An Estimation Procedure for the Joint Distribution of Spatial Direction and Thickness of Flat Bodies Using Vertical Sections Part I: Theoretical Considerations

Abstract

AbstractIn soil micromorphology fissures are considered in vertical sections. To get information about the properties of the soil the joint distribution of spatial direction and width is of interest. The fissures are mathematically generalized to flat bodies which form a stationary weighted surface process with the weight “thickness”. Because of stationarity a joint distribution of spatial direction and thickness exists in a “typical point” of the surface process. A suitable parametric family of distributions is assumed. The corresponding parameters can be estimated from measurements on the vertical sections. But on the sections only the visible thickness and the visible angle of a fissure can be measured. Therefore the joint distribution of these variables is expressed by the joint spatial distribution of spatial direction and thickness. This derived distribution depends on the same parameters. The Chi‐Square method is proposed for the parameter estimation. The estimation procedure is demonstrated using the Bingham‐Mardia distribution for the direction and the lognormal distribution for the thickness and by defining a way to correlate the mean thickness and the direction.