Full Waveform Inversion Guided by Travel Time Tomography (original) (raw)
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Exploration Geophysics, 2017
Full waveform inversion (FWI) is a method that is used to reconstruct velocity models of the subsurface. However, this approach suffers from the local minimum problem during optimisation procedures. The local minimum problem is caused by several issues (e.g. lack of low-frequency information and an inaccurate starting model), which can create obstacles to the practical application of FWI with real field data. We applied a 4-phase FWI in a sequential manner to obtain the correct velocity model when a dataset lacks low-frequency information and the starting velocity model is inaccurate. The first phase is Laplace-domain FWI, which inverts the large-scale velocity model. The second phase is Laplace-Fourier-domain FWI, which generates a large- to mid-scale velocity model. The third phase is a frequency-domain FWI that uses a logarithmic wavefield; the inverted velocity becomes more accurate during this step. The fourth phase is a conventional frequency-domain FWI, which generates an imp...
Earth Sciences Research Journal
This paper investigates the capability of acoustic Full Waveform Inversion (FWI) in building Marmousi velocity model, in time and frequency domain. FWI is an iterative minimization of misfit between observed and calculated data which is generally solved in three segments: forward modeling, which numerically solves the wave equation with an initial model, gradient computation of the objective function, and updating the model parameters, with a valid optimization method. FWI codes developed in MATLAB herein FWISIMAT (Full Waveform Inversion in Seismic Imaging using MATLAB) are successfully implemented using the Marmousi velocity model as the true model. An initial model is obtained by smoothing the true model to initiate FWI procedure. Smoothing ensures an adequate starting model for FWI, as the FWI procedure is known to be sensitive on the starting model. The final model is compared with the true model to review the number of recovered velocities. FWI codes developed in MATLAB herein...
Computational Geosciences
We study the inverse boundary value problem for time-harmonic elastic waves, for the recovery of P-and S-wave speeds from vibroseis data or the Neumann-to-Dirichlet map. Our study is based on our recent result pertaining to the uniqueness and a conditional Lipschitz stability estimate for parametrizations on unstructured tetrahedral meshes of this inverse boundary value problem. With the conditional Lipschitz stability estimate, we design a procedure for full waveform inversion (FWI) with iterative regularization. The iterative regularization is implemented by projecting gradients, after scaling, onto subspaces associated with the mentioned parametrizations yielding Lipschitz stability. The procedure is illustrated in computational experiments using the Continuous Galerkin finite-element method of recovering the rough shapes and wave speeds of geological bodies from simple starting models, near and far from the boundary, that is, the free surface.
Regularized seismic full waveform inversion with prior model information
GEOPHYSICS, 2013
Full Waveform Inversion (FWI) delivers high-resolution quantitative images and is a promising technique to obtain macro-scale physical properties model of the subsurface. In most geophysical applications, prior information, as those collected in wells, is available and should be used to increase the image reliability. For this, we propose to introduce three terms in the definition of the FWI misfit function: the data misfit itself, the first-order Tikhonov regularization term acting as a smoothing operator and a prior model norm term. This last term is the way to introduce smoothly prior information into the FWI workflow. On a selected target of the Marmousi synthetic example, we show the significant improvement obtained when using the prior model term for both noise-free and noisy synthetic data. We illustrate that the prior model term may significantly reduce the inversion sensitivity to incorrect initial conditions. It is highlighted how the limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information, for both multiple-free and multiple-contaminated data. We also demonstrate that prior and initial models play different roles in the inversion scheme. The starting model is used for wave propagation and therefore drives the data-misfit gradient, while the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function. Thus the prior model in not required to follow kinematic properties as precisely as the initial model, except in poor illumination zones. In addition, we investigate the influence of a simple dynamic decreasing weighting of the prior model term. Once the cycle-skipping problem has been solved, the impact of the prior model term is gradually reduced within the misfit function in order to be driven by seismic-data only.
The Journal of the Acoustical Society of America, 2008
Full Waveform Inversion (FWI) is a very general multi-parameter quantitative imaging method originally developed to obtain high resolution images of velocities and attenuations in a natural underground medium. FWI promises relevant performances for civil engineering applications. Performances of the FWI method in seismic exploration are difficult to assess because in situ experimentations the properties of the medium are unknown. Furthermore, characteristics of the source and coupling of receivers are not controlled. In order to appraise the performances of FWI and its adaptability to subsurface applications, small scale physical models are realized and measurements are simulated in a dedicated non contact laser ultrasonic laboratory which allows to simulate seismic reflection measurement configurations. Seismograms well reproduce behaviors of real scale records in terms of waveforms but the use of a piezoelectric source with a large radiation surface area modifies the full seismograms. We applied a FWI algorithm on two synthetic data sets modelised with a punctual and a lineic source. The forward model is based on a frequency domain finite difference method. The inverse problem is solved with a gradient method scaled by the diagonal elements of the Hessian. Influence of the source on the recovered velocity medium is discussed.
Time-Domain Wavefield Reconstruction Inversion for Large-Scale Seismics
2020
Wavefield reconstruction inversion is an imaging technique akin to full-waveform inversion, albeit based on a relaxed version of the wave equation. This relaxation aims to beat the multimodality typical of full-waveform inversion. However it prevents the use of time-marching solvers for the augmented equation and, as a consequence, cannot be straightforwardly employed to large 3D problems. In this work, we formulate a dual version of wavefield reconstruction inversion amenable to explicit time-domain solvers, yielding a robust and scalable inversion technique.
Exploration Geophysics, 2004
Frequency-space domain full-wave tomography is a promising technique for delineating detailed subsurface structure with high resolution. However, this method requires criteria for the selection of a set of optimal temporal frequency components, to achieve stability in the sequence of inversion processes together with computational efficiency. We propose a method of selecting optimal temporal frequencies, based on wavenumber continuity. The proposed method is tested numerically and is shown to be able to select an optimal set of frequency components that are sufficient to image the anomalies.
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...
Seismic Full Waveform Inversion And Modeling
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