Perturbative inversion methods

Perturbative inversion methods for obtaining bottom geoacoustic parameters

Attention is focused primarily on the modal eigenvalue inverse problem for which the theory for determining the compressional  wave speed, compressional wave attenuation, and density is developed in detail. Properties of this technique are studied using synthetic data and include investigations of the dependence of the results on acoustics frequency, number of modes excited, and partial a priori knowledge of the bottom.

Inversion Problem:
The required input data are trapped mode eigenvalues for one or more frequencies, the group velocity dispersion curves for one or more modes, or the cw pressure field versus range (complex field or magnitude only) .

A great explanation of inverse problem:
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in. For example, if we have measurements of the Earth's gravity field, then we might ask the question: "given the data that we have available, what can we say about the density distribution of the Earth in that area?" The solution to this problem (i.e. the density distribution that best matches the data) is useful because it generally tells us something about a physical parameter that we cannot directly observe. Thus, inverse problems are one of the most important, and well-studied mathematical problems in science and mathematics. Inverse problems arise in many branches of science and mathematics, including: computer vision, machine learning, statistics, statistical inference, geophysics, medical imaging (such as computed axial tomographyand EEG/ERP), remote sensing, ocean acoustic tomography, nondestructive testing, astronomy, physics and many other fields.


(Linear Inverse Theory)

The objective of an inverse problem is to find the best model, m, such that (at least approximately)

where G is an operator describing the explicit relationship between the observed data, d, and the model parameters. In various contexts, the operator G is called forward operator,observation operator, or observation function. In the most general context, G represents the governing equations that relate the model parameters to the observed data (i.e. the governing physics).

Mathematical

One central example of a linear inverse problem is provided by a Fredholm first kind integral equation.

Check Fredholm first kind integral equation in next several posts
For sufficiently smooth g the operator defined above is compact on reasonable Banach spaces such as L^p spaces. Even if the mapping is injective its inverse will not be continuous. (However, by the bounded inverse theorem, if the mapping is bijective, then the inverse will be bounded (i.e. continuous).) Thus small errors in the data d are greatly amplified in the solution m. In this sense the inverse problem of inferring m from measured d is ill-posed.


Numerical Scheme (quadrature scheme):
Simpson's Rule
http://en.wikipedia.org/wiki/Simpson's_rule
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:





2 Regularization Method

Regularization:
http://en.wikipedia.org/wiki/Regularization_(mathematics)
Regularization of Inverse Problem
http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-0-7923-4157-4

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that

1. A solution exists
2. The solution is unique
3. The solution depends continuously on the data, in some reasonable topology.

http://en.wikipedia.org/wiki/Ill-posed_problem

In mathematics and statistics, particularly in the fields of machine learning and inverse problems, regularization involves introducing additional information in order to solve an ill-posed problem or to prevent overfitting. This information is usually of the form of a penalty for complexity, such as restrictions for smoothness or bounds on the vector space norm.

Problems that are not well-posed in the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed.

Such continuum problems must often be discretized in order to obtain a numerical solution. While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision, or with errors in the data. Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. An ill-conditioned problem is indicated by a large condition number.

If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as regularization and Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.

Other Keywords:
Normed Vector Space/Vector Space
minimum norm solution
A MatLab Program: Least squires with solving minimum norm solution
http://www.mathworks.com/matlabcentral/fileexchange/17474


Smooth Function:
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.