A Fixed-σ Digital Representation of a Random Scalar Field

Abstract

In this paper, “fixed-σ” refers to a stochastic process that has a constant standard deviation. “Digital representation” refers to a discrete representation of a stochastic process. A good example of a digital representation of a stochastic process field is the earth’s surface, with elevation as the random scalar, commonly referred to as a digital terrain model (DTM). It is widely know that the roughness of terrain is a factor that inhibits DTM accuracy. This implies that accuracy of DTM varies from pixel to pixel; it is not fixed across the landscape because it varies according to the terrain’s vertical variations. Such a nonfixed-σ DTM is not a proper source of topographic data. However, its regular character (pixels are of equal size) is required as a prerequisite for procedures involved in the DTM data acquisition, processing and dissemination. The data acquisition methods may include InSAR and LiDAR technology, and raster techniques for producing a picture on a screen. In this paper, we discuss the possibility of modelling any type of terrain using a DTM which would be characterised by a limiting or fixed standard deviation. A starting point for these considerations is a recently published result of investigations of estimates of the target-induced error of the Shuttle Radar Topography Mission (SRTM) dataset. This target induced-error model connects the standard deviation of the disparities (DTM versus reference data), pixel size and slope of the terrain in an original and straightforward framework. An obvious consequence of such a fixed-σ DTM is the variable pixel size. This paper formulates a number of questions regarding the feasibility for such a DTM and potential advantages or disadvantages. For example, it appears that such a pixel-variable but fixed-σ arrangement of a DTM would better serve many of its purposes and provide opportunities to increase the efficiency of digital image storing and processing. The discussion includes a potential and required change in the data acquisition strategy and feasibility from systematic sampling to an adaptive sampling of the earth’s surface. A new set of algorithms for such a DTM is required to calculate DTM-derived parameters, including aspect and slope. The proposed fixed-σ digital representation of a random scalar field is not restricted in any way to only DTMs. It can be used in many other fields that use digital imagery.