Spatial Analysis of Cell Composition Data

A version of Matheron’s discrete Gaussian model is applied to cell composition data.The examples are for map patterns of felsic metavolcanics in two different areas. Q-Qplots of the model for cell values representing proportion of 10 km x 10 km cell areaunderlain by this rock type are approximately linear, and the line of best fit can be usedto estimate the parameters of the model. It is also shown that felsic metavolcanics in theAbitibi area of the Canadian Shield can be modeled as a fractal

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocksbased on compositional statistics is introduced. It improves and generalizes the commonjoins-count statistics known from map analysis in geographic information systems.Assigning phases independently to objects in RD is modelled by a single-trial multinomialrandom function Z(x), where the probabilities of phases add to one and areexplicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistenciesof the tests based on the conventional joins{count statistics and their possiblycontradictory interpretations are avoided. In practical applications we assume that theprobabilities of phases do not depend on the location but are identical everywhere inthe domain of de nition. Thus, the model involves the sum of r independent identicalmultinomial distributed 1-trial random variables which is an r-trial multinomialdistributed random variable. The probabilities of the distribution of the r counts canbe considered as a composition in the Q-part simplex SQ. They span the so calledHardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This isa generalisation of the well-known Hardy-Weinberg law of genetics. If the assignmentof phases accounts for some kind of spatial dependence, then the r-trial probabilitiesdo not remain on H. This suggests the use of the Aitchison distance between observedprobabilities to H to test dependence. Moreover, when there is a spatial uctuation ofthe multinomial probabilities, the observed r-trial probabilities move on H. This shiftcan be used as to check for these uctuations. A practical procedure and an algorithmto perform the test have been developed. Some cases applied to simulated and realdata are presented.Key words: Spatial distribution of crystals in rocks, spatial distribution of phases,joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinbergmanifold, Aitchison geometry

Rockfall travel distance analysis by using empirical models (Solà d'Andorra la Vella, Central Pyrenees)

The prediction of rockfall travel distance below a rock cliff is an indispensable activity in rockfall susceptibility, hazard and risk assessment. Although the size of the detached rock mass may differ considerably at each specific rock cliff, small rockfall (<100 m3) is the most frequent process. Empirical models may provide us with suitable information for predicting the travel distance of small rockfalls over an extensive area at a medium scale (1:100 000¿1:25 000). "Solà d'Andorra la Vella" is a rocky slope located close to the town of Andorra la Vella, where the government has been documenting rockfalls since 1999. This documentation consists in mapping the release point and the individual fallen blocks immediately after the event. The documentation of historical rockfalls by morphological analysis, eye-witness accounts and historical images serve to increase available information. In total, data from twenty small rockfalls have been gathered which reveal an amount of a hundred individual fallen rock blocks. The data acquired has been used to check the reliability of the main empirical models widely adopted (reach and shadow angle models) and to analyse the influence of parameters which affecting the travel distance (rockfall size, height of fall along the rock cliff and volume of the individual fallen rock block). For predicting travel distances in maps with medium scales, a method has been proposed based on the "reach probability" concept. The accuracy of results has been tested from the line entailing the farthest fallen boulders which represents the maximum travel distance of past rockfalls. The paper concludes with a discussion of the application of both empirical models to other study areas.