Two‐dimensional velocity inversion and synthetic seismogram computation

Geophysics ◽  
1987 ◽  
Vol 52 (1) ◽  
pp. 37-50 ◽  
Author(s):  
Tianfei Zhu ◽  
Larry D. Brown
Keyword(s):  
Least Squares ◽  
Velocity Structure ◽  
Trust Region ◽  
Damping Factor ◽  
Two Dimensional ◽  
Velocity Inversion ◽  
Least Squares Problem ◽  
Velocity Models ◽  
Marquardt Method ◽  
Levenberg Marquardt

A traveltime inversion scheme has been developed to estimate velocity and interface geometries of two‐dimensional media from deep reflection data. The velocity structure is represented by finite elements, and the inversion is formulated as an iterative, constrained, linear least‐squares problem which can be solved by either the singular value truncation method or the Levenberg‐Marquardt method. The damping factor of the Levenberg‐Marquardt method is chosen by the model‐trust region approach. The traveltimes and derivative matrix required to solve the least‐squares problem are computed by ray tracing. To aid seismic interpretation, we also include in the inversion scheme a fast algorithm based on asymptotic ray theory for calculating synthetic seismograms from the derived velocity model. Numerical tests indicate that the inversion scheme is effective, and that the accuracy of inversion results depends upon both noise in the data and the aperture of recording used in data acquisition. Two real examples demonstrate that the new inversion scheme produces velocity models fitting the data better than those estimated by other approaches.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

New insights into structure and seismicity of the Central Alps from 3D Pg and Sg tomography and improved hypocenter relocations

2021 ◽  
Author(s):  
Tobias Diehl ◽  
Edi Kissling ◽  
Marco Herwegh ◽  
Stefan Schmid
Keyword(s):  
Velocity Structure ◽  
Focal Depth ◽  
Small Scale ◽  
Earthquake Catalog ◽  
Depth Range ◽  
Central Alps ◽  
Quartz Content ◽  
Phase Selection ◽  
Velocity Inversion ◽  
Velocity Models

<p>Accuracy of hypocenter location, in particular focal depth, is a precondition for high-resolution seismotectonic analysis of natural and induced seismicity. For instance, linking seismicity with mapped fault segments requires hypocenter accuracy at the sub-kilometer scale. In this study, we demonstrate that inaccurate velocity models and improper phase selection can bias absolute hypocenter locations and location uncertainties, resulting in errors larger than the targeted accuracy. To avoid such bias in densely instrumented seismic networks, we propose a coupled hypocenter-velocity inversion restricted to direct, upper-crustal Pg and Sg phases. The derived three-dimensional velocity models, combined with dynamic phase selection and non-linear location algorithms result in a highly accurate earthquake catalog, including consistent hypocenter uncertainties. We apply this procedure to about 60’000 Pg and 30’000 Sg quality-checked phases of local earthquakes in the Central Alps region. The derived tomographic models image the Vp and Vs velocity structure of the Central Alps’ upper crust at unprecedented resolution, including small-scale anomalies such as those caused by a Permo-Carboniferous trough in the northern foreland, Subalpine Molasse below the Alpine front or crystalline basement units within the Penninic nappes. The external Aar Massif is characterized by low Vp/Vs ratios of about 1.625-1.675 in the depth range of 2-6.5 km, which we relate to a felsic composition of the uplifted crustal block, possibly with increased quartz content. Finally, we discuss along-strike variations imaged by relocated seismicity in the Central Alps and demonstrate how joint interpretation of velocity structure and hypocenters provides additional constraints on lithologies of upper-crustal seismicity.</p>

Comparison of Some Parameter Estimation Techniques Applied to Proton Exchange Membrane Fuel Cell Models

2013 ◽  
Vol 10 (5) ◽  
Author(s):  
Ágnes Havasi ◽  
Róbert Horváth ◽  
Tamás Szabó
Keyword(s):  
Fuel Cell ◽  
Proton Exchange Membrane ◽  
Trust Region Method ◽  
Trust Region ◽  
Proton Exchange ◽  
Unknown Parameters ◽  
Marquardt Method ◽  
Parameter Fitting ◽  
Levenberg Marquardt ◽  
Exchange Membrane

The functioning and the achievable power of a proton exchange membrane fuel cell (PEMFC) are determined by several parameters simultaneously. Part of these cannot be measured directly. They must be estimated with parameter fitting techniques. In order to give reliable estimations for the unknown parameters, we first set up an adequate finite difference numerical solution of the mathematical model of the fuel cell. Then the values of the unknown parameters are calculated by fitting the model results to measurements. In this paper our primary aim is to compare several parameter fitting tools on the model of a PEMFC and give a prescription for the use of these methods. We test three methods together with their variants: the Levenberg–Marquardt method, the trust region method, and the simulated annealing method, among which the Levenberg–Marquardt method turns to be the most efficient one.

scholarly journals Hybrid Levenberg–Marquardt and weak-constraint ensemble Kalman smoother method

2016 ◽  
Vol 23 (2) ◽  
pp. 59-73 ◽  
Author(s):  
J. Mandel ◽  
E. Bergou ◽  
S. Gürol ◽  
S. Gratton ◽  
I. Kasanický
Keyword(s):  
Least Squares ◽  
Newton Method ◽  
Nonlinear Least Squares ◽  
Additional Observation ◽  
Linear Least Squares ◽  
Regularization Term ◽  
Marquardt Method ◽  
Kalman Smoother ◽  
Levenberg Marquardt ◽  
Adjoint Operators

Abstract. The ensemble Kalman smoother (EnKS) is used as a linear least-squares solver in the Gauss–Newton method for the large nonlinear least-squares system in incremental 4DVAR. The ensemble approach is naturally parallel over the ensemble members and no tangent or adjoint operators are needed. Furthermore, adding a regularization term results in replacing the Gauss–Newton method, which may diverge, by the Levenberg–Marquardt method, which is known to be convergent. The regularization is implemented efficiently as an additional observation in the EnKS. The method is illustrated on the Lorenz 63 model and a two-level quasi-geostrophic model.

scholarly journals An inexact Levenberg-Marquardt method for large sparse nonlinear least squres

1985 ◽  
Vol 26 (4) ◽  
pp. 387-403 ◽  
Author(s):  
S. J. Wright ◽  
J. N. Holt
Keyword(s):  
Global Convergence ◽  
Least Squares ◽  
Jacobian Matrix ◽  
Numerical Test ◽  
Convergence Result ◽  
Test Results ◽  
Linear Least Squares ◽  
Optimal Point ◽  
Marquardt Method ◽  
Levenberg Marquardt

AbstractA method for solving problems of the form is presented. The approach of Levenberg and Marquardt is used, except that the linear least squares subproblem arising at each iteration is not solved exactly, but only to within a certain tolerance. The method is most suited to problems in which the Jacobian matrix is sparse. Use is made of the iterative algorithm LSQR of Paige and Saunders for sparse linear least squares.A global convergence result can be proven, and under certain conditions it can be shown that the method converges quadratically when the sum of squares at the optimal point is zero.Numerical test results for problems of varying residual size are given.

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