Research paperatakrig: An R package for multivariate area-to-area and area-to-point kriging predictions
Introduction
Spatial data conversion between different spatial scales is a common challenge in geography applications because of spatial mismatch between different data sources, such as various types of observation sensors, sample collection methods, or administrative units. In human population mapping, for example, national census data are usually reported in terms of administrative units. However, information about downscaled accurate population distributions is very important for making decisions pertaining to public matters, such as environmental pollution mitigation, public health, natural disaster relief operations, and infrastructure allocation (Liu et al., 2008; Dmowska and Stepinski, 2017). The purposes of scaling conversion and prediction might be different for different model inputs or analyses. Geostatistics-based spatial scaling statistics, area-to-point kriging, and area-to-area kriging have been widely used in diverse applications, including remote sensing downscaling (Pardo-Iguzquiza et al., 2010; Wang et al., 2015), crop yield prediction (Brus et al., 2018), determination of soil organic carbon distribution (Kerry et al., 2012), and disease mapping (Goovaerts, 2009). The aforementioned methods can be used to disaggregate areal data into predictions at the levels of points and different areas (Gotway and Young, 2002; Kyriakidis, 2004; Yoo and Kyriakidis, 2006). Schirrmann et al. (2012) mapped soil phosphorus content on a fine scale in coarse samples. Based on co-registered multivariate satellite sensor images, Atkinson et al. (2008) and Pardo-Igúzquiza et al. (2006) proposed area-to-point downscaling cokriging for super-resolution mapping with remotely sensed images, where the pixel size to be predicted is smaller than the pixel sizes of the input images. Jin et al. (2018) proposed geographically weighted area-to-area regression kriging to downscale soil moisture data and obtained 1-km resolution soil moisture products from 25-km resolution soil moisture products. In addition to downscaling, to validate a large-scale remote sensing product, Hu et al. (2015) applied area-to-area kriging to multiple irregular small-scale observations for predicting large-scale sensible heat flux, an important index of land surface water and heat balance. Instead of explicitly obtaining a point-scale variogram model derived from areal samples by deconvolution, Müeller and Thompson (2015) also proposed a topological restricted maximum likelihood method to consider spatial correlations between irregular areas and applied it to predict runoff signatures in ungauged basins.
A myriad of packages and software applications are available for geostatistical prediction, such as gslib (Deutsch and Journel, 1997), gstat (Pebesma and Wesseling, 1998; Pebesma, 2004), geoR (Ribeiro and Diggle, 2018), SAGA (Conrad et al., 2015), and ArcGIS (ESRI, 2018). According to the CRAN task view, in R language alone, more than 10 packages related to geostatistics have been developed (Bivand, 2019). gstat, which offers rich kriging functions, is one of the most frequently used packages in R language (Pebesma, 2004). The package can be used for area-to-point kriging. It discretizes an area into regular grids and each grid has equal weight in calculating areal average value. Moreover, it does not implement the function for fitting a point-scale variogram from areal data, which is necessary for area-to-area kriging. There are also some R packages to handle change of support prediction for spatial or spatial-temporal data, such as spatialCovariance for covariance matrix computing (Clifford, 2015), stcos (Bradley et al., 2015), stUPscales (Torres-Matallana, 2019), and rtop (Skøien et al., 2014). They mainly consider correlation in a single variate. DSCOKRI is another downscaling cokriging program for remote sensing only, and it is written in Fortran-77 language (Pardo-Iguzquiza et al., 2010). Although area-to-area kriging has been applied successfully in many studies, and the associated calculation flow is similar to that of traditional point support kriging, an easily accessible and applicable open-source package for general area-to-area cokriging has not been developed yet. We extended the framework of Goovaerts (2008) with cokriging and implement it in the R environment (R Core Team, 2019).
The remainder of the paper is organized as follows. In Section 2, we present the general theories of area-to-area and area-to-point kriging. The package implementation of the theories is described in Section 3. Then two applications are presented in Section 4. Finally, conclusions are presented in Section 5.
Section snippets
Model description
The theory of area-to-area kriging is well established and solved by many approaches (Goovaerts, 2008; Gottschalk, 1993; Gottschalk et al., 2006; Skøien et al., 2006, Skøien et al., 2014). Although it is similar to the theory of traditional point kriging, one of the most important differences between the two theories is covariance calculation between samples. In ordinary area-to-area kriging, for example, the predictor of an area with unknown value is calculated from a linear combination of
The atakrig package
We developed atakrig in the R environment, a very popular open-source environment for statistics. The package tries to bridge the gap between geostatistical areal prediction and point prediction. The main functions of atakrig include deconvolution of point-scale variograms from irregular/regular spatial areal data and implementation of area-to-area and area-to-point kriging and cokriging. This package was developed as a supplement to geostatistical point-to-point and point-to-area prediction.
River runoff prediction
Runoff prediction at unobserved locations is a fundamental problem in hydrology. In many ungauged or poorly gauged basins, reliable runoff prediction remains a major challenge. In this demonstration, we use data from the rtop package developed by Skøien et al. (2014). This set contains the recorded average summer runoff data of 57 catchment polygons in the federal country of Upper Austria. It is a subset of a full dataset consisting of 134 catchment polygons which can be downloaded from //www.hydro.tuwien.ac.at/fileadmin/mediapool-hydro/Downloads/rtopData.zip
Conclusion
Area-to-area interpolation is being used increasingly nowadays, especially because different supports of samples can be obtained easily from different sources. Herein, we developed an area-to-area kriging interpolation package in the popular R environment under the geostatistical framework. It supplements the existing rich geostatistic packages. The developed package can be distributed on different operating systems and on different computer hardware platforms, ranging from personal laptops to
Author contribution statement
MH designed the package and wrote the manuscript. YH edited various sections of the manuscript and wrote some function of the package.
Declaration of competing interest
We declare that we have no conflicts of interest.
Acknowledgments
This work was supported by the National Science and Technology Major Project (grant number 2017YFA0604804, 2017ZX10201302), the National Natural Science Foundation of China (grant number 41771434, 41601608). We thank Dr. Jon Olav Skøien and two anonymous reviewers for their constructive comments and suggestions which helped to improve the quality of this manuscript.
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