This paper proposes methods for classifying population distribution surfaces according to their topologic characteristics and for measuring overall topologic similarity between two population distribution surfaces. These methods are applied to a comparative study on population distribution surfaces of 20 Japanese cities that range in size from 200, 000 to 400, 000 people.
First, the paper shows a procedure for detecting the topologic structure of a surface. The procedure consists of two steps: i) find critical points (peaks, bottoms, and cols) on a population density surface; and ii) connect the critical points by paths of the steepest ascent or descent. The resulting network is called a “surface network.” The surface network is robust for a broad class of transformations and shows the structural characteristics of the surface. For surface networks, this paper proposes a similarity measure between two surfaces of population distributions, such that two surfaces are structurally similar if their surface networks are identical (isomorphic).
Second, this paper develops a procedure for eliminating trifle peaks and bottoms from a surface network to identify distinctive features of the surface, such as doughnut phenomena. The procedure consists of three steps: i) define the heights and depths of peaks and bottoms; ii) define trifle peaks and bottoms as those for which heights and depths are less than a predetermined threshold; and iii) eliminate the trifle peaks and bottoms in ascending order of their heights and depths.
Third, this paper proposes a method for classifying population density surfaces from which trifle peaks and bottoms are eliminated. Two surfaces are classified into the same category if and only if the surface networks obtained at a given threshold are identical.
Fourth, this paper proposes an overall topologic similarity index between two population distribution surfaces. Although trifle peaks and bottoms are eliminated by the above procedure, there still remains arbitrariness in the choice of the threshold. Therefore this paper regards the threshold as a parameter and proposes a similarity index. The more identical two surface networks are at different thresholds, the higher value the index takes.
Last, the proposed methods are applied to the population density surfaces of 20 middle-sized cities in Japan. From the classification result, it is found that most of the 20 cities have one or two distinct city centers and that cities with more than three centers are exceptional. This paper proposes adistance measure between two population density surfaces that is inversely proportion to the topologic overall similarity, and a distance matrix for the 20 cities is calculated. Multi-dimensional scaling is applied to the matrix so that the interrelationship between the 20 cities can be visually grasped on a plane. From the results, it is revealed that the horizontal axis represents the numbers of the distinctive peaks on the surface networks and the vertical axis represents the complexities of the topologic structures of the surface networks. Furthermore, it is shown that more than half of the cities have similar population distribution structures and that they commonly have one distinct city center.

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In regional science, an area that has special geometric attributes and maintains significant spatial correlation and spatial interaction close to adjacent zones is called a prominent area. The prominence of irregular areas can be measured using a prominence index, which is a stationary distribution of a Markov chain transition matrix identical to a spatial weight matrix. In this paper, to identify the relationship between the prominence of an area and its geometric attributes, an expanded definition for measuring geometric attributes such as size, shape, and location in irregular areas is presented. The expanded definition involves identifying size with area, location with number of adjacent areas, and shape with edge roughness or smoothness and compactness. In this approach, spatial clusters composed of prominent areas are extracted using K-means cluster analysis. A very different relationship between prominence and the geometric attributes of areas is shown by spatial correlation analysis when the prominences are derived from different spatial weight matrices.
The results of this analysis are as follows: 1) Although different prominences can be obtained from different weight matrices, generalized weight matrices are more appropriate to measure the prominence of areas than distance decay and k-order. 2) The prominence measured using the distance-decay matrix is only negatively correlated with the size of areas, but the prominence derived from the generalized matrix is more strongly correlated with the size, location, and shape of areas.

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Applications of remote sensing and geographic information systems (GIS) to hydrology are described. They are important techniques for hydrologic applications and can contribute to human life. The targets of hydrology range from the local to global scale. Global scale involves a number of local regions. Each region has peculiar variability and interrelationships of the components, of a spatial and historic nature, and they form regional characteristics. To achieve the outcomes that contribute to human society, a viewpoint based on the region is necessary. At the same time, the outcomes that will help international exchanges will be achieved by locating the regional experiences in the global context. Such an application will be realized through a knowledge base constructed using GIS.

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The effects of digital elevation model (DEM) resolution on topographic representation need to be evaluated to perform meaningful terrain analysis using a raster DEM. DEMs with a nominal resolution of 50 m, published by the Geographical Survey Institute, have often been used for terrain analysis in Japan. The resolution of the DEMs, however, may be insufficient to characterize complex topography such as finely dissected hilly land. This paper discusses relations between DEM resolutions and topographic parameters derived from DEMs, using data for the Tama area in western Tokyo. The area includes dissected hilly land and adjacent low-altitude mountains. Ten-meter DEMs representing current topography were produced from high-resolution vector digital data including contour lines and information about survey points. Part of the hilly land have undergone extensive artificial landform modification since the mid-1980s to provide flat residential areas. For these areas, analytical aerial photogrammetry was employed to construct 10-m DEMs representing topography in 1984. Coarser DEMs were derived from the 10-m DEMs by sampling elevation values with a certain interval.
Slope angle and profile curvature were calculated from the DEMs with different resolutions. The mean slope angle for a 1 km × 1 km unit area consistently decreases with increasing DEM grid interval. The rate of the decrease in slope angle is mostly larger than that reported in previous studies, reflecting highly complex topography or mosaics of short valleys and ridges in the Tama area. Mean profile curvature tends to converge into null with decreasing DEM resolution, which conforms to the decrease in mean slope angle due to flatter representation of topography. These observations indicate that DEMs with a resolution finer than 50 m are needed for effective terrain analysis in the Tama area.
Because DEMs finer than 50-m resolution are often unavailable in Japan, we examined whether 10-m DEMs interpolated from the 50-m DEMs are useful for terrain analysis in the Tama area. Twenty-seven methods were employed for interpolation, and their accuracy was evaluated using the mean error and the root mean square error of altitude, slope angle, and profile curvature calculated from the original 10-m DEMs. The results revealed that the three most accurate interpolation methods are the radial basis function of thin plate spline, minimum curvature, and modified Shepard's method without a smoothing parameter. Kriging also yields relatively good results if functions corresponding to actual variograms are used. These results are common to hilly land both with and without extensive landform modification. The mean error of slope angle is always negative, showing that the DEMs produced by interpolation can only represent smoothed topography. The 50-m DEMs before interpolation, however, demonstrate further smoothed topography, meaning that interpolation can improve topographic representation. In contrast, profile curvature calculated from the DEMs produced by interpolation seems to be inaccurate, suggesting that real high-resolution DEMs should be used when analysis focuses on curvature.

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