Multi- and Monoscale Attributes for Well

Edges in a medium are the primary source of coherent reflections because they exhibit a large or even diverging amplitude behavior for their derivatives. Generally the medium properties are not only assumed to jump across interfaces, limiting the edge's singular behavior to that of a jump discontinuity, but the interfaces are also assumed to be well separated. Multiscale analysis on well data shows that the model of a jump discontinuity is too limited to account for the scaling behavior displayed by these types of data sets across the seismic scale range. It also demonstrates that the edges are not well separated. These observations coined two generalizations. First the jump is generalized to a wider class of scale exponent indexed transitions of which the jump is a special case. Secondly the edges are allowed to accumulate. The first part of this paper is devoted to the substantiation of these two generalizations. It introduces the necessary tools for the multiscale analysis, which characterizes the individual edges by means of scaling exponents and the overall texture by singularity spectra. The first part is concluded with a discussion on the application to well and seismic data. In the second part a complementary method to obtain information on the scaling is proposed. It is aimed to deal with the unfortunate fact that the scale content of the seismic signal is relatively small, making it difficult to conduct the multiscale analysis. For instance it is hard to obtain estimates for the local scaling exponents, characterizing the different types of transitions via their induced reflectivity. The novel method presented uses fractional differentiations/integrations to estimate the scale exponents at a fixed scale. The estimated scale exponents not only capture the local scaling characteristics but are also related to the local frequency behavior of the reflections. In this capacity they constitute local stratigraphical texture parameterizations. Local texture is relevant for the identification of the major geological markers as well as for localization and characterization of the major channels and barriers for the fluid flow being all important characteristics for the reservoir. Multifractal singularity spectra, on the other hand, provide more general information on the global texture and they are highly relevant for geological sequences and for the properties of the reservoir rock.