Propositions belonging to the PhD. thesis
A scaling medium representation
a discussion on well-logs, fractals and waves
by
Felix Johan Herrmann

Delft, 21 january 1997

Back to preprint page Felix Herrmann

1. The choice of the multiscale analyzing function, the wavelet, determines the part of the singularity spectrum that can be observed.
   [Muzy J. F., Bacry E. and Arneodo A, Physical Review E, 1993; Chapter 2 and 8 of this thesis]
2. Well-log measurements - and hence the earth's heterogeneities along the vertical direction - display a multifractal scaling behaviour across a wide range of scales.
   [Chapter 2 of this thesis]
3. Concepts such as (fine-)layering loose their meaning in cases where the medium properties display a fractal behaviour.
   [Chapter 2 of this thesis]
4. The fact that well-log measurements display a (multi)fractal behaviour has as a consequence that the singular continuous part of the spectral measure, cognate with the Hamiltonian, can not be precluded.
   [Chapter 4 of this thesis]
5. Many (geo)physical phenomena display a fractal behaviour. It is hard to understand why this important observation had, until now, such a limited number of consequences for (geo)physics.
6. The crucial assumption within continuum mechanics, the separation of scales, is difficult to reconcile with a complexity that can not be discerned from being (multi)fractal.
7. Problems with the so-called upscaling (coarse-graining) become manifest when the constitutive parameters display a significant scale dependence within the coarse grained domain.

8. It is noteworthy that the notion of scale and scale dynamics (the occurrence of scale derivatives) did not find its way into the current formulation for the wave motion in those cases where the constitutive parameters display a scale dependence.
9. At the moment that the compressional wavespeed depends on scale one can only issue a statement on the velocity given the definition of a gauge. The most straightforward candidate for this gauge, within the wave problem, would be the wavelength. However, here one ends in a ``Catch 22'' because the wavelength itself depends on the velocity.
10. The current explanation for the observed dissipation with the help of visco-elasticity is nothing but a choice of parametrization for which a true physical explanation in the geophysics is lacking. Despite the fact that such a mechanism gets rid of some problems, such as guaranteeing causality, an explanation with the help of scale dynamics may offer a physical plausible alternative.
11. The inexistence of a dynamic homogenization theory is an indication that the current wave equation is not well equipped to handle the singular reflections and dispersion - both induced by the scaling complexity - in a physical plausible manner.
12. The fact that the Zoeppritz equations are based on an isolated stepfunction in medium properties stands in the way of an adequate explanation of the observed AVO behaviour.
13. Scale exponents are possibly litho-stratigraphical indicators since they characterize the local complexity.
14. Difficult to translate the original proposition from dutch but the general meaning is that solving the inverse problem based on the wave equation will never yield a well-log.
15. The problem with engineers is that they want to be accurate in the wrong way.

 
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