Fourier reconstruction of nonuniformly sampled, aliased seismic data

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. V21-V32 ◽  
Author(s):  
P. M. Zwartjes ◽  
M. D. Sacchi
Keyword(s):  
Seismic Data ◽  
Fourier Coefficients ◽  
A Priori ◽  
Nonuniform Sampling ◽  
Second Step ◽  
Fourier Domain ◽  
Sampled Data ◽  
Underlying Assumption ◽  
Fourier Reconstruction ◽  
Step Algorithm

There are numerous methods for interpolating uniformly sampled, aliased seismic data, but few can handle the combination of nonuniform sampling and aliasing. We combine the principles of Fourier reconstruction of nonaliased, nonuniformly sampled data with the ideas of frequency-wavenumber [Formula: see text] interpolation of aliased, uniformly sampled data in a new two-step algorithm. In the first step, we estimate the Fourier coefficients at the lower nonaliased temporal frequencies from the nonuniformly sampled data. The coefficients are then used in the second step as an a priori model to distinguish between aliased and nonaliased energy at the higher, aliased temporal frequencies. By using a nonquadratic model penalty in the inversion, both the artifacts in the Fourier domain from nonuniform sampling and the aliased energy are suppressed. The underlying assumption is that events are planar; therefore, the algorithm is applied to seismic data in overlapping spatiotemporal windows.

Antileakage Fourier transform for seismic data regularization

Geophysics ◽  
2005 ◽  
Vol 70 (4) ◽  
pp. V87-V95 ◽  
Author(s):  
Sheng Xu ◽  
Yu Zhang ◽  
Don Pham ◽  
Gilles Lambaré
Keyword(s):  
Fourier Transform ◽  
Fourier Coefficient ◽  
Seismic Data ◽  
Fourier Coefficients ◽  
Real Data ◽  
Sampled Data ◽  
Spectral Leakage ◽  
Irregular Grid ◽  
Fourier Basis ◽  
Data Regularization

Seismic data regularization, which spatially transforms irregularly sampled acquired data to regularly sampled data, is a long-standing problem in seismic data processing. Data regularization can be implemented using Fourier theory by using a method that estimates the spatial frequency content on an irregularly sampled grid. The data can then be reconstructed on any desired grid. Difficulties arise from the nonorthogonality of the global Fourier basis functions on an irregular grid, which results in the problem of “spectral leakage”: energy from one Fourier coefficient leaks onto others. We investigate the nonorthogonality of the Fourier basis on an irregularly sampled grid and propose a technique called “antileakage Fourier transform” to overcome the spectral leakage. In the antileakage Fourier transform, we first solve for the most energetic Fourier coefficient, assuming that it causes the most severe leakage. To attenuate all aliases and the leakage of this component onto other Fourier coefficients, the data component corresponding to this most energetic Fourier coefficient is subtracted from the original input on the irregular grid. We then use this new input to solve for the next Fourier coefficient, repeating the procedure until all Fourier coefficients are estimated. This procedure is equivalent to “reorthogonalizing” the global Fourier basis on an irregularly sampled grid. We demonstrate the robustness and effectiveness of this technique with successful applications to both synthetic and real data examples.

Diffraction tomography applied to crosshole and VSP seismic data

1989 ◽  
Vol 20 (2) ◽  
pp. 169
Author(s):  
J.A. Young
Keyword(s):  
Seismic Data ◽  
A Priori ◽  
Tomographic Reconstruction ◽  
Synthetic Data ◽  
Scattered Wave ◽  
Seismic Inversion ◽  
Image Resolution ◽  
Fourier Domain ◽  
Diffraction Tomography ◽  
Data Coverage

Diffraction tomography is an approach to seismic inversion which is analogous to f-k migration. It differs from f-k migration in that it attempts to obtain a more quantitative rather than qualitative image of the Earth's subsurface. Diffraction tomography is based on the generalized projection-slice theorem which relates the scattered wave field to the Fourier spectrum of the scatterer. Factors such as the survey geometry and the source bandwidth determine the data coverage in the spatial Fourier domain which in turn determines the image resolution. Limited view-angles result in regions of the spatial Fourier domain with no data coverage, causing the solution to the tomographic reconstruction problem to be nonunique. The simplistic approach is to assume the missing samples are zero and perform a standard reconstruction but this can result in images with severe artefacts. Additional a priori information can be introduced to the problem in order to reduce the nonuniqueness and increase the stability of the reconstruction. This is the standard approach used in ray tomography but it is not commonly used in diffraction tomography applied to seismic data.This paper shows the application of diffraction tomography to crosshole and VSP seismic data. Using synthetic data, the effects on image resolution of the survey geometry and the finite source bandwidth are examined and techniques for improving image quality are discussed.

scholarly journals Brainwide functional networks associated with anatomically- and functionally-defined hippocampal subfields using ultrahigh-resolution fMRI

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Wei-Tang Chang ◽  
Stephanie K. Langella ◽  
Yichuan Tang ◽  
Sahar Ahmad ◽  
Han Zhang ◽  
...  
Keyword(s):  
Resting State ◽  
Functional Organization ◽  
A Priori ◽  
Longitudinal Axis ◽  
Functional Networks ◽  
Resting State Functional Connectivity ◽  
Underlying Assumption ◽  
Isotropic Resolution ◽  
Hippocampal Subfields ◽  
Fine Grained

AbstractThe hippocampus is critical for learning and memory and may be separated into anatomically-defined hippocampal subfields (aHPSFs). Hippocampal functional networks, particularly during resting state, are generally analyzed using aHPSFs as seed regions, with the underlying assumption that the function within a subfield is homogeneous, yet heterogeneous between subfields. However, several prior studies have observed similar resting-state functional connectivity (FC) profiles between aHPSFs. Alternatively, data-driven approaches investigate hippocampal functional organization without a priori assumptions. However, insufficient spatial resolution may result in a number of caveats concerning the reliability of the results. Hence, we developed a functional Magnetic Resonance Imaging (fMRI) sequence on a 7 T MR scanner achieving 0.94 mm isotropic resolution with a TR of 2 s and brain-wide coverage to (1) investigate the functional organization within hippocampus at rest, and (2) compare the brain-wide FC associated with fine-grained aHPSFs and functionally-defined hippocampal subfields (fHPSFs). This study showed that fHPSFs were arranged along the longitudinal axis that were not comparable to the lamellar structures of aHPSFs. For brain-wide FC, the fHPSFs rather than aHPSFs revealed that a number of fHPSFs connected specifically with some of the functional networks. Different functional networks also showed preferential connections with different portions of hippocampal subfields.

Sensor-Based Estimation of Dim Light Melatonin Onset Using Features of Two Time Scales

2021 ◽  
Vol 2 (3) ◽  
pp. 1-15
Author(s):  
Cheng Wan ◽  
Andrew W. Mchill ◽  
Elizabeth B. Klerman ◽  
Akane Sano
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Light Exposure ◽  
Biological Activities ◽  
Second Step ◽  
Sampled Data ◽  
Dim Light Melatonin Onset ◽  
Dim Light ◽  
Using Data ◽  
Mean Square Errors

Circadian rhythms influence multiple essential biological activities, including sleep, performance, and mood. The dim light melatonin onset (DLMO) is the gold standard for measuring human circadian phase (i.e., timing). The collection of DLMO is expensive and time consuming since multiple saliva or blood samples are required overnight in special conditions, and the samples must then be assayed for melatonin. Recently, several computational approaches have been designed for estimating DLMO. These methods collect daily sampled data (e.g., sleep onset/offset times) or frequently sampled data (e.g., light exposure/skin temperature/physical activity collected every minute) to train learning models for estimating DLMO. One limitation of these studies is that they only leverage one time-scale data. We propose a two-step framework for estimating DLMO using data from both time scales. The first step summarizes data from before the current day, whereas the second step combines this summary with frequently sampled data of the current day. We evaluate three moving average models that input sleep timing data as the first step and use recurrent neural network models as the second step. The results using data from 207 undergraduates show that our two-step model with two time-scale features has statistically significantly lower root-mean-square errors than models that use either daily sampled data or frequently sampled data.

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

2003 ◽  
Vol 10 (3) ◽  
pp. 401-410
Author(s):  
M. S. Agranovich ◽  
B. A. Amosov
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Elliptic Boundary ◽  
A Priori ◽  
Adjoint Problem ◽  
Elliptic Boundary Value ◽  
Right Hand ◽  
The Difference ◽  
The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

scholarly journals Integration of onshore and offshore seismic arrays to study the seismicity of the Calabrian Region: a two steps automatic procedure for the identification of the best stations geometry

2014 ◽  
Vol 36 ◽  
pp. 69-75 ◽  
Author(s):  
A. D'Alessandro ◽  
I. Guerra ◽  
G. D'Anna ◽  
A. Gervasi ◽  
P. Harabaglia ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Network Optimization ◽  
A Priori ◽  
Second Step ◽  
Ocean Bottom ◽  
Broadband Seismometer ◽  
Seismic Stations ◽  
Seismic Arrays ◽  
Monitoring Campaign ◽  
Seismographic Network

Abstract. We plan to deploy in the Taranto Gulf some Ocean Bottom broadband Seismometer with Hydrophones. Our aim is to investigate the offshore seismicity of the Sibari Gulf. The seismographic network optimization consists in the identification of the optimal sites for the installation of the offshore stations, which is a crucial factor for the success of the monitoring campaign. In this paper, we propose a two steps automatic procedure for the identification of the best stations geometry. In the first step, based on the application of a set of a priori criteria, the suitable sites to host the ocean bottom seismic stations are identified. In the second step, the network improvement is evaluated for all the possible stations geometries by means of numerical simulation. The application of this procedure allows us to identify the best stations geometry to be achieved in the monitoring campaign.

scholarly journals Smoothing and Interpolating Noisy GPS Data with Smoothing Splines

2020 ◽  
Vol 37 (3) ◽  
pp. 449-465 ◽  
Author(s):  
Jeffrey J. Early ◽  
Adam M. Sykulski
Keyword(s):  
Gaussian Noise ◽  
A Priori ◽  
Smoothing Splines ◽  
Sampled Data ◽  
Trajectory Data ◽  
Tension Parameter ◽  
Comprehensive Method ◽  
Gps Trajectory Data ◽  
Non Gaussian ◽  
Irregularly Sampled Data

AbstractA comprehensive method is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers that are typical in global positioning system (GPS) signals. We demonstrate the effectiveness of our methods on GPS trajectory data obtained from oceanographic floating instruments known as drifters.

scholarly journals Deep Compressed Sensing for Learning Submodular Functions

Sensors ◽  
2020 ◽  
Vol 20 (9) ◽  
pp. 2591 ◽  
Author(s):  
Yu-Chung Tsai ◽  
Kuo-Shih Tseng
Keyword(s):  
Compressed Sensing ◽  
Fourier Coefficients ◽  
State Of The Art ◽  
The State ◽  
Submodular Functions ◽  
Fourier Domain ◽  
3D Mapping ◽  
Target Search ◽  
Approach To Learning ◽  
Benchmark Approach

The AI community has been paying attention to submodular functions due to their various applications (e.g., target search and 3D mapping). Learning submodular functions is a challenge since the number of a function’s outcomes of N sets is 2 N . The state-of-the-art approach is based on compressed sensing techniques, which are to learn submodular functions in the Fourier domain and then recover the submodular functions in the spatial domain. However, the number of Fourier bases is relevant to the number of sets’ sensing overlapping. To overcome this issue, this research proposed a submodular deep compressed sensing (SDCS) approach to learning submodular functions. The algorithm consists of learning autoencoder networks and Fourier coefficients. The learned networks can be applied to predict 2 N values of submodular functions. Experiments conducted with this approach demonstrate that the algorithm is more efficient than the benchmark approach.

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