Multifractal portrayal of the Swiss population

Fractal geometry is a fundamental approach for describing the complex irregularities of the spatial structure of point patterns. The present research characterizes the spatial structure of the Swiss population distribution in the three Swiss geographical regions (Alps, Plateau and Jura) and at the entire country level. These analyses were carried out using fractal and multifractal measures for point patterns, which enabled the estimation of the spatial degree of clustering of a distribution at different scales. The Swiss population dataset is presented on a grid of points and thus it can be modelled as a "point process" where each point is characterized by its spatial location (geometrical support) and a number of inhabitants (measured variable). The fractal characterization was performed by means of the box-counting dimension and the multifractal analysis was conducted through the Renyi's generalized dimensions and the multifractal spectrum. Results showed that the four population patterns are all multifractals and present different clustering behaviours. Applying multifractal and fractal methods at different geographical regions and at different scales allowed us to quantify and describe the dissimilarities between the four structures and their underlying processes. This paper is the first Swiss geodemographic study applying multifractal methods using high resolution data.

Estimation of the correlation structure of crustal velocity heterogeneity from seismic reflection data

Numerous sources of evidence point to the fact that heterogeneity within the Earth's deep crystalline crust is complex and hence may be best described through stochastic rather than deterministic approaches. As seismic reflection imaging arguably offers the best means of sampling deep crustal rocks in situ, much interest has been expressed in using such data to characterize the stochastic nature of crustal heterogeneity. Previous work on this problem has shown that the spatial statistics of seismic reflection data are indeed related to those of the underlying heterogeneous seismic velocity distribution. As of yet, however, the nature of this relationship has remained elusive due to the fact that most of the work was either strictly empirical or based on incorrect methodological approaches. Here, we introduce a conceptual model, based on the assumption of weak scattering, that allows us to quantitatively link the second-order statistics of a 2-D seismic velocity distribution with those of the corresponding processed and depth-migrated seismic reflection image. We then perform a sensitivity study in order to investigate what information regarding the stochastic model parameters describing crustal velocity heterogeneity might potentially be recovered from the statistics of a seismic reflection image using this model. Finally, we present a Monte Carlo inversion strategy to estimate these parameters and we show examples of its application at two different source frequencies and using two different sets of prior information. Our results indicate that the inverse problem is inherently non-unique and that many different combinations of the vertical and lateral correlation lengths describing the velocity heterogeneity can yield seismic images with the same 2-D autocorrelation structure. The ratio of all of these possible combinations of vertical and lateral correlation lengths, however, remains roughly constant which indicates that, without additional prior information, the aspect ratio is the only parameter describing the stochastic seismic velocity structure that can be reliably recovered.