C Stationary phase approximation

2021 ◽  
pp. 473-480
Keyword(s):  
Stationary Phase ◽  
Phase Approximation

Inelastic eikonal phenomenology in a stationary-phase approximation

1975 ◽  
Vol 12 (7) ◽  
pp. 2031-2036 ◽  
Author(s):  
Hector Moreno ◽  
H. M. Fried
Keyword(s):  
Stationary Phase ◽  
Phase Approximation

The elliptic umbilic diffraction catastrophe

1979 ◽  
Vol 291 (1382) ◽  
pp. 453-484 ◽  
Keyword(s):  
Experimental Study ◽  
Stationary Phase ◽  
Diffraction Pattern ◽  
Close Agreement ◽  
Symmetry Axis ◽  
Three Dimensional ◽  
Phase Approximation ◽  
Caustic Surface ◽  
Diffraction Integral ◽  
Complex Ray

We have made a detailed theoretical and experimental study of the three-dimensional diffraction pattern decorating the geometrical-optics caustic surface whose form is the elliptic umbilic catastrophe in Thom’s classification. This caustic has three sheets joined along three parabolic cusped edges (‘ribs’) which touch at one singular point (the ‘focus’). Experimentally, the diffraction catastrophe was studied in light refracted by a water droplet 'lens’ with triangular perimeter, and photographed in sections perpendicular to the symmetry axis of the pattern. Theoretically, the pattern was represented by a diffraction integral E(x,y,z) , which was studied numerically through computer simulations and analytically by the method of stationary phase. Particular attention was concentrated on the ‘dislocation lines’ where | E | vanishes, since these can be considered as a skeleton on which the whole diffraction pattern is built. Within the region bounded by the caustic surface the interference of four rays produces hexagonal diffraction maxima stacked in space like the atoms of a distorted crystal with space group R3m. The dislocation lines not too close to the ribs form hexagonally puckered rings. On receding from the focus and approaching the ribs, these rings approach one another and eventually join to form ‘hairpins’, each arm of which is a tightly wound sheared helix that develops asymptotically into one of the dislocations of the cusp diffraction catastrophe previously studied by Pearcey. Outside the caustic there are also helical dislocation lines, this time formed by interference involving a complex ray. There is close agreement, down to the finest details, between observation, exact computation of E(x,y,z) , and the four-wave stationary-phase approximation.

3-D prestack migration of common‐azimuth data

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1822-1832 ◽  
Author(s):  
Biondo Biondi ◽  
Gopal Palacharla
Keyword(s):  
Stationary Phase ◽  
Constant Velocity ◽  
Downward Continuation ◽  
Theoretical Interpretation ◽  
Data Sets ◽  
Phase Approximation ◽  
Data Set ◽  
Prestack Migration ◽  
Surface Location ◽  
The Common

In principle, downward continuation of 3-D prestack data should be carried out in the 5-D space of full 3-D prestack geometry (recording time, source surface location, and receiver surface location), even when the data sets to be migrated have fewer dimensions, as in the case of common‐azimuth data sets that are only four dimensional. This increase in dimensionality of the computational space causes a severe increase in the amount of computations required for migrating the data. Unless this computational efficiency issue is solved, 3-D prestack migration methods based on downward continuation cannot compete with Kirchhoff methods. We address this problem by presenting a method for downward continuing common‐azimuth data in the original 4-D space of the common‐azimuth data geometry. The method is based on a new common‐azimuth downward‐continuation operator derived by a stationary‐phase approximation of the full 3-D prestack downward‐continuation operator expressed in the frequency‐wavenumber domain. Although the new common‐azimuth operator is exact only for constant velocity, a ray‐theoretical interpretation of the stationary‐phase approximation enables us to derive an accurate generalization of the method to media with both vertical and lateral velocity variations. The proposed migration method successfully imaged a synthetic data set that was generated assuming strong lateral and vertical velocity gradients. The common‐azimuth downward‐continuation theory also can be applied to the derivation of a computationally efficient constant‐velocity Stolt migration of common‐azimuth data. The Stolt migration formulation leads to the important theoretical result that constant‐velocity common‐azimuth migration can be split into two exact sequential migration processes: 2-D prestack migration along the inline direction, followed by 2-D zero‐offset migration along the cross‐line direction.

Sign in / Sign up

Export Citation Format

Share Document