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The inverse fundamental operator marching method (IFOMM) is suggested to solve Cauchy problems associated with the Helmholtz equation in stratified waveguides. It is observed that the method for large-scale Cauchy problems is computationally efficient, highly accurate, and stable with respect to the noise in the data for the propagating part of a starting field. In further, the application scope of the IFOMM is discussed through providing an error estimation for the method. The estimation indicates that the IFOMM particularly suits to compute wave propagation in long-range and slowly varying stratified waveguides.

In many engineering applications, efficient mathematical methods are often required for the computing of propagation phenomena and transitions in complex systems. Recently, many interesting works on this issue are proposed to improve efficiencies in many different scientific areas, for example, the wavelet-related method for the integrodifferential and integral equations [

Some mathematical problems with their boundary conditions not completely known due to some technical difficulties often happen in many scientific and engineering areas described by the Helmholtz equation, such as ocean acoustics, wave propagation and scattering, and electromagnetic field. With the assistance of additionally supplied data, to determine the boundary conditions on the inaccessible part of the boundary or the source condition constitutes the inverse boundary value problem or the Cauchy problem, which is ill posed in the sense that small perturbations in the data may result in an enormous deviation in the solution. Here, the purpose of this study is to improve the efficiency for the computing of the Cauchy problems.

Some numerical methods for medium-scale problems have been proposed for the Cauchy problem [

Based on the exact one-way reformulation, the “inverse fundamental operator marching method” (IFOMM) [

This paper is arranged as follows. The basic mathematical formulations are described in Section

Consider the two-dimensional Helmholtz equation in a typical ocean and seabed environment with two curved interfaces

Sketch map of a waveguide in ocean acoustics.

We will concentrate on solving the equation for

The forward problem of (

Since the Helmholtz equation under consideration is in a range-dependent stratified waveguide with some curved interfaces, to flatten the curved interfaces of the dependent waveguide, we perform an analytic local orthogonal transform [

By flattening the stratified waveguide with two curved interfaces through coordinate transformation, (

This section reorganizes the operator form of the IFOMM for inverse boundary value problems in multilayered waveguide firstly. An error estimation for the IFOMM is then provided to discuss its properties. In the end, the forward fundamental operator marching method [

Let

The DtN operator

The DtN operator

Equation (

The inverse fundamental operator marching method can be described as follows.

If

Let

If

Generally, the

The error estimation for the IFOMM is provided firstly. Based on the estimation, the utilization scope of the method in stratified waveguides is analyzed.

From the operator marching process (

Then, recurse the formula (

According to (

Suppose that

Let

For weakly range-dependent stratified waveguides, if there exist a constant

In weakly range-dependent waveguides, the reflection waves are very weak which leads to

The operators

A more slim grid may approximate the original problem certainly, while in the same time, it may also amplify the initial errors if the number of discrete points in range direction tend to the infinite. Since large-range step method is used to discretize the computed domain for slowly varied waveguides and overdense grid is of no need for obtaining required accuracy, the IFOMM is stable in weakly range-dependent waveguides without much reflections according to the corollary.

The theorem gives two major factors which affect the IFOMM solutions of (

The formulas in (

Furthermore, we approximate the operator

Let the DtN operator

If

If

Load

Load

If

If similar error analysis of the IFOMM is applied to the FFOMM, similar result can be obtained. The conditions on which the Helmholtz equation can be exactly solved by the FFOMM also demand that the reflection of the propagating modes be small in the waveguide.

A typical numerical example in ocean acoustics is provided here to examine the IFOMM for solving Cauchy problems in stratified waveguides.

Let

Suppose incident wave

The numerical example takes the incident wave at

In practice, the available data is usually contaminated by measurement errors, and the stability of the numerical method is of vital importance for obtaining physically meaningful results. To this end, the simulated noisy data generated by

Figures

The received wave

Comparison with

Comparison with

The received wave

Comparison with

Comparison with

The IFOMM developed in one-layered waveguides is applied to solve inverse boundary value problems associated with the Helmholtz equation in stratified waveguides. Numerical example demonstrates that the IFOMM can be used to compute inverse boundary problems in the stratified waveguides successfully. The scope of the IFOMM's application is then discussed based on an error estimation for the marching scheme. The estimation gives a quantitative estimate of error propagation and shows that IFOMM can only be applied in waveguides where reflection is not strong. In further, the theorem also reveals that the errors may be cumulated greatly in some special circumstances when the grid becomes excessive fine. Numerical examples indicate that the IFOMM is computationally efficient, highly accurate, and stable with respect to the noise in the data for the propagating part of a starting field when the computing domain is long and complex.

There are several further studies related to the IFOMM for solving the Cauchy problems. Firstly, although the IFOMM only considers two-dimensional problems in its current form, the scheme is easily extended to problems in three-dimensional space under cylindrical coordinates, and it can also be extended to solve wave propagation under three-dimensional Cartesian coordinate system. Secondly, when the number of propagating modes varies with the range direction frequently, which is corresponding to some of propagating modes that are totally reflected, whether the IFOMM or FFOMM can be applied through some improvements of them in such waveguides, and how to improve the methods? At least, the parameters for truncating the systems are a little more difficult to be determined. Thirdly, the estimation for error propagation may be improved through more detailed analysis.

This research is supported by the NCET-08-0450 and the 985 II of Xi'an Jiaotong University.