An algorithm to reduce the number of colors in a color image is explained. To bring the appearance frequency of the colors close the same, the number of colors in an image is reduced by dividing three-dimensional space (RGB-space) in which the colors are distributed. The amount of information included in the reduced image is left by this algorithm as much as possible. A software library to reduce the number of colors both in the RGB and palette color images is introduced for the realization of this algorithm. For the application of this library, the UNIX command“tiffreduce”is introduced. This command is available for the color reduction of a TIFF image file.

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Scaling methods are applied to the categorized soil names in drilling-logs for numerically evaluate the ground condition in depth direction. The numerically converted categories are amiable to the succeeding mathematical operations.
The present report outlines the reseach on scaling of soil categories, then shows the application of scale for the cluster analysis of drilling-logs and the constraction of zoning map based on the geodetic coordinates of drilling-logs and their cluster.
We analized 149 drilling-logs in the eastern area of Muroran city, Hokkaido, Japan. Soil names in drilling-logs are summarized into following 13 successive categories: surface soil<highly organic soil<volcanic cohesive soil<organic soil<cohesive soil<silt<sandy soil<sand<gravelly soil<gravel<tuffaceous rock (the Quaternary system) <tuffaceous rock (the Tertiary system) <rock block (except tuffaceous rock) .
Scaling methods are applied to the successive categories. Twenty-two sets of scale values are ex-amined for the categories. The summary of the results are shown below.
1) There is little reason to choose anyone among equal appearing intervals, intentional scales, theoretic scales and semitheoretic scales.
2) The scale evaluation of drilling-log with the equal appearing intervals or the intentional scale have strong correlation with the ones of the theoretical or the semitheoretical scale in statistics.
3) In this study, the intentional scale of 1, 2, 3, …, 10, 11, 16, 17 is adopted for the categories, which is the most amiable scale for interpolation of category and cluster analysis of dilling-logs.
4) The representative drilling-log of each cluster are selected easily. The representative drilling-logs of each cluster gives us concrete image of the soil system of the zoning area.

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In order to formulate the distribution of strata in a survey area Ω in terms of subspaces b₁, b₂, …, b_n bounded by boundary surfaces S₁, S₂, …, S_n-1, we introduce a new concept called C₁ and C₂ type of boundary surface as a mathematical model of conformity and unconformity, respectively. Let S_t be a boundary surface which divides a given successive sequence of subspaces (b_r, b_r+1, …, b_k) into two successive sequences (b_r, …, b_t) and (b_t+1, …, b_k) . Then a surface S_k is called C₁ type of boundary surface if and only if S_t also divides a successive sequence (b_r, …, b_k, b_k+1) into two successive sequences (b_r, ..., b_t) and (b_t+1, …, b_k, b_k+1) . On the other hand, the boundary surface S_k is called C₂ type of boundary surface if and only if S_k divides a successive sequence (b₁, b₂, …, b_k, b_k+1) into a successive sequences (b₁, …, b_k) and a single subspace b_k+1. It is proved that all subspaces b₁, b₂, …, b_n are uniquely defined by boundary surfaces S₁, S₂, …, S_n-1 if subspaces are bounded by either C₁ or C₂ type of boundary surfaces. According to the formulation of strata in terms of subspaces bounded by boundary surfaces, we can define a function g which assigns a label corresponding to a stratum to every point in Ω. The formulation of subspaces and the labeling function provide theoretical bases of the computerized geologic mapping system“CIGMA”.

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