Full three-dimensional tomography: a comparison between the scattering-integral and adjoint-wavefield methods (original) (raw)
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An adjoint tomography is an iterative inversion using an adjoint method to compute a spatially dependent gradient of wavefield misfit (kernel) with respect to model parameters. The gradient therefore provides information, how to modify the trial model in order to get a new one leading to a smaller misfit. The method has been recently successfully applied to many inversion problems at global or regional scales.
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The problem of imaging three-dimensional strong scatterers by means of a two-dimensional sliced tomographic reconstruction algorithm is dealt with. In particular, the focus of the paper is on the experimental validation of the involved inversion algorithm thanks to measurements collected in a controlled environment. A simple strategy exploiting reconstructions obtained at different time instants in order to detect slowly moving scatterers is also experimentally validated.
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Journal of Computational Physics, 2007
The efficient computation of finite-frequency traveltime and amplitude sensitivity kernels for velocity and attenuation perturbations in global seismic tomography poses problems both of numerical precision and of validity of the paraxial approximation used. We investigate these aspects, using a local model parameterization in the form of a tetrahedral grid with linear interpolation in between grid nodes. The matrix coefficients of the linear inverse problem involve a volume integral of the product of the finite-frequency kernel with the basis functions that represent the linear interpolation. We use local and global tests as well as analytical expressions to test the numerical precision of the frequency and spatial quadrature. There is a trade-off between narrowing the bandpass filter and quadrature accuracy and efficiency. Using a minimum step size of 10 km for S waves and 30 km for SS waves, relative errors in the quadrature are of the order of 1% for direct waves such as S, and a few percent for SS waves, which are below data uncertainties in delay time or amplitude anomaly observations in global seismology. Larger errors may occur wherever the sensitivity extends over a large volume and the paraxial approximation breaks down at large distance from the ray. This is especially noticeable for minimax phases such as SS waves with periods >20 s, when kernels become hyperbolic near the reflection point and appreciable sensitivity extends over thousands of km. Errors becomes intolerable at epicentral distance near the antipode when sensitivity extends over all azimuths in the mantle. Effects of such errors may become noticeable at epicentral distances > 140°. We conclude that the paraxial approximation offers an efficient method for computing the matrix system for finite-frequency inversions in global tomography, though care should be taken near reflection points, and alternative methods are needed to compute sensitivity near the antipode.
Geophysical Journal International, 2011
We propose a new approach to computing the sensitivity kernels used in seismic tomography based on a Green's function database. For any perturbation in the Earth's structural model, the waveform Fréchet derivative can be expressed in terms of strain Green's tensors, which are themselves functions of the reference Earth model only. The Fréchet derivative of any seismic observable can then be obtained from waveform Fréchet derivative. Given a reference model, a strain Green's tensor database can be established, thus eliminating the need for repetitive wavefield evaluations in all subsequent synthetic and kernel calculations, and reducing the CPU time. For a spherically symmetric reference Earth model, the strain Green's tensor database can be constructed on a (r, ) grid by normal-mode summation. The stored strain Green's tensors can then be used to quickly evaluate the wavefield between any source and receiver. The generality of the strain Green's tensors makes it possible to compute the Fréchet kernels for any phase on a seismogram (P, S, Pdiff , surface waves, etc.), for any type of data (traveltime, amplitude, splitting intensity, waveform, etc.), and for any parameter (isotropic, anisotropic, attenuation, etc.). The kernel calculation at each point in the medium is reduced to the convolution of two sets of strain Green's tensors extracted from the database, which makes the approach extremely efficient. Efficient Fréchet Kernel calculation-I 923 For regional tomography based upon phase measurements of long period surface waves (a Rayleigh wave of 100-s period has a wavelength of about 300 km), this effect can limit the resolution potential significantly. Efforts have been made more recently to develop efficient numerical methods to compute full-wave sensitivity kernels for seismic tomography based on 1-D reference models (e.g. ). Using the approach of Zhao et al. (2000) and based upon normal-mode coupling to calculate complete Fréchet kernels for the phase of Rayleigh waves, have shown that regional surface wave tomography can indeed resolve structures as small as 100 km, smaller than the size of the first Fresnel zone. However, the application of the normal-mode approach has so far been rather limited owing to a low numerical efficiency.
SEG Technical Program Expanded Abstracts 2019, 2019
Recent years saw a surge of interest in seismic waveform inversion approaches based on quadratic-penalty or augmented-Lagrangian methods, including Wavefield Reconstruction Inversion. These methods typically need to solve a least-squares sub-problem that contains a discretization of the Helmholtz equation. Memory requirements for direct solvers are often prohibitively large in three dimensions, and this limited the examples in the literature to two dimensions. We present an algorithm that uses iterative Helmholtz solvers as a blackbox to solve the least-squares problem corresponding to 3D grids. This algorithm enables Wavefield Reconstruction Inversion and related formulations, in three dimensions. Our new algorithm also includes a root-finding method to convert a penalty into a constraint on the data-misfit without additional computational cost, by reusing precomputed quantities. Numerical experiments show that the cost of parallel communication and other computations are small compared to the main cost of solving one Helmholtz problem per source and one per receiver.
IEEE Transactions on Instrumentation and Measurement, 2000
In this paper, a three-dimensional (3-D) extension of the well-known filtered-backpropagation (FBP) algorithm is presented with the aim of taking into account scattered-field-data measurements obtained using incident directions not restricted in a single plane. The FBP algorithm has been extensively used to solve the two-dimensional inverse-scattering problem under the first-order Born and Rytov approximations for weak scatterers. The extension of this algorithm in three dimensions is not straightforward, because the task of collecting the data needed to obtain a low-pass filtered version of the scattering object, taking into account all spatial frequencies within a radius of √ 2k 0 , and of incorporating these data to the FBP algorithm, needs to be addressed. A simple extension using incident field directions restricted to a single plane (illumination plane) leaves a region of spatial frequencies of the sphere of radius √ 2k 0 undetermined. The locus of these spatial frequencies may be crucial for the accurate reconstruction of objects which do not vary slowly along the axis perpendicular to the illumination plane. The proposed 3-D FBP algorithm presented here is able to incorporate the data collected from more than one illumination plane and to ensure the reliability of the reconstruction results.
Full Waveform Inversion Guided by Travel Time Tomography
SIAM Journal on Scientific Computing, 2017
Full waveform inversion (FWI) is a process in which seismic numerical simulations are fit to observed data by changing the wave velocity model of the medium under investigation. The problem is non-linear, and therefore optimization techniques have been used to find a reasonable solution to the problem. The main problem in fitting the data is the lack of low spatial frequencies. This deficiency often leads to a local minimum and to non-plausible solutions. In this work we explore how to obtain low frequency information for FWI. Our approach involves augmenting FWI with travel time tomography, which has low-frequency features. By jointly inverting these two problems we enrich FWI with information that can replace low frequency data. In addition, we use high order regularization, in a preliminary inversion stage, to prevent high frequency features from polluting our model in the initial stages of the reconstruction. This regularization also promotes the non-dominant low-frequency modes that exist in the FWI sensitivity. By applying a joint FWI and travel time inversion we are able to obtain a smooth model than can later be used to recover a good approximation for the true model. A second contribution of this paper involves the acceleration of the main computational bottleneck in FWI-the solution of the Helmholtz equation. We show that the solution time can be reduced by solving the equation for multiple right hand sides using block multigrid preconditioned Krylov methods.
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Geophysics, 2006
One of the applications of refraction-traveltime tomography is to provide an initial model for waveform inversion and Kirchhoff prestack migration. For such applications, we need a refraction-traveltime tomography method that is robust for complicated and high-velocity-contrast models. Of the many refraction-traveltime tomography methods available, we believe wave-based algorithms to be best suited for dealing with complicated models. We developed a new wave-based, refraction-tomography algorithm using a damped wave equation and a waveform-inversion back-propagation technique. The imaginary part of a complex angular frequency, which is generally introduced in frequency-domain wave modeling, acts as a damping factor. By choosing an optimal damping factor from the numerical-dispersion relation, we can suppress the wavetrains following the first arrival. The objective function of our algorithm consists of residuals between the respective phases of first arrivals in field data and in forward-modeled data. The model-response, firstarrival phases can be obtained by taking the natural logarithm of damped wavefields at a single frequency low enough to yield unwrapped phases, whereas field-data phases are generated by multiplying picked first-arrival traveltimes by the same angular frequency used to compute model-response phases. To compute the steepest-descent direction, we apply a waveform-inversion back-propagation algorithm based on the symmetry of the Green's function for the wave equation ͑i.e., the adjoint state of the wave equation͒, allowing us to avoid directly computing and saving sensitivities ͑Fréchet derivatives͒. From numerical examples of a block-anomaly model and the Marmousi-2 model, we confirm that traveltimes computed from a damped monochromatic wavefield are compatible with those picked from synthetic data, and our refraction-tomography method can provide initial models for Kirchhoff prestack depth migration.
Source Wavefield Reconstruction for Large-scale 3D Elastic Full-waveform Inversion
In elastic full-waveform inversion, the medium parameters are updated iteratively by incrementing them with the derivatives of the misfit functional with respect to each of the medium parameters. The efficient implementation of the derivative computations require large amount of computer memory storage. The large requirements in terms of storage is one the main barriers for the application of elastic full-waveform inversion to large scale 3D problems. In this paper, we propose and test a strategy based on reverse-time wavefield reconstruction using the Kirchhoff integral that effectively reduces the storage requirements, at the cost of a factor of two increase in the computational runtime.
Geophysical Journal International, 2006
In this study, we test the adequacy of 2-D sensitivity kernels for fundamental-mode Rayleigh waves based on the single-scattering (Born) approximation to account for the effects of heterogeneous structure on the wavefield in a regional surface wave study. The calculated phase and amplitude data using the 2-D sensitivity kernels are compared to phase and amplitude data obtained from seismic waveforms synthesized by the pseudo-spectral method for plane Rayleigh waves propagating through heterogeneous structure. We find that the kernels can accurately predict the perturbation of the wavefield even when the size of anomaly is larger than one wavelength. The only exception is a systematic bias in the amplitude within the anomaly itself due to a site response. An inversion method of surface wave tomography based on the sensitivity kernels is developed and applied to synthesized data obtained from a numerical simulation modelling Rayleigh wave propagation over checkerboard structure. By comparing recovered images to input structure, we illustrate that the method can almost completely recover anomalies within an array of stations when the size of the anomalies is larger than or close to one wavelength of the surface waves. Surface wave amplitude contains important information about Earth structure and should be inverted together with phase data in surface wave tomography.