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Geomorphological units (fishcamp)
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9 Geomorphological units (fishcamp) 9.1 Introduction The purpose of this exercise is to: (1) generate and ﬁlter a DEM from point data, and use it to derive various DEM parameters; (2) extract landform classes using an objective procedure (fuzzy kmeans algorithm); and (3) improve the accuracy of soil mapping units using an existing map. We will also use geostatistical tools to assess variogram parameters for various landform classes to see if there are signiﬁcant differences between them. For geographical analysis and visualization, we will exclusively use (Brenning, 2008); an 1 equally good alternative to run a similar analysis is (Neteler and Mitasova, 2008). We will use three standard elevation data sets common for contemporary geomorphometry applications: pointsampled elevations (LiDAR), contours lines digitized from a topo map, and a raster of elevations sampled using a remote sensing system. All three elevation sources (lidar.shp, contours.shp and DEMSRTM1.asc) refer to the same geographical area — a 1×2 km case study fishcamp located in the eastern part of California (Fig. 9.4). This area is largely covered with forests; the elevations range from 1400 to 1800 meters. The data set was obtained from the USGS National Map seamless server2 . The map of soil mapping units was obtained from the Natural Resources Conservation Service (NRCS) Soil Data Mart3 . There are six soil mapping units: (1) Holland family, 35 to 65% slopes; (2) Chaixchawanakee familyrock outcrop complex; (3) Chaix family, deep, 5 to 25% slopes; (4) Chaix family, deep, 15 to 45% slopes, (5) Holland family, 5 to 65% slopes, valleys; (6) Chaixchawanakee familiesrock outcrop complex, hilltops. The complete data set shown in this chapter is available via the geomorphometry.org website4 ; the scripts used to predict soil mapping units and extract landforms are available via the book’s homepage. There are basically two inputs to a supervised extraction of landforms (shown in Fig. 9.5): (1) raw elevation measurements (either points or unﬁltered rasters); (2) existing polygon map i.e. the expert knowledge. The raw elevations are used to generate the initial DEM, which typically needs to be ﬁltered for artifacts. An expert then also needs to deﬁne a set of suitable Land Surface Parameters (LSPs) that can be used to parameterize the features of interest. In practice, this is not trivial. On one hand, classes from the geomorphological or soil map legend are often determined by their morphology; hence we can easily derive DEM parameters that describe shape (curvatures, wetness index), hydrologic context (distance from the streams, height above the drainage network) or climatic conditions (incoming solar radiation). On the other hand, many classes are deﬁned by land surface and subsurface (geological) parameters that are difﬁcult to obtain and often not at our disposal. Hence, the results of mapping soil and landform units will often be of limited success, if based only on the DEM and its derivatives. Please keep that in mind when running similar types of analysis with your own data. This chapter is largely based on the most recent book chapter for the Geomorphological Mapping handbook by Seijmonsbergen et al. (2010). An introduction to some theoretical considerations connected with the GRASS GIS SAGA GIS http://grass.itc.it http://seamless.usgs.gov 3 http://soildatamart.nrcs.usda.gov 4 http://geomorphometry.org/content/fishcamp 1 2 207 208 Geomorphological units (fishcamp) geostatistical modeling of land surface topography is given in section 2.7. 9.2 Data download and exploration Open the script (fishcamp.R) and start preparing the data. First, download the fischamp.zip complete data set and layers of interest: R > + # > > + > > # > > + > > # > > + > > # > > # > > download.file("http://geomorphometry.org/system/files/fishcamp.zip", destfile=paste(getwd(), "fishcamp.zip", sep="/")) LiDAR points: for(j in list(".shp", ".shx", ".dbf")){ fname < zip.file.extract(file=paste("lidar", j, sep=""), zipname="fishcamp.zip") file.copy(fname, paste("./lidar", j, sep=""), overwrite=TRUE) } contour lines: for(j in list(".shp", ".shx", ".dbf")){ fname < zip.file.extract(file=paste("contours", j, sep=""), zipname="fishcamp.zip") file.copy(fname, paste("./contours", j, sep=""), overwrite=TRUE) } streams: for(j in list(".shp", ".shx", ".dbf")){ fname < zip.file.extract(file=paste("streams", j, sep=""), zipname="fishcamp.zip") file.copy(fname, paste("./streams", j, sep=""), overwrite=TRUE) } SRTM DEM: fname < zip.file.extract(file="DEMSRTM1.asc", zipname="fishcamp.zip") file.copy(fname, "./DEMSRTM1.asc", overwrite=TRUE) soil map: fname < zip.file.extract(file="soilmu.asc", zipname="fishcamp.zip") file.copy(fname, "./soilmu.asc", overwrite=TRUE) where lidar.shp is a point map (LiDAR ground reﬂections), contours.shp is a map of contours (lines) digitized from a topo map, and DEMSRTM1.asc is the 1 arcsec (25 m) Shuttle Radar Topography Mission (SRTM) DEM. To load LiDAR point data set, SRTM DEM, and the soil map to , we use functionality of the package: R rgdal > lidar < readOGR("lidar.shp", "lidar") OGR data source with driver: ESRI Shapefile Source: "lidar.shp", layer: "lidar" with 273028 rows and 1 columns Feature type: wkbPoint with 2 dimensions > grids25m < readGDAL("DEMSRTM1.asc") DEMSRTM1.asc has GDAL driver AAIGrid and has 40 rows and 80 columns > grids5m < readGDAL("soilmu.asc") soilmu.asc has GDAL driver AAIGrid and has 200 rows and 400 columns this shows that lidar.shp contains 273,028 densely sampled points. The original LiDAR data set consist of, in fact, over 5 million of points; these points were subsampled for the purpose of this exercise, i.e. to speed up the processing. Next, we can estimate the approximate grid cell size5 based on the density of points in the area of interest: 5 We can take a rule of thumb that there should be at least 2 points per grid cell. 9.3 DEM generation 209 > pixelsize < round(2*sqrt(areaSpatialGrid(grids25m)/length(lidar$Z)),0) > pixelsize [1] 5 and then also attach the correct coordinate system to each spatial object: > proj4string(lidar) < CRS("+init=epsg:26911") > proj4string(lidar) [1] " +init=epsg:26911 +proj=utm +zone=11 +ellps=GRS80 + +datum=NAD83 +units=m +no_defs +towgs84=0,0,0" which is the UTM coordinate system with North American Datum 83 and GRS80 ellipsoid6 . This allows us to obtain the geographical coordinates of the study area: > # > > proj4string(grids25m) < CRS("+init=epsg:26911") coordinates of the center: grids25m.ll < spTransform(grids25m, CRS("+proj=longlat +ellps=WGS84")) grids25m.ll@bbox min max x 119.6232 119.60060 y 37.4589 37.46817 > clon < mean(grids25m.ll@bbox[1,]) > clat < mean(grids25m.ll@bbox[2,]) 9.3 DEM generation 9.3.1 Variogram modeling The ﬁrst data analysis step is generation of a DEM from the LiDAR point data. Here geostatistics can provide a lot of information. First, we can determine how smoothly elevation varies in space, are the measurements noisy, is the feature of interest anisotropic? This type of analysis can be run by using e.g. the package. However, ﬁtting a variogram model with such a large point data set will take a long time on standard desktop PC. Instead, an equally reliable variogram can be produced by taking a random subsample of the measurements: gstat > > + > > + > # > > lidar.sample < lidar[runif(length(lidar$Z))<0.05,] varmap.plt < plot(variogram(Z ∼ 1, lidar.sample, map=TRUE, cutoff=50*pixelsize, width=pixelsize), col.regions=grey(rev(seq(0,1,0.025)))) Z.svar < variogram(Z ∼ 1, lidar.sample, alpha=c(45,135)) # cutoff=50*dem.pixelsize Z.vgm < fit.variogram(Z.svar, vgm(psill=var(lidar.sample$Z), "Gau", sqrt(areaSpatialGrid(grids25m))/4, nugget=0, anis=c(p=135, s=0.6))) vgm.plt < plot(Z.svar, Z.vgm, plot.nu=F, cex=2, pch="+", col="black") plot the two variograms next to each other: print(varmap.plt, split=c(1,1,2,1), more=T) print(vgm.plt, split=c(2,1,2,1), more=F) which results in Fig. 9.1. This variogram map provides a clear picture of how semivariances change in every compass direction. This allows one to more easily ﬁnd the appropriate principal axis for deﬁning the anisotropic variogram model. In this case, elevations are more spatially ‘continuous’ in the direction NW–SE. > Z.vgm 1 2 model psill range ang1 anis1 Nug 64.34422 0.0000 0 1.0 Gau 11309.80343 963.0828 135 0.6 The resulting variogram shows that the feature of interest varies smoothly in the area of interest (Fig. 9.1), which is typical for elevation data. Nugget variation is insigniﬁcant, but =0. This information can help us determine the amount of ﬁltering needed to reduce manmade objects and artifacts in the LiDAR DEM. 6 http://spatialreference.org/ref/epsg/26911/ 210 Geomorphological units (fishcamp) 200 400 600 45 135 + var1 200 4000 3500 10000 + + 3000 100 2500 8000 + semivariance + 6000 dy 2000 0 1500 −100 1000 500 −200 0 −200 −100 0 100 200 + + + + + + + + + + 4000 + + + + + ++ + 200 400 600 2000 dx ++ ++ distance + + + Fig. 9.1: Variogram map (left) and ﬁtted anisotropic variogram model (right) for LiDARbased elevations. 9.3.2 DEM ﬁltering Although the original LiDAR product is suppose to contain only ground reﬂection measurements, we can easily notice that there are still many artiﬁcial spikes and isolated pixels, with much higher elevation values than the neighboring pixels. For example, we can quickly generate a DEM in by converting the point map to a raster map: SAGA > rsaga.geoprocessor(lib="grid_gridding", module=0, + param=list(GRID="DEM5LIDAR.sgrd", INPUT="lidar.shp", FIELD=0, LINE_TYPE=0, + USER_CELL_SIZE=pixelsize, USER_X_EXTENT_MIN=grids5m@bbox[1,1]+pixelsize/2, + USER_X_EXTENT_MAX=grids5m@bbox[1,2]pixelsize/2, + USER_Y_EXTENT_MIN=grids5m@bbox[2,1]+pixelsize/2, + USER_Y_EXTENT_MAX=grids5m@bbox[2,2]pixelsize/2)) which will contain many missing pixels (Fig. 9.2a). In addition, this DEM will show many small pixels with 10–20 m higher elevations from the neighbors. Spikes, roads and similar artifacts are not really connected with the geomorphology and need to be ﬁltered before we can use the DEM for geomorphological mapping. Spikes7 can be detected using, for example, difference from the mean value, given a search radius (see ‘residual analysis’ in ): SAGA > + + + # > + > > > > rsaga.geoprocessor(lib="geostatistics_grid", 0, param=list(INPUT="DEM5LIDAR.sgrd", MEAN="tmp.sgrd", STDDEV="tmp.sgrd", RANGE="tmp.sgrd", DEVMEAN="tmp.sgrd", PERCENTILE="tmp.sgrd", RADIUS=5, DIFF="dif_lidar.sgrd")) read back into R and mask out all areas: rsaga.sgrd.to.esri(in.sgrd=c("dif_lidar.sgrd", "DEM5LIDAR.sgrd"), out.grids=c("dif_lidar.asc", "DEM5LIDAR.asc"), out.path=getwd(), prec=1) grids5m$DEM5LIDAR < readGDAL("DEM5LIDAR.asc")$band1 grids5m$dif < readGDAL("dif_lidar.asc")$band1 lim.dif < quantile(grids5m$dif, c(0.025,0.975), na.rm=TRUE) lim.dif 7 These individual pixels are most probably dense patches of forest, which are very difﬁcult for LiDAR to penetrate. 9.3 DEM generation 211 2.5% 97.5% 3.9 3.4 > grids5m$DEM5LIDARf < ifelse(grids5m$dif<=lim.dif[[1]]grids5m$dif>=lim.dif[[2]], + NA, grids5m$DEM5LIDAR) > summary(grids5m$DEM5LIDARf)[7]/length(grids5m@data[[1]]) # 15% pixels have been masked out which will remove about 15% of ‘suspicious’ pixels. The remaining missing pixels can be ﬁltered/reinterpolated8 from the neighboring pixels (see ‘close gaps’ method in ; the resulting map is shown in Fig. 9.2b): SAGA rsaga.geoprocessor(lib="grid_tools", module=7, param=list(INPUT="DEM5LIDARf.sgrd", + RESULT="DEM5LIDARf.sgrd")) # we write to the same file! Fig. 9.2: Initial 5 m DEM (a) generated directly from the LiDAR points, and after ﬁltering (b). In comparison with the 25 m DEM (c) derived from the contour lines. Seen from the western side. 9.3.3 DEM generation from contour data We can also generate DEM surfaces from digitized contour lines (contours.shp) using a spline interpolation, which is often recommended as the most suited DEM gridding technique for contour data (Conrad, 2007; Neteler and Mitasova, 2008). In : SAGA > rsaga.geoprocessor(lib="grid_spline", module=1, param=list(GRID="DEM25TPS.sgrd", + SHAPES="contours.shp", TARGET=0, SELECT=1, MAXPOINTS=10, + USER_CELL_SIZE=25, USER_FIT_EXTENT=T)) This looks for the nearest 10 points in a local search radius and ﬁts the Thin Plate Spline9 over a 25 grid. This initial DEM can be hydrologically adjusted using the deepen drainage route: > rsaga.geoprocessor(lib="ta_preprocessor", module=1, + param=list(DEM="DEM25TPS.sgrd", DEM_PREPROC="DEM25TPSf.sgrd", METHOD=0)) The resulting DEM surface can be seen in Fig. 9.2(c). We can compare the LiDARbased and topomap based DEMs and estimate the accuracy10 of the DEM derived from the contour lines. First, we need to aggregate the 5 m resolution DEM5LIDAR to 25 m resolution: This can then be considered a void ﬁlling type of DEM ﬁltering (Hengl and Reuter, 2008, p.104–106). SAGA implements the algorithm of Donato and Belongie (2003). 10 Assuming that the LiDAR DEM is the ‘true’ DEM. 9 8 212 Geomorphological units (fishcamp) # > + + + # > + + create empty grid: rsaga.geoprocessor(lib="grid_tools", module=23, param=list(GRID="DEM25LIDAR.sgrd", M_EXTENT=0, XMIN=grids5m@bbox[1,1]+pixelsize/2, YMIN=grids5m@bbox[2,1]+pixelsize/2, NX=grids25m@grid@cells.dim[1], NY=grids25m@grid@cells.dim[2], CELLSIZE=25)) resample to 25 m: rsaga.geoprocessor(lib="grid_tools", module=0, param=list(INPUT="DEM5LIDARf.sgrd", GRID="DEM25LIDAR.sgrd", GRID_GRID="DEM25LIDAR.sgrd", METHOD=2, KEEP_TYPE=FALSE, SCALE_UP_METHOD=5)) The difference between the two DEMs is then: > + > > > rsaga.sgrd.to.esri(in.sgrd=c("DEM25LIDAR.sgrd", "DEM25TPS.sgrd"), out.grids=c("DEM25LIDAR.asc", "DEM25TPS.asc"), out.path=getwd(), prec=1) grids25m$DEM25LIDAR < readGDAL("DEM25LIDAR.asc")$band1 grids25m$DEM25TPS < readGDAL("DEM25TPS.asc")$band1 sqrt(sum((grids25m$DEM25LIDARgrids25m$DEM25TPS)^2)/length(grids25m$DEM25LIDAR)) [1] 5.398652 which means that the average error of the DEM derived using contour lines (topo map) is about 5 m, which is well within the accuracy standards for this scale. 9.4 Extraction of Land Surface Parameters We proceed with the extraction of LSPs that will be used to explain the distribution of soil mapping units. can derive over 100 LSPs given an input DEM. There is no need of course to use all of them; instead we need to try to list the LSPs that are relevant to the mapping objectives, study area characteristics and scale of application. For example, because the area is of high relief we can derive some representative DEM parameters that can explain hydrological, climatic and morphological properties of a terrain. For example: (1) Topographic Wetness Index (TWI), (2) Valley depth (VDEPTH), (3) Solar Insolation (INSOLAT), and (4) Convergence index (CONVI): SAGA SAGA # > + + # > + + # > + + # > + + Topographic Wetness Index: rsaga.geoprocessor(lib="ta_hydrology", module=15, param=list(DEM="DEM5LIDARf.sgrd", C="catharea.sgrd", GN="catchslope.sgrd", CS="modcatharea.sgrd", SB="TWI.sgrd", T=10)) valley depth: rsaga.geoprocessor(lib="ta_morphometry", module=14, param=list(DEM="DEM5LIDARf.sgrd", HO="tmp.sgrd", HU="VDEPTH.sgrd", NH="tmp.sgrd", SH="tmp.sgrd", MS="tmp.sgrd", W=12, T=120, E=4)) incoming solar radiation: rsaga.geoprocessor(lib="ta_lighting", module=2, param=list(ELEVATION="DEM5LIDARf.sgrd", INSOLAT="INSOLAT.sgrd", DURATION="durat.sgrd", LATITUDE=clat, HOUR_STEP=2, TIMESPAN=2, DAY_STEP=5)) convergence index: rsaga.geoprocessor(lib="ta_morphometry", module=2, param=list(ELEVATION="DEM5LIDARf.sgrd", RESULT="CONVI.sgrd", RADIUS=3, METHOD=0, SLOPE=TRUE)) Note that, because the features of interest (soil mapping units) are geographically continuous and smooth, we should use a wide search radius to derive the LSPs. In this case we use arbitrary parameters to derive speciﬁc LSPs — these are not as easy to determine objectively11 . Now that we have prepared a list of DEM parameters that can be used to describe geomorphology of the terrain, we can proceed with the extraction of landsurface objects. Before we can proceed, we need to read the maps into : R 11 Note also that the resulting LSPs can also differ largely for different combination of parameters 9.5 Unsupervised extraction of landforms 213 > > + > > > LSP.list < c("TWI.asc", "VDEPTH.asc", "INSOLAT.asc", "CONVI.asc") rsaga.sgrd.to.esri(in.sgrds=set.file.extension(LSP.list, ".sgrd"), out.grids=LSP.list, prec=1, out.path=getwd()) for(i in 1:length(LSP.list)){ grids5m@data[strsplit(LSP.list[i], ".asc")[[1]]] < readGDAL(LSP.list[i])$band1 } TWI.asc has GDAL driver AAIGrid and has 200 rows and 400 columns ... CONVI.asc has GDAL driver AAIGrid and has 200 rows and 400 columns 9.5 Unsupervised extraction of landforms 9.5.1 Fuzzy kmeans clustering Geomorphological classes can be optimally extracted using, for example, the fuzzy kmeans clustering approach as implemented in that package (Venables and Ripley, 2002). This will optimally assign each individual pixel to an abstract class; the class centers will be selected in such way that the within groups sum of squares is minimized. In statistical terms, this is the cluster analysis approach to extraction of features. We can start by converting the LSPs to independent components by using Principal Component analysis (Fig. 9.3): stats > pc.dem < prcomp( ∼ DEM5LIDARf+TWI+VDEPTH+ + INSOLAT+CONVI, scale=TRUE, grids5m@data) > biplot(pc.dem, arrow.len=0.1, + xlabs=rep(".", length(pc.dem$x[,1])), + main="PCA biplot") which shows that the LSPs are relatively independent. To be statistically correct, we will proceed with clustering the Principal Components instead of using the original predictors. Next we can try to obtain the optimal number of classes for fuzzy kmeans clustering by using (Venables and Ripley, 2002)12 : Fig. 9.3: The PCA biplot showing ﬁrst two components derived using ﬁve LSPs. > demdata < as.data.frame(pc.dem$x) > wss < (nrow(demdata)1)*sum(apply(demdata,2,var)) > for (i in 2:20) {wss[i] < sum(kmeans(demdata, centers=i)$withinss)} Warning messages: 1: did not converge in 10 iterations which unfortunately did not converge13 . For practical reasons, we will assume that 12 classes are sufﬁcient: > > > > kmeans.dem < kmeans(demdata, 12) grids5m$kmeans.dem < kmeans.dem$cluster grids5m$landform < as.factor(kmeans.dem$cluster) summary(grids5m$landform) 12 An alternative to kmeans clustering is to use the Partitioning Around Medoids (PAM) method, which is generally more robust to ‘messy’ data, and will always return the same clusters. 13 Which also means that increasing the number of classes above 20 will still result in smaller within groups sum of squares. 214 Geomorphological units (fishcamp) 1 2 3 4 2996 3718 6785 7014 7 8 9 10 7367 13232 2795 6032 5 6 4578 7895 11 12 7924 9664 Fig. 9.4: Results of unsupervised classiﬁcation (12 classes) visualized in Google Earth. The map of predicted classes can be seen in Fig. 9.4. The size of polygons is well distributed and the polygons are spatially continuous. The remaining issue is what do these classes really mean? Are these really geomorphological units and could different classes be combined? Note also that there is a number of object segmentation algorithms that could be combined with extraction of (homogenous) landform units (Seijmonsbergen et al., 2010). 9.5.2 Fitting variograms for different landform classes Now that we have extracted landform classes, we can see if there are differences between the variograms for different landform units (Lloyd and Atkinson, 1998). To be efﬁcient, we can automate variogram ﬁtting by running a loop. The best way to achieve this is to make an empty data frame and then ﬁll it in with the results of ﬁtting: > > # > # > > + + # > > + > + > > > > lidar.sample.ov < overlay(grids5m["landform"], lidar.sample) lidar.sample.ov$Z < lidar.sample$Z number of classes: landform.no < length(levels(lidar.sample.ov$landform)) empty dataframes: landform.vgm < as.list(rep(NA, landform.no)) landform.par < data.frame(landform=as.factor(levels(lidar.sample.ov$landform)), Nug=rep(NA, landform.no), Sill=rep(NA, landform.no), range=rep(NA, landform.no)) fit the variograms: for(i in 1:length(levels(lidar.sample.ov$landform))) { tmp < subset(lidar.sample.ov, lidar.sample.ov$landform==levels(lidar.sample.ov$landform)[i]) landform.vgm[[i]] < fit.variogram(variogram(Z ∼ 1, tmp, cutoff=50*pixelsize), vgm(psill=var(tmp$Z), "Gau", sqrt(areaSpatialGrid(grids25m))/4, nugget=0)) landform.par$Nug[i] < round(landform.vgm[[i]]$psill[1], 1) landform.par$Sill[i] < round(landform.vgm[[i]]$psill[2], 1) landform.par$range[i] < round(landform.vgm[[i]]$range[2], 1) } 9.6 Spatial prediction of soil mapping units and we can print the results of variogram ﬁtting in a table: 215 > landform.par 1 2 3 4 5 6 7 8 9 10 11 12 landform 1 2 3 4 5 6 7 8 9 10 11 12 Nug 4.0 0.5 0.1 8.6 0.0 7.6 3.3 5.8 6.4 5.0 5.1 7.4 Sill 5932.4 871.1 6713.6 678.7 6867.0 2913.6 1874.2 945.6 845.6 3175.5 961.6 1149.9 range 412.6 195.1 708.2 102.0 556.8 245.5 270.9 174.1 156.0 281.2 149.5 194.4 which shows that there are distinct differences in variograms between different landform classes (see also Fig. 9.4). This can be interpreted as follows: the variograms differ mainly because there are differences in the surface roughness between various terrains, which is also due to different tree coverage. Consider also that there are possibly still many artiﬁcial spikes/trees that have not been ﬁltered. Also, many landforms are ‘patchy’ i.e. represented by isolated pixels, which might lead to large differences in the way the variograms are ﬁtted. It would be interesting to try to ﬁt local variograms14 , i.e. variograms for each grid cell and then see if there are real discrete jumps in the variogram parameters. 9.6 Spatial prediction of soil mapping units 9.6.1 Multinomial logistic regression Next, we will use the extracted LSPs to try to improve the spatial detail of an existing traditional15 soil map. A suitable technique for this type of analysis is the multinomial logistic regression algorithm, as implemented in the multinom method of the package (Venables and Ripley, 2002, p.203). This method iteratively ﬁts logistic models for a number of classes given a set of training pixels. The output predictions can then be evaluated versus the complete geomorphological map to see how well the two maps match and where the most problematic areas are. We will follow the iterative computational framework shown in Fig. 9.5. In principle, the best results can be obtained if the selection of LSPs and parameters used to derive LSPs are iteratively adjusted until maximum mapping accuracy is achieved. nnet 9.6.2 Selection of training pixels Because the objective here is to reﬁne the existing soil map, we use a selection of pixels from the map to ﬁt the model. A simple approach would be to randomly sample points from the existing maps and then use them to train the model, but this has a disadvantage of (wrongly) assuming that the map is absolutely the same quality in all parts of the area. Instead, we can place the training pixels along the medial axes for polygons of interest. The medials axes can be derived in , but we need to convert the gridded map ﬁrst to a polygon map, then extract lines, and then derive the buffer distance map: SAGA # > + > + # > + convert the raster map to polygon map: rsaga.esri.to.sgrd(in.grids="soilmu.asc", out.sgrd="soilmu.sgrd", in.path=getwd()) rsaga.geoprocessor(lib="shapes_grid", module=6, param=list(GRID="soilmu.sgrd", SHAPES="soilmu.shp", CLASS_ALL=1)) convert the polygon to line map: rsaga.geoprocessor(lib="shapes_lines", module=0, param=list(POLYGONS="soilmu.shp", LINES="soilmu_l.shp")) 14 15 Local variograms for altitude data can be derived in the Digeman software provided by Bishop et al. (2006). Mapping units drawn manually, by doing photointerpretation or following some similar procedure. 216 Geomorphological units (fishcamp) # > + + + + + + # > + + + # > + + derive the buffer map using the shapefile: rsaga.geoprocessor(lib="grid_gridding", module=0, param=list(GRID="soilmu_r.sgrd", INPUT="soilmu_l.shp", FIELD=0, LINE_TYPE=0, TARGET_TYPE=0, USER_CELL_SIZE=pixelsize, USER_X_EXTENT_MIN=grids5m@bbox[1,1]+pixelsize/2, USER_X_EXTENT_MAX=grids5m@bbox[1,2]pixelsize/2, USER_Y_EXTENT_MIN=grids5m@bbox[2,1]+pixelsize/2, USER_Y_EXTENT_MAX=grids5m@bbox[2,2]pixelsize/2)) buffer distance: rsaga.geoprocessor(lib="grid_tools", module=10, param=list(SOURCE="soilmu_r.sgrd", DISTANCE="soilmu_dist.sgrd", ALLOC="tmp.sgrd", BUFFER="tmp.sgrd", DIST=sqrt(areaSpatialGrid(grids25m))/3, IVAL=pixelsize)) surface specific points (medial axes!): rsaga.geoprocessor(lib="ta_morphometry", module=3, param=list(ELEVATION="soilmu_dist.sgrd", RESULT="soilmu_medial.sgrd", METHOD=1)) YES Experts knowledge (existing map) A Redesign the selected LSPs Raw measurements (elevation) + + + + + ++ + + ++ + ++ + ++ + ++ + + Training pixels (class centres) B C Poorly predicted class? NO DEM Select suitable LSPs based on the legend description library(mda) Accuracy assessment Filtering needed? NO SAGA GIS Terrain analysis modules library(nnet) Multinomial Logistic Regression Initial output Revised output YES filtered DEM List of Land Surface Parameters Fig. 9.5: Data analysis scheme and connected R packages: supervised extraction of geomorphological classes using the existing geomorphological map — a hybrid expert/statistical based approach. The map showing medial axes can then be used as a weight map to randomize the sampling (see further Fig. 9.6a). The sampling design can be generated using the rpoint method16 of the package: spatstat # > + > # > > # read into R: rsaga.sgrd.to.esri(in.sgrds="soilmu_medial.sgrd", out.grids="soilmu_medial.asc", prec=0, out.path=getwd()) grids5m$soilmu_medial < readGDAL("soilmu_medial.asc")$band1 generate the training pixels: grids5m$weight < abs(ifelse(grids5m$soilmu_medial>=0, 0, grids5m$soilmu_medial)) dens.weight < as.im(as.image.SpatialGridDataFrame(grids5m["weight"])) image(dens.weight) 16 This will generate a point pattern given a prior probability i.e. a mask map. 9.6 Spatial prediction of soil mapping units 217 > # > + > training.pix < rpoint(length(grids5m$weight)/10, f=dens.weight) plot(training.pix) training.pix < data.frame(x=training.pix$x, y=training.pix$y, no=1:length(training.pix$x)) coordinates(training.pix) < ∼ x+y This reﬂects the idea of sampling the class centers, at least in the geographical sense. The advantage of using the medial axes is that also relatively small polygons will be represented in the training pixels set (or in other words — large polygons will be underrepresented, which is beneﬁcial for the regression modeling). The most important is that the algorithm will minimize selection of transitional pixels that might well be in either of the two neighboring classes. Fig. 9.6: Results of predicting soil mapping units using DEMderived LSPs: (a) original soil mapping units and training pixels among medial axes, (b) soil mapping units predicted using multinomial logistic regression. Once we have allocated the training pixels, we can ﬁt a logistic regression model using the and then predict the mapping units for the whole area of interest: nnet package, # > > > overlay the training points and grids: training.pix.ov < overlay(grids5m, training.pix) library(nnet) mlr.soilmu < multinom(soilmu.c ∼ DEM5LIDARf+TWI+VDEPTH+INSOLAT+CONVI, training.pix.ov) # weights: 42 (30 variable) initial value 14334.075754 iter 10 value 8914.610698 iter 20 value 8092.253630 ... iter 100 value 3030.721321 final value 3030.721321 stopped after 100 iterations # make predictions: > grids5m$soilmu.mlr < predict(mlr.soilmu, newdata=grids5m) Finally, we can compare the map generated using multinomial logistic regression versus the existing map (Fig. 9.6). To compare the overall ﬁt between the two maps we can use the package: mda > library(mda) # kappa statistics > sel < !is.na(grids5m$soilmu.c) > Kappa(confusion(grids5m$soilmu.c[sel], grids5m$soilmu.mlr[sel])) 218 Geomorphological units (fishcamp) value ASE Unweighted 0.6740377 0.002504416 Weighted 0.5115962 0.003207276 which shows that the matching between the two maps is 51–67%. A relatively low kappa is typical for soil and/or geomorphological mapping applications17 . We have also ignored that these map units represent suites of soil types, stratiﬁed by prominence of rock outcroppings and by slope classes and NOT uniform soil bodies. Nevertheless, the advantage of using a statistical procedure is that it reﬂects the experts knowledge more objectively. The results will typically show more spatial detail (small patches) than the hand drawn maps. Note also that the multinom method implemented in the package is a fairly robust technique in the sense that it generates few artifacts. Further reﬁnement of existing statistical models (regressiontrees and/or machinelearning algorithms) could also improve the mapping of landform categories. nnet 9.7 Extraction of memberships We can also extend the analysis and extract memberships for the given soil mapping units, following the fuzzy kmeans algorithm described in Hengl et al. (2004c). For this purpose we can use the same training pixels, but then associate the pixels to classes just by standardizing the distances in feature space. This is a more trivial approach than the multinomial logistic regression approach used in the previous exercise. The advantage of using membership, on the other hand, is that one can observe how crisp certain classes are, and where the confusion of classes is the highest. This way, the analyst has an opportunity to focus on mapping a single geomorphological unit, adjust training pixels where needed and increase the quality of the ﬁnal maps (Fisher et al., 2005). A supervised fuzzy kmeans algorithm is not implemented in any package (yet), so we will derive memberships stepbystep. First, we need to estimate the class centers (mean and standard deviation) for each class of interest: R Fig. 9.7: Membership values for soil mapping unit: Chaix family, deep, 15 to 45% slopes. # maskout classes with <5 points: mask.c < as.integer(attr(summary(training.pix.ov$soilmu.c + [summary(training.pix.ov$soilmu.c)<5]), "names")) # fuzzy exponent: > fuzzy.e < 1.2 # extract the class centroids: > class.c < aggregate(training.pix.ov@data[c("DEM5LIDARf", "TWI", "VDEPTH", + "INSOLAT", "CONVI")], by=list(training.pix.ov$soilmu.c), FUN="mean") > class.sd < aggregate(training.pix.ov@data[c("DEM5LIDARf", "TWI", "VDEPTH", + "INSOLAT", "CONVI")], by=list(training.pix.ov$soilmu.c), FUN="sd") which allows us to derive diagonal/standardized distances between the class centers and all individual pixels: # > > > > + > > derive distances in feature space: distmaps < as.list(levels(grids5m$soilmu.c)[mask.c]) tmp < rep(NA, length(grids5m@data[[1]])) for(c in (1:length(levels(grids5m$soilmu.c)))[mask.c]){ distmaps[[c]] < data.frame(DEM5LIDARf=tmp, TWI=tmp, VDEPTH=tmp, INSOLAT=tmp, CONVI=tmp) for(j in list("DEM5LIDARf", "TWI", "VDEPTH", "INSOLAT", "CONVI")){ distmaps[[c]][j] < ((grids5m@data[j]class.c[c,j])/class.sd[c,j])^2 17 An indepth discussion can be followed in Kempen et al. (2009). 9.7 Extraction of memberships 219 > > # > > > > > } } sum up distances per class: distsum < data.frame(tmp) for(c in (1:length(levels(grids5m$soilmu.c)))[mask.c]){ distsum[paste(c)] < sqrt(rowSums(distmaps[[c]], na.rm=T, dims=1)) } str(distsum) 'data.frame': $ $ $ $ $ $ 1: 2: 3: 4: 5: 6: num num num num num num 80000 obs. of 6 variables: 1.53 1.56 1.75 2.38 3.32 ... 4.4 4.41 4.53 4.88 5.48 ... 37.5 37.5 37.5 37.5 37.6 ... 10 10 10.1 10.2 10.5 ... 2.64 2.54 3.18 4.16 5.35 ... 8.23 8.32 8.28 8.06 7.58 ... # total sum of distances for all pixels: > totsum < rowSums(distsum^(2/(fuzzy.e1)), na.rm=T, dims=1) Once we have estimated the standardized distances, we can then derive memberships using formula (Sokal and Sneath, 1963): 1 − (q−1) µc (i) = 2 dc (i) k c=1 2 dc (i) 1 − (q−1) c = 1, 2, ...k i = 1, 2, ...n (9.7.1) µc (i) ∈ [0, 1] (9.7.2) where µc (i) is a fuzzy membership value of the ith object in the cth cluster, d is the similarity (diagonal) distance, k is the number of clusters and q is the fuzzy exponent determining the amount of fuzziness. Or in syntax: R > for(c in (1:length(levels(grids5m$soilmu.c)))[mask.c]){ > grids5m@data[paste("mu_", c, sep="")] <+ (distsum[paste(c)]^(2/(fuzzy.e1))/totsum)[,1] > } The resulting map of memberships for class “Chaix family, deep, 15 to 45% slopes” is shown in Fig. 9.7. Compare also with Fig. 9.6. Selfstudy exercises: (1.) How much is elevation correlated with the TWI map? (derive correlation coefﬁcient between the two maps) (2.) Which soil mapping unit in Fig. 9.6 is the most correlated to the original map? (HINT: convert to indicators and then correlate the maps.) (3.) Derive the variogram for the ﬁltered LIDAR DEM and compare it to the variogram for elevations derived using the (LiDAR) point measurements. (HINT: compare nugget, sill, range parameter and anisotropy parameters.) 220 Geomorphological units (fishcamp) (4.) Extract landform classes using the pam algorithm as implemented in the package and compare it with the results of kmeans. Which polygons/classes are the same in >50% of the area? (HINT: overlay the two maps and derive summary statistics.) (5.) Extract the landform classes using unsupervised classiﬁcation and the coarse SRTM DEM and identify if there are signiﬁcant differences between the LiDAR based and coarse DEM. (HINT: compare the average area of landform unit; derive a correlation coefﬁcient between the two maps.) (6.) Try to add ﬁve more LSPs and then rerun multinomial logistic regression. Did the ﬁtting improve and how much? (HINT: Compare the resulting AIC for ﬁtted models.) (7.) Extract membership maps (see section 9.7) for all classes and derive the confusion index. Where is the confusion index the highest? Is it correlated with any input LSP? cluster Further reading: Brenning, A., 2008. Statistical Geocomputing combining R and SAGA: The Example of Landslide susceptibility Analysis with generalized additive Models. In: J. Böhner, T. Blaschke & L. Montanarella (eds.), SAGA — Seconds Out (Hamburger Beiträge zur Physischen Geographie und Landschaftsökologie, 19), 23–32. Burrough, P A., van Gaans, P F. M., MacMillan, R. A., 2000. Highresolution landform classiﬁcation . . using fuzzy kmeans. Fuzzy Sets and Systems 113, 37–52. Conrad, O., 2007. SAGA — Entwurf, Funktionsumfang und Anwendung eines Systems für Automatisierte Geowissenschaftliche Analysen. Ph.D. thesis, University of Göttingen, Göttingen. Fisher, P F., Wood, J., Cheng, T., 2005. Fuzziness and ambiguity in multiscale analysis of landscape mor. phometry. In: Petry, F. E., Robinson, V B., Cobb, M. A. (Eds.), Fuzzy Modeling with Spatial Information . for Geographic Problems. SpringerVerlag, Berlin, pp. 209–232. Smith, M. J., Paron, P and Grifﬁths, J. (eds) 2010. Geomorphological Mapping: a professional hand. book of techniques and applications. Developments in Earth Surface Processes, Elsevier, in preparation.

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