and …, 2012
Surface wave methods consist of the extraction and inversion of the Rayleigh wave phase-velocity dispersion curve to recover the (usually 1D) shear-wave velocity profile. In the literature, uncertainty due to data error has not received much attention, but the discussion about uncertainty due to model error is even poorer. Even with an unrealistic noise-free dataset and an exact forward model, an inappropriate parameterization can generate solutions very far from the actual soil structure. In general, the model used for the dispersion curve interpretation is 1D. Hence, when the velocity distribution is laterally heterogeneous, model errors can have significant consequences on the reliability of the resulting shear-wave velocity distribution. From a poor velocity reconstruction, an unsatisfactory, and often dangerous site response analysis follows. In fact, shear wave measurements play a relevant role in seismic ground motion amplification estimation. In this paper, we discuss the possibility of processing the seismograms using a multi-offset phase analysis (MOPA), in order to derive soil elastic parameters for weak motion predictions. This technique allows the detection and location of the lateral discontinuities, and a better model parameterization. In fact, once the discontinuities are identified, we can split the profile into several, truly 1D, parts. The use of the standard 1D dispersion curve extraction and inversion for each side of the heterogeneity generates velocity profiles that we can put side by side to get correct 2D reconstructions of the shear-wave distributions. From 2D velocity reconstruction, we can calculate the site response that may be significantly different from the site response generated from a traditional 1D analysis of the same seismograms. In this work, we discuss the site responses of two synthetic examples with lateral heterogeneities. We show how misleading a 1D analysis may be if applied to a truly 2D velocity distribution, particularly in terms of site response prediction.
2009
We propose a novel technique for seismic waveform tomography on continental scales. This is based on the fully numerical simulation of wave propagation in complex Earth models, the inversion of complete waveforms and the quantification of the waveform discrepancies through a specially designed phase misfit. The numerical solution of the equations of motion allows us to overcome the limitations of ray theory and of finite normal mode summations. Thus, we can expect the tomographic models to be more realistic and physically consistent. Moreover, inverting entire waveforms reduces the non-uniqueness of the tomographic problem. Following the theoretical descriptions of the forward and inverse problem solutions, we present preliminary results for the upper mantle structure in the Australasian region.
Near surface 2D Full waveform inversion using seismic refraction data.
The evaluation of the shallow soil parameters happens to have a huge importance for earthquake engineering purposes. We proposed a new method to estimate the 2D inhomogeneous Shear wave velocity profile of shallow soils using a waveform inversion of P-SV refraction data. The numerical part of this method is based on a 2.5D finite difference staggered grid materializing the propagation wave field, a vertical point source simulates a plank hammering which generates the P-SV waves. In data processing before the inversion, a waveform deconvolution was applied to the refracted data to get rid of the source influence. The algorithm used for the inversion is the Hybrid Heuristic Search method proposed by Yamanaka (2007). Numerical experiments were conducted using synthetic observed waves. The inverted results shows that we succeeded to reconstruct a 2D soil profile with irregular layer interface and a soil model with a blind layer, which is impossible to detect with the conventional seismic refraction method. This approach is inexpensive and allows obtaining more information about the soil structure using little number of receivers.
Seismic-velocity inversion using surface-wave tomography
SEG Technical Program Expanded Abstracts 2018, 2018
In this paper, we present the workflow of the application of surface wave tomography to characterize the near surface. A near surface model is simulated with a shallow unconsolidated layer that has a variable thickness and a deep consolidated zone. Data is filtered by bandpass filters and surface wave phase arrival times are picked, and surface wave velocities are inverted using a ray theoretical tomographic approach. A good agreement is found between the inverted results and the original model. In another model, a thin fault zone is detected. Agreement between the tomographic results and the model suggests that surface wave tomography is well suited for characterizing shallow geology with the capability of identifying low velocity anomalies and general lateral variations.
Elastic Full Waveform Inversion of Near-Surface Seismic Data Incorporating Topography
2017
Elastic full waveform inversion (FWI) is an imaging tool that can yield subsurface models of seismic velocities and density at sub-wavelength resolution. For near-surface applications (tens to hundreds of metres depth penetration), FWI is particularly valuable, because it requires no separation of different seismic phases, such as direct waves, reflections and surface waves, which is a difficult task at this scale. In contrast to conventional methods of seismic data analysis, FWI utilises and interprets the full wavefield. However, real data applications are still scarce. This is due to (i) the non-linearity of the inversion problem, (ii) the high computational costs and (iii) systematic errors that are not taken care of by the FWI algorithm. Although considerable progress has been made during the past few years, there are still a number of issues that remain to be resolved. In my thesis I have tackled three of these problems. Surface waves often dominate shallow seismic data. With their high amplitudes they dominate the misfit functional and control the model update. Due to their limited depth penetration they are mainly sensitive to shallow parts, such that model updates at greater depth are often very small. In order to balance sensitivities and to increase model updates at depth, I have introduced a novel scaling technique and I have demonstrated its efficiency on synthetic models of varying complexity. The scaling technique involves normalising the squared column sums of the Jacobian matrix prior to adding regularisation and updating the model. This leads to significantly improved velocity images at depth. Although the technique is introduced on near-surface FWI, it is rather general and can be applied to all kind of geophysical inversion problems such as the inversion of geoelectric or electromagnetic data. Investigating unstable slopes threatened by landslides is a typical near-surface application among many others, where significant topographic undulations are present. It is inevitable to account for such topography in FWI. Through the adaption of SPECFEM2D, a well-established forward solver incorporating irregular grids, I have made it possible IV Abstract to run FWI on profiles featuring arbitrary surface topography. I have demonstrated the capability to handle considerable topography in the presence of a complex subsurface model including stochastic fluctuations and several block anomalies. Furthermore, I have investigated the effects of neglecting such topography during inversion. It has turned out that topographic undulations with wavelengths or amplitudes similar to the minimum seismic wavelengths have a detrimental effect on model reconstruction. Seismic survey setups are typically governed by the needs of reflection seismology processing, that is, high fold and dense spatial sampling are required. Using tools of experimental design I have optimised the survey setup for the needs of FWI. I have established a clear recipe consisting of the following points: (i) use horizontally directed sources; (ii) multi-component geophones clearly outperform single-component receivers; (iii) a receiver spacing in the order of the minimum seismic wavelength is sufficient; (iv) the sources employed can be reduced to a few well-selected positions. In this way the costs of a survey can be drastically reduced while the quality of the obtained subsurface images is only slightly affected. The topics addressed in my thesis shall be a step forward towards successful and efficient FWI of real data. It is anticipated that in a foreseeable future FWI will become a standard tool for the analysis of near-surface seismic data. V Zusammenfassung Elastische Wellenfeld-Inversion (WFI) ist eine bildgebende Methode, mit welcher man eine Auflösung kleiner als die minimale seismische Wellenlänge erzielen kann. Für Anwendungen im Bereich der nahen Oberfläche (bis zu einigen hundert Metern Eindringtiefe) ist WFI besonders nützlich, weil die verschiedenen seismischen Phasen nicht separiert werden müssen, was sich auf dieser Skala schwierig gestalten würde. Im Gegensatz zu herkömmlichen Analyse-Methoden seismischer Daten verwendet und interpretiert WFI das gesamte Wellenfeld. Trotzdem gibt es bisher nur wenige Studien mit echten Daten. Die Gründe dafür liegen (i) in der Nichtlinearität des Inversionsproblems, (ii) im grossen rechnerischen Aufwand und (iii) in systematischen Abweichungen, die vom WFI Algorithmus nicht berücksichtigt werden. Obwohl in den letzten Jahren grosse Fortschritte erzielt wurden, bleiben einige Fragen offen. In meiner Doktorarbeit möchte ich drei dieser Probleme angehen. Seismogramme der nahen Oberfläche werden oft von Oberflächenwellen dominiert. Mit ihren hohen Amplituden dominieren sie die Misfit-Funktion und kontrollieren den Modell-Update. Wegen ihrer limitierten Penetrationstiefe wird vor allem die nahe Oberfläche aufgelöst während der tiefere Untergrund verborgen bleibt. Um die Sensitivitäten auszugleichen und den Modell-Update in der Tiefe zu vergrössen habe ich eine neue Skalierungsmethode eingeführt und deren Effizienz an verschiedenen synthetischen Modellen demonstriert. Die Skalierungsmethode basiert auf Normalisierung der quadratischen Spaltensummen der Jacobi-Matrix bevor Regularisierungsterme dazuaddiert werden und das Modell neu aufgesetzt wird. So wird die Auflösung der seismischen Geschwindigkeiten in der Tiefe massgeblich verbessert. Die Skalierungsmethode wird hier anhand eines WFI-Beispiels eingeführt, kann aber auf jegliche Inversionsprobleme angewandt werden, wie zum Beispiel die Inversion elektromagnetischer oder geoelektrischer Daten. VI Zusammenfassung Untersuchungen eines instabilen Hangs, der von Erdrutschen bedroht wird, sind nur ein Beispiel unter vielen, bei welchem Oberflächen-Topographie eine grosse Rolle spielt. Es ist unumgänglich, solche Topographie bei der WFI zu berücksichtigen. Durch die Adaption von SPECFEM2D, einem etablierten Vorwärtslöser für unregelmässige Gitter, habe ich es ermöglicht, willkürliche Topographie zu berücksichtigen. Dies demonstriere ich an einem komplexen Untergrund-Modell mit beträchtlicher Topographie und mit stochastischen Fluktuationen und mehreren Block-Anomalien. Des Weiteren untersuche ich die Effekte, wenn Topographie vernachlässigt wird. Es hat sich gezeigt, dass Topographie mit einer Wellenläge oder Amplitude ähnlich wie die minimale seismische Wellenlänge nicht vernachlässigt werden sollte. Der Aufbau seismischer Messungen wird typischerweise an die Anforderungen der Reflexionsseismik angepasst, sprich, es wird eine hohe Dichte an Geophonen aufgewendet. Ich habe den Aufbau optimiert für die Anforderungen der WFI. Ich habe ein klares Rezept hergeleitet, bestehend aus folgenden Punkten: (i) horizontale Quellen sollen verwendet werden; (ii) Multi-Komponenten-Geophone liefern bedeutend bessere Resultate als Ein-Komponenten-Geophone; (iii) die Geophon-Abstände sollen ungefähr einer seismische Wellenlänge entsprechen; (iv) es reicht, wenn Quellen nur an wenigen, ausgewählten Standorten verwendet werden. So können drastisch Kosten gespart werden, während die Qualität der erhaltenen Abbildungen des Untergrundes nur wenig beeinträchtigt wird. Die Themen, die ich in meiner Doktorarbeit behandle, sollen ein Schritt sein in Richtung erfolgreicher und effizienter WFI echter Daten. In meinen Augen ist es absehbar, dass WFI zur Standardmethode wird für die Analyse seismischer Daten der nahen Oberfläche. There are three basic variants on the seismic method that exploit different wave types. In reflection seismic exploration the phases reflected at layer interfaces (Fig. 1.1, yellow arrow) are exploited and interpreted to delineate impedance contrasts in the subsurface (e.g., Yilmaz, 2001). In refraction seismic exploration (or travel time tomography), the first arrivals due to direct, critically refracted and / or diving waves (Fig. 1.1, dispersion curves. The method is therefore mainly suitable for layered media. If there are variations in the second or third direction, pseudo-2D or-3D methods are applied, i.e., 1-D profiles are strung together and laterally constrained by the neighbouring profiles. The conventional methods summarised above have in common that only a small part of the wavefield (reflected events, first arrival times or surface waves) is utilised while ω ω ω = u f S , (1.1) where ω denotes the angular frequency, S(ω) is the impedance matrix containing the model parameters, u(ω) is the wavefield and f(ω) contains the source terms. Once having inverted S, which is computationally the most demanding task, u can be calculated 1.3.2 Inversion As mentioned above, the goal of FWI is to find a subsurface model that best explains the measured data. In principle, this problem could be solved with global optimisation, which immediately yields the global minimum of the misfit functional. This requires the 1.4 Thesis Objectives and Structure The research questions above are addressed in the framework of my thesis. Surface waves play a crucial role in near-surface seismic data sets. Due to their high amplitudes (Fig. 1.1b), they dominate the misfit functional. However, due to their limited penetration depth, the sensitivities rapidly decay with depth and the information about the deeper structure mainly stems from other events, such as reflections and refractions. In Chapter 2, I present a strategy of upscaling sensitivities at depth, such that the inversion frequency-domain FWI (Brossier et al., 2009; Pratt, 1999). Further challenges occur when inverting for multiple model parameters simultaneously. Typically, elastic FWI seeks to recover the P-and S-wave velocities Vp and Vs, as well as density ρ. These parameters have different sensitivity (Frećhet derivative) magnitudes, that is, the effect on the waveforms, caused by small changes of individual parameters, can be quite different (Operto et al., 2013). This can lead to trade-offs between the...
Waveform inversion for lateral heterogeneities using multimode surface waves
Geophysical Journal International, 2002
We propose a waveform inversion scheme that can be used to invert strong structural heterogeneities. The method is based on a ray approximation for surface waves. Strong structural lateral variations are modelled by vertical discontinuities. Therefore, a geologically heterogeneous region is partitioned into a number of lateral homogeneous subregions. Synthetic seismograms are calculated by modal summation over incident and transmitted modes. With this method the complete waveforms of surface waves, including mode cross-branch coupling, multipathing and scattering, are considered in the inversion. We test the method with several geophysically realistic structural models. An example of waveform inversion for real data-a wave path across the Iberian peninsula and the oceanic structure off the French coast owing to the Gibraltar earthquake (M s = 5.7)-is presented. The method allows us to obtain the information represented by different set of structures along the same source-receiver minor arc through which the waves have propagated. The method is applicable to different models of lateral heterogeneity. In its present development, it is most appropriate for inverting structures around tectonic scenarios such as continental margins, grabens and discontinuities between major plates.
Physics of the Earth and Planetary Interiors, 2000
We investigate the impact of the theoretical limitations brought by asymptotic methods on upper-mantle tomographic Ž. models deduced from long-period surface wave data period) 80 s , by performing a synthetic test using a non-asymptotic formalism. This methodology incorporates the effects of back and multiple forward scattering on the wave field by summing normal modes computed to third order of perturbations directly in the 3D Earth, and models the sensitivity to scatterers away from the great-circle path. We first compare the methods we used for the forward problem, both theoretically and numerically. Then we present results from the computation of 7849 synthetic Love waveforms in an upper mantle model consisting of two heterogeneities with power up to spherical harmonic degree 12. The waveforms are subsequently inverted Ž. using a 0th order asymptotic formalism equivalent to a path-average approximation in the surface waves domain. We show that the main structures are retrieved, but that the theoretical noise on the output model is of the same order as the noise due to the path-coverage and a priori constraints.