Journal of Geographical Sciences ›› 2015, Vol. 25 ›› Issue (2): 196210.doi: 10.1007/s1144201511622
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BERGONSE Rafaello(), REIS Eusébio
Received:
20130605
Accepted:
20140420
Online:
20150215
Published:
20150215
About author:
Author: Rafaello Bergonse, Email:
BERGONSE Rafaello, REIS Eusébio. Reconstructing preerosion topography using spatial interpolation techniques: A validationbased approach[J].Journal of Geographical Sciences, 2015, 25(2): 196210.
Table 1
Contexts and methodologies of some published preerosion 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  Undefined^{1} 
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 preerosion topography to gully system properties  Linear interpolation (manual^{2}) 
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 10^{1}, 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 Andes^{2}: Eastman, 2006)  None  Local  No 
Inverse Distance Weighted  Interpolated values are a function of the values of the nearest points (quantity is userdefined), 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 userdefined 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 Global^{3}  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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