Surface analysis
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Surface simulation and analysis in GIS
Digital elevation model (DEM) maps are fundamental to almost all analysis of spatial data in both GIS and remote sensing applications. Software tools for processing DEM data must produce maps of slope and aspect (compass direction of slope), and typically provide features for watershed and hydrologic flow modeling. There is a surprising variety of algorithms for even the fundamental operation of deriving a map of aspect and slope. There are also a variety of common artifacts and defects in DEM data, which, in combination with each algorithm's peculiarities, can produce a great variety of artifacts in products which are often contradicted by field observation. It's important to sort out these artifacts to identify their source -- at least whether the origin is in the DEM data, or in subsequent processing.
In 2005 I began work on this issue with Gil Pontius at Clark University, with a study of comparison of vector valued maps. The aspect angle and slope of a surface, considered together as its gradient, or mathematical first derivative, is a two dimensional vector value which can be expressed either as a (easting-slope, northing-slope) pair, or as (aspect, slope) pair. The second is typical in GIS, with the aspect angle measured in degrees clockwise from north. However, many basic GIS algorithms for aspect mapping use a greatly simplified encoding of the aspect of a pixel in a surface, as one of the eight cardinal points of the compass rather than degrees from north. This "D8" encoding produces a simplified map of aspect very quickly; it is often used on broad scale mapping of aspect from DEMs, because of its processing speed compared to the trigonometric calculation of aspect angle. The information in such a map is useful for agronomy and planning, but is often inadequate for hydrology due to accumulation of error resulting from the simple encoding -- though this is indeed what is often done in an algorithm due to Jensen and Domingue (1988) which uses the D8 encoding of aspect as flow direction at each pixel location.
Our work in "Comparison of maps of vector valued variables to identify components of information in scaling of spatial resolution and encoding precision" compared aspect and flow maps to quantify this loss of information. Since the gradient at a point on a surface, and its flow direction approximation, are both vector quantities, it makes sense to compute a vector difference. In the Jensen and Domingue model algorithm, gradient only determines the flow destination and the slope value does not play a direct role, so a full gradient encoding is not available. The appropriate comparison is then between only the direction angles of aspect and flow, without magnitude of slope. These angles, discrete and continuous positive values from 0 to 360 degrees (0 to 2*pi radians), are still vector quantities on the unit circle, and should not be treated as scalar values. A difference of vectors on the unit circle is a vector of magnitude between 0 and 2, with a small magnitude indicating similar vectors. (The locus of differences on the unit circle, created by varying one angle, is an arc around the first vector's point on the circle.) Aggregation of this difference magnitude over all points by averaging produces a quantity with exactly the same properties as Mean Absolue Error, which in this case can be taken as a measure of information loss due to encoding.
Using an aerial LIDAR DEM of a study area (which showed approximately uniform distribution of aspect) we compared trigonometric aspect and D8 flow variable encoding in terms of information loss. By repeated comparison over a range of encoding precisions ("D4", "D8", "D16") and spatial resolution (also stepping by factors of two) we demonstrated that the loss of information by categorical encoding of flow far exceeds loss of information due to reduction of spatial resolution. The map information content due to encoding, measured as the mean absolute error figure of the difference of aspect and flow maps, changes by a factor of two with a corresponding factor of two in encoding precision, suggesting that information content due to encoding of vector information in maps can be measured in the familiar units of "bits", or base-two logarithm of the number of possible encoding values.
In practial application terms, this study demonstrated that a spatial model algorithm's fidelity to observed data can be far more important than spatial resolution of that data, to the success of that model. Encoding of data and derived values is an essential part of a model algorithm, and the selection of floating point, fixed point, byte, or categorical encoding for each datum and intermediate product, is critical -- as is detailed recording of encoding in metadata. In terms of hydrology modelling, we can observe that J&D and other early algorithms, chosen to simplify and accelerate computing in a time of very limited storage and computing resources, should be abandoned for more accurate physical models wherever computing resources provide the full floating point computational power of modern systems.
One major weakness of this study was the obvious dependence of observed error and loss of information on anisotropy, or non-uniform distribution of aspect angle over the terrain: for example, the error in D8 encoding of aspect of a tilted flat surface is zero if the direction of its tilt is to one of the eight points of the compass. To complete this work, I am investigating simulation of an ensemble of surfaces with controlled anisotropy, which then can be used as test data in the vector difference procedure for evaluating information loss in categorical encoding of aspect.
- For more information on this work see this web page.
- A Powerpoint presentation is available here.
- Below: Scatterplots of vector differences of aspect and encoding, varying Dn and spatial resolution, with MAE figures
