# Verification of slopes

## Introduction

This article aims to provide external verification of the results provided by the package. So far only one verification dataset has been used, but we hope to find others. If you know of verification datasets, please let us know — initially we planned to use a dataset from a paper on river slopes (Cohen et al. 2018), but we could find no way of extracting the underlying data to do the calculation.

For this article we primarily used the following packages, although others are loaded in subsequent code chunks.

    library(slopes)
    library(sf)

The results are reproducible (requires downloading input data manually and installing additional packages). To keep package build times low, only the results are presented below.

## Case studies

### Roads in the Netherlands

    u = "https://downloads.rijkswaterstaatdata.nl/nwb-wegen/geogegevens/shapefile/NWB_hoogtebestand/01-10-2024%20Hoogtes_Rijkswegen.zip"
    f_zip = "NWB_hoogtebestand.zip"
    if (!file.exists(f_zip)) {
      download.file(u, f_zip)
      unzip(f_zip)
      # Datasets in
      list.files("NWB3D_resultaten_oktober_2024.gdb")
      roads_nl_layers = sf::st_layers("NWB3D_resultaten_oktober_2024.gdb")
    #   Driver: OpenFileGDB
    # Available layers:
    #                                   layer_name        geometry_type features
    # 1    NWB3D_resultaten_oktober_2024_wegvakken 3D Multi Line String    16456
    # 2 NWB3D_resultaten_oktober_2024_vertexpunten             3D Point  1095615
    #   fields            crs_name
    # 1     59 Amersfoort / RD New
    # 2     59 Amersfoort / RD New
      roads_nl = sf::st_read("NWB3D_resultaten_oktober_2024.gdb", layer = "NWB3D_resultaten_oktober_2024_wegvakken")
      names(roads_nl)
    #    [1] "WVK_ID"        "BST_CODE"      "WVK_BEGDAT"    "JTE_ID_BEG"
    #  [5] "JTE_ID_END"    "WEGBEHSRT"     "WEGNUMMER"     "WEGDEELLTR"
    #  [9] "HECTO_LTTR"    "RPE_CODE"      "ADMRICHTNG"    "RIJRICHTNG"
    # [13] "STT_NAAM"      "STT_BRON"      "WPSNAAM"       "GME_ID"
    # [17] "GME_NAAM"      "HNRSTRLNKS"    "HNRSTRRHTS"    "E_HNR_LNKS"
    # [21] "E_HNR_RHTS"    "L_HNR_LNKS"    "L_HNR_RHTS"    "BEGAFSTAND"
    # [25] "ENDAFSTAND"    "BEGINKM"       "EINDKM"        "POS_TV_WOL"
    # [29] "WEGBEHCODE"    "WEGBEHNAAM"    "DISTRCODE"     "DISTRNAAM"
    # [33] "DIENSTCODE"    "DIENSTNAAM"    "WEGTYPE"       "WGTYPE_OMS"
    # [37] "ROUTELTR"      "ROUTENR"       "ROUTELTR2"     "ROUTENR2"
    # [41] "ROUTELTR3"     "ROUTENR3"      "ROUTELTR4"     "ROUTENR4"
    # [45] "WEGNR_AW"      "WEGNR_HMP"     "GEOBRON_ID"    "GEOBRON_NM"
    # [49] "BRONJAAR"      "OPENLR"        "BAG_ORL"       "FRC"
    # [53] "FOW"           "ALT_NAAM"      "ALT_NR"        "REL_HOOGTE"
    # [57] "Hoogte_bron"   "Kwaliteitlaag" "SHAPE_Length"  "SHAPE"
    # Plot the slopes (variable called REL_HOOGTE):
      summary(roads_nl$REL_HOOGTE)
      # xyz
      roads_nl_xyz = sf::st_coordinates(roads_nl)
      head(roads_nl_xyz)
      hist(roads_nl_xyz[, "Z"], breaks = 50, main = "Histogram of elevation values (m)", xlab = "Elevation (m)")
      plot(roads_nl["REL_HOOGTE"])
      summary(sf::st_geometry_type(roads_nl))
      roads_nl$slope = roads_nl |>
        sf::st_cast("LINESTRING") |>
        slopes::slope_xyz() * 100

      summary(roads_nl$slope)
      library(tmap)
      library(tmap.mapgl)
      m = tm_shape(roads_nl) +
        tm_lines(
          col = "slope",
          col.scale = tm_scale_intervals(
            breaks = c(-1, 0.5, 1, 2, 20),
            labels = c("0-0.5%", "0.5-1%", "1-2%", "2%+"),
            values = cols4all::c4a("-brewer.rd_yl_gn")
          ),
          lwd = 5
      )
      tmap_mode("maplibre")
      m
    }

### Comparison of DEM-derived and survey-grade Z values

The NL NWB Hoogtebestand contains 3D linestring geometries with Z coordinates derived from high-resolution Dutch elevation models (AHN4/AHN5, with sub-10 cm vertical accuracy). These provide a ground truth against which the `slopes` package’s DEM-derived elevation estimates can be compared, as proposed in issue #71.

The following code demonstrates the comparison, using a sample of 30 road segments from South Limburg (the hilliest region of the Netherlands, with elevations ranging from 32 m to 181 m above NAP) and an independent DEM sourced from AWS Open Data terrain tiles via the `elevatr` package (zoom level 10, approximately 120 m resolution).

    # Load the NL data (see code above) and focus on South Limburg
    roads_wgs84 = sf::st_transform(roads_nl, 4326)
    coords_centroid = sf::st_coordinates(sf::st_centroid(st_geometry(roads_wgs84)))
    in_limburg = coords_centroid[, "Y"] > 50.75 &
                 coords_centroid[, "Y"] < 51.0 &
                 coords_centroid[, "X"] > 5.6 &
                 coords_centroid[, "X"] < 6.1
    roads_limburg = roads_nl[in_limburg, ]

    # Pick 30 segments with most Z variation
    get_z_range = function(geom) {
      c = sf::st_coordinates(geom)
      if (nrow(c) < 2) return(0)
      diff(range(c[, "Z"]))
    }
    z_ranges = sapply(sf::st_geometry(roads_limburg), get_z_range)
    top_idx = order(z_ranges, decreasing = TRUE)[1:30]
    roads_sample = roads_limburg[top_idx, ]

    # Download DEM via elevatr (AWS Open Data, no API key required)
    library(elevatr)
    roads_sample_wgs84 = sf::st_transform(roads_sample, 4326)
    bb = sf::st_bbox(roads_sample_wgs84)
    # Buffer the bbox by ~1 km to ensure all vertices are covered
    bb_mat = matrix(c(
      bb["xmin"] - 0.01, bb["ymin"] - 0.01,
      bb["xmax"] + 0.01, bb["ymin"] - 0.01,
      bb["xmax"] + 0.01, bb["ymax"] + 0.01,
      bb["xmin"] - 0.01, bb["ymax"] + 0.01,
      bb["xmin"] - 0.01, bb["ymin"] - 0.01
    ), ncol = 2, byrow = TRUE)
    bb_sf = sf::st_sf(geometry = sf::st_sfc(sf::st_polygon(list(bb_mat)), crs = 4326))
    dem_elevatr = get_elev_raster(bb_sf, z = 10, clip = "bbox")
    # Project DEM to match the NL data CRS (EPSG:28992)
    dem_28992 = terra::project(terra::rast(dem_elevatr), "EPSG:28992", method = "bilinear")

    # Prepare roads: cast to LINESTRING and record actual Z values
    roads_ls = sf::st_cast(roads_sample, "LINESTRING")
    coords_before = sf::st_coordinates(roads_ls)
    actual_z = coords_before[, "Z"]

    # Add elevation from the DEM using the slopes package
    roads_with_z = slopes::elevation_add(roads_ls, dem = dem_28992)
    coords_after = sf::st_coordinates(roads_with_z)
    est_z = coords_after[, "Z"]

    # Compare Z values
    z_diff = est_z - actual_z
    cat("Z RMSE:", round(sqrt(mean(z_diff^2)), 2), "m\n")
    cat("Z MAE:", round(mean(abs(z_diff)), 2), "m\n")
    cat("Z correlation (r):", round(cor(est_z, actual_z), 3), "\n")
    cat("Z mean bias:", round(mean(z_diff), 2), "m\n")

    # Compare slopes (both in EPSG:28992, so lonlat = FALSE)
    slopes_actual = slopes::slope_xyz(roads_ls, lonlat = FALSE) * 100
    slopes_est = slopes::slope_xyz(roads_with_z, lonlat = FALSE) * 100
    cat("Slope correlation (r):",
        round(cor(slopes_actual, slopes_est, use = "complete.obs"), 3), "\n")
    cat("Slope RMSE:",
        round(sqrt(mean((slopes_est - slopes_actual)^2, na.rm = TRUE)), 3), "%\n")

The results of this comparison, based on 7458 vertices across 30 road segments in South Limburg, are summarised below.

Metric | Elevation (Z) | Slope (%)
--- | --- | ---
RMSE | 4.78 m | 0.805
MAE | 3.59 m | 0.586
Correlation (r) | 0.989 | 0.802
Mean bias | -0.67 m | +0.14

The elevation estimates from the DEM are very strongly correlated with the survey-grade NL road heights (r = 0.989, RMSE = 4.78 m), demonstrating that the `slopes` package’s `elevation_add()` function accurately captures the general terrain. The slight negative bias (-0.67 m) and the RMSE of under 5 m are consistent with the expected vertical accuracy of the AWS terrain tiles at this zoom level.

The slope estimates are also well correlated (r = 0.802), with a modest RMSE of 0.81%. The DEM-derived slopes have a lower maximum (5.5% vs 7.3%) and lower standard deviation (1.16% vs 1.34%) than the NL data, reflecting the smoothing effect of the coarser DEM resolution. This is consistent with the findings from the GNSS trace comparison above: the package provides less noisy slope estimates than raw GPS data, but may underestimate steeper gradients due to DEM resolution limitations.

# DEM for eu:

install.packages(“CopernicusDEM”) system(“msiexec.exe /i https://awscli.amazonaws.com/AWSCLIV2.msi”) # Test it’s installed: system(“aws –version”) # Find location of aws cli from powershell (equivalent of `which aws` on Linux): # Get-Command aws | Select-Object -ExpandProperty Source # C:Files.exe current_path = Sys.getenv(“PATH”)

aws_path = “C:\Program Files\Amazon\AWSCLIV2” new_path = paste(aws_path, current_path, sep = “;”) Sys.setenv(PATH = new_path) # Now aws should work system(“aws –version”) # Download DEM for Brussels 6 km from center zones = zonebuilder::zb_zone(“Brussels”, n_circles = 3) region = sf::st_union(zones\(geometry) |> sf::st_make_valid() sf::sf_use_s2(TRUE) # disable s2 for this operation region_plus_100m = sf::st_buffer(region, dist = 100) mapview::mapview(region) + mapview::mapview(region_plus_100m) sf::sf_use_s2(FALSE) # disable s2 for this operation dir_save_tifs = "dems-brussels" dem = CopernicusDEM::aoi_geom_save_tif_matches( sf_or_file = region_plus_100m, dir_save_tifs = dir_save_tifs, resolution = 30, crs_value = 4326, threads = parallel::detectCores(), verbose = TRUE ) dems = list.files(dir_save_tifs, full.names = TRUE) dem_terra = terra::rast(dems) names(dem_terra) dem_cropped = terra::crop(dem_terra, region_plus_100m) mapview::mapview(dem_cropped) travel_network = osmactive::get_travel_network( region, boundary = region, boundary_type = "clipsrc" ) plot(travel_network\)geometry) travel_network = travel_network |> sf::st_filter(region, .predicate = sf::st_within)

cycle_net = osmactive::get_cycling_network(travel_network) drive_net = osmactive::get_driving_network(travel_network) cycle_net = osmactive::distance_to_road(cycle_net, drive_net) cycle_net = osmactive::classify_cycle_infrastructure(cycle_net, include_mixed_traffic = TRUE) names(cycle_net) mapview::mapview(cycle_net, zcol = “cycle_segregation”) nrow(cycle_net) # Calculate slopes with the slopes package: # Add elevation to cycle network segments

sf::st_crs(dem_cropped) == sf::st_crs(cycle_net) # check the extents of both: mapview::mapview(dem_cropped) + mapview::mapview(cycle_net)

cycle_net_clean = sf::st_cast(cycle_net, “LINESTRING”) cycle_net_xyz = elevation_add(cycle_net_clean, dem = dem_cropped) summary(sf::st_geometry_type(cycle_net_xyz))

# Calculate slopes for each segment

cycle_net_xyz\(slope = slope_xyz(cycle_net_xyz, lonlat = TRUE, fun = slope_matrix_weighted) # cycle_net_xyz\)slope = slope_xyz(cycle_net_xyz, lonlat = TRUE, fun = slope_matrix_mean) summary(cycle_net_xyz\(slope) # Convert to percentage: cycle_net_xyz = cycle_net_xyz |> # Convert to factor with greater than 5 being "5+" dplyr::mutate( slope_percent = dplyr::case_when( slope < 0.02 ~ as.character("0-2"), slope < 0.05 ~ as.character("2-5"), slope < 0.08 ~ as.character("5-8"), TRUE ~ "8+" ) ) table(cycle_net_xyz\)slope_percent)

# Drop z dimension

cycle_net_slopes = sf::st_zm(cycle_net_xyz) |> dplyr::transmute(osm_id, highway, cycle_segregation, slope = round(slope, 3), slope_percent) summary(duplicated(cycle_net_slopes$geometry)) mapview::mapview(cycle_net_slopes, zcol = “slope_percent”, legend = TRUE) sf::write_sf(cycle_net_slopes, “cycle_net_slopes_brussels.gpkg”, delete_dsn = TRUE) system(“gh release upload v1.0.1 cycle_net_slopes_brussels.gpkg –clobber”) cycle_net_slopes = sf::read_sf(“cycle_net_slopes_brussels.gpkg”)

# install cran version

remotes::install_dev(“tmap”) # Save with tmap library(tmap) v = cols4all::c4a(“brewer.rd_yl_gn”, n = 4) |> rev() m = tm_shape(cycle_net_slopes) + tm_lines( col = “slope_percent”, col.scale = tm_scale(values = v), lwd = 2, popup.vars = FALSE ) m tmap_save(m, “cycle_net_slopes_brussels.html”) browseURL(“cycle_net_slopes_brussels.html”) system(“gh release upload v1.0.1 cycle_net_slopes_brussels.html –clobber”) # url of the file: u_release = “https://github.com/ropensci/slopes/releases/download/v1.0.1/cycle_net_slopes_brussels.html” download.file(u_release, “cycle_net_slopes_brussels.html”)

    ![Example from the Netherlands](example-nl.png)

    # Comparison with results from ArcMap 3D Analyst

    # Three-dimensional traces of roads dataset

    <!-- todo: make the segments -->

    An input dataset, comprising a 3D linestring recorded using a dual frequency GNSS receiver (a [Leica 1200](https://gef.nerc.ac.uk/equipment/gnss/)) with a vertical accuracy of 20 mm
    <!-- 138 GPS 3D traces of a hilly road from a peer reviewed journal article -->
    [@ariza-lopez_dataset_2019] was downloaded from the
    <!-- [figshare website as a .zip file](https://ndownloader.figshare.com/files/14331197) - raw data -->
    [figshare website as a .zip file](https://ndownloader.figshare.com/files/14331185)
    and unzipped and inflated in the working directory as follows (not evaluated to reduce package build times):

    ``` r
    download.file("https://ndownloader.figshare.com/files/14331185", "3DGRT_AXIS_EPSG25830_v2.zip")
    unzip("3DGRT_AXIS_EPSG25830_v2.zip")
    trace = sf::read_sf("3DGRT_AXIS_EPSG25830_v2.shp")
    plot(trace)
    nrow(trace)
    #> 11304
    summary(trace$X3DGRT_h)
    #>  Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   642.9   690.3   751.4   759.9   834.3   884.9

To verify our estimates of hilliness, we generated slope estimates for each segment and compared them with Table 7 in Ariza-López et al. (2019). The absolute gradient measure published in that paper were:

    res_gps = c(0.00, 4.58, 1136.36, 6.97)
    res_final = c(0.00, 4.96, 40.70, 3.41)
    res = data.frame(cbind(
      c("GPS", "Dual frequency GNSS receiver"),
      rbind(res_gps, res_final)
    ))
    names(res) = c("Source", "min", " mean", " max", " stdev")
    knitr::kable(res, row.names = FALSE)

Source | min | mean | max | stdev
--- | --- | --- | --- | ---
GPS | 0 | 4.58 | 1136.36 | 6.97
Dual frequency GNSS receiver | 0 | 4.96 | 40.7 | 3.41

    # mapview::mapview(trace) # check extent: it's above 6km in height
    # remotes::install_github("hypertidy/ceramic")
    loc = colMeans(sf::st_coordinates(sf::st_transform(trace, 4326)))
    e = ceramic::cc_elevation(loc = loc[1:2], buffer = 3000)
    trace_projected = sf::st_transform(trace, 3857)
    plot(e)
    plot(trace_projected$geometry, add = TRUE)

The slopes were estimated as follows:

    # source: https://www.robinlovelace.net/presentations/munster.html#31
    points2line_trajectory = function(p) {
      c = st_coordinates(p)
      i = seq(nrow(p) - 2)
      l = purrr::map(i, ~ sf::st_linestring(c[.x:(.x + 1), ]))
      lfc = sf::st_sfc(l)
      a = seq(length(lfc)) + 1 # sequence to subset
      p_data = cbind(sf::st_set_geometry(p[a, ], NULL))
      sf::st_sf(p_data, geometry = lfc)
    }
    r = points2line_trajectory(trace_projected)
    # summary(st_length(r)) # mean distance is 1m! Doesn't make sense, need to create segments
    s = slope_raster(r, e = e)
    slope_summary = data.frame(min = min(s), mean = mean(s), max = max(s), stdev = sd(s))
    slope_summary = slope_summary * 100
    knitr::kable(slope_summary, digits = 1)

min | mean | max | stdev
--- | --- | --- | ---
0 | 6.2 | 48.2 | 5.6

Combined with the previous table from Ariza-López et al. (2019), these results can be compared with those obtained from mainstream GPS, and an accurate GNSS receiver:

Source | min | mean | max | stdev
--- | --- | --- | --- | ---
GPS | 0 | 4.58 | 1136.36 | 6.97
Dual frequency GNSS receiver | 0 | 4.96 | 40.7 | 3.41
Slopes R package | 0 | 6.2 | 48.2 | 5.6

It is notable that the package substantially overestimates the gradient, perhaps due to the low resolution of the underlying elevation raster. However, the slopes package seems to provide less noisy slope estimates than the GPS approach, with lower maximum values and low standard deviation.

# References

Ariza-López, Francisco Javier, Antonio Tomás Mozas-Calvache, Manuel Antonio Ureña-Cámara, and Paula Gil de la Vega. 2019. “Dataset of Three-Dimensional Traces of Roads.” Scientific Data 6 (1): 1–10. https://doi.org/10.1038/s41597-019-0147-x.
Cohen, Sagy, Tong Wan, Md Tazmul Islam, and J. P. M. Syvitski. 2018. “Global River Slope: A New Geospatial Dataset and Global-Scale Analysis.” Journal of Hydrology 563 (August): 1057–67. https://doi.org/10.1016/j.jhydrol.2018.06.066.