Spatial and spatio-temporal analysis of stable isotopes in biogeochemical processes

From a geostatistical point of view, both time and space variability characterise many subjects of interest of Earth and environmental sciences. Datasets often include such spatial- and temporal variability and while some established tools exist for spatial interpolation and time series analysis, mixing these techniques calls for compromise: researchers are often forced to choose which is the main source of variation, neglecting the other (Cressie 1990, Cressie and Wikle 2011). We developed Timescape, a simple algorithm, which can be used in many fields of environmental sciences when both time and space variability must be considered on equal grounds. Stable isotopes of Hydrogen, Oxygen, Carbon and Nitrogen are involved in biogeochemical cycles (West et al. 2010) and their concentration varies within the substrates depending on the environmental processes. Isotopic fractionation occurs in every reaction and it is a well-known tracer of the biochemical details involved. Analysing the isotopic variability from both a time and space point of view highlights the patterns of change of the concentration values and allows the researchers to find a possible relationship between the sample values and the time and place of collection. As some studies relating the site of production to the isotopic content of food commodities suggest (Camin et al. 2010a, Camin et al. 2010b, Chiocchini et al. 2016, Portarena et al. 2014, 2015, van der Veer 2013, West et al. 2007), finding an isotopic signature of such commodities can be of great help in contrasting the frauds related to a pretended geographical origin. We present as first case study, a "flat" space-only interpolation of Carbon stable isotopes concentrations for the Ogliarola campana olive cultivar; this is a regular Isoscape (West et al. 2010). Isoscapes (short for Isotopic landscape) are thematic GIS raster layers, which relate the stable isotopes ratios values ?nX with the space coordinates. Stable isotopes bring a lot of information about the environment where the olives growth took place (soil ?15N and ?13C) and precipitation waters (?2H and ?18O) so they can be used to relate a final consumer product to the actual zone of production. Regular Isoscapes, however, do not capture the temporal variability of the isotopic datasets. The GIS standard procedure delivers a stack of Isoscapes, one per each relevant time or time interval. This can be acceptable if the dataset consistency (number of samples) allows the interpolation of such many layers but this is not always the case. Furthermore, the samples could have been collected sparsely both in time and in space, or could be the result of the merging of different research projects. This was the main goal in the development of the Timescape algorithm which produces, in a broad sense, a time-aware three-dimensional extension of regular Isoscapes, relating the dependent ?nX ratio variable to the independent space and time coordinates. Two geostatistical software tools have been developed based on the Timescape algorithm: TimescapeGlobal, which uses geographical coordinates and TimescapeLocal, which uses projected coordinates. The global version has already been published as free software with the open license GNU GPLv3.0 (the whole package is available at https://sourceforge.net/projects/timescapeglobal/). The local version is currently under development, some results in the field of stable isotopes modelling have been obtained with an older, unpublished version, available upon request to the authors. The second case study shows the interpolation over space and time of the mycorrhizal ?15N derived from a host-mycorrhiza symbiosis study in Umbria (Gavrichkova et al. 2016). The 15N fractionation between mycorrhiza and host trees occurred all