The existing digital models can be classified into spatial interpolation models and digital terrain models. Spatial interpolation can be defined as the procedure of estimating the value of properties at unsampled sites within the area covered by existing point observation. The methods for spatial interpolation include interpolation by drawing boundaries

trend surface analysis

spline functions

moving averages and Kriging interpolation. A digital terrain model is an ordered array of numbers that represent the spatial distribution of terrain attributes. Three exiting principal ways of structuring a digital terrain model are triangulated irregular networks

regular grid networks and contour|based networks. Issues on errors of digital terrain models and spatial interpolation models have been important research topics since the late 1960s and many methods for analyzing and measuring errors have been developed. However

the error problem is not attacked at the root. Although relative studies have found that slope

aspect and curvature are the most important variables for a surface

these variables are not used in formulation of the relative models. In fact

the first and the second fundamental forms are determinants of a surface

while the slope

aspect and curvature only are determinants of the thalweg of a surface according to differential geometry. The digital model of the 4th generation GIS is based on the fundamental theorem of surfaces and approach of remote sensing inversion

which can integrate theory of differential geometry with expertise

remote sensing information over an area and monitoring information at points.