Journal of Geographical Sciences ›› 2015, Vol. 25 ›› Issue (2): 196-210.doi: 10.1007/s11442-015-1162-2
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BERGONSE Rafaello(), REIS Eusébio
Received:
2013-06-05
Accepted:
2014-04-20
Online:
2015-02-15
Published:
2015-02-15
About author:
Author: Rafaello Bergonse, E-mail:
BERGONSE Rafaello, REIS Eusébio. Reconstructing pre-erosion topography using spatial interpolation techniques: A validation-based approach[J].Journal of Geographical Sciences, 2015, 25(2): 196-210.
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Table 1
Contexts and methodologies of some published pre-erosion surface reconstructions"
Author | Purpose of reconstruction | Interpolation method |
---|---|---|
Wells and Gutiérrez (1982) | Estimation of eroded volumes and combination of results with current mean erosion rates in order to date badland initiation | Undefined |
Daba et al. (2003) | Estimation of eroded volume in a large gully system for two different dates; comparison of results in order to quantify temporal evolution | Undefined1 |
Alexander et al. (2008) | Understanding the geomorphic evolution of a badland site from a set of remnant surfaces, and estimating rates of denudation | Linear interpolation (Triangulated Irregular Networks) |
Perroy et al. (2010) | Estimating volumetric soil loss from a set of gully channels | Linear interpolation (grid based) |
Buccolini et al. (2012) | Estimating the volume eroded by a set of gully systems (calanchi), and relating pre-erosion topography to gully system properties | Linear interpolation (manual2) |
Table 2
General properties of the most common commercially available exact spatial interpolation methods"
Method | General features | Smoothing | Proximity | Geostatistical assumptions |
---|---|---|---|---|
Linear interpolation | May be based on a previous Delaunay triangulation, with the value for each cell being defined by the linear surface of the triangle it overlays (e.g. Surfer 101, ArcGIS 9.1). In other cases, estimations are obtained simply as a function of the nearest known values and the respective distances (e.g. IDRISI Andes2: Eastman, 2006) | None | Local | No |
Inverse Distance Weighted | Interpolated values are a function of the values of the nearest points (quantity is user-defined), with the weight of each in the result being a function of distance. | None | Local to Global | No |
Splines | Generated surface results from fitting a polynomial to a quantity of user-defined known values, subjected to two constraints: (1) surface passes exactly through the known data points; (2) curvature of generated surface is minimized. Has problems representing discrete transitions (e.g. limits of flood plains, slope breaks), sometimes ‘overshooting’ the true surface ( | Elevated | Local to Global | No |
Topo to Raster | Similar to Spline, but modified in order to produce a hydrologically correct surface and incorporate slope breaks. Conceived to use points, lines and polygons as input. | Elevated | Local to Global3 | No |
Ordinary Kriging | Based on preliminary analysis and statistical modelling of the variation of differences between all known values with spatial distance and/or direction. For each location, the functions thus defined are used to estimate values from surrounding data points of known value. | Medium | Local to Global | Yes |
Natural neighbour | Based on the construction of a network of Voronoy polygons incorporating all known data points. Each point to be estimated is inserted on the network, and the latter is modified in order to incorporate it. Each estimated value is the average of all known surrounding points of known value, weighted by the proportion of the new Voronoy polygon overlaying each of the initial polygons. | None | Local | No |
Table 3
The interpolation methods and parameterizations adopted"
Method (different parameter sets used) | Parameters |
---|---|
Linear interpolation (1) | Obtained through triangulation and conversion of a TIN model (points as input) |
Topo to Raster (2) | Two parameterizations: points as input and contours as input. Further parameters were set as default. |
Spline (10) | Spline Regularized: w = 0; 0.001; 0.01; 0.1; 0.5 |
Spline Tension: w = 0, 1, 4, 7, 10. |
Table 4
Characteristics of the distributions of absolute error obtained for each interpolation method (i.e. square root of the square of the difference between real and interpolated values). Methods are ordered by ascending mean absolute error (MAE). Spline Reg and Spline Ten respectively identify the Regularized and Tension methods; w = weight parameter; P50 and P80 are the 50th and 80th percentiles; SD - standard deviation. All values are in metres."
Method | MAE | Min | Max | P50 | P80 | SD |
---|---|---|---|---|---|---|
Topo to Raster (contours) | 0.752 | 0.000 | 5.399 | 0.440 | 1.203 | 0.872 |
Spline Reg w=0.01 | 0.767 | 0.000 | 5.001 | 0.463 | 1.176 | 0.889 |
Spline Reg w=0.1 | 0.771 | 0.000 | 5.395 | 0.470 | 1.218 | 0.913 |
Spline Reg w=0.001 | 0.810 | 0.000 | 5.338 | 0.493 | 1.239 | 0.904 |
Spline Reg w=0.5 | 0.813 | 0.000 | 5.827 | 0.473 | 1.242 | 1.000 |
Spline Reg w=0 | 0.834 | 0.000 | 5.456 | 0.526 | 1.288 | 0.912 |
Spline Ten w=1 | 0.887 | 0.000 | 5.375 | 0.540 | 1.496 | 0.939 |
Spline Ten w=4 | 0.954 | 0.000 | 5.354 | 0.587 | 1.654 | 0.976 |
Spline Ten w=7 | 0.998 | 0.000 | 5.349 | 0.619 | 1.758 | 1.004 |
Spline Ten w=10 | 1.034 | 0.000 | 5.350 | 0.639 | 1.828 | 1.028 |
Linear | 1.214 | 0.000 | 6.908 | 0.815 | 1.962 | 1.239 |
Topo to Raster (points) | 1.589 | 0.000 | 12.773 | 1.094 | 2.383 | 1.795 |
Spline Ten w=0 | 3.463 | 0.001 | 72.642 | 1.084 | 3.494 | 7.500 |
Table 5
Characteristics of the distributions of absolute error obtained during parameter optimization. Methods are ordered by ascending mean absolute error (MAE).; w = weight parameter in the regularized (Reg) Spline method; R = roughness penalty in the Topo to Raster method; P50 and P80 are the 50th and 80th percentiles; SD - standard deviation. All values are in metres."
Method | Mean | Min | Max | P50 | P80 | SD |
---|---|---|---|---|---|---|
Spline Reg w=0.033 | 0.758 | 0.001 | 5.206 | 0.458 | 1.160 | 0.890 |
Spline Reg w=0.055 | 0.762 | 0.001 | 5.291 | 0.468 | 1.191 | 0.896 |
Spline Reg w=0.078 | 0.766 | 0.000 | 5.349 | 0.466 | 1.190 | 0.905 |
Spline Reg w=0.008 | 0.771 | 0.000 | 5.042 | 0.463 | 1.171 | 0.890 |
Spline Reg w=0.006 | 0.776 | 0.000 | 5.095 | 0.466 | 1.175 | 0.892 |
Spline Reg w=0.003 | 0.791 | 0.000 | 5.210 | 0.480 | 1.191 | 0.897 |
Topo to Raster R=0.4 | 0.825 | 0.000 | 5.499 | 0.496 | 1.318 | 0.922 |
Topo to Raster R=0.3 | 0.871 | 0.000 | 4.920 | 0.547 | 1.442 | 0.923 |
Topo to Raster R=0.2 | 0.938 | 0.000 | 5.585 | 0.618 | 1.519 | 0.980 |
Topo to Raster R=0.1 | 0.987 | 0.000 | 5.634 | 0.664 | 1.599 | 1.007 |
Topo to Raster R=0.5 | 1.020 | 0.000 | 5.716 | 0.701 | 1.635 | 1.029 |
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