Figure 1 – uploaded by Tomislav Hengl

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Fig. 1. Upscaling and downscaling in a grid-based GIS. S indicates scale: S~ are smaller scales and S* are larger scales. Based on McBratney (1998). EMA eee BE Se eee eee: BE EM ee ee ee ge et the grid model (Rossiter and Hengl, 2002). Grid cell can be also related (but should not be confused) with the support size, which is typically a fixed area or volume of the land that is being sampled. Support size can be increased by using composite samples or by averaging point-sampled values belonging to the same blocks of land. In geostatistics, one can also control the support size of the output models by averaging multiple predictions per regular blocks of land, which is known as “block kriging’ (Heuvelink and Pebesma, 1999). This means that we can sample at point locations, then make predictions for blocks of 10 x 10m. The latter often confuses GIS users because we can produce predictions at regular point locations (point kriging) and then display them using a raster map, but we can also make predictions for blocks of land (block kriging) and display them using the same raster model (Bishop et_al., 2001). This distinction is especially important for the validation of the spatial prediction models because it can lead to serious misconceptions—validating a point Although the raster structure has a number of serious disadvantages such as of under- and over- sampling in different parts of the study area and large data storage requirements, it will remain the most popular format for spatial modelling in the coming years (DeMers, 2001). What makes it especially attractive is that most of the technical characteristics are controlled by a single measure: grid resolution, expressed as ground resolution in meters. The enlargement of grid resolution leads to aggregation or upscaling and decrease of grid resolution leads to disaggregation or downscaling. As grid becomes coarser, the overall information content in the map will progressively decrease and vice versa (McBratney, 1998; Kuo et al., 1999; Stein et al., 2001). In cartography, coarser grid resolu- tions are connected with smaller scales and larger study areas, and finer grid resolutions are connected with larger scales and smaller study areas. The former definition often confuses non-cartographers because bigger pixel means smaller scale, which usually means Jarger study area (Fig. 1). Note in Fig. 1, that both aggregation and disaggregation can be done before or after geo-computation. If the — Figure 1 Upscaling and downscaling in a grid-based GIS. S indicates scale: S~ are smaller scales and S* are larger scales. Based on McBratney (1998). EMA eee BE Se eee eee: BE EM ee ee ee ge et the grid model (Rossiter and Hengl, 2002). Grid cell can be also related (but should not be confused) with the support size, which is typically a fixed area or volume of the land that is being sampled. Support size can be increased by using composite samples or by averaging point-sampled values belonging to the same blocks of land. In geostatistics, one can also control the support size of the output models by averaging multiple predictions per regular blocks of land, which is known as “block kriging’ (Heuvelink and Pebesma, 1999). This means that we can sample at point locations, then make predictions for blocks of 10 x 10m. The latter often confuses GIS users because we can produce predictions at regular point locations (point kriging) and then display them using a raster map, but we can also make predictions for blocks of land (block kriging) and display them using the same raster model (Bishop et_al., 2001). This distinction is especially important for the validation of the spatial prediction models because it can lead to serious misconceptions—validating a point Although the raster structure has a number of serious disadvantages such as of under- and over- sampling in different parts of the study area and large data storage requirements, it will remain the most popular format for spatial modelling in the coming years (DeMers, 2001). What makes it especially attractive is that most of the technical characteristics are controlled by a single measure: grid resolution, expressed as ground resolution in meters. The enlargement of grid resolution leads to aggregation or upscaling and decrease of grid resolution leads to disaggregation or downscaling. As grid becomes coarser, the overall information content in the map will progressively decrease and vice versa (McBratney, 1998; Kuo et al., 1999; Stein et al., 2001). In cartography, coarser grid resolu- tions are connected with smaller scales and larger study areas, and finer grid resolutions are connected with larger scales and smaller study areas. The former definition often confuses non-cartographers because bigger pixel means smaller scale, which usually means Jarger study area (Fig. 1). Note in Fig. 1, that both aggregation and disaggregation can be done before or after geo-computation. If the

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Fig. 2. Popular cartographic rules to select grid resolution: (a) relationship between grid resolution and cartographic scale; (b) relationshir between grid resolution and density of observation points for soil mapping applications. So for 1:50K scale, the smallest legible grid resolution is 12.5m (5m). Resolutions finer than 5m truly do not make sense as it will be hard to visualize or print them at this scale of work. Following notions in microscopy, and the Nyquist frequency concept from signal processing (Shannon, 1949), which states that the original signal can be reconstructed if sampling frequency is two times the original frequency, McBratney et al. (2003, Table 1), suggested that there should be at least 2 x 2 pixels to represent smallest rounded objects of interest and at least two pixels to represent the width of elongated objects. The smallest objects are typically of size 1x 1mm on the map, so that the grid resolution can be determined using the p = 0.5mm rule (Fig. 2a). The former can be used as the The grid resolution can also be estimated by dividing the size of the study area by the number of pixels that computer can handle. For example, if the size of area is about 100000km? and computer can handle 10’ pixels, we should probably work with resolutions of 100 or more meters. Based on this principle, we can also derive the coarsest global standard resolution. This is the resolution of an

Fig. 3. (a) Growth of standard image size in pixels also follows Moore’s Law. (b) By year 2040, images of almost all Earth should be available in resolutions of 25m or better. Note inflection point at year 2026.

Fig. 4. Some important concepts for evaluation of positioning methods: (a) distribution of error vectors follows log-normal distribution; (b) confidence radius where most of measurements fall should not have larger area than grid node.

Fig. 5. Impact of local contrast on visibility of objects for point, line and polygon features: (a) high contrast between target object and surrounding (rr = 0.4%, rs = 99.6%); and (b) low contrast between target object and surrounding (rr = 78.4%,rs = 94.1%). Fine resolution images have been resampled to coarser grids using bilinear resampling. to make predictions. To be consistent, every mapping project should have approximately an equal density of samples per area, also called inspection density. Obviously, the denser the ob- servation points, the larger the scale of mapping. A cartographic rule, used for example in soil mapping, is that there should be at least one (ideally four) observation per lcm? of the map (Avery, 1987, Table 1). This principle can be used to estimate the effective scale of a data set consisting of sampled points only. For example, 10 observations per km?” corresponds to the scale of about 1:50K. The same principle can be also expressed mathematically resolution would be 25m (see also Fig. 2b), which can also be expressed as 160 pixels per point sample.

Fig. 6. Selection of grid resolution in relation to different point sample patterns: (a) in situation of regular sampling, coarsest grid resolution is half average spacing to closest point pair; (b) in situation of random sampling, coarsest grid resolution should be about two times finer; (c) probability of finding one point in neighborhood for regular and point samples.

Fig. 7. Schematic example showing effect of grid resolution on representation of topography: too coarse grid resolution (p = 2.5m) will misrepresent topography; whereas finer grid resolution (p = 0.5m) will be more effective in representing all peaks and channels.

Fig. 8. Selecting grid resolution based on confidence radius of positioning method: (a) 100 single-fix GPS measurements around true location of point; (b) histogram of error vectors, average error vector and 95% probability confidence radius. I will now demonstrate how to select a remote sensing sensor based on the size of agricultural plots (Fig. 9a). The polygon map consists of 121 polygons in total. The smallest polygon is 0.005ha, the In this example, I will first demonstrate how to select a grid resolution based on the evaluation of the GPS positioning method. In this case, 100 positioning fixes were recorded using the single-fix GPS positioning method (Arnaud and Flori, 1998) at the control point with a known location. The fluctuation of the GPS readings can be seen in Fig. 8a. The errors ranged from 0.7 to 23.9m, average error was 8.5m with a standard distribution of 5.2m. The error vectors seem to follow the

Fig. 9. Selection of grid resolution based on average size of objects observed: (a) agricultural plots; (b) distribution of surfaces for compact plots (S<3) and related grid resolution.

Fig. 10. Selection of grid resolution based on point pattern analysis: (a) a set of 552 soil profile observations; (b) probability of finding one point in neighborhood and graph of distances to closest point; (c) variogram and parameters fitted automatically in gstat (http://www.gstat.org), (d) interpolated map using ordinary kriging at grid resolution of 10m. Automated fitting of the variogram using a global model further gave a nugget parameter (Co) of 0.042, a sill (Co +C,) of 0.097 and a range parameter (R) of 175.2m, which means that the variable is correlated up to the distance of about 525m (hr). There are 11 807 point pairs within this range, which finally means that the optimal lag/grid size would be about 23m. The pattern analysis of the point data set further shows that there is clear regularity in the point geometry: most of the distances are grouped at 180m. The final inter- polated map in resolution of 10m can be seen in Fig. 10d. If we apply block-kriging, the nugget variation, which seems to be significant in this case (Co = 0.042), would be diminished and the model would explain 71% of variation at the support size of 10m and 78% at the support size of 60m. Note In this case study, I will demonstrate how a grid resolution can be selected from a map of contours, ie. a dataset consisting of lines digitized from a topo-map. The study area is described in detail in Heng! et al. (2004). Contour lines were extracted

Figure 1 – uploaded by Tomislav Hengl

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