Stationary Phase Points Research Articles

A Nonlinear Stationary Phase Method for Oscillatory Riemann-Hilbert Problems

We study the asymptotic behavior of oscillatory Riemann–Hilbert problems (RHPs) arising in the Ablowitz-Kaup-Newell-Segur hierarchy of integrable nonlinear partial differential equations. Our method is based on the Deift–Zhou nonlinear steepest descent method in which the given RHP localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently, Varzugin extended the Deift–Zhou method to a certain restricted class of nonanalytic phase functions. In this paper, we extend Varzugin’s method to a substantially more general class of nonanalytic phase functions. In our work, real variable methods play a key role.

3-D Radome-Enclosed Aperture Antenna Analyses and Far-Side Radiation

Physical optics integral representations of the fields are given for the 3-D model of the aperture antenna with specified ampliphase excitation law enclosed in a radome. The problem is reduced to finding fields of a plane wave diffracted on the “symmetrized” radome. This model is used for calculations of radiation patterns and for analyses of far-field radiation. Peculiarities of the ray pattern and caustics are analyzed using geometrical optics (geometrical theory of diffraction) method. The contribution of the stationary phase points in the aperture to the far-side radiation has been investigated. Results of numerical calculations are presented.

A New Fast Physical Optics for Smooth Surfaces by Means of a Numerical Theory of Diffraction

A new technique to compute the physical optics (PO) integral is presented. The technique consists of a blind code that computes the different contributions (stationary phase points, end points, etc.) numerically. This technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plane. This algorithm permits a fast and accurate evaluation of the PO integral for smooth large surfaces. The CPU time is almost independent of frequency.

Analysis of scattering from buried object by using the geometrical optics

This paper is concerned with a theoretical solution to the problem of TE-Scattering radio waves by a buried object in a three-layered medium. The region of interest consists of three homogenous layered medium, the object of an inhomogeneous material with respect to its layer. With the aid of geometrical optics technique, this paper analyzes the influence of the buried object on the scattered magnetic field from the layered medium. Closed-form expressions for the far-field of a VMD, located above the ground surface, are obtained due to the buried object under the ground. This technique takes into consideration all the probabilities of plane waves under and above the ground to calculate the total reflection coefficient. The extra-scattered field due to the perturbation in the earth magnetic field has been obtained by solving Sommerfeld integrals (SI). The solution of SI has been based on the stationary-phase point method. The obtained results are compared with those mentioned in three- layered free buried objects.

The virtual refraction: Useful spurious energy in seismic interferometry

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.

ANALYSIS OF SHADOWING PROCESSING TECHNIQUE BASED ON MODELING USING NURBS

The shadowing relationship between facet elements can be determined rapidly through analytical expressions. On the basis of modeling using curved surfaces, an effective shadowing processing algorithm is proposed which is in combination with that used in the shadowing judgement of facet elements. Firstly several sampling points are taken on the ergodic curved surface element to construct a group of facet elements, which can replace the curved surface element. Then the shadowing processing between the stationary phase point and the ergodic curved surface element is converted to that between the stationary phase point and several facet elements, thus avoiding utilizing optimization method and can increase the computation speed. Similarly, the shadowing processing between the stationary phase segment and the ergodic curved surface element is converted to that between the stationary phase segment and several facet elements. And the trimming algorithm is used to accurately find the visible part of the stationary phase segment, which gets rid of the rough shadowing processing technique that determines the visibility of the whole stationary phase segment through the visibility of the center of the stationary phase segment. Therefore, the computation precision is greatly improved. When there exists a huge number of curved surfaces, maximum-minimum preprocessing is utilized to increase the computation speed. Examples show that this novel algorithm is superior to the traditional one in both computation speed and precision.

Analysis of Ground Wave Propagation over Land-to-Sea Mixed-Path by Using Equivalent Current Source on Aperture Plane

In this paper, we have obtained the integral representation for the ground wave propagation over land-to-sea mixed-paths which uses the equivalent current source on an aperture plane. By extending the integral to the complex plane and deforming the integration path into the steepest descent path, we have derived a simple integral representation for the mixed-path ground wave propagation. We have also derived the hybrid numerical and asymptotic representation for an efficient calculation of the ground wave and for easy understanding of the diffraction phenomena. By using the method of the stationary phase applicable uniformly as the stationary phase point approaches the endpoint, we have derived the high-frequency asymptotic solution for the ground wave propagation over the mixed-path. We have confirmed the validity of the various representations by comparing both with the conventional mixed-path theory and with the experimental results performed in Kanto areas including the sea near Tokyo bay. By examining the asymptotic solution in detail, we have found out the cause or the mechanism of the recovery effect occurring on the portion of the sea over the land-to-sea mixed-path.

A Comparison of Point Target Spectra Derived for Bistatic SAR Processing

The existence of a double hyperbola in the bistatic range equation makes it difficult to find an exact analytical solution for the 2D point target spectrum. Several approximate solutions for the spectrum have been derived and used to focus bistatic synthetic aperture radar data. In this paper, we establish the relationship between three independently derived bistatic point target spectra. The first spectrum is Loffeld's bistatic formula, which consists of a quasi-monostatic and a bistatic phase term. The second spectrum makes use of Rocca's smile operator, which transforms bistatic data in a defined configuration to a monostatic equivalent. The third spectrum is derived using a power series - called the method of series reversion (MSR). The MSR spectrum is the most general among the three. This paper shows that this spectrum can be reduced to the same formulation as the former two when certain conditions are met. In addition, a new approximate spectrum is derived using a Taylor series expansion about the two stationary phase points of the transmitter and receiver. We also give an alternative geometrical proof of the relationship between Rocca's smile operator and Loffeld's bistatic deformation term. The accuracies of the point target spectra are demonstrated using simulations of an X-band bistatic airborne radar with a fixed baseline.

Weighted LBF for spaceborne general bistatic SAR processing

Loffeld’s bistatic formula (LBF) is the first two-dimensional analytic point target reference spectrum derived for general bistatic SAR frequency domain focusing. The phase history is expanded in Taylor series around the individual points of stationary phase of the transmitter-target and target-receiver phase histories, respectively, and thus the common bistatic stationary phase point can be obtained using the method of stationary phase. Unfortunately, it shows limitations for extreme bistatic configurations, namely the highly squinted mode and space-surface application. The weighted LBF (WLBF) is proposed in this paper based on the different contributions of total phase modulation from the transmitter and receiver. The formulae we derived are compared with that of the original literature. The extreme bistatic stripmap SAR data can be focused using WLBF, which accommodates the spaceborne squint geometry using the modified effective velocity solution. A point target simulation example is presented to verify the accuracy of the new WLBF spectrum.

A Bistatic Point Target Reference Spectrum for General Bistatic SAR Processing

A bistatic point target reference spectrum (BPTRS) based on Loffeld's bistatic formula (LBF) is derived in this letter. For LBF, the same contributions of the transmitter and receiver to the total azimuth modulation are assumed. This assumption results in the failure of LBF in the extreme configuration (i.e., spaceborne/airborne configuration). For general bistatic configurations, the azimuth modulations are unequal for the transmitter and receiver due to the different slant ranges and velocities. Therefore, the azimuth time-bandwidth products (TBPs) from the transmitter and receiver are different; in some cases (e.g., spaceborne/airborne case), one of them might be very small, which might even result in a serious error of the principle of stationary phase. This letter uses TBP to weight the azimuth phase modulation contributions of the transmitter and receiver to the common azimuth spectrum to approximately obtain the point of stationary phase of the total azimuth phase history. Simulations show that the proposed BPTRS can work well for spaceborne/airborne configurations.

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

The unique concept of the modified edge representation (MER) was proposed for the surface to the line integral reduction of the physical optic (PO). The equivalence between the MER line and the PO surface integration was analytically derived by using the Stokes theorem relations as well as asymptotic treatments, for the smooth scattering surfaces without inner stationary phase points (SPP). Later on, for the planar surface, the MER line integration around the inner SPP was investigated, and it was identified as the scattering geometrical optics (SGO). In this paper, findings related with the MER line integration around the inner SPP are extended to curved surfaces. The accuracy and the applicability of the SGO extraction in terms of the MER line integration are numerically investigated for different radii of curvatures of the scattering surfaces. Authors introduce a geometrical criterion for the applicability of the method. The MER line integration provides an alternative way to the stationary phase method or the classical geometrical optics for calculating SGO. In addition, this numerical result indirectly identifies the entity of the MER line integration along the periphery of the scattering illuminated region, irrespective of the position of observer, as not other than diffraction.

Asymptotic Fields in Frequency and Time Domains Generated by a Point Source at the Horizontal Interface Between Vertically Uniaxial Media

For a point current source placed at the interface between two half-space media, investigation is made of the far-zone asymptotic fields in the frequency domain and their transient waveforms. The media are electrically uniaxial with the uniaxis normal to the interface. The leading-order fields contributed by the stationary-phase points serve as good approximations well off the critical angles. Near the critical angles, the uniform asymptotic techniques need to be employed for a proper approximation. Unlike the ordinary wave, the extraordinary wave cannot be correctly described merely by taking into account the proximity of the stationary and branch points, except in some rare cases. Consideration must also be given to the combined effect associated with the pole introduced in the formulation. The validities of the asymptotic approximations are examined in the time domain by comparing the asymptotic-field waveforms with the wavenumber-synthetic waveforms, around the critical angle of the extraordinary wave.

Wavefront modelling of pulse propagation

Wavefront modelling has been developed for the rapid calculation of receiver waveforms in underwater sound pulse propagation. The method uses a solution of the wave equation which expresses the field as a sum over phase integrals. Each phase integral corresponds to a particular sequence of reflections or turning points and is evaluated analytically by using stationary phase approximations. Each point of stationary phase corresponds to a ray path. Pairs or triplets of stationary phase points correspond to caustics or foci and the phase and amplitude of the field is found directly. The method can handle rapid range dependence such as occurs with surface waves in shallow water and double scattering from the same surface. The method is fast and accurate and examples will be discussed. [Work supported by ONR.]

Frequency- and time-domain asymptotic fields near the critical cone in fluid-fluid configuration

Near the critical cone of a point source placed at the interface between two half-space fluid media, investigation is made of the asymptotic fields in the frequency domain and their synthetic wave forms in the time domain. While the leading-order (and the head-wave) components give a good description of the true fields well off the critical cone, the uniform asymptotic (UA) analysis has to be made for the approximation near the critical cone. The UA analysis splits into two cases, depending on the medium densities and wave speeds. For Case 1, the UA1 approximation is employed that takes into account the proximity of the stationary-phase point to the branch point. In Case 2, the UA2 approximation is employed in which consideration is also given to the proximity of the stationary point to the pole and to the combined effect of the stationary point, the branch point, and the pole. The validities of the asymptotic fields are checked in the time domain by comparing the asymptotic field wave forms against the wave-number-synthetic wave forms. The UA fields show good accuracy and causal behaviors, with the causality of the UA2 fields previously unreported in the literature.

Multifold phase space path integral synthetic seismograms

SUMMARY We consider a medium consisting of homogeneous layers separated by curved interfaces. In order to evaluate the response of a single generalized ray, the source and receiver wavefields are expanded in a series of plane waves. The coupling of these plane waves at each point of the surfaces of material discontinuity is determined by means of a Kirchhoff integral using generalized reflection and transmission coefficients. The resulting integral, called the multifold phase space path integral (PSPI) consists of a series of integrals over ray parameters and over interfaces touched by the generalized ray on its way from the source to the receiver. This approach is a generalization of the multifold configuration space path integral (CSPI) to which it reduces by successive application of the stationary phase point method over the ray-parameter integrals. The PSPI like the CSPI automatically includes diffractions from corners. In addition classical head waves are included, although for curved interfaces the head waves are only approximate. 2-D synthetic seismograms are converted to equivalent approximate point-source responses by assuming cylindrical symmetry about source and/or receiver. The waveforms and amplitude of PSPI synthetic seismograms compare well with those computed by generalized ray theory for a 1-D model, and with finite difference synthetics for a 2-D model.

The Stationary Phase Point Method for Transitional Scattering: Diffractive Radio Scintillation

The stationary phase point (SPP) method is introduced to treat the diffractive scintillation. From weak scattering, where the SPP number N = 1, to strong scattering (N>>1), via transitional scattering regime (N∼2, 3), we find that the modulation index of intensity experiences the monotonically increasing from 0 to 1 with the scattering strength, characterized by the ratio of Fresnel scale rF to diffractive scale rdiff.

Dislocation lines in the swallowtail diffraction catastrophe

The three-dimensional distribution of amplitude and phase represented by the swallowtail catastrophe diffraction integral is based on a network of null lines for amplitude (wave dislocations or optical vortices), where the phase is singular. In the tail region there is four-wave interference, which results in an approximately repeating pattern of amplitude based on the monoclinic space group C 2/ m and also an approximately repeating pattern of wave dislocations based on the black–white monoclinic space group C 2/ m ′. Helical dislocations spring from the plane of symmetry and gradually straighten out to be parallel to the two riblines of the caustic; eventually they become the straight dislocations of the Pearcey pattern for the cusp catastrophe. In the front region, where there are zero or two points of stationary phase, each dark Airy fringe surface associated with the fold surface condenses into a single dislocation in the plane of symmetry.

Scintillation of Radio Sources: The Role of Caustics

Scintillation of a radio source can be treated relatively simply by making the stationary phase point (SPP) approximation to the Fresnel-Kirchhoff integral, but the range of validity of the SPP approximation is not well determined. We identify three idealized models for the phase variation for which there are accurate approximations (exact results), and we compare these exact results with the SPP approximations to them. For two of the models—the coherent wave (CW) model and the single-phase pulse model—the integral can be evaluated in closed form, involving infinite sums that can be truncated without loss of accuracy, and for the third model the integral is determined accurately using fast Fourier transforms. We find that for the majority of observer positions the SPP method gives excellent results. However, near a caustic the SPP approximation breaks down and can be positively misleading. We find that for moderate screen strengths the putative caustic lines are the loci of regions where the intensity oscillates rapidly. The conventional caustic behavior is exhibited only as an envelope of rapid diffractive fluctuations for sufficiently large phase variations, with amplitude 0 103 rad. We discuss a suggested application of the single phase pulse to extreme scattering events and compare our exact results with existing models that are based on geometric optics. We also suggest that the CW model might be applied to the highly localized, anisotropic turbulence inferred for intraday variable sources and for the secondary spectra in pulsars.

Closed-Form Expressions of Quantum Electron Transfer Rate Based on the Stationary-Phase Approximation

Closed-form rate expressions are derived on the basis of the stationary-phase approximation for the Fermi golden rule expression of the quantum electron-transfer (ET) rate. First, on the basis of approximate solutions of the stationary-phase points near DeltaG = 0, -lambda, and lambda, where DeltaG is the reaction free energy and lambda is the reorganization energy, three closed-form rate expressions are derived, which are respectively valid near each value of DeltaG. Numerical tests for a model Ohmic spectral density with an exponential cutoff demonstrate good performance of the derived expressions in the respective regions of their validity. In particular, the expression near DeltaG = -lambda, which differs from the semiclassical approximation only by a prefactor quadratic in DeltaG, works substantially better than the latter. Then, a unified formula is suggested, which interpolates the three approximate expressions and serves as a good approximation in all three regions. We have also demonstrated that the interpolation formula can serve as a good quantitative means for understanding the temperature dependence of the quantum ET rate.

An [formula omitted] integration scheme for three-dimensional surface scattering problems

We present an accurate method of O ( 1 ) -complexity with respect to frequency (i.e., a method that, to achieve a prescribed error tolerance, requires a bounded computational cost for arbitrarily high frequencies) for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three-dimensional space. Like the two-dimensional counterpart of this algorithm, which we introduced recently and which is applicable to scattering by curves in the plane, the present method is based on a combination of two main elements: (1) a high-frequency ansatz for the unknown density in a boundary integral formulation of the problem, and (2) an extension of the ideas of the method of stationary phase to allow for O ( 1 ) (high-order-accurate) integration of oscillatory functions. The techniques we introduce to implement an efficient O ( 1 ) integrator in the present three-dimensional context differ significantly from those used in the earlier two-dimensional algorithm. In particular, in the present text, we introduce an efficient “canonical” (hybrid analytic-numerical) algorithm which, in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase, can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two-dimensional scattering surface. We include numerical results illustrating the behavior of the integration algorithm on sound-soft spheres with diameters of up to 5000 wavelengths: in such cases, for a single integral, the algorithm yields accuracies of the order of three digits in computational times of less than two seconds. In a preliminary full scattering simulation we present, a solution with two digits of accuracy in the surface density was obtained in about three hours running time, in a single 1.5 GHz AMD Athlon processor, for a sphere of 500 wavelengths in diameter.