Elliptic differential‐difference operators with degeneration and the Kato square root problem

2018 ◽  
Vol 291 (17-18) ◽  
pp. 2660-2692 ◽  
Author(s):  
A. L. Skubachevskii
Keyword(s):  
Difference Operators ◽  
Square Root ◽  
Kato Square Root Problem

Seismic depth migration with the Dirac equation

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 925-933 ◽  
Author(s):  
Ketil Hokstad ◽  
Rune Mittet
Keyword(s):  
Dirac Equation ◽  
Taylor Series ◽  
Rapid Expansion ◽  
Difference Operators ◽  
Square Root ◽  
Exact Linearization ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Spatial Derivatives ◽  
Velocity Variations

We demonstrate the applicability of the Dirac equation in seismic wavefield extrapolation by presenting a new explicit one‐way prestack depth migration scheme. The method is in principle accurate up to 90° from the vertical, and it tolerates lateral velocity variations. This is achieved by performing the extrapolation step of migration with the Dirac equation, implemented in the space‐frequency domain. The Dirac equation is an exact linearization of the square‐root wave equation and is equivalent to keeping infinitely many terms in a Taylor series or continued‐fraction expansion of the square‐root operator. An important property of the new method is that the local velocity and the spatial derivatives decouple in separate terms within the extrapolation operator. Therefore, we do not need to precompute and store large tables of convolutional extrapolator coefficients depending on velocity. The main drawback of the explicit scheme is that evanescent energy must be removed at each depth step to obtain numerical stability. We have tested two numerical implementations of the migration scheme. In the first implementation, we perform depth stepping using the Taylor series approximation and compute spatial derivatives with high‐order finite difference operators. In the second implementation, we perform depth stepping with the Rapid expansion method and numerical differentiation with the pseudospectral method. The imaging condition is a generalization of Claerbout’s U / D principle. For both implementations, the impulse response is accurate up to 80° from the vertical. Using synthetic data from a simple fault model, we test the depth migration scheme in the presence of lateral velocity variations. The results show that the proposed migration scheme images dipping reflectors and the fault plane in the correct positions.

3-D implicit finite‐difference migration by multiway splitting

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 554-567 ◽  
Author(s):  
Dietrich Ristow ◽  
Thomas Rühl
Keyword(s):  
Finite Difference ◽  
Power Series ◽  
Taylor Expansion ◽  
Optimization Technique ◽  
Optimization Procedure ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Square Root ◽  
Implicit Finite Difference ◽  
Root Operator

We show that 3-D implicit finite‐difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite‐difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher‐degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the opfimization procedure.

scholarly journals The Kato Square Root Problem follows from an extrapolation property of the Laplacian

2016 ◽  
Vol 60 ◽  
pp. 451-483 ◽  
Author(s):  
M. Egert ◽  
R. Haller-Dintelmann ◽  
P. Tolksdorf
Keyword(s):  
Square Root ◽  
Kato Square Root Problem

scholarly journals Kato square root problem with unbounded leading coefficients

2018 ◽  
Vol 146 (12) ◽  
pp. 5295-5310 ◽  
Author(s):  
Luis Escauriaza ◽  
Steve Hofmann
Keyword(s):  
Square Root ◽  
Kato Square Root Problem

The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on \Bbb R n

2002 ◽  
Vol 156 (2) ◽  
pp. 633 ◽  
Author(s):  
Pascal Auscher ◽  
Steve Hofmann ◽  
Michael Lacey ◽  
Alan McIntosh ◽  
Ph. Tchamitchian
Keyword(s):  
Elliptic Operators ◽  
Second Order ◽  
Square Root ◽  
Second Order Elliptic Operators ◽  
Kato Square Root Problem

The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

2001 ◽  
Vol 1 (4) ◽  
pp. 361-385 ◽  
Author(s):  
Pascal Auscher ◽  
Steve Hofmann ◽  
Alan McIntosh ◽  
Philippe Tchamitchian
Keyword(s):  
Higher Order ◽  
Elliptic Operators ◽  
Square Root ◽  
Kato Square Root Problem
Sign in / Sign up

Export Citation Format

Share Document