WKBJ Method Research Articles

Time-lapse difference seismic inversion based on scattering theory

Time-lapse seismic monitoring technology is an effective tool for quantitatively estimating subsurface medium variations and plays a crucial role in resource and engineering explorations. Among existing time-lapse inversion methods, the difference inversion technique offers higher accuracy but requires linear inversion operators. During linearization, the background is often approximated as a homogeneous medium, resulting in the loss of effective information. Additionally, merely using reflection wave information is not conducive to improving inversion accuracy. However, the true structure of the subsurface is often complex and inhomogeneous, which can easily generate complex wavefields, such as scattering waves, thereby hindering the resolution of differences between time-lapse models. Therefore, incorporating more comprehensive wavefield information is essential for enhancing the accuracy of time-lapse seismic inversion. To address this issue, a wave-equation-based time-lapse difference inversion method is proposed to retrieve the production-induced property perturbation. In the proposed method, the forward operator is formulated through the scattering wave equation, the operator is linearized through the Born approximation, and the WKBJ method is used to approximate Green's function with high accuracy for the inhomogeneous background. In this study, time-lapse seismic parallel inversion and difference inversion are conducted using the proposed forward operator. Both numerical experiments and application of field seismic data indicate that the proposed time-lapse difference inversion method can realize a more accurate quantitative characterization of time-lapse variations.

WKB solutions of the Schrödinger equation with a quartic potential

The Schrödinger equation with a quartic potential is a fundamental equation describing the physics of stimulated side scattering, but no exact, even approximate solution has been formulated yet. Here, the approximate solutions to such a Schrödinger equation are derived using the WKBJ method under different variables. The solutions are accurate enough in most cases. Eigenvalues and eigenfunctions, which are the most impressive properties of such equations, are also discussed.

A generalization of WKBJ method for solving a system describing propagation of coupled modes in underwater acoustics

A generalization of the WKBJ method for the case of coupled modes propagation in problems of underwater acoustics is proposed. The generalized WKBJ ansatz has the form of a product of a matrix of phase factors and vector of amplitudes. The transport equation in the matrix form is derived, and the phase factors are computed exactly. The similarities and differences with other forms of mode coupling equations are discussed. A numerical example demonstrating the accuracy of the generalized WKBJ solution is presented.

Explicit complex-valued solutions of the 2D eikonal equation

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of the Helmholtz equation by the WKBJ or Evanescent Wave Tracking methods. We obtain the complex-valued solutions (called eikonals) as parametrizations in a complex variable. We consider both the cases of constant and non-constant index of refraction. In both cases, the relevant parametrizations depend on some holomorphic function. In the case of a non-constant index of refraction, the parametrization also depends on some extra exponential complex-valued function and on a quasi-conformal homeomorphism. This is due to the use of the theory of pseudo-analytic functions and the related similarity principle. The parametrizations give information about the formation of caustics and the light and shadow regions for the relevant eikonals.

Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems

<p style='text-indent:20px;'>Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.</p>

Fundamental problems of wave dynamics stratified medium modelling

In paper fundamental problems of wave dynamic stratified medium modeling (ocean, atmosphere) are considered. The basic mathematical models describing the processes of excitation, the propagation of internal gravity waves in vertical stratified horizontally inhomogeneous and non-stationary medium are presented. The uniform asymptotic forms of the internal gravity waves in horizontally inhomogeneous and non-stationary stratified medium are obtained. A modified spatio-temporal ray method is proposed, which belongs to the class of geometrical optics methods (WKBJ method). Analytical and numerical algorithms of internal gravity wave calculations for the real stratified medium parameters are presented. Some results of internal gravity waves measurements in real stratified medium (ocean, atmosphere) and it’s interpretations are discussed. The universal nature of the developed asymptotic methods of modeling makes it possible to efficiently calculate the wave fields and qualitatively analyze the solutions obtained, which makes it possible to analyze wave patterns in general, to correctly set up mathematical models of wave dynamics, and to carry out express estimates in actual measurements of wave fields in natural stratified medium.

Hydrodynamic convection in accretion discs

The prevalence and consequences of convection perpendicular to the plane of accretion discs have been discussed for several decades. Recent simulations combining convection and the magnetorotational instability have given fresh impetus to the debate, as the interplay of the two processes can enhance angular momentum transport, at least in the optically thick outburst stage of dwarf novae. In this paper we seek to isolate and understand the most generic features of disc convection, and so undertake its study in purely hydrodynamical models. First, we investigate the linear phase of the instability, obtaining estimates of the growth rates both semi-analytically, using one-dimensional spectral computations, as well as analytically, using WKBJ methods. Next we perform three-dimensional, vertically stratified, shearing box simulations with the conservative, finite-volume code PLUTO, both with and without explicit diffusion coefficients. We find that hydrodynamic convection can, in general, drive outward angular momentum transport, a result that we confirm with ATHENA, an alternative finite-volume code. Moreover, we establish that the sign of the angular momentum flux is sensitive to the diffusivity of the numerical scheme. Finally, in sustained convection, whereby the system is continuously forced to an unstable state, we observe the formation of various coherent structures, including large- scale and oscillatory convective cells, zonal flows, and small vortices.

An analysis of the renormalization group method for asymptotic expansions with logarithmic switchback terms

The renormalization group method of Chen, Goldenfeld, and Oono offers a comprehensive approach to formally computing asymptotic expansions of the solutions to singular perturbation problems and multi-scale problems. In particular, the RG method applies to a broad array of problems customarily treated with disparate methods, such as the method of multiple scales, boundary layer theory, matched asymptotic expansions, Poincare-Lindstedt theory, the WKBJ method with and without turning points, the method of averaging, and others. For problems in which the expansions are in powers of the small parameter, it has been shown that the RG method leads to uniformly valid asymptotic expansions of the solutions. In addition, it has been shown that the RG condition constitutes an invariance condition, that the RG method is effectively a resummation technique, and that it is equivalent to normal form theory for certain broad classes of perturbed ordinary differential equations. However, there has not yet been an analysis of the validity of the RG method, i.e.,i> a demonstration that it leads to uniformly valid asymptotic expansions, for problems in which the expansions also involve logarithms of the small parameter as gauge functions. The aim of this article is to provide this justification of the RG method. In particular, we extend the approach developed in DeVille, et ali>. Physica D 237 no. 8, 1029 {-}1052] from the classes of autonomous and non-autonomous perturbations considered there to include the non-autonomous systems subject to singular perturbations for which the solutions involve logarithmic gauge functions. This framework is built upon the relationship between the RG method and normal form theory. We apply the RG method to three successively-more complex examples and also elucidate the common general features.

Asymptotical analysis of internal gravity wave dynamics in stratified medium

In paper fundamental problems of internal gravity waves dynamics are considered. Wave dynamics of stratified medium (ocean, atmosphere) is highly dependent on density distribution and bottom topography. The exact analytical solution is obtained only if the water distribution density and bottom shape described by some model functions. The solution of this problem is expressed in terms of the Green’s function and the asymptotic representations of the solutions are considered. The uniform asymptotic forms of the internal gravity waves in horizontally inhomogeneous and non-stationary stratified ocean are obtained. A modified spatio-temporal ray method is proposed, which belongs to the class of geometrical optics methods (WKBJ method). The solution is proposed in terms of wave modes, propagating independently at the adiabatic approximation, and described as a non-integral degree series of a small parameter characterizing the stratified medium properties. Analytical and numerical algorithms of internal gravity wave calculations for the real ocean parameters are presented.

Fundamental Problems of Internal Gravity Waves Dynamics in Ocean

In paper fundamental problems of internal gravity waves dynamics are considered. The solution of this problem is expressed in terms of the Green’s function and the asymptotic representations of the solutions are considered. The uniform asymptotic forms of the internal gravity waves in horizontally inhomogeneous and non-stationary stratified ocean are obtained. A modified spatio-temporal ray method is proposed, which belongs to the class of geometrical optics methods (WKBJ method). Analytical and numerical algorithms of internal gravity wave calculations for the real ocean parameters are presented.

LINEAR STABILITY ANALYSIS OF FLUVIAL BARS WITH BED AGGRADATION AND DEGRADATION

In the existing linear stability analysis of fluvial bars, it is assumed that the river bed is in equilibrium, and the bed slope and flow are uniform in the streamwise direction. Though it is generally said that the bed becomes more unstable under bed aggradation, and becomes more stable under bed degradation, the existing analysis cannot shed further light on their effects. In this paper, linear stability analysis incorporating weakly non-equilibrium processes of bed aggradation/degradaion is performed by the use of the WKBJ method in order to clarify their effects. Bed aggradation or degradation is assumed to be sufficiently slow compared with the bed evolution due to instability, and its non-dimensional speed is used as a small parameter. The analysis explains that river beds become stable and sand bars tend not to be formed under degradation, and that river beds become unstable under aggradation. In addition, alternate bars are more strongly affected by bed aggradation/degradation than multiple bars.

The accuracy of models derived by WKBJ waveform inversion

Summary. In this paper the accuracy of velocity-depth profiles derived by matching WKBJ seismograms to observations is quantitatively evaluated. Seismograms computed with the WKBJ method are generally quite reliable but possess predictable, systematic inaccuracies in the presence of strong velocity gradients. The effects of these inaccuracies on models derived through WKBJ waveform inversion are studied, using reflectivity seismograms as ‘data'. The velocity structure used is an oceanic lithosphere model that contains several transition regions separated by relatively homogeneous layers, producing partially-reflected reverberations in the reflectivity synthetics that are absent from the WKBJ seismograms. The inversion incorporates the ‘jumping’ strategy to solve for the smoothest models consistent with the data. We find these solutions to be independent of the starting model and to have a stable basic structure that agrees well with the correct model. The differences, everywhere less than a seismic wavelength, depend on the frequency content of the seismograms. Reverberations in the reflectivity seismograms that are well separated from WKBJ arrivals are treated as ‘noise’ in the inversion.

The time dependent Ginzburg–Landau equation in fractal space–time

We use the hydrodynamic formulation of Scale Relativity Theory to analyze the TDGL equation. As a result, London equations come naturally from the system, when equating to zero the real velocity, the imaginary one turns real, the superconducting fluid act as a subquantum medium energy accumulator, the vector potential, the real and the imaginary velocity are all written in terms of the elliptic function. When solving the resulted system by means of WKBJ method, we get tunneling and quantization. In other words, scale transformation laws produce, on the motion equation of particles governed by the TDGL equation, under some peculiar assumptions, effects which are analogous to those of a “macroscopic quantum mechanics”.

The excitation of spiral density waves through turbulent fluctuations in accretion discs - I. WKBJ theory

We study and elucidate the mechanism of spiral density wave excitation in a differentially rotating turbulent flow. We formulate a set of wave equations with sources that are only non-zero in the presence of turbulent fluctuations. We solve these in a shearing box domain using a WKBJ method. It is found that, for a particular azimuthal wave length, the wave excitation occurs through a sequence of regularly spaced swings during which the wave changes from leading to trailing form. This is a generic process that is expected to occur in shearing discs with turbulence. Trailing waves of equal amplitude propagating in opposite directions are produced, both of which produce an outward angular momentum flux that we give expressions for as functions of the disc parameters and azimuthal wave length. By solving the wave amplitude equations numerically we justify the WKBJ approach for a Keplerian rotation law for all parameter regimes of interest. In order to quantify the wave excitation completely the important wave source terms need to be specified. Assuming conditions of weak nonlinearity, these can be identified and are associated with a quantity related to the potential vorticity, being the only survivors in the linear regime. Under the additional assumption that the source has a flat power spectrum at long azimuthal wave lengths, the optimal azimuthal wave length produced is found to be determined solely by the WKBJ response and is estimated to be 2 pi H, with H being the nominal disc scale height.

Imaging the lowermost mantle (D″) and the core-mantle boundary withSKKScoda waves

SUMMARY In our previous studies we developed a method for imaging heterogeneity at and near the core– mantle boundary (CMB) with a generalized Radon transform (GRT) of (transverse component, broad-band) ScS data, and we developed a statistical model for producing images of the D �� discontinuity with variable confidence levels. In these applications, the background is smooth and perturbations are represented as contrasts. Here we extend the theory to allow (known) discontinuities, such as the CMB, in the background model. The resulting imaging operator, which is formally not a GRT, can be used, either alone or along with ScS, for the imaging of lowermost mantle structure and, in particular, the D �� discontinuity with the scattered SKKS wavefield. Synthetic seismograms calculated with the WKBJ method are used to test the performance of our approach. As a proof of concept, we transform ∼38 000 radial component SKKS waveforms into image gathers of a CMB patch beneath Central America. The SKKS image gathers and image traces are in good agreement with the image traces obtained from the GRT transform of ScS data.

Erratum: Volume operator for spin networks with planar or cylindrical symmetry [Phys. Rev. D73, 124004 (2006)

This paper constructs a kinematic basis for spin networks with planar or cylindrical symmetry, by exploiting the fact that the basis elements are representations of an O(3) subgroup of O(4). The action of the volume operator on this basis gives a difference equation for the eigenvalues and eigenvectors of the volume operator. For basis elements of low spin, the difference equation can be solved readily on a computer. For higher spins, I solve for the eigenvalues using a WKBJ method. This paper considers only the case where the gravitational wave can have both polarizations. The single polarization case is considered in a spearate paper.

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502–4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré–Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré–Birkhoff normal forms for these systems up to and including terms of O ( ϵ 2 ) , where ϵ is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O ( 1 / ϵ ) .

Trapped edge waves in stratified rotating fluids: numerical and asymptotic results

The existence of trapped edge waves in a rotating stratified fluid with non-constant topography is studied using asymptotic and numerical techniques. A refinement of the classical WKBJ method is employed that is uniform at both the shoreline and caustic, where the classical approximation is singular, and is also uniform over long distances from the shore. This approach requires the use of comparison equations and it is shown that the two used previously in the literature are asymptotically equivalent in terms of the wave amplitude, but have small differences in the predicted wave frequencies. These asymptotic results, and results using shallow-water theory, are then compared to results from a careful numerical study of the nonlinear differential eigenvalue problem, allowing their range of practical applicability to be assessed. This numerical approach is also used to investigate whether trapping occurs in non-trivial and realistic geometries in the internal gravity wave band, which has been an open question for some time.

Mode formulas for shallow water waveguides using a modified asymptotic approximation

Sound speed profiles in shallow water waveguides are small deviations from isospeed and often decrease monotonically with depth. Approximation formulas for the propagating modes are found using a modified form of the classical WKBJ method. The ocean bottom is taken as homogeneous. The approach is accurate for modes with phase speeds greater than, and even slightly less than, the maximum water sound speed. The validity and accuracy of the approximations over frequency and mode number are illustrated using benchmark numerical calculations. Comparisons with approximations from other methods, including previous WKBJ approaches, are described. The formulas here are typically more compact and convenient. The results demonstrate how changes in parameters, such as frequency, bottom sound speed, and water sound speed profile quantitatively affect the mode functions. Applications for representative waveguide profiles are provided. [Work supported by the Office of Naval Research.]

Photovoltaic spatial soliton pairs in two-photon photorefractive materials

We report the existence of incoherently coupled bright–bright steady state photovoltaic soliton pairs in two-photon photorefractive material under open circuit conditions. Based on WKBJ method and paraxial ray approximation, we have obtained coupled equations describing dynamical evolution of spatial soliton pairs. In the steady state regime, the present analysis leads to the identification of existence equation of bright–bright solitons, which captures a plethora of soliton pairs. We have undertaken linear stability analysis which shows that these solitons are stable.