Heavy-tailed, Non-gaussian Nature of Terrain and Its Implications for Terrain Modeling by L1 Splines

This paper presents the first step in establishing a link between the heavy-tailed nature of terrain and a new terrain modeling technique, L1 splines, that is, splines based on minimizing the L1 norm rather than the square of the L2 norm. To establish this link, we focus on the heavy-tailed nature of the second derivatives that occur in the L1 spline minimization principles. For one urbanterrain data set (Baltimore, Maryland) and two naturalterrain data sets (Killeen, Texas), the second derivatives behave asymptotically like c|x−μ| for −α between –4 and −2 rather than like exponential functions. Similar results for first derivatives minus first differences are presented. The distributions investigated here are not directly related to the spatial frequency spectra that have been the topic of most previous investigations of the heavy-tailed nature of terrain. The heavy-tailed nature of the frequency spectra of terrain has not resulted in any major impact on modeling of large terrain datasets (although it has had significant positive impact on modeling of vehicle-terrain interaction, where the data sets are local and smaller). The investigation of the heavy-tailed nature of the derivatives of terrain will have significant impact by providing the theoretical underpinnings for the current observation that L1 splines provide better terrain modeling than alternative techniques.