Numerical Tests on tomography

Tomography, which is derived from the Greek word “tomos” that is “slice”, denotes

forming an image of an object from measurements made from slices or rays through

it. In other words, forming the structure of an object based on a collection of slices

through it.

In principle, it is usually an inverse problem where we have the data, we formulate the

geometry of the ray paths and invert for the model parameters that defines the

structure.

d = Gm

where, d is the vector containing the data, m is the vector containing the model

parameters and G is the kernel which maps the data into the model and vice-versa.

If there are N number of data and M number of model parameters then the size of the

matrix G is M×N.

The first step of this problem is to formulate a theoretical relationship between the

data and model spaces and the underlying physics. Once, this theoretical relationship

is established or the matrix G is formulated, we can address the above problem in two

ways – a forward problem or an inverse problem.

The forward problem can be taken as a trial and error process or as an exhaustive

search through the model space for the best set of model parameters which can reduce

the misfit between the observed and the predicted data. For this case, we guess the

model based on the simple real world geology and then determine the data based on

the theoretical relationship between data and models. In contrast, in the inverse

problem we invert for the model parameters using the data and considering the

theoretical relationship between the data and model parameters. Then we interpret for

the real world geology.

The solution of the linear tomographic inverse problem depends on the relationship

between Nand M. If N < M, then we have more number of model parameters or

variables to obtained and lesser number of data or equations to solve. In this case, the

equation d = Gm does not provide enough information to determine uniquely all the

model parameters and thus the problem is called underdetermined. We usually need

extra constraints to solve this kind of problem, for example we take the a priori

constraint that the solution to this problem is simple. If N = M, then there is ideally

enough information to determine the model parameters. But in real world, data is

usually inaccurate and inconsistent to invert for the best model parameters in this

case. If N > M, there is too much information contained in the equation d = Gm for

it to possess an exact solution. This case is called the overdetermined.

overdetermined

Overdetermined Problem

equal_determined

Equal – Determined Problem

mixed_determined

Mixed-Determined Problem

Additional Matlab codes:

Calculating G

Creating random G matrix of any size

—Utpal Kumar (IES,  Academia Sinica)