Robust inversion of seismic data using the Huber norm

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1310-1319 ◽  
Author(s):  
Antoine Guitton ◽  
William W. Symes
Keyword(s):  
Seismic Data ◽  
Field Data ◽  
Gradient Methods ◽  
Nonlinear Problems ◽  
Velocity Analysis ◽  
Error Measures ◽  
Robust Model ◽  
Least Squares Fit ◽  
Gradient Based ◽  
Quasi Newton

The “Huber function” (or “Huber norm” ) is one of several robust error measures which interpolates between smooth (ℓ2) treatment of small residuals and robust (ℓ1) treatment of large residuals. Since the Huber function is differentiable, it may be minimized reliably with a standard gradient‐based optimizer. We propose to minimize the Huber function with a quasi‐Newton method that has the potential of being faster and more robust than conjugate‐gradient methods when solving nonlinear problems. Tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the Huber function gives far more robust model estimates than does a least‐squares fit with or without damping.

Poststack velocity analysis by separation and imaging of seismic diffractions

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. U89-U94 ◽  
Author(s):  
Sergey Fomel ◽  
Evgeny Landa ◽  
M. Turhan Taner
Keyword(s):  
High Resolution ◽  
Seismic Data ◽  
Field Data ◽  
Small Scale ◽  
Velocity Analysis ◽  
High Resolution Imaging ◽  
Diffracted Waves ◽  
Resolution Imaging ◽  
Seismic Reflections ◽  
Optimal Focusing

Small geologic features manifest themselves in seismic data in the form of diffracted waves, which are fundamentally different from seismic reflections. Using two field-data examples and one synthetic example, we demonstrate the possibility of separating seismic diffractions in the data and imaging them with optimally chosen migration velocities. Our criteria for separating reflection and diffraction events are the smoothness and continuity of local event slopes that correspond to reflection events. For optimal focusing, we develop the local varimax measure. The objectives of this work are velocity analysis implemented in the poststack domain and high-resolution imaging of small-scale heterogeneities. Our examples demonstrate the effectiveness of the proposed method for high-resolution imaging of such geologic features as faults, channels, and salt boundaries.

scholarly journals High-resolution coherency functionals for velocity analysis: An application for subbasalt seismic exploration

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. U53-U63 ◽  
Author(s):  
Andrea Tognarelli ◽  
Eusebio Stucchi ◽  
Alessia Ravasio ◽  
Alfredo Mazzotti
Keyword(s):  
High Resolution ◽  
Seismic Data ◽  
Field Data ◽  
Signal To Noise Ratio ◽  
Synthetic Seismograms ◽  
Seismic Exploration ◽  
Velocity Analysis ◽  
Signal To Noise ◽  
Geologic Setting ◽  
Analytic Signals

We tested the properties of three different coherency functionals for the velocity analysis of seismic data relative to subbasalt exploration. We evaluated the performance of the standard semblance algorithm and two high-resolution coherency functionals based on the use of analytic signals and of the covariance estimation along hyperbolic traveltime trajectories. Approximate knowledge of the wavelet was exploited to design appropriate filters that matched the primary reflections, thereby further improving the ability of the functionals to highlight the events of interest. The tests were carried out on two synthetic seismograms computed on models reproducing the geologic setting of basaltic intrusions and on common midpoint gathers from a 3D survey. Synthetic and field data had a very low signal-to-noise ratio, strong multiple contamination, and weak primary subbasalt signals. The results revealed that high-resolution coherency functionals were more suitable than semblance algorithms to detect primary signals and to distinguish them from multiples and other interfering events. This early discrimination between primaries and multiples could help to target specific signal enhancement and demultiple operations.

scholarly journals Accelerating Symmetric Rank-1 Quasi-Newton Method with Nesterov’s Gradient for Training Neural Networks

2021 ◽  
Author(s):  
S. Indrapriyadarsini ◽  
Shahrzad Mahboubi ◽  
Hiroshi Ninomiya ◽  
Takeshi Kamio ◽  
Hideki Asai
Keyword(s):  
Neural Networks ◽  
Newton Method ◽  
Trust Region ◽  
Nonlinear Problems ◽  
Classification Problem ◽  
Second Order ◽  
Highly Nonlinear ◽  
Gradient Based ◽  
Quasi Newton ◽  
Second Order Methods

Gradient based methods are popularly used in training neural networks and can be broadly categorized into first and second order methods. Second order methods have shown to have better convergence compared to first order methods, especially in solving highly nonlinear problems. The BFGS quasi-Newton method is the most commonly studied second order method for neural network training. Recent methods have shown to speed up the convergence of the BFGS method using the Nesterov’s acclerated gradient and momentum terms. The SR1 quasi-Newton method though less commonly used in training neural networks, are known to have interesting properties and provide good Hessian approximations when used with a trust-region approach. Thus, this paper aims to investigate accelerating the Symmetric Rank-1 (SR1) quasi-Newton method with the Nesterov’s gradient for training neural networks and briefly discuss its convergence. The performance of the proposed method is evaluated on a function approximation and image classification problem.

scholarly journals Accelerating Symmetric Rank-1 Quasi-Newton Method with Nesterov’s Gradient for Training Neural Networks

2021 ◽  
Author(s):  
S. Indrapriyadarsini ◽  
Shahrzad Mahboubi ◽  
Hiroshi Ninomiya ◽  
Takeshi Kamio ◽  
Hideki Asai
Keyword(s):  
Neural Networks ◽  
Newton Method ◽  
Trust Region ◽  
Nonlinear Problems ◽  
Classification Problem ◽  
Second Order ◽  
Highly Nonlinear ◽  
Gradient Based ◽  
Quasi Newton ◽  
Second Order Methods

Gradient based methods are popularly used in training neural networks and can be broadly categorized into first and second order methods. Second order methods have shown to have better convergence compared to first order methods, especially in solving highly nonlinear problems. The BFGS quasi-Newton method is the most commonly studied second order method for neural network training. Recent methods have shown to speed up the convergence of the BFGS method using the Nesterov’s acclerated gradient and momentum terms. The SR1 quasi-Newton method though less commonly used in training neural networks, are known to have interesting properties and provide good Hessian approximations when used with a trust-region approach. Thus, this paper aims to investigate accelerating the Symmetric Rank-1 (SR1) quasi-Newton method with the Nesterov’s gradient for training neural networks and briefly discuss its convergence. The performance of the proposed method is evaluated on a function approximation and image classification problem.

Rayleigh-Marchenko redatuming for target-oriented, true-amplitude imaging

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. S439-S452 ◽  
Author(s):  
Matteo Ravasi
Keyword(s):  
Seismic Data ◽  
Field Data ◽  
Scaling Factor ◽  
Velocity Analysis ◽  
Ocean Bottom ◽  
Structural Imaging ◽  
Dual Sensor ◽  
Band Limited ◽  
Borehole Seismic ◽  
Amplitude Versus

Marchenko redatuming is a revolutionary technique to estimate Green’s functions from virtual sources in the subsurface using only data measured at the earth’s surface, without having to place either sources or receivers in the subsurface. This goal is achieved by crafting special wavefields (so-called focusing functions) that can focus energy at a chosen point in the subsurface. Despite its great potential, strict requirements on the reflection response such as knowledge and accurate deconvolution of the source wavelet (including absolute scaling factor) and co-location of sources and receivers have so far challenged the application of Marchenko redatuming to real-world scenarios. I combine the coupled Marchenko equations with a one-way version of the Rayleigh integral representation to obtain a new redatuming scheme that handles internal as well as free-surface multiples using dual-sensor, band-limited seismic data (with an unknown source signature) from any acquisition system that presents arbitrarily located sources above a line of regularly sampled receivers—for example, ocean-bottom, source-over-spread streamer, and horizontal borehole seismic data. The redatuming scheme is validated using synthetic and field data, and the retrieved subsurface wavefields are used for improved structural imaging and taken as input for the computation of true-amplitude angle gathers, which can lead to more accurate amplitude-versus-angle interpretation and velocity analysis.

scholarly journals Accelerating Symmetric Rank-1 Quasi-Newton Method with Nesterov’s Gradient for Training Neural Networks

Algorithms ◽  
2021 ◽  
Vol 15 (1) ◽  
pp. 6
Author(s):  
S. Indrapriyadarsini ◽  
Shahrzad Mahboubi ◽  
Hiroshi Ninomiya ◽  
Takeshi Kamio ◽  
Hideki Asai
Keyword(s):  
Neural Networks ◽  
Newton Method ◽  
Trust Region ◽  
Nonlinear Problems ◽  
Classification Problem ◽  
Second Order ◽  
Highly Nonlinear ◽  
Gradient Based ◽  
Quasi Newton ◽  
Second Order Methods

Gradient-based methods are popularly used in training neural networks and can be broadly categorized into first and second order methods. Second order methods have shown to have better convergence compared to first order methods, especially in solving highly nonlinear problems. The BFGS quasi-Newton method is the most commonly studied second order method for neural network training. Recent methods have been shown to speed up the convergence of the BFGS method using the Nesterov’s acclerated gradient and momentum terms. The SR1 quasi-Newton method, though less commonly used in training neural networks, is known to have interesting properties and provide good Hessian approximations when used with a trust-region approach. Thus, this paper aims to investigate accelerating the Symmetric Rank-1 (SR1) quasi-Newton method with the Nesterov’s gradient for training neural networks, and to briefly discuss its convergence. The performance of the proposed method is evaluated on a function approximation and image classification problem.

scholarly journals Optimization techniques for tree-structured nonlinear problems

2020 ◽  
Vol 17 (3) ◽  
pp. 409-436
Author(s):  
Jens Hübner ◽  
Martin Schmidt ◽  
Marc C. Steinbach
Keyword(s):  
Predictive Control ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Nonlinear Problems ◽  
Optimization Techniques ◽  
Active Set ◽  
Robust Model Predictive Control ◽  
Robust Model ◽  
Quasi Newton ◽  
Structured Optimization

Abstract Robust model predictive control approaches and other applications lead to nonlinear optimization problems defined on (scenario) trees. We present structure-preserving Quasi-Newton update formulas as well as structured inertia correction techniques that allow to solve these problems by interior-point methods with specialized KKT solvers for tree-structured optimization problems. The same type of KKT solvers could be used in active-set based SQP methods. The viability of our approach is demonstrated by two robust control problems.

Source-side spatial wavefield gradients in land seismic exploration

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. P73-P85 ◽  
Author(s):  
Cédéric Van Renterghem ◽  
Cédric Schmelzbach ◽  
David Sollberger ◽  
Mauro Häusler ◽  
Johan Olof Anders Robertsson
Keyword(s):  
Seismic Data ◽  
Field Data ◽  
Seismic Exploration ◽  
Gradient Estimates ◽  
Spatial Gradient ◽  
Receiver Side ◽  
S Wave ◽  
Wave Source ◽  
Rotational Components ◽  
Gradient Based

The recording of seismic data using arrays of densely spaced receivers enables the estimation of the spatial gradient components of the wavefield, in addition to the acquisition of conventional translational motion. We have extended the concept of array-based receiver-side gradiometry to the source-side and investigated the potential of combining source- and receiver-side gradient estimates for land seismic exploration. The robustness of array-based gradient source formation is demonstrated with a field data reciprocity experiment. We apply a gradient-based elastic wavefield decomposition technique to small arrays of densely spaced vertically and horizontally oriented force sources and determine with synthetic and field data examples that the processing of data obtained from multicomponent source arrays allows us to simulate a composite source that theoretically only emits S-waves at all emergence angles. A promising application of the gradient-based S-wave source is downhole S-wave imaging. Finally, by combining source- and receiver-side gradient estimates, 49C seismic data can be obtained comprising three translational components, three rotational components, and one divergence component on the source and receiver side. This concept could have a significant potential to enhance the acquisition and processing of data from locally dense arrays in land seismic exploration.

A Comprehensive Investigation of Ensembles of Gaussian-Based and Gradient-Based Transformed Reliability Analyses: When and How to Use Them

2016 ◽  
Author(s):  
Po Ting Lin ◽  
Wei-Hao Lu ◽  
Shu-Ping Lin
Keyword(s):  
Design Optimization ◽  
Design Space ◽  
Optimization Problems ◽  
Nonlinear Problems ◽  
Kernel Functions ◽  
Sensitivity Analyses ◽  
Normal Space ◽  
Highly Nonlinear ◽  
Standard Normal ◽  
Gradient Based

In the past few years, researchers have begun to investigate the existence of arbitrary uncertainties in the design optimization problems. Most traditional reliability-based design optimization (RBDO) methods transform the design space to the standard normal space for reliability analysis but may not work well when the random variables are arbitrarily distributed. It is because that the transformation to the standard normal space cannot be determined or the distribution type is unknown. The methods of Ensemble of Gaussian-based Reliability Analyses (EoGRA) and Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) have been developed to estimate the joint probability density function using the ensemble of kernel functions. EoGRA performs a series of Gaussian-based kernel reliability analyses and merged them together to compute the reliability of the design point. EGTRA transforms the design space to the single-variate design space toward the constraint gradient, where the kernel reliability analyses become much less costly. In this paper, a series of comprehensive investigations were performed to study the similarities and differences between EoGRA and EGTRA. The results showed that EGTRA performs accurate and effective reliability analyses for both linear and nonlinear problems. When the constraints are highly nonlinear, EGTRA may have little problem but still can be effective in terms of starting from deterministic optimal points. On the other hands, the sensitivity analyses of EoGRA may be ineffective when the random distribution is completely inside the feasible space or infeasible space. However, EoGRA can find acceptable design points when starting from deterministic optimal points. Moreover, EoGRA is capable of delivering estimated failure probability of each constraint during the optimization processes, which may be convenient for some applications.

THE DETERMINATION OF DIGITAL WIENER FILTERS BY MEANS OF GRADIENT METHODS

Geophysics ◽  
1973 ◽  
Vol 38 (2) ◽  
pp. 310-326 ◽  
Author(s):  
R. J. Wang ◽  
S. Treitel
Keyword(s):  
Seismic Data ◽  
High Speed ◽  
Gradient Methods ◽  
Wiener Filter ◽  
Gradient Algorithm ◽  
Rapid Convergence ◽  
True Solution ◽  
Peripheral Device ◽  
Wiener Filters

The normal equations for the discrete Wiener filter are conventionally solved with Levinson’s algorithm. The resultant solutions are exact except for numerical roundoff. In many instances, approximate rather than exact solutions satisfy seismologists’ requirements. The so‐called “gradient” or “steepest descent” iteration techniques can be used to produce approximate filters at computing speeds significantly higher than those achievable with Levinson’s method. Moreover, gradient schemes are well suited for implementation on a digital computer provided with a floating‐point array processor (i.e., a high‐speed peripheral device designed to carry out a specific set of multiply‐and‐add operations). Levinson’s method (1947) cannot be programmed efficiently for such special‐purpose hardware, and this consideration renders the use of gradient schemes even more attractive. It is, of course, advisable to utilize a gradient algorithm which generally provides rapid convergence to the true solution. The “conjugate‐gradient” method of Hestenes (1956) is one of a family of algorithms having this property. Experimental calculations performed with real seismic data indicate that adequate filter approximations are obtainable at a fraction of the computer cost required for use of Levinson’s algorithm.

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