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Traveltime tomography using transmission data has been widely used for static corrections and for obtaining near-surface models for seismic depth imaging. More recently, it is also being used to build initial models for full-waveform inversion. The classic traveltime tomography approach based on ray tracing has difficulties in handling large data sets arising from current seismic acquisition surveys. Some of these difficulties can be addressed using the adjoint-state method, due to its low memory requirement and numerical efficiency. By coupling the gradient computation to nonlinear optimization, it avoids the need for explicit computation of the Fréchet derivative matrix. Furthermore, its cost is equivalent to twice the solution of the forward-modeling problem, irrespective of the size of the input data. The presence of anisotropy in the subsurface has been well established during the past few decades. The improved seismic images obtained by incorporating anisotropy into the seismic processing workflow justify the effort. However, previous literature on the adjoint-state method has only addressed the isotropic approximation of the subsurface. We have extended the adjoint-state technique for first-arrival traveltime tomography to vertical transversely isotropic (VTI) media. Because [Formula: see text] is weakly resolvable from surface seismic alone, we have developed the mathematical framework and procedure to invert for [Formula: see text] and [Formula: see text]. Our numerical tests on the VTI SEAM model demonstrate the ability of the algorithm to invert for near-surface model parameters and reveal the accuracy achievable by the algorithm.

Computation of first-arrival traveltimes for quasi-P waves in the presence of anisotropy is important for high-end near-surface modeling, microseismic-source localization, and fractured-reservoir characterization — and it requires solving an anisotropic eikonal equation. Anisotropy deviating from elliptical anisotropy introduces higher order nonlinearity into the eikonal equation, which makes solving the eikonal equation a challenge. We addressed this challenge by iteratively solving a sequence of simpler tilted elliptically anisotropic eikonal equations. At each iteration, the source function was updated to capture the effects of the higher order nonlinear terms. We used Aitken’s extrapolation to speed up convergence rate of the iterative algorithm. The result is an algorithm for computing first-arrival traveltimes in tilted anisotropic media. We evaluated the applicability and usefulness of our method on tilted transversely isotropic media and tilted orthorhombic media. Our numerical tests determined that the proposed method matches the first arrivals obtained by wavefield extrapolation, even for strongly anisotropic and highly complex subsurface structures. Thus, for the cases where two-point ray tracing fails, our method can be a potential substitute for computing traveltimes. The approach presented here can be easily extended to compute first-arrival traveltimes for anisotropic media with lower symmetries, such as monoclinic or even the triclinic media.

We introduce an approach to H ∞ model reduction that is founded on ideas originating in realization theory, interpolatory H 2 -optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. By employing Loewner “data-driven” partial realizations within each optimization cycle, computationally demanding H ∞ norm calculations can be completely avoided. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better H ∞ performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal Hankel norm approximation.

The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equations. In this paper we show that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods. We will call these shifts pseudo H"2-optimal shifts. These shifts are also optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to the rational Krylov projection subspace. Via several examples, we show that the pseudo H"2-optimal shifts consistently yield nearly optimal low rank approximations to the solutions of the Lyapunov equations.

We present a new localized planewave decomposition method in time-space domain for multicomponent measurements to be utilized in migration and model building. In order to achieve this, we have: (i) developed the concept of multicomponent semblance to estimate the dips; (ii) used block orthogonal matching-pursuit with backward substitution for estimation of the wavelets for the upgoing and downgoing wavefields corresponding to each estimated dip. We demonstrate our method on synthetic examples.

SUMMARY Computing first-arrival traveltimes of quasi-P waves in the presence of anisotropy is important for high-end nearsurface modeling, microseismic-source localization, and fractured-reservoir characterization, and requires solving an anisotropic eikonal equation. Anisotropy deviating from elliptical anisotropy introduces higher-order nonlinearity into the eikonal equation, which makes solving the eikonal equation a challenge. We address this challenge by iteratively solving a sequence of simpler tilted elliptically anisotropic eikonal equations. At each iteration, the source function is updated to capture the effects of the higher order nonlinear terms. We use Aitken extrapolation to speed up the convergence rate of the iterative algorithm. The result is an algorithm for firstarrival traveltime computations in tilted anisotropic media. We demonstrate our method on tilted transversely isotropic media and tilted orthorhombic media. Our numerical tests demonstrate that the proposed method can match the first arrivals obtained by wavefield extrapolation, even for strong anisotropy and complex structures. Therefore, for the cases where oneor two-point ray tracing fails, our method may be a potential substitute for computing traveltimes. Our approach can be extended to anisotropic media with lower symmetries, such as monoclinic or even triclinic media.

In this paper, we focus on model reduction of large-scale bilinear systems. The main contributions are threefold. First, we introduce a new framework for interpolatory model reduction of bilinear systems. In contrast to the existing methods where interpolation is forced on some of the leading subsystem transfer functions, the new framework shows how to enforce multipoint interpolation of the underlying Volterra series. Then, we show that the first-order conditions for optimal H2 model reduction of bilinear systems require multivariate Hermite interpolation in terms of the new Volterra series interpolation framework; and thus we extend the interpolation-based first-order necessary conditions for H2 optimality of LTI systems to the bilinear case. Finally, we show that multipoint interpolation on the truncated Volterra series representation of a bilinear system leads to an asymptotically optimal approach to H2 optimal model reduction, leading to an efficient model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach.

The iterative rational Krylov algorithm ( IRKA ) of Gugercin et al. (2008) [8] is an interpolatory model reduction approach to the optimal H 2 approximation problem. Even though the method has been illustrated to show rapid convergence in various examples, a proof of convergence has not been provided yet. In this note, we show that in the case of state-space-symmetric systems, IRKA is a locally convergent fixed-point iteration to a local minimum of the underlying H 2 approximation problem.

We introduce an interpolatory approach to ℋ ∞ model reduction for large-scale dynamical systems. Guided by the optimality conditions of [26] for best uniform rational approximants on the unit disk, our proposed method uses the freedom in choosing the d-term in the reduced order model to enforce 2r + 1 interpolation conditions in the right-half plane for any given reduction order, r. 2r of these points are initialized by the Iterative Rational Krylov Algorithm of [16]; and then the d-term is chosen to minimize the ℋ ∞ error for this initial set of interpolation points. Several numerical examples illustrate the effectiveness of the proposed method. It consistently yields better results than balanced truncation. In all cases examined its performance is very close to or better than that of Hankel norm approximation. For the special case of state-space symmetric systems, important properties are established. Finally, we examine ℋ ∞ model reduction from a potential theoretic perspective and present a second methodology for choosing interpolation points.

Computing first-arrival traveltimes in the presence of anisotropy is important for high-end near surface modeling, microseismic source localization, and fractured reservoir characterization. Anisotropy deviating from elliptical anisotropy introduces higher-order nonlinearity into the eikonal equation, which makes solving the equation a challenging task. We address this challenge by iteratively solving a sequence of simpler tilted elliptically anisotropic eikonal equations. At each iteration, the source function is updated to capture the effects due to the higher order nonlinear terms in the anisotropy. We use Aitken extrapolation to speed up the convergence rate of the iterative algorithm. The result is an efficient algorithm for first-arrival traveltime computations in tilted anisotropic media. We demonstrate the proposed method for the tilted transversely isotropic media and the tilted orthorhombic media. Numerical tests show that the proposed method is feasible and produces results that are comparable to wavefield extrapolation, even for strongly anisotropic and complex structures. Therefore, for the cases where one or two-point ray tracing fails, our method may be a potential substitute for computing traveltimes.