Tomography (2):Model

What is Model?

Model is the process by which we can reproduce certain phenomenon.  It can be used to predict the phenomena based on certain model parameters (forward modeling). It can also be used to determine the model parameters based on measurements of the phenomenon.  For instance, Ohm’ law can be regarded as a simple example of a model.  The model parameter here is the resistance, whereas the potential difference (V) is the measured phenomena according to the form:

V=IR

Where ‘I’ is  the current intensity and R is the resistance in ohm. In forward modeling, we can predict V, whereas for inversion we estimate R based on a number of measurements of the pair V and I.  The inversion is simply a linear regression or least squares curve fitting.

Models can be subdivided into analog and mathematical. Analog modeling was extensively used before the availability of both powerful computers and advanced software.  Mathematical modeling was available a long time ago to study natural phenomena. The partial differential equations in any classical mathematical physics textbook belong to the later subdivision.  This latter subdivision is the one related to the present discussion.

The mathematical or computational model is by itself subdivided into two main classes; namely, analytic and numerical. Analytical modeling is exact (error free solution), however, it can be applied to simple cases only.  Numerical modeling, on the other hand, can be applied to complex cases, however, errors are introduced (why?!!!). To answer this question, we need to simply understand how the numerical solution is performed. For simplicity, in the numerical solution, we represent the original function by another basis functions that are easier for computation.  The Taylor expansion series is an example for this. The representation of the original function by basis function is not exact. In Taylor series, one can choose a number of terms only and neglect the rest. The choice of the number of terms depends on the problem at hand.  For problems that require high accuracy, we usually choose more terms, however, the computational time and cost will increase. Of course, other types of errors are present and not related directly to the computational process.  Examples are measuring errors, roundoff error …etc.

In tomography, the analytic model is approximated by the so-called high-frequency ray series asymptote.  In such a case, the wavefield is approximated by rays. The solution using ray theory involves the estimation of travel time from the source to the receiver in addition to the amplitudes.  For simplicity, I will focus on the travel time estimation for tomography modeling (the case widely used in seismology).

For tomography calculation, we need to determine the path followed by the ray (ray tracing), 2D or 3D block representation of the system (media under investigation), A priori model and an inversion technique (usually non-linear iterative one).