Inner stationary phase point contribution of physical optic in terms of the modified edge representation line integrals (curved surfaces)

Radio Science ◽  
2007 ◽  
Vol 42 (6) ◽  
Author(s):  
Luis Rodriguez ◽  
Koji Yukimasa ◽  
Tetsu Shijo ◽  
Makoto Ando
Keyword(s):  
Stationary Phase ◽  
Phase Point ◽  
Curved Surfaces ◽  
Physical Optic

Phase retrieval from spectral interferograms including a stationary-phase point

2012 ◽  
Vol 285 (24) ◽  
pp. 4733-4738 ◽  
Author(s):  
Petr Hlubina ◽  
Jacek Olszewski
Keyword(s):  
Stationary Phase ◽  
Phase Retrieval ◽  
Phase Point

scholarly journals Asymptotic expansion of integrals occurring in linear wave theory

1972 ◽  
Vol 7 (1) ◽  
pp. 121-130 ◽  
Author(s):  
P. van den Driessche ◽  
R.D. Braddock
Keyword(s):  
Asymptotic Expansion ◽  
Stationary Phase ◽  
Wave Theory ◽  
Linear Wave ◽  
Phase Point ◽  
One Dimensional ◽  
Special Cases ◽  
Order Of Magnitude ◽  
Leading Term ◽  
Wave Problems

The asymptotic expansion of an integral of the type , is derived in terms of the large parameter t. Functions Φ(k) and ψ(k) are assumed analytic, and ψ(k) may have zeros at a stationary phase point. The usual one dimensional stationary phase and Airy integral terms are found as special cases of a more general result. The result is used to find the leading term of the asymptotic expansion of the double integral. A particular two dimensional Φ(k) relevant to surface water wave problems is considered in detail, and the order of magnitude of the integral is shown to depend on the nature of ψ(k) at the stationary phase point.

The asymptotic theory of rapidly rotating sound fields

1992 ◽  
Vol 436 (1898) ◽  
pp. 511-526 ◽  
Keyword(s):  
Stationary Phase ◽  
Acoustic Field ◽  
Airy Function ◽  
Sound Field ◽  
Stationary Points ◽  
Simple Type ◽  
Phase Point ◽  
Sound Fields ◽  
Asymptotic Formulae ◽  
Leading Term

A rapidly rotating sound field, such as that produced by a supersonic propeller, may contain several types of diffraction pattern, each in a different region of space. This paper determines these patterns and their locations for the field around a propeller of simple type ; the method used is stationary phase analysis of a certain double integral, and leads to asymptotic formulae valid when the number of blades is not too small or the high harmonics are being investigated. Physically, the results describe propagation along rays : each stationary phase point is a ‘ loud spot ’, producing a ray which points directly at the observer; most of the noise comes from these loud spots, because extensive cancellation takes place everywhere else. At most two interior and two boundary stationary points may be present: the number and type depend on the position of the observer in relation to a cusped torus and two hyperboloids of one sheet. As these surfaces are crossed, the acoustic field changes in character. For example, when two stationary points coalesce and annihilate each other, as they do at a caustic, an Airy function describes the transition from a loud zone of rapid oscillation to a quiet zone of exponential decay; and when an interior stationary point crosses the boundary of the disc the transition region is described either by a Fresnel integral or by a generalized Airy function. Separate analyses are given for regions close to and well away from the transition surfaces, and inner and outer limits are calculated for use in the method of matched asymptotic expansions. In all cases, an overlap region is found in which the leading term s agree. The results of the paper determine completely the geometry of the acoustic field, because the different regions have boundaries at known positions and cover the whole of space.

scholarly journals The virtual refraction: Useful spurious energy in seismic interferometry

Geophysics ◽  
2009 ◽  
Vol 74 (3) ◽  
pp. A13-A17 ◽  
Author(s):  
Dylan Mikesell ◽  
Kasper van Wijk ◽  
Alexander Calvert ◽  
Matt Haney
Keyword(s):  
Energy Distribution ◽  
Stationary Phase ◽  
Critical Time ◽  
Seismic Interferometry ◽  
Phase Point ◽  
Wave Speeds ◽  
The Real ◽  
Spurious Wave ◽  
Established Technique ◽  
Spurious Event

Seismic interferometry is rapidly becoming an established technique to recover the Green’s function between receivers, but practical limitations in the source-energy distribution inevitably lead to spurious energy in the results. Instead of attempting to suppress all such energy, we use a spurious wave associated with the crosscorrelation of refracted energy at both receivers to infer estimates of subsurface parameters. We named this spurious event the virtual refraction. Illustrated by a numerical two-layer example, we found that the slope of the virtual refraction defines the velocity of the faster medium and that the stationary-phase point in the correlation gather provides the critical offset. With the associated critical time derived from the real shot record, this approach includes all of the necessary information to estimate wave speeds and interface depth without the need of inferences from other wave types.

Elastic-wavefield interferometry using P-wave source VSPs at the Wamsutter field, Wyoming

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. Q27-Q36 ◽  
Author(s):  
James Gaiser ◽  
Ivan Vasconcelos ◽  
Rosemarie Geetan ◽  
John Faragher
Keyword(s):  
Stationary Phase ◽  
Wave Velocity ◽  
P Wave ◽  
Phase Point ◽  
S Wave ◽  
Wave Source ◽  
S Waves ◽  
Wave Splitting ◽  
Zero Offset ◽  
Tube Wave

In this study, elastic-wavefield interferometry was used to recover P- and S-waves from the 3D P-wave vibrator VSP data at Wamsutter field in Wyoming. S-wave velocity and birefringence is of particular interest for the geophysical objectives of lithology discrimination and fracture characterization in naturally fractured tight gas sand reservoirs. Because we rely on deconvolution interferometry for retrieving interreceiver P- and S-waves in the subsurface, the output fields are suitable for high-resolution, local reservoir characterization. In 1D media where the borehole is nearly vertical, data at the stationary-phase point is not conducive to conventional interferometry. Strong tube-wave noise generated by physical sources near the borehole interfere with S-wave splitting analyses. Also, converted P- to S-wave (PS-wave) polarity reversals occur at zero offset and cancel their recovery. We developed methods to eliminate tube-wave noise by removing physical sources at the stationary-phase point and perturbing the integration path in the integrand based on P-wave NMO velocity of the direct-arrival. This results in using nonphysical energy outside a Fresnel radius that could not have propagated between receivers. To limit the response near the stationary-phase point, we also applied a weighting condition to suppress energy from large offsets. For PS-waves, a derivative-like operator was applied to the physical sources at zero offset in the form of a polarity reversal. These methods resulted in effectively recovering P-wave dipole and PS-wave quadrupole pseudosource VSPs. The retrieved wavefields kinematically correspond to a vertical incidence representation of reflectivity/transmissivity and can be used for conventional P- and S-wave velocity analyses. Four-component PS-wave VSPs retrieve S-wave splitting in transmitted converted waves that provide calibration for PS-wave and P-wave azimuthal anisotropy measurements from surface-seismic data.

scholarly journals Monitoring the dominance of higher-order chromatic dispersion with spectral interferometry using the stationary phase point method

2015 ◽  
Vol 338 ◽  
pp. 292-299 ◽  
Author(s):  
Tímea Grósz ◽  
Attila P. Kovács ◽  
Katalin Mecseki ◽  
Lénárd Gulyás ◽  
Róbert Szipőcs
Keyword(s):  
Stationary Phase ◽  
Chromatic Dispersion ◽  
Higher Order ◽  
Point Method ◽  
Phase Point ◽  
Spectral Interferometry

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

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