It is essentially necessary to calculate square of the difference for every pair of given data in order to obtain a variogram, which is one of the most important functions in geostatistics. If data number is N, the number of pairs is N(N-1)/2. Accordingly, the calculation time increases proportionally to square of N. As general statistical data processing, it is possible to calculate a variogram using a part of data randomly collected from all data (say partial calculation using a part of data). In addition to this, variograms have been calculated using all data of a randomly rearranged sequence but using a part of their pairs of the following categories: 1) every datum pairs only limited data leading the sequence (partial calculation giving leading data priority), 2) every datum pairs only data neighboring in the sequence (partial calculation giving neighboring data priority), and 3) all data are divided into several groups of a similar size, and every datum pairs only the data belonging to the same group (partial calculation grouping data). When the similarity between two variograms are evaluated by correlation coefficient between them, variograms obtained by the partial calculations giving neighbors priority and using grouped data have high similarity to the variogram obtained by the complete calculation in the present data set, which is residues from a linear regression plane of submarine topography near Bonin islands (N = 8891). The correlation coefficient is higher than 0.99, if the partial calculation giving neighbor priority is carried out where the pair number is 1/100 of that of the complete calculation. If a correlation coefficient between two variograms whose neighbor numbers are different by 1 in the partial calculation is 0.9s (s denotes number of 9 below the decimal point), it is expected that the correlation coefficient between the variograms in the partial and complete calculations is higher than 0.9s/2.

View full abstract

Logical model of geologic structure has been described strictly by a set of mathematical formulae and/or conventionally by a relation table to define a logical relation between geologic units and surfaces. The present paper proposes a graphical expression of the logical model in terms of a labeled binary tree. A root shows a whole space under consideration. Each branch node shows a subspace divided by surfaces. Two branches leaving from a branch nodes show a subdivision by a surface. A terminal node or a leaf shows the distribution of a geologic unit. The labels of branches on a path from a root to a leaf represent the relation between a geologic unit and surfaces. As for the logical model that can be expressed by a binary tree, the graphical expression is equivalent to the relation table of a logical model; one can be drawn from the other without loss of information. There exists a simple algorithm to derive a graphical expression for a logical model of geologic structure formed through a sequence of sedimentation and erosion based on its recursive definition. It is confirmed by simple examples that the graphical expression is useful as a visual tool to understand a nature of geologic structure expressed by a logical model and basic concepts related to three-dimensional geologic modeling such as a function to assign a geologic units to a point within a space.

View full abstract