Artificial Neural Networks as a Method of Spatial Interpolation for Digital Elevation Models

This paper examines the performance of artificial neural networks (ANNs) as a method of spatial interpolation, when presented with irregular and regular samples of elevation data. The results of the ANN interpolation are compared with results obtained by kriging. Tests of spatial bias in the systematic errors contained in each of the neural network-derived DEMs were conducted using four attributes: slope, aspect, average direction and average distance from the nearest sampled value. Based on RMS and other evaluation measures, the accuracy of estimated DEMs from regular and irregular sample distributions using neural networks is lower than the accuracy level derived from kriging. The accuracy level of the ANN interpolators also decreases as the range of elevation values in DEMs increases. As reported in the literature, ANNs are approximate interpolators, and the pattern of under-prediction and over-prediction of elevation values in this study revealed that all estimated values fell within the range of sample elevations. Neural networks cannot predict values outside the range of elevation values contained in the sample, a property shared by other interpolators such as inverse weighted distance.     

On the Uncertainty of Local Shape of Lines and Surfaces

Modeling line or surface phenomena digitally involves two tasks: discretization of the phenomenon, which yields a finite set of data, and subsequent interpolation, which reconstructs the continuum. Many mathematical techniques exist for the latter task, and most methods require a number of parameters to be specified. The shape of digital line or surface models between the data points (that is, the local shape) and the information derived from these models both depend on the selected method and, possibly, on the specification of parameters. The reconstruction of the continuum thus introduces uncertainty. This paper examines the sources and effects of this type of uncertainty. For this purpose, the modeling of lines and surfaces is separated into an abstraction, an implementation, and measurement. The individual factors affecting uncertainty of local shape in each step are identified and discussed. The paper concludes that local shape uncertainty, unlike positional uncertainty of given data, cannot be numerically assessed. Instead, measures of plausibility have to be used to denote the quality of digital models of lines and surfaces. Finally, the concept and potential problems of future empirical investigations are discussed.     

Toward a Differential Calculus for Temporal Map Analysis

Investigators in many fields are analyzing temporal change in spatial data. Such analyses are typically conducted by comparing the value of some metric (e. g., area, contagion, or diversity indices) measured at time T₁ with the value of the same metric measured at time T₂ . These comparisons typically include the use of simple interpolation models to estimate the value of the metric of interest at points in time between observations, followed by applications of differential calculus to investigate the rates at which the metric is changing. Unfortunately, these techniques treat the values of the metrics being analyzed as if they were observed values, when in fact the metrics are derived from more fundamental spatial data. The consequence of treating metrics as observed values is a significant reduction in the degrees of freedom in spatial change over time. This results in an oversimplified view of spatio-temporal change. A more accurate view can be produced by (1) applying temporal interpolation models to observed spatial data rather than derived spatial metrics; (2) expanding the metric of interest's computational equation by replacing the terms relating to the observed spatial data with their temporal interpolation equations; and (3) differentiating the expanded computational equation. This alternative, three-step spatio-temporal analysis technique will be described and justified. The alternative technique will be compared to the conventional approach using common metrics and a sample data set.     

Interpolation of Point Values From Isoline Maps

A Bidirectional Hermitian Spline (BHS) method for the estimation of point values from isoline maps is presented and compared to three other methods. Hermitian splines are used and first derivatives are estimated by either Akima's method or by a clamped cubic spline, if Akima's method returns a zero first derivative. Every desired point value is interpolated twice, once by each of two orthogonally-directed splines. The two spline estimates are then averaged using the error formula for Hermitian splines. In addition, a periodic Hermitian spline is constructed around the study-area perimeter (representing a cross-sectional profile of the edge) to damp undesirable edge effects. Point values can be estimated from small-scale isoline maps drawn in spherical coordinates or from large-scale isoline maps drawn in Cartesian coordinates.     

Cartographic Notes

On a previously-digitized map, a series of 685 points were screen-digitized so as to represent major spatial features of each polygon composing 17 forest types having inherently uncertain boundaries; three forest types having precise boundaries were treated as such. From these points and three exact types, surfaces were generated which provided weights of the certainty of having each of the 20 forest types at a location. These surfaces were “reconstituted” into a thematic map by assigning each location to the type having the highest weight. When compared to the precisely digitized and rasterized map, areal and locational inaccuracy were 5% and 18%, respectively, for the reconstituted thematic map. For certain situations—such as the preparation of long-term forest management plans—these results show promise for adequately representing maps of natural features while reducing digitizing efforts.     

Rapid K-Neighbor Identification Algorithm for the Sphere

Spatial searching, such as the identification of k-nearest neighbors to a point, is one of the most time-intensive tasks in vector-based geographic information systems transformational or analytical operations, and as such continues to impede many studies. While a number of computationally efficient k-neighbor searching algorithms have been developed for d-dimensional monotonic coordinate axes, these methods are inappropriate for spherical coordinates necessary in many global studies. This article briefly examines the assumptions and resulting limitations of k-neighbor searching algorithms with spherical coordinates. One of the simplest, yet most efficient k-neighbor searching algorithms is applicable to spherical applications, if constrained. Comparisons between the processing efficiency of the brute-force searching method commonly in use, a constrained heuristic k-neighbor searching algorithm, and a modified k-neighbor algorithm indicate processing times may be decreased by as much as 99% using such rapid searching methods in global geographic applications.     

Medial Axis Generalization of River Networks

We examine some benefits of using the medial axis as a centerline for rivers and lakes. One benefit, automatic centerline generation, has been used for many years. We show that additional benefits can be derived from the geometric relationships between the medial axis and the riverbanks or lakeshores. These include area estimates, association of centerline analysis to banks, and definition of opposite for riverbanks. We also report on our experience at approximating the medial axis with a Voronoi diagram of point sites.     

Numerical Evaluation of the Robinson Projection

The Robinson projection is one of the most preferred projections for world reference maps in atlas cartography. The projection is constructed from Robinson's look-up table since there are no analytical formulas. This deficiency has led to a number of requests for the plotting formulas to which cartographers have responded by deriving analytical equations using different interpolation algorithms applied to Robinson's table values. The Robinson projection was examined with regard to its deformations calculated by four different algorithms, including the multiquadratic method. The numerical evaluations were then used to compare the algorithms. Solutions have been presented including some criticisms about this projection. The latitudes along which the scale is true and on which the maximum angular distortion equals zero have been determined.     

Producing Intermediate Contours from Digitized Contours

Contour lines on existing maps are often the only source of information about the terrain relief. In an effective digital mapping environment, the generation of intermediate contours is one of many essential capabilities. These intermediate contours can be generated in the original units or in some other units. During the digitization process, errors are introduced into the cartographic data, and data manipulation processes may introduce additional errors. A new six-step method is presented here to reduce the data manipulation errors during the process of generating intermediate contours. In our method, the original digitized contours serve not only as a data source, but also as a control on the interpolation process. This process utilizes one- and two-dimensional filters to determine the differences between original and modeled contours. The interpolated contours are adjusted to the original contours thus preserving both general and local shape of the original relief representations. The most frequent application of this method likely will be in the conversion of “foot” contours to “meter” contours.