Tomislav Hengl

About regression-kriging: From equations to case studies

This paper discusses the characteristics of regression-kriging (RK), its strengths and limitations, and illustrates these with a simple example and three case studies. RK is a spatial interpolation technique that combines a regression of the dependent variable on auxiliary variables (such as land surface parameters, remote sensing imagery and thematic maps) with simple kriging of the regression residuals. It is mathematically equivalent to the interpolation method variously called Universal Kriging (UK) and Kriging with External Drift (KED), where auxiliary predictors are used directly to solve the kriging weights. The advantage of RK is the ability to extend the method to a broader range of regression techniques and to allow separate interpretation of the two interpolated components. Data processing and interpretation of results are illustrated with three case studies covering the national territory of Croatia. The case studies use land surface parameters derived from combined Shuttle Radar Topography Mission and contour-based digital elevation models and multitemporal-enhanced vegetation indices derived from the MODIS imagery as auxiliary predictors. These are used to improve mapping of two continuous variables (soil organic matter content and mean annual land surface temperature) and one binary variable (presence of yew). In the case of mapping temperature, a physical model is used to estimate values of temperature at unvisited locations and RK is then used to calibrate the model with ground observations. The discussion addresses pragmatic issues: implementation of RK in existing software packages, comparison of RK with alternative interpolation techniques, and practical limitations to using RK. The most serious constraint to wider use of RK is that the analyst must carry out various steps in different software environments, both statistical and GIS.

Finding the right pixel size

This paper discusses empirical and analytical rules to select a suitable grid resolution for output maps and based on the inherent properties of the input data. The choice of grid resolution was related with the cartographic and statistical concepts: scale, computer processing power, positional accuracy, size of delineations, inspection density, spatial autocorrelation structure and complexity of terrain. These were further related with the concepts from the general statistics and information theory such as Nyquist frequency concept from signal processing and equations to estimate the probability density function. Selection of grid resolution was demonstrated using four datasets: (1) GPS positioning data-the grid resolution was related to the area of circle described by the error radius, (2) map of agricultural plots-the grid resolution was related to the size of smallest and narrowest plots, (3) point dataset from soil mapping-the grid resolution was related to the inspection density, nugget variation and range of spatial autocorrelation and (4) contour map used for production of digital elevation model-the grid resolution was related with the spacing between the contour lines i.e. complexity of terrain. It was concluded that no ideal grid resolution exists, but rather a range of suitable resolutions. One should at least try to avoid using resolutions that do not comply with the effective scale or inherent properties of the input dataset. Three standard grid resolutions for output maps were finally recommended: (a) the coarsest legible grid resolution-this is the largest resolution that we should use in order to respect the scale of work and properties of a dataset; (b) the finest legible grid resolution-this is the smallest grid resolution that represents 95% of spatial objects or topography; and (c) recommended grid resolution-a compromise between the two. Objective procedures to derive the true optimal grid resolution that maximizes the predictive capabilities or information content of a map are further discussed. This methodology can now be integrated within a GIS package to help inexperienced users select a suitable grid resolution without doing extensive data preprocessing.

SoilGrids250m: Global gridded soil information based on machine learning

This paper describes the technical development and accuracy assessment of the most recent and improved version of the SoilGrids system at 250m resolution (June 2016 update). SoilGrids provides global predictions for standard numeric soil properties (organic carbon, bulk density, Cation Exchange Capacity (CEC), pH, soil texture fractions and coarse fragments) at seven standard depths (0, 5, 15, 30, 60, 100 and 200 cm), in addition to predictions of depth to bedrock and distribution of soil classes based on the World Reference Base (WRB) and USDA classification systems (ca. 280 raster layers in total). Predictions were based on ca. 150,000 soil profiles used for training and a stack of 158 remote sensing-based soil covariates (primarily derived from MODIS land products, SRTM DEM derivatives, climatic images and global landform and lithology maps), which were used to fit an ensemble of machine learning methods—random forest and gradient boosting and/or multinomial logistic regression—as implemented in the R packages ranger, xgboost, nnet and caret. The results of 10–fold cross-validation show that the ensemble models explain between 56% (coarse fragments) and 83% (pH) of variation with an overall average of 61%. Improvements in the relative accuracy considering the amount of variation explained, in comparison to the previous version of SoilGrids at 1 km spatial resolution, range from 60 to 230%. Improvements can be attributed to: (1) the use of machine learning instead of linear regression, (2) to considerable investments in preparing finer resolution covariate layers and (3) to insertion of additional soil profiles. Further development of SoilGrids could include refinement of methods to incorporate input uncertainties and derivation of posterior probability distributions (per pixel), and further automation of spatial modeling so that soil maps can be generated for potentially hundreds of soil variables. Another area of future research is the development of methods for multiscale merging of SoilGrids predictions with local and/or national gridded soil products (e.g. up to 50 m spatial resolution) so that increasingly more accurate, complete and consistent global soil information can be produced. SoilGrids are available under the Open Data Base License.

Mapping Soil Properties of Africa at 250 m Resolution: Random Forests Significantly Improve Current Predictions

80% of arable land in Africa has low soil fertility and suffers from physical soil problems. Additionally, significant amounts of nutrients are lost every year due to unsustainable soil management practices. This is partially the result of insufficient use of soil management knowledge. To help bridge the soil information gap in Africa, the Africa Soil Information Service (AfSIS) project was established in 2008. Over the period 2008–2014, the AfSIS project compiled two point data sets: the Africa Soil Profiles (legacy) database and the AfSIS Sentinel Site database. These data sets contain over 28 thousand sampling locations and represent the most comprehensive soil sample data sets of the African continent to date. Utilizing these point data sets in combination with a large number of covariates, we have generated a series of spatial predictions of soil properties relevant to the agricultural management—organic carbon, pH, sand, silt and clay fractions, bulk density, cation-exchange capacity, total nitrogen, exchangeable acidity, Al content and exchangeable bases (Ca, K, Mg, Na). We specifically investigate differences between two predictive approaches: random forests and linear regression. Results of 5-fold cross-validation demonstrate that the random forests algorithm consistently outperforms the linear regression algorithm, with average decreases of 15–75% in Root Mean Squared Error (RMSE) across soil properties and depths. Fitting and running random forests models takes an order of magnitude more time and the modelling success is sensitive to artifacts in the input data, but as long as quality-controlled point data are provided, an increase in soil mapping accuracy can be expected. Results also indicate that globally predicted soil classes (USDA Soil Taxonomy, especially Alfisols and Mollisols) help improve continental scale soil property mapping, and are among the most important predictors. This indicates a promising potential for transferring pedological knowledge from data rich countries to countries with limited soil data.

Islands as model systems in ecology and evolution: prospects fifty years after MacArthur‐Wilson

The study of islands as model systems has played an important role in the development of evolutionary and ecological theory. The 50th anniversary of MacArthur and Wilsons (December 1963) article, An equilibrium theory of insular zoogeography, was a recent milestone for this theme. Since 1963, island systems have provided new insights into the formation of ecological communities. Here, building on such developments, we highlight prospects for research on islands to improve our understanding of the ecology and evolution of communities in general. Throughout, we emphasise how attributes of islands combine to provide unusual research opportunities, the implications of which stretch far beyond islands. Molecular tools and increasing data acquisition now permit re-assessment of some fundamental issues that interested MacArthur and Wilson. These include the formation of ecological networks, species abundance distributions, and the contribution of evolution to community assembly. We also extend our prospects to other fields of ecology and evolution - understanding ecosystem functioning, speciation and diversification - frequently employing assets of oceanic islands in inferring the geographic area within which evolution has occurred, and potential barriers to gene flow. Although island-based theory is continually being enriched, incorporating non-equilibrium dynamics is identified as a major challenge for the future.

Soil sampling strategies for spatial prediction by correlation with auxiliary maps

The paper evaluates spreading of observations in feature and geographical spaces as a key to sampling optimisation for spatial prediction by correlation with auxiliary maps. Although auxiliary data are commonly used for mapping soil variables, problems associated with the design of sampling strategies are rarely examined. When generalised least-squares estimation is used, the overall prediction error depends upon spreading of points in both feature and geographical space. Allocation of points uniformly over the feature space range proportionally to the distribution of predictor (equal range stratification, or ER design) is suggested as a prudent sampling strategy when the regression model between the soil and auxiliary variables is unknown. An existing 100-observation sample from a 50 by 50 km soil survey in central Croatia was used to illustrate these concepts. It was re-sampled to 25-point datasets using different experimental designs: ER and 2 response surface designs. The designs were compared for their performance in predicting soil organic matter from elevation (univariate example) using the overall prediction error as an evaluation criterion. The ER design gave overall prediction error similar to the minmax design, suggesting that it is a good compromise between accurate model estimation and minimisation of spatial autocorrelation of residuals. In addition, the ER design was extended to the multivariate case. Four predictors (elevation, temperature, wetness index, and NDVI) were transformed to standardised principal components. The sampling points were then assigned to the components in proportion to the variance explained by a principal component analysis and following the ER design. Since stratification of the feature space results in a large number of possible points in each cluster, the spreading in geographical space can also be maximised by selecting the best of several realisations.

Chapter 4 Preparation of DEMs for Geomorphometric Analysis

Publisher Summary This chapter gives guidance on how to prepare elevation data for geomorphometric analysis. It outlines some common errors in raw digital elevation model (DEM) data sources and then suggests approaches for systematically improving the quality of DEMs. It starts with height samples and ends with the final DEMs used for geomorphometric analysis. More precisely, the chapter describes both simple and more advanced algorithms that can be used to reduce systematic and random errors, and enrich the quality of DEMs by incorporating auxiliary information on land cover and the hydrological properties of an area. These algorithms are implemented and described further in the software packages. The nature of DEM-preprocessing algorithms very much depends on the type1 of input data. For this reason, not all algorithms are applicable to raw DEMs. From all the approaches to preprocessing DEMs to reduce errors, three different groups are distinguished: The empirical methods, the filtering methods, and the simulation methods.

Spatio-temporal interpolation of daily temperatures for global land areas at 1 km resolution

Combined Global Surface Summary of Day and European Climate Assessment and Dataset daily meteorological data sets (around 9000 stations) were used to build spatio-temporal geostatistical models and predict daily air temperature at ground resolution of 1km for the global land mass. Predictions in space and time were made for the mean, maximum, and minimum temperatures using spatio-temporal regression-kriging with a time series of Moderate Resolution Imaging Spectroradiometer (MODIS) 8 day images, topographic layers (digital elevation model and topographic wetness index), and a geometric temperature trend as covariates. The accuracy of predicting daily temperatures was assessed using leave-one-out cross validation. To account for geographical point clustering of station data and get a more representative cross-validation accuracy, predicted values were aggregated to blocks of land of size 500x500km. Results show that the average accuracy for predicting mean, maximum, and minimum daily temperatures is root-mean-square error (RMSE) =2 degrees C for areas densely covered with stations and between 2 degrees C and 4 degrees C for areas with lower station density. The lowest prediction accuracy was observed at high altitudes (>1000m) and in Antarctica with an RMSE around 6 degrees C. The model and predictions were built for the year 2011 only, but the same methodology could be extended for the whole range of the MODIS land surface temperature images (2001 to today), i.e., to produce global archives of daily temperatures (a next-generation repository) and to feed various global environmental models. Key Points Global spatio-temporal regression-kriging daily temperature interpolation Fitting of global spatio-temporal models for the mean, maximum, and minimum temperatures Time series of MODIS 8 day images as explanatory variables in regression part

Mathematical and digital models of the land surface

Publisher Summary This chapter introduces the land-surface concept from both the geodetic and statistical perspectives, and reviews ways to represent it. It also discusses ways of producing models of the land-surface, from sampling procedures to digital elevation model (DEM) gridding techniques. An extensive comparison of the methods used to derive first and second order derivatives from DEMs have been presented. Mathematical models of the land surface have their uses, but it can be dangerous to regard them as being universally applicable, or even as capturing the essence of a real land surface. Understanding the concept of the land surface and its specific properties is a first step toward successful geomorphometric analysis. Ignoring aspects, such as the correct definition of a reference vertical datum, the density and distribution of the initial height observations, and the accuracy of measurement, can lead to serious artefacts and inaccuracies in the outputs of geomorphometric analysis.

Soil carbon debt of 12,000 years of human land use

Significance Land use and land cover change has resulted in substantial losses of carbon from soils globally, but credible estimates of how much soil carbon has been lost have been difficult to generate. Using a data-driven statistical model and the History Database of the Global Environment v3.2 historic land-use dataset, we estimated that agricultural land uses have resulted in the loss of 133 Pg C from the soil. Importantly, our maps indicate hotspots of soil carbon loss, often associated with major cropping regions and degraded grazing lands, suggesting that there are identifiable regions that should be targets for soil carbon restoration efforts. Human appropriation of land for agriculture has greatly altered the terrestrial carbon balance, creating a large but uncertain carbon debt in soils. Estimating the size and spatial distribution of soil organic carbon (SOC) loss due to land use and land cover change has been difficult but is a critical step in understanding whether SOC sequestration can be an effective climate mitigation strategy. In this study, a machine learning-based model was fitted using a global compilation of SOC data and the History Database of the Global Environment (HYDE) land use data in combination with climatic, landform and lithology covariates. Model results compared favorably with a global compilation of paired plot studies. Projection of this model onto a world without agriculture indicated a global carbon debt due to agriculture of 133 Pg C for the top 2 m of soil, with the rate of loss increasing dramatically in the past 200 years. The HYDE classes “grazing” and “cropland” contributed nearly equally to the loss of SOC. There were higher percent SOC losses on cropland but since more than twice as much land is grazed, slightly higher total losses were found from grazing land. Important spatial patterns of SOC loss were found: Hotspots of SOC loss coincided with some major cropping regions as well as semiarid grazing regions, while other major agricultural zones showed small losses and even net gains in SOC. This analysis has demonstrated that there are identifiable regions which can be targeted for SOC restoration efforts.