Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

Chapter 2
Inverse theory
2.1 Introduction
The laws of physics allow us to simulate the action and interaction of physical parameters within
a given system called the model. This model is a representation of a natural system (the earth,
the subsurface,... the atom...) and is described by a set of model parameters (velocity, density,
conductivity....). This model, once excited, yields a set of measurable quantities: the data. The
system of equations relating the data d to the model m is called the forward problem and is
noted
g (m) = d (2.1)
where g(m) is a set of mathematical equations dependent on m.
The reverse action consists in finding the model m that explains a set of observed data d obs
and is called the inverse problem. The complexity of the inverse problem is closely related to
the complexity of the forward problem. The latter must be known in order to determine the
inverse solution. Often, this answer is an estimate and differs from the true model because
the forward modeling provides with incomplete information on the model. Furthermore, the
forward problem is an approximation and the model is a simplified representation of the true
system. In addition, the observed data are recorded by instruments and thus contain noise.
In waveform inversion, we aim to recover a quantitative representation of a model of the
subsurface. The model is the most often characterized by the P-wave velocity varying with depth
although other quantities may be accounted for (S-wave, quality factor....). The information
used as data are the seismic traces recorded at the earth’s surface by receivers. These receivers
13