rtop: An R package for interpolation of data with a variable spatial support, with an example from river networks

Highlights

• R package for geostatistical interpolation of observations with a support.

• Consistent predictions along stream networks.

• Methods for exploratory data analysis.

Abstract

Geostatistical methods have been applied only to a limited extent for spatial interpolation in applications where the observations have an irregular support, such as runoff characteristics along a river network and population health data. Several studies have shown the potential of such methods, but these developments have so far not led to easily accessible, versatile, easy to apply and open source software. Based on the top-kriging approach suggested by Skøien et al. (2006), we will here present the package rtop, which has been implemented in the statistical environment R (R Core Team, 2013). Taking advantage of the existing methods in R for analysis of spatial objects (Bivand et al., 2013), and the extensive possibilities for visualizing the results, rtop makes it easy to apply geostatistical interpolation methods when observations have a non-point spatial support. The package also offers integration with the intamap package for automatic interpolation and the possibility to run rtop through a Web Service.

Introduction

For many applications where a process has been observed, it is also necessary to have predictions of the process at locations without observations; this could be for visualization purposes, for input in a different model, or for further analysis. There exists an appealing set of methods for such predictions, based on a model of spatial correlation between random variables defined at observation and prediction locations. These are commonly referred to as geostatistical methods (Isaaks and Srivastava, 1989) due to the background from mining, although similar methods were also developed within meteorology (Gandin, 1963) and referred to as objective methods. Kriging is defined as the best linear unbiased estimator for spatial data, i.e. a linear interpolation method where the expected bias is zero and the expected interpolation error is minimized. One major advantage of these methods is that they can give an estimate of the prediction uncertainty in addition to the prediction itself.

Whereas geostatistical and similar methods mostly have been developed for and applied to observations with point support or a regular support (e.g. pixels from satellite images), and also to make predictions for areas or volumes (Journel and Huijbregts, 1978), many data sources have a more irregular support in time and/or space, such as runoff related data and aggregated health data. The literature shows some different approaches to solve the interpolation problem for observations with non-point support (Goovaerts, 2006, Goovaerts, 2008, Gottschalk, 1993, Gottschalk et al., 2006, Gotway and Young, 2002, Kyriakidis, 2004, Sauquet et al., 2000, Skøien et al., 2006). Although all these studies make similar assumptions and solve the problem in a similar way, there has not yet been any easily accessible, easily applicable and open source software that is able to interpolate such data.

Based on the top-kriging approach suggested in Skøien et al. (2006), and extended with suggestions by Gottschalk (1993) and Gottschalk et al. (2011), we present the package rtop, which has been implemented in the open source statistical environment R (R Core Team, 2013). The R environment includes a large range of tools for data analysis and visualization and is a platform where it is easy to extend the existing system with new methods in a package system. Several packages already deal with spatial data within R (Bivand et al., 2013), but none of these can do geostatistical interpolation with a variable spatial support of observations taken into account. rtop therefore makes it considerably easier to do such interpolation and to use visualize results, in comparison to former implementations of the method. And although the top-kriging method was originally developed for interpolation of runoff characteristics, we also suggest that the package can be useful for a number of other analyses such as health statistics from administrative regions, aggregated forest data and remote sensing images.

There are several other R-packages that can do some similar operations. The dissever-package (Malone et al., 2012) can for example do downscaling of remote sensing images, but it is based on regular areas and is not available on CRAN. The SSN-package (VerHoef and Peterson, 2010) is able to do interpolation along stream networks, but this is based on a different theory which does not take the areal support into account. It also requires preprocessing of the stream network with the STARS geoprocessing toolset running under ArcGIS.

The constrainedKriging-package (Hofer and Papritz, 2011) can do interpolation to irregular blocks, but not from observations with irregular supports, and does not include variogram fitting tools comparable to rtop. It is also possible to make extensions to the gstat-package (Pebesma, 2004) for interpolation between different supports, but this is currently not in the code itself, and it is not possible to fit variograms for such observations. It is also likely that the framework of the INLA-package (Rue et al., 2013) can deal with the same type of problems as rtop, but such functionality has, to our knowledge, not yet been implemented. Also the C-code of the psgp-package (Barillec et al., 2011) can in theory interpolate observations and predictions with a spatial support, but this is not available from the R-package.

rtop is currently the only package which provides all necessary functionality for interpolating data with variable spatial support. In this paper we demonstrate the package with an example from hydrology: the interpolation of streamflow along river networks, where it was earlier necessary to combine the use of spreadsheets, analyses tools, graphical tools and GIS to perform the interpolation. We show how rtop can be applied to carry out this chain of analyses within R, to get an easy solution for the river network problem and other problems of spatial interpolation with a variable support.