One‐pass 3-D seismic extrapolation with the 45° wave equation

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1817-1824 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Seismic Data ◽  
Correction Term ◽  
Synthetic Data ◽  
Second Order ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
Partial Differential ◽  
Numerical Tests ◽  
And Migration

One‐pass 3-D modeling and migration for poststack seismic data may be implemented by replacing the traditional 45° one‐way wave equation (a third‐order partial differential equation) with a pair of second‐ and first‐order partial differential equations. Except for an extra correction term, the resulting second‐order equation has a form similar to the Claerbout 15° one‐way wave equation, which is known to have a nearly circular impulse response. In this approach, there is no need to compensate for splitting errors. Numerical tests on synthetic data show that this algorithm has the desirable attributes of being second order in accuracy and economical to solve. A modification of the Crank‐Nicholson implementation maintains stability.

scholarly journals A note on the nonlocal boundary value problem for a third order partial differential equation

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 801-808 ◽  
Author(s):  
Kh. Belakroum ◽  
A. Ashyralyev ◽  
A. Guezane-Lakoud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Order Partial Differential Equation ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

Flux-normalized wavefield decomposition and migration of seismic data

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. S83-S92 ◽  
Author(s):  
Bjørge Ursin ◽  
Ørjan Pedersen ◽  
Børge Arntsen
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Transmission Loss ◽  
Correction Term ◽  
Eigenvalue Decomposition ◽  
System Matrix ◽  
Interaction Terms ◽  
Reflectivity Function ◽  
And Migration ◽  
Wavefield Decomposition

Separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix. In conventional pressure-normalized wavefield decomposition, the resulting one-way wave equations contain an interaction term which depends on the reflectivity function. Applying directional wavefield decomposition using flux-normalized eigenvalue decomposition, and disregarding interaction between up- and downgoing wavefields, these interaction terms were absent. By also applying a correction term for transmission loss, the result was an improved estimate of the up- and downgoing wavefields. In the wave equation angle transform, a crosscorrelation function in local offset coordinates was Fourier-transformed to produce an estimate of reflectivity as a function of slowness or angle. We normalized this wave equation angle transform with an estimate of the plane-wave reflection coefficient. The flux-normalized one-way wave-propagation scheme was applied to imaging and to the normalized wave equation angle-transform on synthetic and field data; this proved the effectiveness of the new methods.

scholarly journals The crossdip correction as a tool to improve imaging of crooked line seismic data: A case study from the post-glacial Burträsk fault, Sweden

2018 ◽  
Author(s):  
Ruth A. Beckel ◽  
Christopher Juhlin
Keyword(s):  
Seismic Data ◽  
Correction Term ◽  
Regional Scale ◽  
Synthetic Data ◽  
Shear Zones ◽  
Seismic Profile ◽  
Line Geometry ◽  
Processing Methods ◽  
Processing Scheme ◽  
Noise Tolerant

Abstract. Understanding the development of post-glacial faults and their associated seismic activity is crucial for risk assessment in Scandinavia. However, imaging these features and their geological environment is complicated due to special challenges of their hardrock setting, such as weak impedance contrasts, sometimes high noise levels and crooked acquisition lines. A crooked line geometry can cause time shifts that seriously de-focus and deform reflections containing a crossdip component. Advanced processing methods like swath 3D processing and 3D pre-stack migration can, in principle, handle the crooked line geometry, but may fail when the noise level is too high. For these cases, the effects of reflector crossdip can be compensated for by introducing a linear correction term into the standard processing flow. However, existing implementations of the crossdip correction rely on a slant stack approach which can, for some geometries, lead to a duplication of reflections. Here we present a module for the crossdip correction that avoids the reflection duplication problem by shifting the reflections prior to stacking. Based on tests with synthetic data, we developed an iterative processing scheme where a sequence consisting of crossdip correction, velocity analysis and DMO correction is repeated until the stacked image converges. Using our new module to reprocess a reflection seismic profile over the post-glacial Burträsk Fault in Northern Sweden increased the image quality significantly. Strike and dip information extracted from the crossdip analysis helped to interpret a set of southeast dipping reflections as shear zones belonging to the regional scale Burträsk Shear Zone (BSZ), implying that the BSZ itself is not a vertical, but a southeast dipping feature. Our results demonstrate that the crossdip correction is a highly useful alternative to more sophisticated processing methods for noisy datasets. This highlights the often underestimated potential of rather simple, but noise-tolerant methods, in processing hardrock seismic data.

scholarly journals The cross-dip correction as a tool to improve imaging of crooked-line seismic data: a case study from the post-glacial Burträsk fault, Sweden

Solid Earth ◽  
2019 ◽  
Vol 10 (2) ◽  
pp. 581-598
Author(s):  
Ruth A. Beckel ◽  
Christopher Juhlin
Keyword(s):  
Seismic Data ◽  
Correction Term ◽  
Regional Scale ◽  
Synthetic Data ◽  
Shear Zones ◽  
Seismic Profile ◽  
Line Geometry ◽  
Processing Methods ◽  
Processing Scheme ◽  
The Cross

Abstract. Understanding the development of post-glacial faults and their associated seismic activity is crucial for risk assessment in Scandinavia. However, imaging these features and their geological environment is complicated due to special challenges of their hardrock setting, such as weak impedance contrasts, often high noise levels and crooked acquisition lines. A crooked-line geometry can cause time shifts that seriously de-focus and deform reflections containing a cross-dip component. Advanced processing methods like swath 3-D processing and 3-D pre-stack migration can, in principle, handle the crooked-line geometry but may fail when the noise level is too high. For these cases, the effects of reflector cross-dip can be compensated for by introducing a linear correction term into the standard processing flow. However, existing implementations of the cross-dip correction rely on a slant stack approach which can, for some geometries, lead to a duplication of reflections. Here, we present a module for the cross-dip correction that avoids the reflection duplication problem by shifting the reflections prior to stacking. Based on tests with synthetic data, we developed an iterative processing scheme where a sequence consisting of cross-dip correction, velocity analysis and dip-moveout (DMO) correction is repeated until the stacked image converges. Using our new module to reprocess a reflection seismic profile over the post-glacial Burträsk fault in northern Sweden increased the image quality significantly. Strike and dip information extracted from the cross-dip analysis helped to interpret a set of southeast-dipping reflections as shear zones belonging to the regional-scale Burträsk Shear Zone (BSZ), implying that the BSZ itself is not a vertical but a southeast-dipping feature. Our results demonstrate that the cross-dip correction is a highly useful alternative to more sophisticated processing methods for noisy datasets. This highlights the often underestimated potential of rather simple but noise-tolerant methods in processing hardrock seismic data.

On a Linear Partial Differential Equation of Hyperbolic Type

1922 ◽  
Vol 41 ◽  
pp. 76-81
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Order Partial Differential Equation ◽  
Sound Waves ◽  
Partial Differential ◽  
Method Of Solution ◽  
Image Position

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equationwhere α, β, γ are functions of x and y.

scholarly journals A segmented Adomian algorithm for the boundary value problem of a second-order partial differential equation on a plane triangle area

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yin-shan Yun ◽  
Ying Wen ◽  
Temuer Chaolu ◽  
Randolph Rach
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Groundwater Flow ◽  
Boundary Value ◽  
Flow Region ◽  
Second Order ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Adomian Decomposition

Abstract For the boundary value problem (BVP) of a second-order partial differential equation on a plane triangle area, we propose a new algorithm based on the Adomian decomposition method (ADM) combined with a segmented technique. In addition, we present a new theorem that ensures the convergence of the algorithm. By this algorithm, the model for the effect of regional recharge on the plane triangle groundwater flow region is solved, from which we obtain the segmented exact solution of the problem, which satisfies the governing equation and all of the specified boundary conditions. Then, by the algorithm combined with Taylor’s formula, the heterogeneous aquifer model on the plane triangle groundwater flow region is considered, from which we obtain the segmented high-precision approximate solution of the problem.

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