A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.

Linearized formulations and approximate solutions for the complex eikonal equation in orthorhombic media and applications of complex seismic traveltime

We have developed linearized complex eikonal equations for orthorhombic media and the analytic solution for homogeneous acoustic media. The linearized formulations are developed from the highly nonlinear complex eikonal equation using a perturbation analysis. The analytic solution is obtained by expanding the complex traveltime in a Taylor series and constructing its coefficients. A critical factor for deriving this analytic expression is to apply the complex point-source method to the analytic expression for the background anisotropic medium. The analytic expressions show that the complex traveltime is a function of the anisotropic parameters, wave velocity, and spatial coordinates. We give the numerical results of the complex traveltime for different anisotropic parameters and initial beam widths. Furthermore, numerical examples are investigated for various coefficients as a function of the coordinates.

An acoustic eikonal equation for attenuating orthorhombic media

Attenuating orthorhombic models are often used to describe the azimuthal variation of the seismic wave velocity and attenuation in finely layered hydrocarbon reservoirs with vertical fractures. In addition to the P-wave related medium parameters, S-wave parameters are also present in the complex eikonal equation needed to describe the P-wave complex-valued traveltime in an attenuating orthorhombic medium, which increases the complexity of using the P-wave traveltime to invert for the medium parameters in practice. We have used the acoustic assumption to derive an acoustic eikonal equation that approximately governs the complex-valued traveltime of P-waves in an attenuating orthorhombic medium. For a homogeneous attenuating orthorhombic media, we solve the eikonal equation using a combination of the perturbation method and Shanks transform. For a horizontal attenuating orthorhombic layer, the real and imaginary parts of the complex-valued reflection traveltime have nonhyperbolic behaviors in terms of the source-receiver offset. Similar to the roles of normal moveout (NMO) velocity and anellipticity, the attenuation NMO velocity and the attenuation anellipticity characterize the variation of the imaginary part of the complex-valued reflection traveltime around zero source-receiver offset.

Maxwell, Yang–Mills, Weyl and eikonal fields defined by any null shear-free congruence

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, [Formula: see text] Yang–Mills and complex Maxwell fields, the latter produced by integer-valued electric charges (“elementary” for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding “particle-like” field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit nontrivial collective dynamics simulating physical interactions.

Scientific Review on the Ionospheric Absorption and Research Prospects of a Complex Eikonal Model for One-Layer Ionosphere

The present paper conducts a scientific review on ionospheric absorption, extrapolating the research prospects of a complex eikonal model for one-layer ionosphere. As regards the scientific review, here a quasi-longitudinal (QL) approximation for nondeviative absorption is deduced which is more refined than the corresponding equation reported by Davies (1990). As regards the research prospects, a complex eikonal model for one-layer ionosphere is analyzed in depth here, already discussed by Settimi et al. (2013). A simple formula is deduced for a simplified problem. A flat, layered ionospheric medium is considered, without any horizontal gradient. The authors prove that the QL nondeviative amplitude absorption according to the complex eikonal model is more accurate than Rawer’s theory (1976) in the range of middle critical frequencies.