Quantification Algorithm of Topography based on Bi-Cubic B-Spline Surface

Geomorphologic and hydrologic analyses are necessary for disaster prevention, environment preservation and resource management. In these analyses, it is quite important to quantify topography. Over the past few decades, a considerable number of studies have been conducted on generation algorithm of DEM (Digital Elevation Model) that represents the geomorphic and geologic surface and on quantification algorithms of topography based on DEM. In quantifications based on DEM, the numerical differentiations and numerical integrations are generally used to calculate the partial derivatives or definite integration of the surface because DEM is a set of discrete values that give heights of the surface at grid nodes. However such a discrete valuable calculation method does not avoid some approximation errors. The problems in data and calculation become a great obstacle to quantitative analyses related to the surface. In order to solve the problems as mentioned above, the present study proposed a new surface estimation algorithm using a bi-cubic B-spline function. The algorithm is designed to express the surfaces in a form of a bi-cubic B-spline function, to determine an optimal surface based on exterior penalty function method using not only equality information but also inequality information and slope information as constraints of the surface, and to produce a bi-cubic B-spline function of the optimal surface as well as DEM. In addition, as an example of practical uses of this algorithm, the present study proposed another algorithm designed to calculate some geomorphologic characteristic values defined with the aid of the partial derivatives and definite integration of the surface, utilizing the feature of the cubic B-spline function. An effectiveness of the algorithm was confirmed through some kinds of the geomorphologic characteristic values calculations. The advantages of the present ideas are (1) the estimated surface has the continuity up to the partial derivatives of second order, (2) the quantitative values always can be calculated with high accuracy and (3) the amount of computer memory needed in analysis can be saved. We consider these advantages not only contribute to improvement of the quantitative techniques in the field of geomorphology and hydrology but also lead to development of a new analysis technique, which is never realized by DEM, in the field of geomorphology, hydrology and geology. A further direction of this study is to explore the utilization of the bi-cubic B-spline surface in the field of geology, especially in three dimensional geologic structure analyses.