S33E-01 INVITED
Multi-Frequency Diffraction Tomography and Imaging: Resolution and Fidelity Enhancement
Classic diffraction tomography is generalized to inhomogeneous background with finite data aperture (finite frequency band and finite spatial aperture of acquisition). Tomographic inversions can be summarized as a process of backpropagation plus filtering in the local angle domain. The backpropagation is a doubly focusing process, similar to the imaging principle in migration/imaging with modified imaging condition in the local angle domain. The filtering is a deconvolution in the local angle domain, which includes the correction factor for acquisition aperture and propagation effects in arbitrarily heterogeneous media. It is shown that multi-frequency diffraction tomography can enhance the resolution and the fidelity of the tomographic inversion, hence the image quality and inversion accuracy. The low frequency component can help in recovering the low wavenumber spectra of the structure (large-scale, smooth background), especially with the use of evanescent waves. Numerical examples of multi-frequency illumination and aperture effects, resolving kernel, spectral recovery and imaging enhancement will be shown to demonstrate the theory and method.
S33E-02 INVITED
Isolated Task Inverse Scattering Sub-series and Seismic Processing Objectives: Concepts, Algorithms and Data Requirements
The inverse scattering series and task specific sub-series(1) provide the opportunity of addressing the most daunting and pressing challenges in exploration seismology. Those challenges often derive from the inability to provide an adequate subsurface velocity model needed for accurate imaging in the presence of rapid varying boundaries and/or highly heterogeneous media. The inverse scattering series has the property of allowing all processing objectives (e.g., multiple removal, depth imaging, and non-linear AVO) to be achieved without in principle or practice requiring or needing the velocity. There is no proxy, or surrogate for the velocity nor a search or invariance principle nor model matching of any kind. The algorithms are direct, and distinct algorithms remove free surface multiples, remove internal multiples or depth image primaries. We will describe and exemplify these methods, and describe data requirements, plans and open issues. (1) Weglein et al., 2003: "Topical review: inverse scattering series and seismic exploration", Inverse Problems, R27-R83.
S33E-03
Extending high-frequency wave asymptotics to low frequencies
Recent successful applications of full waveform solutions have renewed the interest in using low frequency information in seismology. In this paper, we attempt to fill the gap between the ray theory or the high-frequency asymptotic description of wave propagation and the full waveform theory. We show that it is possible to extend the traditional high-frequency solutions to lower frequencies by including additional terms in the phase function that provide a non-liner dependence of phase on frequency. We suggest a particular functional form of this dependence: u(\mathbf{x},ω) ≈ A(\mathbf{x}) ei \sqrt{ω2 T2(\mathbf{x) + B(\mathbf{x}) T(\mathbf{x})}}, where u is the wavefield, \mathbf{x} is a space coordinate, ω is frequency, A is the wave amplitude, T is the traveltime, and B is a new correction factor connected with other parameters by special differential equations. We compare our approximation with full waveform solutions using both exact analytical techniques and numerical experiments. The comparison shows that, although the suggested approximation does not extend all the way to zero frequency, it provides a reasonably accurate extension of the ray theory for describing low-frequency effects in the range of frequencies used in seismic exploration. We discuss possible applications of our theory in seismic imaging, tomography, and full waveform inversion.
S33E-04 INVITED
A multi-scale approach to imaging and Frechet derivatives in wave-equation tomography
We discuss a common (PDE) framework for wave-equation transmission and reflection tomography. The development of the associated imaging procedures and optimization, via adjoint states, involves the Frechet derivatives of the solution operators modelling the different types of data. We present the principles of a multi- scale approach to constructing these Frechet derivatives, using curvelets, in velocity models of limited smoothness. The Frechet derivatives would directly appear in the adjoint state formulation based on `fitting' the data, but here we focus on different functionals for optimization, the sensitivity kernels of which can be viewed as generalizations of their ray-geometric counterparts. We discuss commonalities with the multi-scale decomposition of linearized inverse scattering with the generalized Radon transform. The constructions can be naturally integrated with sparsity constrained optimization via multi-scale representations of the model perturbation.
S33E-05
On the recovery of missing low and high frequency information from bandlimited reflectivity data
During the last two decades, an important effort in the seismic exploration community has been made to retrieve broad-band seismic data by means of deconvolution and inversion. In general, the problem can be stated as a spectral reconstruction problem. In other words, given limited spectral information about the earth's reflectivity sequence, one attempts to create a broadband estimate of the Fourier spectra of the unknown reflectivity. Techniques based on the principle of parsimony can be effectively used to retrieve a sparse spike sequence and, consequently, a broad band signal. Alternatively, continuation methods, e.g., autoregressive modeling, can be used to extrapolate the recorded bandwidth of the seismic signal. The goal of this paper is to examine under what conditions the recovery of low and high frequencies from band-limited and noisy signals is possible. At the heart of the methods we discuss, is the celebrated non-Gaussian assumption so important in many modern signal processing methods, such as ICA, for example. Spectral recovery from limited information tends to work when the reflectivity consist of a few well isolated events. Results degrade with the number of reflectors, decreasing SNR and decreasing bandwidth of the source wavelet. Constrains and information-based priors can be used to stabilize the recovery but, as in all inverse problems, the solution is nonunique and effort is required to understand the level of recovery that is achievable, always keeping the physics of the problem in mind. We provide in this paper, a survey of methods to recover broad-band reflectivity sequences and examine the role that these techniques can play in the processing and inversion as applied to exploration and global seismology.
S33E-06
Implications of Enhanced Low Frequency Data for Linear and Non-linear Inverse Scattering Methods in Absorptive-Dispersive Media
Linear and non-linear inverse scattering series methods(1) for application to absorptive-dispersive (AD) seismic data are being developed and progressed(2,3). The methods are scattering-based approaches to the solution of two well-known seismic inverse problems: Q-estimation and Q-compensation. The linear output, interpretable either as an end goal, if the perturbations are small, or the input to higher order methods, extracts Q information from data via the dispersive reflection coefficient. This has been presented for a non- absorptive reference medium(3); we describe the extension to AD reference media. Meanwhile, the non- linear inverse scattering series component involves collecting series terms to create a Q compensation operator that involves only the data and reference medium properties. Numerical and analytic results outline recent progress. Densely sampled, broadband data will benefit these methods. We describe two implications of a strong low frequency component in the reflected seismic primaries. First, although the error associated with transmission through perturbed regions (which impacts all linear inverse scattering procedures) is aggravated in the AD case, weights that emphasize low-frequency data more aggressively are seen to be a strong mitigating factor. Second, non-linear terms acting on primaries to compensate for Q are seen to share the response to missing low-frequency reported(4) in the case of imaging; a continuous increase in effectiveness is seen as the low end of the input data is enhanced. (1) Weglein et al., 2003: `Topical review: inverse scattering series and seismic exploration', Inverse Problems, R27-R83. (2) Innanen and Weglein, 2005: `Towards non-linear construction of a Q-compensation operator directly from measured seismic reflection data', SEG. (3) Innanen and Weglein, 2007: `On the construction of an absorptive-dispersive medium model via direct linear inversion of reflected seismic primaries', Inverse Problems, to appear. (4) Shaw and Weglein, 2004: `A leading order imaging series for prestack data acquired over a laterally invariant acoustic medium: analysis for bandlimited input data', SEG.
S33E-07
SEAM: The SEG Advanced Modeling Project, Phase I
The SEG Advanced Modeling Project (SEAM) is a consortium being run by the Society of Exploration Geophysicists whose goal is to use numerical modeling to develop geophysical datasets that mimic those used for exploration and characterization of petroleum resources. Phase I of the project is underway and involves the calculation of a large 3D seismic exploration dataset in deepwater using acoustic-wave modeling. There are several aspects of this phase that include: (1) the development of a realistic geological and geophysical model of a region that contains a salt body with subsalt petroleum reservoirs. The salt structure is derived from a well- characterized salt body in the Gulf of Mexico. Surrounding sediments and structures within the sediments are based on geological and seismological information that represent typical Gulf of Mexico geology. The initial model will be acoustic with variable density and compressional-wave velocities; a subsequent model will be elastic and possibly contain anisotropy. EM and gravity models are also being considered (2) Characterization and development of a suitable numerical modeling scheme. The scheme must be capable of reproducing reliable amplitude and traveltime information for wave propagation in the realistic model for frequencies up to approximately 20 Hz and propagation distances of several hundred wavelengths. (3) Development of an acquisition scheme that includes shotpoints and locations of receivers on both the surface and within the Earth's subsurface. (4) Numerical modeling for tens of thousands shot points located on the Earth's surface, and (5) Storage of data and its distribution to industrial, academic and government laboratory investigators who will use the data for a variety of individually-developed research projects. The overall goal of Phase I the project is to provide a large, relatively high-frequency, high-fidelity dataset for a realistic petroleum resource located in deep water that researchers can use for development of improved approaches for analyzing seismic data using multiple approaches for petroleum exploration and reservoir characterization. http://www.seg.org/SEAM
S33E-08
Imaging Seismic Attenuation in the Crust and Upper Mantle by Ambient Noise Correlation
The cross correlation of ambient noise data yields an estimate of the Green's function (EGF) corresponding to a point source located at one station and recorded at the other. As the distance between the two stations goes to zero (the autocorrelation) this is equivalent to the zero-offset record of the "source". The energy in this "source" can be used to normalize the amplitude of the EGF computed between station pairs to get a measurement of seismic attenuation (Q). Here, we present a 2D tomogram of Q in the crust and upper mantle beneath the Ristra PASSCAL deployment in New Mexico. The Ristra experiment consisted of 54 STS2 instruments with an average interstation spacing of 18 km and was fully deployed for 18 months between 1999 to 2001. Arranged along a line stretching from Texas to Utah, it crossed the western Great Plains, the Rio Grande Rift and into the Colorado Plateau. The dense, regular interstation spacing, and uniform instrumentation along a 950 km line makes the data from this array ideal for studying crust and upper mantle seismic structure and for investigating the resolution of new techniques such as noise correlation. We calculated the EGFs using a year's worth of data for each pair of stations in the array. The raw data were instrument corrected and converted to displacement, then whitened and finally converted to single-bit signals. These were then crosscorrelated to produce the final EGFs. We inverted the resulting waveforms to create a 2D velocity profile. The data are particularly sensitive to structure at mid-crustal depths, and resolve heterogeneities into the upper mantle. The broad features of our velocity model match those of a traditional surface wave inversion of the same profile (West et al. [2004]), although the noise-correlation result has sharper features with higher velocity contrasts. In the course of the study, we observed that there is a distinct and predictable decrease in amplitude of the EGFs with respect to distance when normalized by the number of days in the stack. Key to our analysis is the assumption that enough data has been correlated that the equipartition assumption is valid. On the Ristra profile, the interstation amplitude, normalized by the source term, becomes stable after a few months of data have been stacked. Once the data were properly normalized, we were able to measure finer details related to the subsurface geology - for example the rapid loss of amplitude along paths crossing through the Rio Grande Rift. By combining all our measurements we are able to create a profile of Q beneath the array.