A Helmholtz iterative solver for 3D seismic-imaging problems

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM185-SM194 ◽  
Author(s):  
R.-E. Plessix
Keyword(s):  
Wave Equation ◽  
Wave Form ◽  
Iterative Solver ◽  
Grid Spacing ◽  
Scale Separation ◽  
Approximate Inverse ◽  
Full Wave ◽  
Order Of Magnitude ◽  
Preconditioned Iterative ◽  
The Time Domain

A preconditioned iterative solver for the 3D frequency-domain wave equation applied to seismic problems is evaluated. The preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the (undamped) wave equation. The approximate inverse is computed with one multigrid cycle. Numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength. To evaluate the relevance of this iterative solver, the number of floating-point operations required for two imaging problems are roughly evaluated. This rough estimate indicates that the time-domain migration approach is more than one order of magnitude faster. The full-wave-form tomography, based on a least-squares formulation and a scale separation approach, has the same complexity in both domains.

scholarly journals A new iterative solver for the time-harmonic wave equation

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E57-E63 ◽  
Author(s):  
C. D. Riyanti ◽  
Y. A. Erlangga ◽  
R.-E. Plessix ◽  
W. A. Mulder ◽  
C. Vuik ◽  
...  
Keyword(s):  
Wave Equation ◽  
Linear System ◽  
Time Domain ◽  
Multigrid Method ◽  
Harmonic Wave ◽  
Computational Time ◽  
Full Time ◽  
Iterative Solver ◽  
Time Harmonic ◽  
The Time Domain

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

scholarly journals Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

2015 ◽  
Vol 25 (10) ◽  
pp. 1927-1960 ◽  
Author(s):  
Hoai-Minh Nguyen ◽  
Linh Viet Nguyen
Keyword(s):  
Wave Equation ◽  
Three Dimensions ◽  
Impedance Boundary ◽  
Change Of Variables ◽  
Full Wave ◽  
Time Harmonic ◽  
Time Regime ◽  
Frequency Regime ◽  
The Difference ◽  
The Time Domain

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latter is frequency-dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well controlled. This idea is originally due to Nguyen and Vogelius [Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], we revise and extend the work of Haddar, Joly, and Nguyen, [Generalized impedance boundary condition for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci.15 (2005) 1273–1300], where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variable, SIAM J. Math. Anal.44 (2012) 769–807] does not fit directly in our setting. Our proof makes use of a result by Hörmander in [Lp estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967) 65–78]. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.

Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration

Geophysics ◽  
2005 ◽  
Vol 70 (4) ◽  
pp. E1-E10 ◽  
Author(s):  
Yu Zhang ◽  
Guanquan Zhang ◽  
Norman Bleistein
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Forward Modeling ◽  
Inversion Method ◽  
Wave Form ◽  
Wave Operators ◽  
Full Wave ◽  
Migration Imaging ◽  
Standard Wave ◽  
And Migration

One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in the standard wave-equation migration imaging condition. The boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation. When the full wave-form solutions are replaced by their ray-theoretic approximations, the imaging formula reduces to the common-shot Kirchhoff inversion formula. In this sense, the migration is true amplitude as well. On the other hand, this new method retains all of the fidelity features of wave equation migration. Computer output using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data must be collected from a single common-shot experiment. Multiexperiment data, such as common-offset data, cannot be used with this method as presently formulated.

scholarly journals In-situ monitoring for liquid metal jetting using a millimeter-wave impedance diagnostic

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Tammy Chang ◽  
Saptarshi Mukherjee ◽  
Nicholas N. Watkins ◽  
David M. Stobbe ◽  
Owen Mays ◽  
...  
Keyword(s):  
Liquid Metal ◽  
Millimeter Wave ◽  
Manufacturing Systems ◽  
Wave Impedance ◽  
In Situ Monitoring ◽  
Full Wave ◽  
Drop On Demand ◽  
The Time Domain ◽  
Metal Droplets

AbstractThis article presents a millimeter-wave diagnostic for the in-situ monitoring of liquid metal jetting additive manufacturing systems. The diagnostic leverages a T-junction waveguide device to monitor impedance changes due to jetted metal droplets in real time. An analytical formulation for the time-domain T-junction operation is presented and supported with a quasi-static full-wave electromagnetic simulation model. The approach is evaluated experimentally with metallic spheres of known diameters ranging from 0.79 to 3.18 mm. It is then demonstrated in a custom drop-on-demand liquid metal jetting system where effective droplet diameters ranging from 0.8 to 1.6 mm are detected. Experimental results demonstrate that this approach can provide information about droplet size, timing, and motion by monitoring a single parameter, the reflection coefficient amplitude at the input port. These results show the promise of the impedance diagnostic as a reliable in-situ characterization method for metal droplets in an advanced manufacturing system.

scholarly journals Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography

2021 ◽  
Vol 11 (7) ◽  
pp. 3010
Author(s):  
Hao Liu ◽  
Xuewei Liu
Keyword(s):  
Wave Equation ◽  
Partial Derivative ◽  
Model Experiment ◽  
Surface Velocity ◽  
Seismic Exploration ◽  
Initial Condition ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Full Wave ◽  
Time Migration

The lack of an initial condition is one of the major challenges in full-wave-equation depth extrapolation. This initial condition is the vertical partial derivative of the surface wavefield and cannot be provided by the conventional seismic acquisition system. The traditional solution is to use the wavefield value of the surface to calculate the vertical partial derivative by assuming that the surface velocity is constant. However, for seismic exploration on land, the surface velocity is often not uniform. To solve this problem, we propose a new method for calculating the vertical partial derivative from the surface wavefield without making any assumptions about the surface conditions. Based on the calculated derivative, we implemented a depth-extrapolation-based full-wave-equation migration from topography using the direct downward continuation. We tested the imaging performance of our proposed method with several experiments. The results of the Marmousi model experiment show that our proposed method is superior to the conventional reverse time migration (RTM) algorithm in terms of imaging accuracy and amplitude-preserving performance at medium and deep depths. In the Canadian Foothills model experiment, we proved that our method can still accurately image complex structures and maintain amplitude under topographic scenario.

scholarly journals GPU-Based Fast Decoupled Power Flow With Preconditioned Iterative Solver and Inexact Newton Method

2017 ◽  
Vol 32 (4) ◽  
pp. 2695-2703 ◽  
Author(s):  
Xue Li ◽  
Fangxing Li ◽  
Haoyu Yuan ◽  
Hantao Cui ◽  
Qinran Hu
Keyword(s):  
Newton Method ◽  
Power Flow ◽  
Inexact Newton Method ◽  
Iterative Solver ◽  
Preconditioned Iterative

Migration‐based traveltime waveform inversion of 2-D simple structures: A synthetic example

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 845-860 ◽  
Author(s):  
François Clément ◽  
Guy Chavent ◽  
Susana Gómez
Keyword(s):  
Least Squares ◽  
Waveform Inversion ◽  
Domain Of Attraction ◽  
Optimization Methods ◽  
Data Sets ◽  
Seismic Reflection Data ◽  
Prestack Migration ◽  
Reflection Data ◽  
Full Wave ◽  
The Time Domain

Migration‐based traveltime (MBTT) formulation provides algorithms for automatically determining background velocities from full‐waveform surface seismic reflection data using local optimization methods. In particular, it addresses the difficulty of the nonconvexity of the least‐squares data misfit function. The method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity. We present an implementation of the MBTT approach for a 2-D finite‐difference (FD) full‐wave acoustic model. Numerical analysis on a 2-D synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least‐squares approach, even when starting from very poor initial guesses. This enlargement of the domain of attraction for the global minima of the least‐squares misfit has a price: each evaluation of the new objective function requires, besides the usual FD full‐wave forward modeling, an additional full‐wave prestack migration. Hence, the FD implementation of the MBTT approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size.

Randomized HSS acceleration for full-wave-equation depth stepping migration

2014 ◽  
Author(s):  
Lina Miao* ◽  
Polina Zheglova ◽  
Felix J. Herrmann
Keyword(s):  
Wave Equation ◽  
Full Wave
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