A Scaling Medium Representation And Its Implication For Acoustic Wave Propagation

The main subject of the research reported in this paper is to u nderstand how the complexity of the earth’s subsurface is tra nsferred to the wavefield. It will be shown that the apparent com plexity can be captured by means of a scaling medium represen tation. In this multiscale representation the Hölder expo nents and the singularity spectrum constitute useful order of mag nitude estimates mathematically quantifying the local and gl obal aspects of the singularity structure delineating the scali ng. At this point the question arises whether the current formulat ion for acoustic wave motion is capable of describing the transport of the singularity structure, displayed by the medium’s heter og neity, to the acoustic wavefield and vice versa. The ultimate go al of this new approach is to come up with an alternative descrip tion for the global dynamic effects, the dispersion, and the local dynamic effects, the frequency dependent AVO, both of which can be seen as manifestation of the scaling complexity displ ayed by the heterogeneities in the constitutive parameters as th ey are being observed from well-logs. Introduction As a consequence of the ever increasing demand for hydrocarbons and the steadily decline of newly explored resources an uprise has taken place in the production oriented seismic te chnology. The ambitions set by this new orientation are far mor e ambitious since they aim for the petrophysical and litholog ical information, sometimes even in a 4-D setting. For this reaso n the integration of different types of data, acquired within the disciplines of seismology and petrophysics, will be all importan t in the quest to further optimize the current production techni ques. Given this ambitious goal it will be necessary to interrelat measurements obtained at varying scales. This has been and stil l is the primary goal of this research project where I try to improve my understanding of the dynamics of waves interacting with a medium the properties of which display a wild fluctuating behaviour across a wide range of scales. The main questions emerging in this project are at best paraphrased by: Is it possible to come up with a dynamic wave theory explaining for the apparent dispersion and reflection induced by scaling/n o differentiable medium heterogeneities? Is it possible to c ome up with a “dynamic wave theory” allowing for a better integration of data acquired at different scale ranges? Before trying to answer these eloquent questions it is of primary importance to first find an adequate, (mathematically) proper, representation for the the medium’s erratic heterogeneity. As I will make clear the con-