Regularization of the AVO inverse problem by means of a ...

CHAPTER 1. INTRODUCTION 4
1.3 AVO inversion
As defined above, the amplitude variation with offset (AVO) describes the change in am-
plitude of a seismic signal incident on an interface between two adjacent layers of different
contrast as a function offset. AVO inversion is one of the inversion techniques that is be-
coming most popular in oil and gas industries. It uses pre-stack seismic data that has
been processed in amplitude preserving fashion. The inversion helps to extract information
about subsurface physical parameters such as P-wave velocity, S-wave velocity, and density
or their attributes. The combination of these parameters is a powerful tool to distinguish
hydrocarbon reservoirs from other subsurface materials. The basis for forward modeling
in AVO inversion are the Zoeppritz equations. These equations provide a non-linear link
between elastic parameters and amplitudes of waves transmitted and reflected at a given
interface (Aki and Richards, 1980). The non-linearity of this equation makes it complicated
for inversion. The first approximation to the Zoeppritz equation was provided by Bortfeld
(1962). This equation shows clear relationship between the reflection coefficient and the
earth model parameters but non-linear with respect to those parameters. Following the
first approximation, other researchers have made linearized approximations including Aki
and Richards’s, Shuey’s, and Smith and Gidlow’s, Fatii’s approximations. Using these ap-
proximations as a physical model, one can construct an inverse problem in order to retrieve
the model parameters associated to the physical model. Combining the information from
the processed data with the results of the inversion, makes it possible to produce accurate
information about the subsurface. AVO inversion is an ill-posed problem which makes the
solution of the inversion unreliable and therefore it needs regularization. Bayesian method
is a statistical approach that can be used for many geophysical problems including AVO to
formulate inverse problems with prior information (Tarantola, 1987; Ulrych et al., 2001). It
combines the observed data with the knowledge we have about the model parameters be-
fore measuring the data. This could increase stability of the inversion and leads to reliable
result. Before I go into the details of prior information, I would like to explain some of the
empirical relationships among P-wave velocity, S-wave velocity, and density.
1.4 Empirical relationship between AVO parameters
It has been shown that P-wave velocity and S-wave velocity, P-wave velocity and density, and
S-wave velocity and density are correlated. These also imply strong correlation among their
corresponding adjacent layer contrast parameters; P-wave reflectivity, S-wave reflectivity,
and density reflectivity.