Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

A three-dimensional(3D) parabolic equation(PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.     

A single-scattering correction for large contrasts in elastic layers

The single-scattering solution is implemented in a formulation that makes it possible to accurately handle solid-solid interfaces with the parabolic equation method. Problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems. The approach can handle complex layering and is applicable to a large class of seismic problems. The solution of the scattering problem is based on an iteration formula, which has improved convergence in the new formulation, and the transverse operator of the parabolic wave equation, which is implemented efficiently in terms of banded matrices. Accurate solutions can often be obtained by using only one iteration.     

A parabolic linear evolution equation for cellular detonation instability

Using the combined limits of a large activation energy and a ratio of specific heats close to unity, a dispersion relation has recently been derived which governs the stability of a steady Chapman - Jouguet detonation wave to two-dimensional linear disturbances. The analysis considers instability evolution time scales that are long on the time scale of fluid particle passage through the main reaction layer. In the following, a simplified polynomial form of the dispersion relation is derived under an appropriate choice of a distinguished limit between an instability evolution time scale that is long on the time scale of particle passage through the induction zone and a transverse disturbance wavelength that is long compared to the hydrodynamic thickness of the induction zone. A third order in time, sixth order in space, parabolic linear evolution equation is derived which governs the initial dynamics of cellular detonation formation. The linear dispersion relation is shown to have the properties of a most unstable wavenumber, leading to a theoretical prediction of the initial detonation cell spacing and a wavenumber above which all disturbances decay, eliminating the growth of small-wavelength perturbations. The role played by the curvature of the detonation front in the dynamics of the cellular instability is also highlighted.     

A two-way parabolic equation that accounts for multiple scattering

A two-way parabolic equation that accounts for multiple scattering is derived and tested. A range-dependent medium is divided into a sequence of range-independent regions. The field is decomposed into outgoing and incoming fields in each region. The conditions between vertical interfaces are implemented using rational approximations for the square root of an operator. Rational approximations are also used to relate fields between neighboring interfaces. An iteration scheme is used to solve for the outgoing and incoming fields at the vertical interfaces. The approach is useful for solving problems involving scattering from waveguide features and compact objects.     

Impedance-matched absorbers for finite-difference parabolic equation algorithms

In this paper, a perfectly matched layer (PML) absorber, recently introduced into the electromagnetic propagation literature by Berenger [J. Comput. Phys. 114, 185-200 (1994)], is adapted for use with both paraxial and wide-angle acoustic parabolic equations (PEs). Our procedure incorporates an imaginary component into the transverse coordinate that mimics the introduction of a fictitious absorber on the edge of the computational grid. Use of such an impedance-matched layer can significantly reduce spurious reflections compared to physical absorbing layer methods and thus allows a smaller number of boundary points to be employed in PE calculations. Numerical results obtained with several higher-order propagator approximations confirm that such impedance-matched absorbers efficiently eliminate reflections.     

The Riemann problem for a forward-backward parabolic equation

The Riemann problem for a forward-backward parabolic equation of interest in physical and biological models is studied. A complete classification of suitably defined entropy solutions is provided. Thereafter, the existence and uniqueness of a solution is proven, and its type is identified.     

A wide angle and high Mach number parabolic equation

Various parabolic equations for advected acoustic waves have been derived based on the assumptions of small Mach number and narrow propagation angles, which are of limited validity in atmospheric acoustics. A parabolic equation solution that does not require these assumptions is derived in the weak shear limit, which is appropriate for frequencies of about 0.1 Hz and above for atmospheric acoustics. When the variables are scaled appropriately in this limit, terms involving derivatives of the sound speed, density, and wind speed are small but can have significant cumulative effects. To obtain a solution that is valid at large distances from the source, it is necessary to account for linear terms in the first derivatives of these quantities [A. D. Pierce, J. Acoust. Soc. Am. 87, 2292-2299 (1990)]. This approach is used to obtain a scalar wave equation for advected waves. Since this equation contains two depth operators that do not commute with each other, it does not readily factor into outgoing and incoming solutions. An approximate factorization is obtained that is correct to first order in the commutator of the depth operators.     

Variational iteration method for solving an inverse parabolic equation

The variational iteration method is applied to solving an inverse problem of determining an unknown parameter in a linear parabolic equation. This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a functional. We get a rapid convergent sequence tending to the exact solution of the inverse problem. To show the efficiency of the present method, some interesting examples are presented.     

抛物方程远场近似条件分析

研究了抛物方程(PE)远场近似条件及起始场的计算,得到了容许误差和最小参考距离之间的关系。结果表明在最小参考距离以内,PE远场近似条件会通过PE传播算子的传递而引起较大的计算误差,因此,PE传播算子不能用于自起始场的计算。在此基础上,给出了没有远场近似条件限制的传播算子,该算子可以将起始场和远场计算统一起来,便于数值实现。对Pekeris波导的声传播问题(JASA标准PE测试问题)的数值计算结果表明,利用该算子可以精确地计算出PE自起始场和传播损失曲线。     

A parabolic equation for penetrable rough surfaces: using the Foldy-Wouthuysen transformation to buffer density jumps

The goal of this effort is to design a parabolic equation (PE) that can be fully integrated into modern rough surface scattering theory. The Foldy-Wouthuysen transformation is used to design a PE that addresses this challenge. The paradigm employed is based on a non-relativistic theory of the quantum Lamb shift, in which a parabolic equation (the Schrödinger equation) was used to model a field near a rough surface (the world line of the hydrogen nucleus advected by vacuum fluctuations). With the acoustic field serving as the prototypical classical field, the PE derived using the Foldy-Wouthuysen transformation exploits higher-order boundary conditions to buffer density discontinuities in a manner precisely dictated by the formalism. This is significant because the techniques used in rough surface scattering theory ultimately rely on perturbation theory, conformal mappings, a local method of images, or some similar distortion of the range-independent problem. This type of distortion is not possible with techniques currently employed at a density jump, and so ad hoc rules have been used instead. The new PE allows interfaces where the density jumps (such as the ocean bottom) to be distorted into rough ones, and so it is fully compatible with rough surface scattering theory.     

Estimates of homogenization for a parabolic equation with periodic coefficients

The asymptotic behavior of the operator exponent related to the Cauchy problem for a parabolic equation with periodic coefficients is studied either under the reduction of the periodicity cell or for large times. Estimates for the closeness of the operator exponentials (the original and the limit) with respect to the L ²-operator norm and the related H ¹-estimates are obtained under minimal assumptions concerning the smoothness of the heat matrix and of the initial data. Financially supported by RFBR under grant no. 05-01-00621.     

改进的单次散射相函数解析表达式

单次散射相函数对电磁辐射传输模拟过程的准确性和计算效率有重要的影响.基于电磁散射与辐射传输中的基本理论,对单次散射相函数的解析表达式进行了研究,提出了一种新的单次散射相函数解析表达式.比较了单个粒子的Henyey-Greenstein相函数、Henyey-Greenstein*相函数与新的相函数随角度的分布,发现新的散射相函数提高了后向散射峰值,可以更合理地描述单个粒子的散射特性.按三种气溶胶粒子谱分布模式计算了Henyey-Greenstein*相函数和新的相函数对应的数值结果,并与多分散系Mie散射相函数进行对比,发现新的相函数提高了与多分散系Mie散射相函数的符合程度.研究表明,对于大角度(大于90°)后向散射,新的相函数与Mie散射相函数均方根差较小的占73.3%,高于Henyey-Greenstein*相函数的26.7%,证明了新的相函数可以显著提高后向散射峰值.新的相函数对准确模拟辐射传输过程具有重要意义.     

Experimental testing of the variable rotated elastic parabolic equation

A series of laboratory experiments was conducted to obtain high-quality data for acoustic propagation in shallow water waveguides with sloping elastic bottoms. Accurate modeling of transmission loss in these waveguides can be performed with the variable rotated parabolic equation method. Results from an earlier experiment with a flat or sloped slab of polyvinyl chloride (PVC) demonstrated the necessity of accounting for elasticity in the bottom and the ability of the model to produce benchmark-quality agreement with experimental data [J. M. Collis et al., J. Acoust. Soc. Am. 122, 1987-1993 (2007)]. This paper presents results of a second experiment, using two PVC slabs joined at an angle to create a waveguide with variable bottom slope. Acoustic transmissions over the 100-300 kHz band were received on synthetic horizontal arrays for two source positions. The PVC slabs were oriented to produce three different simulated waveguides: flat bottom followed by downslope, upslope followed by flat bottom, and upslope followed by downslope. Parabolic equation solutions for treating variable slopes are benchmarked against the data.     

Generalization of the rotated parabolic equation to variable slopes

The rotated parabolic equation [J. Acoust. Soc. Am. 87, 1035-1037 (1990)] is generalized to problems involving ocean-sediment interfaces of variable slope. The approach is based on approximating a variable slope in terms of a series of constant slope regions. The original rotated parabolic equation algorithm is used to march the field through each region. An interpolation-extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source. For the elastic case, a series of operators is applied to rotate the dependent variable vector along with the coordinate system. The variable rotated parabolic equation should provide accurate solutions to a large class of range-dependent seismo-acoustics problems. For the fluid case, the accuracy of the approach is confirmed through comparisons with reference solutions. For the elastic case, variable rotated parabolic equation solutions are compared with energy-conserving and mapping solutions.     

Higher order parabolic approximations of the reduced wave equation

Asymptotic solutions of order k⁻ⁿ are developed for the reduced wave equation. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is applied to the example of radiation from a piston, and it is demonstrated that the higher order approximations are more accurate for decreasing values of k.     

A single-scattering approximation for infrared radiative transfer in limb geometry in the Martian atmosphere

We present a single-scattering approximation for infrared radiative transfer in limb geometry in the Martian atmosphere. It is based on the assumption that the upwelling internal radiation field is dominated by a surface with a uniform brightness temperature. It allows the calculation of the scattering source function for individual aerosol types, mixtures of aerosol types, and mixtures of gas and aerosol. The approximation can be applied in a Curtis-Godson radiative transfer code and is used for operational retrievals from Mars Climate Sounder measurements. Radiance comparisons with a multiple scattering model show good agreement in the mid- and far-infrared although the approximate model tends to underestimate the radiances in realistic conditions of the Martian atmosphere. Relative radiance differences are found to be about 2% in the lowermost atmosphere, increasing to ∼10% in the middle atmosphere of Mars. The increasing differences with altitude are mostly due to the increasing contribution to limb radiance of scattering relative to emission at the colder, higher atmospheric levels. This effect becomes smaller toward longer wavelengths at typical Martian temperatures. The relative radiance differences are expected to produce systematic errors of similar magnitude in retrieved opacity profiles.     

Mode equation and its solution for dielectric waveguide with parabolic cross-section

The mode equation for the dielectric waveguide with parabolic cross-section is derived using both the effective index method and WKB theory, and the expression of the mode propagation constant or effective index is solved directly from this mode equation, so that the mode-propagation properties of this kind of waveguide can be analysed conveniently with comparative accuracy.