Surface estimation program BS-Horizon developed by Nonogaki et al. (2008) has the capability to save the estimated cubic B-Spline surface not only as a DEM but also as a cubic B-Spline function. The purpose of this study is to apply the surface in a form of cubic B-Spline function produced by BS-Horizon into the analysis in the fields of geomorphology and geology, and to develop a new analysis technique that is distinct from the one based on DEM.
As an example of practical use of the surface derived from BS-Horizon, we developed a new algorithm to exactly estimate some geomorphological characteristics, which are estimated by approximate calculation in the case of DEM, based on a set of coefficients of the cubic B-Spline function defined by BS-Horizon, and coded it in a FORTRAN program APP4OPT. Through the calculation of geomorphological characteristics using a mathematical function model, it was confirmed that APP4OPT estimates the target values exactly. Through the calculation using topographic map, it was confirmed that the new algorithm is available in the case of real topography. Furthermore, through the comparison between the results of analysis based on DEM and the ones based on the cubic B-Spline surface derived from BS-Horizon, it was proved that the present algorithm realizes the exact and continuous analysis using first derivative, second derivative and definite integration of the surface, which was never possible in the case of DEM.

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The examination of method to describe mathematically geological information observed in outcrops is necessary to construct the outcrop database. For the purpose, we propose the structure graph, the structure matrix, and the structure formula by using the graph theory and the binary relation, and examine their theoretical backgrounds. The structure graph represents an outcrop structure and a geological sequence as a labeled directed graph G = (V, R, ∅), where V is a set of vertexes (geological objects), R is a set of arcs (contact relation between geological objects), and ∅ ; is a function that maps every ordered pair (v_i, v_j) in E to a type of contact. The arc is directed from the lower object to the upper one. The structure matrix (s_ij) is refined from the adjacency matrix (a_ij); s_ij = ## if (v_i, v_j) ∈ E and s_ij=0 if otherwise, where s_ij is a(i,j) element of the structure matrix, and ## is the symbol to show the type of contact between objects v_i and v_j. The structure formula expresses an outcrop structure and a geological sequence as a formula in a form of v₁##₁v₂##₂…##_n-1v_n. Further we express a complicated structure as a formula with parentheses (), [ ) or (] such as (v₁##₁v₂##₂…##_n-2v_n-1)##_n-1v_n. Thus we can mathematically express the geologic structure of the outcrop and the geological sequence.

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