In conventional seismic surveys, there is a waiting time between sequentially fired shots. This time is determined such that the deepest reflection of interest is recorded before the following source is fired. In a survey with simultaneous or blended sources, the waiting time between the firing of shots is not dependent on the deepest reflection of interest, it is usually much shorter and/or can have random time delays. Thus, the wavefields due to independent sources are overlapped in the records.
The blended data exhibit strong discontinuities in the source direction, in contrast to the coherency expected from seismic measurements. A strategy for deblending could then be to suppress these discontinuities. In this paper, we propose to do this by designing an energy functional that uses a combination of individual functionals that penalize deviations from local plane waves in the reconstructed (deblended) data, as well as a least squares term that penalizes discrepancies between the deblended and the measured data. In this way, we derive a set of coupled nonlinear partial differential equations that we use for the deblending procedure.
We construct a variational formulation for the problem of interpolating seismic data in the case of missing traces. We assume that we have derivative information available at the traces. The variational problem is essentially the minimization of the integral over the smallest eigenvalue of the structure tensor associated with the interpolated data. This has the physical meaning of penalizing the local presence of more than one direction in the interpolation. The variational problem is used to justify the solutions of a non-standard anisotropic diffusion problem as reasonable interpolated images. We show existence and uniqueness for this type of anisotropic diffusion. In particular, the uniqueness property is important as it guarantees that the solution can be obtained by the numerical schemes we propose.
In this paper we prove that the recently introduced method of signal apparition optimally separates signals from interfering sources recorded during simultaneous source seismic data acquisition. By utilizing a periodic sequence of source signatures along one source line, that wavefield becomes separately partially visible in the spectral domain where it can be isolated from interfering signals, processed, and subtracted from the original recordings, thereby separating the wavefields from each other. Whereas other methods for simultaneous source separation can recover data in triangle-shaped region in the spectral domain, signal apparition allows for the exact separation of data in a diamond-shaped region that is twice as large thereby enabling superior reconstruction of separated wavefields throughout the entire data bandwidth.
In this paper we prove that the recently introduced method of signal apparition optimally separates signals from interfering sources recorded during simultaneous source seismic data acquisition. By utilizing a periodic sequence of source signatures along one source line, that wavefield becomes separately partially visible in the spectral domain where it can be isolated from interfering signals, processed, and subtracted from the original recordings, thereby separating the wavefields from each other. Whereas other methods for simultaneous source separation can recover data in a triangle-shaped region in the spectral domain, signal apparition allows for the exact separation of data in a diamond-shaped region that is twice as large thereby enabling superior reconstruction of separated wavefields throughout the entire data bandwidth.
Signal apparition offers a fundamentally new perspective on simultaneous source separation. Whereas the method exactly separates the signal from interfering sources in diamond-shaped regions of the f-k space, signal from simultaneous source still overlap outside these regions. We present a method based on using local phase functions and the analytic part of the blended data to reconstruct the separated data throughout the full data bandwidth.
Signal apparition offers a fundamentally new perspective on simultaneous source separation. Whereas the method exactly separates the signal from interfering sources in diamond-shaped regions of the frequency-wavenumber domain, signals from simultaneous sources still overlap outside these regions. We present a method based on using quaternion representations of the blended data that reconstruct the separated data throughout the full data bandwidth by iteratively using reconstructions for lower frequencies to recover the higher frequency content. Presentation Date: Tuesday, September 26, 2017 Start Time: 9:20 AM Location: Exhibit Hall C, E-P Station 2 Presentation Type: EPOSTER
We provide a mathematical account of the recent letter by Tarnopolsky, Kruchkov and Vishwanath (Phys. Rev. Lett.122:10 (2019), art. id. 106405). The new contributions are a spectral characterization of magic angles, its accurate numerical implementation and an exponential estimate on the squeezing of all bands as the angle decreases. Pseudospectral phenomena due to the nonhermitian nature of operators appearing in the model considered in the letter of Tarnopolsky et al. play a crucial role in our analysis.
We revisit the classical problem of when a given function, which is analytic in the upper half plane C+, can be written as the Fourier transform of a function or distribution with support on a half axis (-infinity, b], b is an element of R. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as H-P (C+).
In this paper the Dirichlet problem for a class of standard weighted Laplace operators in the upper half plane is solved by means of a counterpart of the classical Poisson integral formula. Boundary limits and representations of the associated solutions are studied within a framework of weighted spaces of distributions. Special attention is given to the development of a, suitable uniqueness theory for the Dirichlet problem under appropriate growth constraints at infinity. (C) 2015 Elsevier Inc. All rights reserved.
The unfolding of the COVID-19 pandemic has been very difficult to predict using mathematical models for infectious diseases. While it has been demonstrated that variations in susceptibility have a damping effect on key quantities such as the incidence peak, the herd-immunity threshold and the final size of the pandemic, this complex phenomenon is almost impossible to measure or quantify, and it remains unclear how to incorporate it for modeling and prediction. In this work we show that, from a modeling perspective, variability in susceptibility on an individual level is equivalent with a fraction θ of the population having an “artificial” sterilizing immunity. We also derive novel formulas for the herd-immunity threshold and the final size of the pandemic, and show that these values are substantially lower than predicted by the classical formulas, in the presence of variable susceptibility. In the particular case of SARS-CoV-2, there is by now undoubtedly variable susceptibility due to waning immunity from both vaccines and previous infections, and our findings may be used to greatly simplify models. If such variations were also present prior to the first wave, as indicated by a number of studies, these findings can help explain why the magnitude of the initial waves of SARS-CoV-2 was relatively low, compared to what one may have expected based on standard models.
In this paper we provide quenched central limit theorems, large deviation principles and local central limit theorems for random U(1) extensions of expanding maps on the torus. The results are obtained as special cases of corresponding theorems that we establish for abstract random dynamical systems. We do so by extending a recent spectral approach developed for quenched limit theorems for expanding and hyperbolic maps to be applicable also to partially hyperbolic dynamics.
The purpose of this paper is to study microlocal conditions for inclusion relations between the ranges of square systems of pseudodifferential operators which fail to be locally solvable. The work is an extension of earlier results for the scalar case in this direction, where analogues of results by L. Hormander about inclusion relations between the ranges of first order differential operators with coefficients in C which fail to be locally solvable were obtained. We shall study the properties of the range of systems of principal type with constant characteristics for which condition () is known to be equivalent to microlocal solvability.
Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the unit circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.
We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.
Finite-difference (FD) modelling of seismic waves in the vicinity of dipping interfaces gives rise to artefacts. Examples are phase and amplitude errors, as well as staircase diffractions. Such errors can be reduced in two general ways. In the first approach, the interface can be anti-aliased (i.e. with an anti-aliased step-function, or a lowpass filter). Alternatively, the interface may be replaced with an equivalent medium (i.e. using Schoenberg & Muir (SM) calculus or orthorhombic averaging). We test these strategies in acoustic, elastic isotropic, and elastic anisotropic settings. Computed FD solutions are compared to analytical solutions. We find that in acoustic media, anti-aliasing methods lead to the smallest errors. Conversely, in elastic media, the SM calculus provides the best accuracy. The downside of the SM calculus is that it requires an anisotropic FD solver even to model an interface between two isotropic materials. As a result, the computational cost increases compared to when using isotropic FD solvers. However, since coarser grid spacings can be used to represent the dipping interfaces, the two effects (an expensive FD solver on a coarser FD grid) equal out. Hence, the SM calculus can provide an efficient means to reduce errors, also in elastic isotropic media.
In recent years, it has been recognized that the seismic wave equation solved with a finite-difference method in time causes a predictable and removable error through the use of so-called time-dispersion transforms. These transforms were thought not to apply to visco-elastic media. However, in this paper we demonstrate that the time-dispersion transforms remain applicable when the visco-elastic wave equation is solved with memory variables, as is commonly done. The crucial insight is that both the wave equation and the memory variables are computed with the same time-dispersion error. We show how the time-dispersion transforms can be implemented in, for example, MATLAB, and demonstrate the developed theory on a visco-elastic version of the Marmousi model. Then, the time-dispersion transforms allow computation of the visco-elastic wave equation with large steps in time without significant loss of accuracy, and without having to make any modifications to the model.
This paper concerns the compact group extension f:T2→T2,f(x,s)=(E(x),s+τ(x) mod 1) of an expanding map E : S^{1}→S^{1}. The dynamics of f and its stochastic perturbations have previously been studied under the so-called partial captivity condition. Here we prove a supplementary result that shows that partial captivity is a C^{r} generic condition on τ, once we fix E.
We consider quenched random perturbations of skew products of rotations on the unit circle over uniformly expanding maps on the unit circle. It is known that if the skew product satisfies a certain condition (shown to be generic in the case of linear expanding maps), then the transfer operator of the skew product has a spectral gap. Using semiclassical analysis we show that the spectral gap is preserved under small random perturbations. This implies exponential decay of quenched random correlation functions for smooth observables at small noise levels.
In this paper a counterpart of the classical Poisson integral formula is found for a class of standard weighted Laplace differential operators in the unit disc. In the process the corresponding Dirichlet boundary value problem is solved for arbitrary distributional boundary data. Boundary limits and representations of the associated solutions are studied within a framework of homogeneous Banach spaces. Special emphasis is put on the so-called relative completion of a homogeneous Banach space.
We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.
Signal apparition is a recent signal processing advance that has numerous applications in seismic data acquisition and processing. In this paper we review the basic principles of signal apparition and discuss applications related to simultaneous source acquisition. We discuss the generalization of the technique to large number of sources and the application in a full 3D configuration enabling large productivity gains and the acquisition of broad band seismic data.
We study a counterpart of the classical Poisson integral for a family of weighted Laplace differential equations in Euclidean half space, solutions of which are known as generalized axially symmetric potentials. These potentials appear naturally in the study of hyperbolic Brownian motion with drift. We determine the optimal class of tempered distributions which by means of the so-called l'-convolution can be extended to generalized axially symmetric potentials. In the process, the associated Dirichlet boundary value problem is solved, and we obtain sharp order relations for the asymptotic growth of these extensions. (C) 2015 Elsevier Masson SAS. All rights reserved.
We obtain microlocal analogues of results by L. Hörmander about inclusion relations between the ranges of first order differential operators with coefficients in C∞ that fail to be locally solvable. Using similar techniques, we study the properties of the range of classical pseudodifferential operators of principal type that fail to satisfy condition (Ψ).
The purpose of this thesis is to obtain microlocal analogues of results by L. Hörmander about inclusion relations between the ranges of first order differential operators with smooth coefficients which fail to be locally solvable. Using similar techniques, we shall study the properties of the range of classical pseudo-differential operators of principal type which fail to satisfy condition (ψ).
The recently introduced method of ‘signal apparition’ offers a fundamentally different approach to separation of multiple interfering sources, by using a periodic sequence of source signatures along one source line. This leads to exact separation of signals in diamond-shaped regions of the frequency-wavenumber domain which are twice as large compared to those recovered by other methods. In this paper we investigate the method’s sensitivity to the appearance of white noise in the periodic sequence, and show that signal apparition is stable by using a probabilistic model. We also demonstrate the stability by numerical simulations on a finite-difference synthetic data set generated over a complex salt model.
A major focus of research in the seismic industry of the past two decades has been the acquisition and subsequent separation of seismic data using multiple sources fired simultaneously. The recently introduced method of signal apparition provides a new take on the problem by replacing the random time-shifts usually employed to encode the different sources by fully deterministic periodic time-shifts. In this paper, we give a mathematical proof showing that the signal apparition method results in optimally large regions in the frequency–wavenumber space where exact separation of sources is achieved. These regions are diamond shaped and we prove that using any other method of source encoding results in strictly smaller regions of exact separation. The results are valid for arbitrary number of sources. Numerical examples for different number of sources (three, respectively, four sources) demonstrate the exact recovery of these diamond-shaped regions. The implementation of the theoretical proofs in the field is illustrated by the results of a conducted field test.
In the field of exploration geophysics, the method known as "signal apparition" offers a different perspective on how to separate signals acquired from simultaneously fired seismic sources. The method uses a periodic sequence of small time shift variations to encode the different sources, and this choice leads to exact separation of signals in diamond-shaped regions of the frequency-wavenumber domain which are twice as large compared to those recovered by other methods. In this paper we investigate the signal apparition method's sensitivity to perturbations of the periodic time shift sequence. We model (measured or unknown) perturbations in a stochastic fashion and prove that the resulting inverse problem of separating the data is still well-posed, and we demonstrate the stability by numerical simulations on a finite-difference synthetic data set generated over a complex salt model.
We describe a method for removing the numerical errors in the modeling of linear evolution equations that are caused by approximating the time derivative by a finite difference operator. The method is based on integral transforms realized as certain Fourier integral operators, called time dispersion transforms, and we prove that, under an assumption about the frequency content, it yields a solution with correct evolution throughout the entire lifespan. We demonstrate the method on a model equation as well as on the simulation of elastic and viscoelastic wave propagation.