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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Abdul JJ (1977) The Shannon sampling theorem – its various extensions and applications: a tutorial review. Proc IEEE 65:1565–1595
Ahmadzadeh MR, Petrou M (2001) Error statistics for slope and aspect when derived from interpolated data. IEEE Trans Geosci Remote Sens 39(9):1823–1833
Ayeni OO (1982) Optimum sampling for digital terrain models: a trend towards automation. Photogr Eng Remote Sens 48:1687–1694
Becek K (2007) Comparison of decimation and averaging methods of DEM’s resampling. In: Proceedings of the MapAsia 2007 conference, Kuala Lumpur. http://www.gisdevelopment.net/technology/ip/ma07267.htm. Accessed Dec 2008
Becek K (2008) Investigating error structure of shuttle radar topography mission elevation data product. Geophys Res Lett 35:L15403. doi:10.1029/2008GL034592
Borkowski A (2002) On the optimal grid cell size for digital terrain models interpolated from contour lines maps, Scientarum Polonorum. Geodesia et descriptio Terrarum 1(1–2):15–22
De Berg M, van Kreveld M, Overmars M, Schwarzkopf O (2000) Computational geometry, 2nd edn. Springer, Heidelberg. ISBN 3-540-65620-0
Gray RM, Davisson LD (2003) An introduction to statistical signal processing. Cambridge University Press, Cambridge
Smith JO (2007) Fourier theorems for the DFT. In: Mathematics of the discrete fourier transform (dft) with audio applications [electronic], 2nd edn. W3K, Menlo Park, CA. http://ccrma.stanford.edu/_jos/mdft/Fourier_Theorems_DFT.html
Smith B, Sandwell D (2003) Accuracy and resolution of shuttle radar topography mission data. Geophys Res Lett 30(9)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Becek, K. (2012). A Fixed-σ Digital Representation of a Random Scalar Field. In: Kenyon, S., Pacino, M., Marti, U. (eds) Geodesy for Planet Earth. International Association of Geodesy Symposia, vol 136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20338-1_90
Download citation
DOI: https://doi.org/10.1007/978-3-642-20338-1_90
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20337-4
Online ISBN: 978-3-642-20338-1
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)