Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we propose a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. The proposed methods can be useful for migration and tomography in attenuating media.

Eikonal Equations for Null Radial Geodesics in the Schwarzschild Metric

We study the eikonal function φ corresponding to outgoing and ingoing radial null geodesics (light rays in the short wave length limit) in the Schwarzschild spacetime. Contrary to the behavior of the expansion scalar θ at the singularities (past and future), φ turns out to be finite at r = 0 (except for light travelling along the horizons) and inversely proportional to M, the mass of the black hole, and so proportional to the Hawking temperature.

THE METHOD OF CALCULATION OF COEFFICIENTS OF THE PHASE ABERRATIONS OF AXIALLY ASYMMETRIC DUAL REFLECTOR ANTENNAS

Aerodrome-enroute surveillance radar systems (ARSR) of civil aviation are designed for detection of aerial objects and determination of their range and movement parameters. Meanwhile, the reflector-type antennas are still the main type of_used antenna systems. Radars (ARSR) have, as a rule, antennas with a bidirectional beam, structurally reflector-type antenna is made in the form of a truss welded structure. The use of active phased antenna arrays (APAR) is a promising direction of radars (ARSR) development. Radars (ARSR) based on the phased antenna arrays can significantly improve such critical tactical and technical criteria as: the accuracy of the coordinates determination, the number of simultaneously followed targets. However, the use of APAR does not solve one of the main problems - the presence of a support-rotary device in the radar (ARSR). The principal necessity of a support-rotary device leads to the need of the use of movable joints of feeder lines, which, with a significant number of antenna elements used in the PAR, significantly reduces the potential time before failure. It seems promising to use in the radar antenna systems based on several stationary multi-beam dual-reflector antennas made by axially asymmetric scheme. Methodology of estimation of aberrations of a dual-reflector sensimetrics antenna based on a polynomial expansion of the eikonal function in the plane of the aperture of the main reflector is proposed in the article. Analytical expressions of the functions of display and rotation of the scanning plane of a dual-reflector axially asymmetric antenna with arbitrary parameters, designed based on clippings from centrally symmetric surfaces of rotation of the second order, are obtained. It is shown that the aplanatic antennas lose their scanning features during the transition to the axially asymmetric variant.