Detecting Characteristic Scales of Slope Gradient

Detecting Characteristic Scales of Slope Gradient Clemens EISANK and Lucian DRĂGUŢ Abstract Very high resolution (VHR) DEMs such as obtained from LiDAR (Light Detection And Ranging) often present too much detail for various applications. Finding the right spatial scale for analysis is a challenging task, especially in landscape ecology. It still seems to be unsettled, whether scales in digital representations of the land surface are explicitly detectable. We applied the statistical method of local variance (LV) to explore the datainherent scale structure. Scale levels of slope gradient were produced by 1) degrading the initial LiDAR DEM at 1 m resolution through resampling to successively coarser resolutions (further referred to as scale levels) and then calculating slopes from the degraded DEMs, and 2) performing multi-resolution segmentation on slope gradient in Object-Based Image Analysis (OBIA). LV was calculated as mean standard deviation. Values of LV were plotted against scales to derive scale signatures. Obtained graphs exhibited peaks and stepwise changes. These thresholds can be interpreted as characteristic spatial scales. Results demonstrated the potential of LV for identifying characteristic scales of continuous terrain data within multi-scale analysis of slope gradient. For the object- and the cell-based approach we could identify 4 respectively 5 characteristic scales. We found out that scale detection in OBIA-based LV graphs is easier to perform than in the one obtained from resampling, due to differences in underlying aggregation techniques. 1 Introduction In this paper we address the important question of finding characteristic spatial scale(s) for analysis. From a landscape ecology perspective, scale refers to both grain and extent. Grain is defined as the finest level of spatial resolution possible with a given data set, e.g. pixel size for raster data (TURNER et al. 1989). Extent indicates the size of the study area. Characteristic scale may be defined as level, where the grain size of the digital representation allows for meaningful mapping of real-world units with similar properties. Thus, the term meaningful is user-dependent, as it relates to individual modelling objectives. In landscape ecology and related fields such as remote sensing and geomorphometry, scale issues have been widely discussed among scientists, since phenomena and processes on the earth s surface operate at specific spatial scales (EVANS 2003). Modelling outcomes are restricted to the scale of investigation, which in many cases is not chosen appropriately (DRĂGUŢ et al. 2009a; DRĂGUŢ et al. 2009b). The problem of selecting characteristic scales mainly arises from ever finer spatial resolutions of digital elevation models (DEMs) and derived parameters (e.g. for Austria whole provinces have already been covered by LiDAR data at 1 m resolution). Landsurface models are increasingly used as input for ecological modelling, but in many cases they hold too much detail (i.e. noise) for given applications. If the original data set is too

C. Eisank and L. Drăguţ detailed, the grain size has to be degraded to coarser ones. Recently, LE COZ et al. (2009) have demonstrated the need for aggregating even the relatively low resolution SRTM to address the scale specificity of large basins. Hence, comprehensive methods for revealing changes in the structure of continuous terrain data as a function of grain size, and that allow fast detection of characteristic scale levels for a given application, are required. There are several authors who promoted the use of statistical methods to obtain scale signatures, graphs that show the behaviour of statistical indicators across a defined range of scale (WOOD 2009). These signatures exhibit some thresholds that indicate more appropriate scales for analysis, i.e. where groups of real-world units with similar spatial, temporal, and statistical properties occur (WU & LI 2006). HILL (1973) was one of the first who recognizes the suitability of variance measures for scale detection in landscape ecology. CARLILE et al. (1989) suggested the use of spatial variance and, in addition, correlation estimates to determine the inherent scale of ecological processes. Based on this, appropriate levels of resolution for measuring plant distribution could be identified. Later, CULLINAN et al. (1997) compared the methods developed by the afore-mentioned authors. They used satellite images to investigate scales of vegetation patterns. Therefore, they calculated statistics for ever larger window sizes and plotted the values against scales. All obtained graphs showed thresholds, whereas peaks in the variance and correlation curves, and troughs in graphs from Hill s method corresponded with characteristic scales of vegetation pattern. The statistical approaches for detecting characteristic scales presented in the previous section are mostly applied on digital images. Of course, images from remote sensing are continuous digital representations of the earth s surface. However, they exhibit more pronounced boundaries and transitions between spatial units than land-surface models such as DEMs and slope gradient that show much smoother characteristics (HENGL & EVANS 2009). As has been proved, the identification of characteristic scales works well on satellite images, since there are more differences in spectral and spatial properties of image units resulting in higher variations of statistical estimates across scales. Though, little efforts have been undertaken to transfer these approaches to smoother land-surface models. Still, it seems to be unsettled if spatial scales in digital representations of the land-surface are explicitly detectable or if scale is just a window of perception (MARCEAU & HAY 1999). The presented research was based on WOODCOCK & STRAHLER (1987), who measured local variance (LV) as a function of resolution to detect characteristic spatial scales in digital images. We went one step further by measuring LV on continuous land-surface models rather than digital images. Slope gradient is one prominent example for a continuous land-surface model that influences ecological patterns (e.g. vegetation) and processes (e.g. erosion).the main objectives of this study were 1) to investigate the potential of LV to detect characteristic scales in a multi-scale analysis of slope gradient, and 2) to compare between cell- and object-based scaling methods. Following the work of DRĂGUŢ & BLASCHKE (2006) we applied multi-resolution segmentation in OBIA. Additionally, we simulated scale levels using the method of resampling. Finally, we plot local variance graphs as examples for scale signatures to identify scale thresholds that might correspond with characteristic spatial scale levels.

Detecting Characteristic Scales of Slope Gradient 2 Material and Methods 2.1 Study Area and Input Data The study area Schlossalm is located in the Gasteinertal, an alpine valley in the southern part of the province of Salzburg, Austria (Fig. 1). It is 3 x 3 km in size and characterized by mountainous terrain features such as glacial cirques, ridges, gullies and steep slopes. The elevation ranges from 1,635 to 2,578 m a.s.l. The federal government of Salzburg provided us with a VHR digital elevation model, namely a LiDAR DEM at 1 m resolution acquired during a flight campaign in 2006. Fig. 1: Location and topography of the study area

C. Eisank and L. Drăguţ 2.2 Local Variance Local variance is a statistical indicator that measures the spatial variation of values in a scene. It is calculated as the mean value of standard deviation in a defined neighbourhood over a scene. The higher the LV value, the higher the variation. One example for a scale signature is a graph showing how LV changes across scales. Again, we consider scale a function of grain size (mean object size and moving window size respectively) that specifies the neighborhood size for calculating LV. More details about the method can be found in WOODCOCK & STRAHLER (1987). 2.3 Scaling Scaling was performed in a cell- and an object-based environment (Fig. 2). 2.3.1 Cell-based Cell-based: In previous studies traditional pixel-based techniques have been applied to simulate multiple scale levels of the same input data to evaluate the effects of scale and to find at least one characteristic scale level for analysis (BIAN & WALSH 1993; WOODCOCK & STRAHLER 1987). We used the technique of resampling. Resampling changes the proportion of a raster data set by transforming the input cell size to a user-specified cell size without altering the extent. For the transformation process several algorithms are implemented in standard GIS software. We decided on bilinear interpolation, which determines the new value of a cell based on a weighted distance average of the four nearest input cell centers. The bilinear option is useful for continuous data and causes some smoothing of the data. Fig. 2: Spatial scale levels from resampling (top) and multi-resolution segmentation (OBIA, bottom). Values indicate resolution in m (top) and scale parameter, as used in OBIA (bottom).

Detecting Characteristic Scales of Slope Gradient 2.3.2 Object-based Object-based: Since several years, the object-based approach offers a powerful framework to overcome some of the cell-based limitations in multi-scale analysis of complex systems. The basic processing units in OBIA are segments, so called image objects. Through image segmentation the input layer is subdivided into regions of minimum heterogeneity based on several user-defined parameters. Thus, input layers are segmented into more realistic irregular-sized objects rather than in regular-sized pixels. Heterogeneity refers to both spectral and shape properties and threshold for each must be set. The value assigned to an image object is the mean of the aggregated pixel values. The most crucial factor influencing the segmentation result is the scale parameter. Its value defines the threshold of the maximum increase in heterogeneity when two objects are merged (BENZ et al. 2004). A well-designed software package we used for OBIA is provided by Definiens AG (http://www.definiens.com). Originally, OBIA was introduced for the use with remote sensing and aerial images (BAATZ & SCHÄPE 2000; BLASCHKE & HAY 2001; LANG & LANGANKE 2006). We applied multi-resolution segmentation to generate multiple scale levels of slope gradient. 3 Implementation In order to derive scale signatures for further scale detection, we generated a wide range of scales from the initial VHR data. In the first step, cell-based scaling was performed applying the technique of resampling. Within multiple resampling operations the initial high resolution LiDAR DEM at 1 m was constantly generalized by an increment of 2. The process was stopped at a cell size of 49. This threshold is in line with WOODCOCK & STRAHLER (1987) who suggested at least 60 pixels at each side of a raster to get significant results for LV. For each of the 25 transformed DEMs slope gradient was derived using the 3 x 3 moving window algorithm implemented in a standard GIS. Then, LV was measured as the mean layer value of standard deviations of cells, whereas standard deviation was calculated from slope values within a 3 x 3 neighbourhood. The size of the neighbourhood for calculating slope gradient and LV values ranged from 3 m (3 x 1 m) to 147 m (3 x 49 m). Consequently, the area of the neighbourhood increased from 9 to 21,609 m². In the second step, the slope gradient layer, as derived from the initial LiDAR DEM at 1 m, served as input for scaling in OBIA. Multiple scale levels were produced by increasing the scale parameter from 1 up to 170 within a multi-resolution segmentation process using an increment of 1 (DRĂGUŢ et al. 2010). We selected 170 as the upper threshold for comparison reasons, because at this level the mean object size is similar to the size of the maximum moving window in cell-based scaling. For each level LV was measured as the mean value of standard deviations of objects, whereas standard deviation was derived from slope gradient values within each object. For both methods values of LV were plotted against scale levels. In order to make the resulting graphs comparable, we introduced a common denominator for scale: the mean area size of reference units for LV derivation, i.e. the size of the moving window and the mean size of objects, respectively. For assessing the LV dynamics from a scale level to the

C. Eisank and L. Drăguţ next higher level, we introduced a measure called rate of change of LV (ROC-LV; DRĂGUŢ et al. 2010). ROC-LV was plotted against scale levels as well. Figure 3 presents the operational workflow of our study. Fig. 3: Operational workflow 4 Results In general, LV graphs of slope gradient layers obtained from cell- and object-based scaling displayed ascendant trends with more generalized scale levels (Fig. 4). For both methods

Detecting Characteristic Scales of Slope Gradient the rise in LV values from one level to another was much higher at finer scales and significantly lower for coarser ones. Except for finest scale levels, where LV values are nearly the same, OBIA levels of slope gradient showed higher LV than cell-based levels and this margin increased with scales. The maximum LV values at the coarsest scale (mean area size = 22,000 m²) were 6.3 and 4.3 respectively. This is one remarkable difference between the two scaling approaches. Despite the fact that slope gradient is smoother than digital images, LV graphs exhibited some steps in the spatial scale continuum, even though they were not well pronounced and a few were not yet observable. However, in ROC-LV curves these steps were better indicated as peaks or plateaus. The measure ROC-LV was introduced to calculate the difference in LV values from one level to the next higher level, thus showing the variation across scales. In ROC-LV graphs, we were able to identify 5 thresholds for the cell-based approach at window sizes of 3.5, 4.5, 10, 12.5 and 16.5 thousand m² corresponding to resolutions of 19, 23, 33, 37 and 43 m. The resulting ROC-LV obtained from multiresolution segmentation in OBIA indicated 4 thresholds at mean object size of 2, 10, 14 and 18 thousand m² corresponding to scale parameters of 46, 80, 108 and 151. Most of the detected peaks in ROC were represented in LV curves as well. Only two thresholds one in the object- and one in the cell-based LV graph could not be identified. As we have assumed, in most cases thresholds in cell- and object-based graphs did not occur at the same scale. Only at mean area size of 10 thousand m² detected thresholds were identical, though they were not very pronounced at this level. Fig. 4: LV graphs from cell- and object-based scaling and correspondent ROC-LV curves. Black circles mark observed scale thresholds.

C. Eisank and L. Drăguţ 5 Discussion In this experimental research we tested the potential and usability of LV for revealing the inherent scale structure of continuous terrain models for further detection of characteristic scales. The thematic focus was on ecological terrain-related analysis. In landscape ecology, breaks in statistical properties of land-surface parameters such as slope gradient across scales might reveal levels of organization in the structure of data as a consequence of the occurrence of similar sized spatial objects (DRĂGUŢ et al. 2010). We simulated scale levels using both a cell-based method (resampling) and an object-based approach (multiresolution segmentation). Detection of characteristic scales has become necessary due to increased integration of VHR DEMs into ecological modeling. Digital elevation data often hold too much detail for the application of interest. Hence, there is the need for more generalized scale levels, i.e. coarser resolutions in cell-based approaches and larger objects in multi-resolution segmentation. In our study we proved that scale signatures such as the suggested LV graphs indicate scale thresholds. These breaks mark characteristic spatial scales, where groups of real-world objects are more appropriately imaged than for other scale levels. Therefore, these levels could be more appropriate for analysis. Both methods resampling and segmentation try to emulate real-world units by aggregating cells. While resampling operates locally on the basis of quadratic windows that do not account for spatial anisotropy of real-world units in the scaling process (SCHMIDT & ANDREW 2005), segmentation produces more realistic objects based on minimum heterogeneity. Due to the smoothing of DEMs through resampling, variation of slope gradients decreases with coarser scales, while this variation is preserved in objects, regardless their size. This difference between object-based and cell-based method eventually leads to significant differences of LV values for the coarsest scales (/Fig. 4). Visual evaluation of segmentation results for the detected scale levels confirms good matching of slope values and their aggregation into objects (Fig. 5). Fig. 5: Characteristic scale levels of slope gradient at scale parameters of 46, 108 and 151 (left to right) as derived from the OBIA-based LV graph.

Detecting Characteristic Scales of Slope Gradient 6 Conclusion and Outlook As suggested by LI (2008) we applied LV method on land-surface models and demonstrated the usability of LV for the detection of characteristic scales within a multiscale analysis of slope gradient. Due to differences in the underlying aggregation techniques, scale detection works better in LV graphs obtained from OBIA scales than LV graphs obtained from resampling-based levels. Our future work will apply the same methodology to other areas. We have already defined three additional test sites. Each of them shows different terrain characteristics ranging from flat to mountain. We will aim for comparing detected scales across terrain types. For both scaling approaches, we will increase the scale ranges, since we assume that we have not captured all the characteristic scales with the chosen range. Acknowledgements: Work has been carried out within the frame of the research project SCALA (Scales and Hierarchies in Landform Classification, Einzelprojekt, No. P20777) funded by the Austrian Science Fund (FWF). References BAATZ, M., & SCHÄPE, A. (2000), Multiresolution Segmentation - an optimization approach for high quality multi-scale image segmentation. In STROBL et al. (Eds.), Angewandte Geographische Informationsverarbeitung. Wichmann-Verlag, Heidelberg, pp. 12-23. BENZ, U.C., HOFMANN, P., WILLHAUCK, G., LINGENFELDER, I., & HEYNEN, M. (2004), Multi-resolution, object-oriented fuzzy analysis of remote sensing data for GIS-ready information. ISPRS Journal of Photogrammetry and Remote Sensing, 58(3-4), pp. 239-258. BIAN, L., & WALSH, S.J. (1993), Scale Dependencies of Vegetation and Topography in a Mountainous Environment of Montana. The Professional Geographer, 45 (1), pp. 1-11. BLASCHKE, T., & HAY, G. (2001), Object-oriented image analysis and scale-space: theory and methods for modeling and evaluating multiscale landscape structure. International Archives of Photogrammetry and Remote Sensing, 34 (4), pp. 22 29. CARLILE, D.W., SKALSKI, J.R., BATKER, J.E., THOMAS, J.M., & CULLINAN, V.I. (1989), Determination of ecological scale. Landscape Ecology, 2 (4), pp. 203-213. CULLINAN, V.I., SIMMONS, M.A., & THOMAS, J.M. (1997), A Bayesian test of hierarchy theory: scaling up variability in plant cover from field to remotely sensed data. Landscape Ecology, 12 (5), pp. 273-285. DRĂGUŢ, L., & BLASCHKE, T. (2006), Automated classification of landform elements using object-based image analysis. Geomorphology, 81 (3-4), pp. 330-344.

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