Stable soft extrapolation of entire functions

Soft extrapolation refers to the problem of recovering a function from its samples, multiplied by a fast-decaying window and perturbed by an additive noise, over an interval which is potentially larger than the essential support of the window. A core theoretical question is to provide bounds on the possible amount of extrapolation, depending on the sample perturbation level and the function prior. In this paper we consider soft extrapolation of entire functions of finite order and type (containing the class of bandlimited functions as a special case), multiplied by a super-exponentially decaying window (such as a Gaussian). We consider a weighted least-squares polynomial approximation with judiciously chosen number of terms and a number of samples which scales linearly with the degree of approximation. It is shown that this simple procedure provides stable recovery with an extrapolation factor which scales logarithmically with the perturbation level and is inversely proportional to the characteristic lengthscale of the function. The pointwise extrapolation error exhibits a H\"{o}lder-type continuity with an exponent derived from weighted potential theory, which changes from 1 near the available samples, to 0 when the extrapolation distance reaches the characteristic smoothness length scale of the function. The algorithm is asymptotically minimax, in the sense that there is essentially no better algorithm yielding meaningfully lower error over the same smoothness class. When viewed in the dual domain, the above problem corresponds to (stable) simultaneous de-convolution and super-resolution for objects of small space/time extent. Our results then show that the amount of achievable super-resolution is inversely proportional to the object size, and therefore can be significant for small objects

Extrapolated full waveform inversion with deep learning

The lack of low frequency information and a good initial model can seriously affect the success of full waveform inversion (FWI), due to the inherent cycle skipping problem. Computational low frequency extrapolation is in principle the most direct way to address this issue. By considering bandwidth extension as a regression problem in machine learning, we propose an architecture of convolutional neural network (CNN) to automatically extrapolate the missing low frequencies without preprocessing and post-processing steps. The bandlimited recordings are the inputs of the CNN and, in our numerical experiments, a neural network trained from enough samples can predict a reasonable approximation to the seismograms in the unobserved low frequency band, both in phase and in amplitude. The numerical experiments considered are set up on simulated P-wave data. In extrapolated FWI (EFWI), the low-wavenumber components of the model are determined from the extrapolated low frequencies, before proceeding with a frequency sweep of the bandlimited data. The proposed deep-learning method of low-frequency extrapolation shows adequate generalizability for the initialization step of EFWI. Numerical examples show that the neural network trained on several submodels of the Marmousi model is able to predict the low frequencies for the BP 2004 benchmark model. Additionally, the neural network can robustly process seismic data with uncertainties due to the existence of noise, poorly-known source wavelet, and different finite-difference scheme in the forward modeling operator. Finally, this approach is not subject to the structural limitations of other methods for bandwidth extension, and seems to offer a tantalizing solution to the problem of properly initializing FWI.Comment: 30 pages, 22 figure

Estimates for the SVD of the truncated fourier transform on L2(cosh(b.)) and stable analytic continuation

The Fourier transform truncated on [−c, c] is usually analyzed when acting on L2(−1/b, 1/b) and its right-singular vectors are the prolate spheroidal wave functions. This paper considers the operator acting on the larger space L2(cosh(b·)) on which it remains injective. We give nonasymptotic upper and lower bounds on the singular values with similar qualitative behavior in m (the index), b, and c. The lower bounds are used to obtain rates of convergence for stable analytic continuation of possibly nonbandlimited functions which Fourier transform belongs to L2(cosh(b·)). We also derive bounds on the sup-norm of the singular functions. Finally, we provide a numerical method to compute the SVD and apply it to stable analytic continuation when the function is observed with error on an interval

Feature augmentation for the inversion of the Fourier transform with limited data

We investigate an interpolation/extrapolation method that, given scattered observations of the Fourier transform, approximates its inverse. The interpolation algorithm takes advantage of modeling the available data via a shape-driven interpolation based on variably scaled Kernels (VSKs), whose implementation is here tailored for inverse problems. The so-constructed interpolants are used as inputs for a standard iterative inversion scheme. After providing theoretical results concerning the spectrum of the VSK collocation matrix, we test the method on astrophysical imaging benchmarks

Acoustic impedance estimation from combined harmonic reconstruction and interval velocity

Low-frequency components of reflection seismic data are of paramount importance for acoustic impedance inversion, but they typically suffer from a poor signal-to-noise ratio. The estimation of low frequencies of the acoustic impedance can benefit from the combination of a harmonic reconstruction method (based on autoregressive models) and a seismic-derived interval velocity field. We propose the construction of a convex cost-function that accounts for the velocity field, together with geologic a priori information on acoustic impedance and its uncertainty, during the autoregressive reconstruction of the low frequencies. The minimization of this function allows one to reconstruct sensible estimates of low-frequency components of the subsurface reflectivity, which lead to an estimation of acoustic impedance model via a recursive formulation. In particular, the method is suited for an initial and computationally inexpensive assessment of the absolute value of acoustic impedance even when no well log data are available. We first tested the method on layered synthetic models, then we analyzed its applicability and limitations on a real marine seismic dataset that included tomographic velocity information. Despite a strong trace-to-trace variability in the results, which could partially be mitigated by multi-trace inversion, the method demonstrates its capability to highlight lateral variations of acoustic impedance that cannot be detected when the low frequencies only come from well log information