391 research outputs found
Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals
Slepian functions provide a solution to the optimization problem of joint
time-frequency localization. Here, this concept is extended by using a
generalized optimization criterion that favors energy concentration in one
interval while penalizing energy in another interval, leading to the
"augmented" Slepian functions. Mathematical foundations together with examples
are presented in order to illustrate the most interesting properties that these
generalized Slepian functions show. Also the relevance of this novel
energy-concentration criterion is discussed along with some of its
applications
Full waveform inversion with extrapolated low frequency data
The availability of low frequency data is an important factor in the success
of full waveform inversion (FWI) in the acoustic regime. The low frequencies
help determine the kinematically relevant, low-wavenumber components of the
velocity model, which are in turn needed to avoid convergence of FWI to
spurious local minima. However, acquiring data below 2 or 3 Hz from the field
is a challenging and expensive task. In this paper we explore the possibility
of synthesizing the low frequencies computationally from high-frequency data,
and use the resulting prediction of the missing data to seed the frequency
sweep of FWI. As a signal processing problem, bandwidth extension is a very
nonlinear and delicate operation. It requires a high-level interpretation of
bandlimited seismic records into individual events, each of which is
extrapolable to a lower (or higher) frequency band from the non-dispersive
nature of the wave propagation model. We propose to use the phase tracking
method for the event separation task. The fidelity of the resulting
extrapolation method is typically higher in phase than in amplitude. To
demonstrate the reliability of bandwidth extension in the context of FWI, we
first use the low frequencies in the extrapolated band as data substitute, in
order to create the low-wavenumber background velocity model, and then switch
to recorded data in the available band for the rest of the iterations. The
resulting method, EFWI for short, demonstrates surprising robustness to the
inaccuracies in the extrapolated low frequency data. With two synthetic
examples calibrated so that regular FWI needs to be initialized at 1 Hz to
avoid local minima, we demonstrate that FWI based on an extrapolated [1, 5] Hz
band, itself generated from data available in the [5, 15] Hz band, can produce
reasonable estimations of the low wavenumber velocity models
Fast Algorithms for the computation of Fourier Extensions of arbitrary length
Fourier series of smooth, non-periodic functions on are known to
exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of
overcoming these problems is by using a Fourier series on a larger domain, say
with , a technique called Fourier extension or Fourier
continuation. When constructed as the discrete least squares minimizer in
equidistant points, the Fourier extension has been shown shown to converge
geometrically in the truncation parameter . A fast algorithm has been described to compute Fourier extensions for the case
where , compared to for solving the dense discrete
least squares problem. We present two algorithms for
the computation of these approximations for the case of general , made
possible by exploiting the connection between Fourier extensions and Prolate
Spheroidal Wave theory. The first algorithm is based on the explicit
computation of so-called periodic discrete prolate spheroidal sequences, while
the second algorithm is purely algebraic and only implicitly based on the
theory
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