Cross-Covariance Functions for Multivariate Geostatistics

Continuously indexed datasets with multiple variables have become ubiquitous in the geophysical, ecological, environmental and climate sciences, and pose substantial analysis challenges to scientists and statisticians. For many years, scientists developed models that aimed at capturing the spatial behavior for an individual process; only within the last few decades has it become commonplace to model multiple processes jointly. The key difficulty is in specifying the cross-covariance function, that is, the function responsible for the relationship between distinct variables. Indeed, these cross-covariance functions must be chosen to be consistent with marginal covariance functions in such a way that the second-order structure always yields a nonnegative definite covariance matrix. We review the main approaches to building cross-covariance models, including the linear model of coregionalization, convolution methods, the multivariate Mat\'{e}rn and nonstationary and space-time extensions of these among others. We additionally cover specialized constructions, including those designed for asymmetry, compact support and spherical domains, with a review of physics-constrained models. We illustrate select models on a bivariate regional climate model output example for temperature and pressure, along with a bivariate minimum and maximum temperature observational dataset; we compare models by likelihood value as well as via cross-validation co-kriging studies. The article closes with a discussion of unsolved problems.Comment: Published at http://dx.doi.org/10.1214/14-STS487 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Geostatistics as a tool to improve the natural background level definition: An application in groundwater

The Natural Background Level (NBL), suggested by UE BRIDGE project, is suited for spatially distributed datasets providing a regional value that could be higher than the Threshold Value (TV) set by every country. In hydrogeochemically dis-homogeneous areas, the use of a unique regional NBL, higher than TV, could arise problems to distinguish between natural occurrences and anthropogenic contaminant sources. Hence, the goal of this study is to improve the NBL definition employing a geostatistical approach, which reconstructs the contaminant spatial structure accounting geochemical and hydrogeological relationships. This integrated mapping is fundamental to evaluate the contaminant's distribution impact on the NBL, giving indications to improve it. We decided to test this method on the Drainage Basin of Venice Lagoon (DBVL, NE Italy), where the existing NBL is seven times higher than the TV. This area is notoriously affected by naturally occurring arsenic contamination. An available geochemical dataset collected by 50 piezometers was used to reconstruct the spatial distribution of arsenic in the densely populated area of the DBVL. A cokriging approachwas applied exploiting the geochemical relationships among As, Fe andNH4+. The obtained spatial predictions of arsenic concentrationswere divided into three different zones: i) areas with an As concentration lower than the TV, ii) areas with an As concentration between the TV and the median of the values higher than the TV, and iii) areas with an As concentration higher than the median. Following the BRIDGE suggestions, where enough samples were available, the 90th percentile for each zone was calculated to obtain a local NBL (LNBL). Differently from the original NBL, this local value gives more detailed water quality information accounting the hydrogeological and geochemical setting, and contaminant spatial variation. Hence, the LNBL could give more indications about the distinction between natural occurrence and anthropogenic contamination

Use of geostatistical Bayesian updating to integrate airborne radiometrics and soil geochemistry to improve mapping for mineral exploration

Mineral exploration programmes around the world use data from remote sensing, geophysics, and direct sampling. On a regional scale, the combination of airborne geophysics and ground-based geochemical sampling can aid geological mapping and mineral exploration. Since airborne geophysical and traditional soil-sampling data are generated at different spatial resolutions, they are not immediately comparable due to their different sampling density. Several geostatistical techniques, including indicator cokriging and collocated cokriging, can be used to integrate different types of data into a geostatistical model. However, with increasing numbers of variables the inference of the crosscovariance model required for cokriging can be demanding in terms of effort and computational time. In this paper a Gaussian-based Bayesian updating approach is applied to integrate airborne radiometric data and ground-sampled geochemical soil data to maximize information generated from the soil survey, enabling more accurate geological interpretation for the exploration and development of natural resources. The Bayesian updating technique decomposes the collocated estimate into two models: prior and likelihood models. The prior model is built from primary information and the likelihood model is built from secondary information. The prior model is then updated with the likelihood model to build the final model. The approach allows multiple secondary variables to be simultaneously integrated into the mapping of the primary variable. The Bayesian updating approach is demonstrated using a case study from Northern Ireland. The geostatistical technique was used to improve the resolution of soil geochemistry, at a density of one sample per 2 km2, by integrating more closely measured airborne geophysical data from the GSNI Tellus Survey, measured over a footprint of 65 x 200 m. The directly measured geochemistry data were considered as primary data and the airborne radiometric data were used as secondary data. The approach produced more detailed updated maps and in particular enhanced information on the mapped distributions of zinc, copper, and lead. The enhanced delineation of an elongated northwest/southeast trending zone in the updated maps strengthened the potential for discovering stratabound base metal deposits

Using stochastic space-time models to map extreme precipitation in southern Portugal

The topographic characteristics and spatial climatic diversity are significant in the South of continental Portugal where the rainfall regime is typically Mediterranean. Direct sequential cosimulation is proposed for mapping an extreme precipitation index in southern Portugal using elevation as auxiliary information. The analysed index (R5D) can be considered a flood indicator because it provides a measure of medium-term precipitation total. The methodology accounts for local data variability and incorporates space-time models that allow capturing long-term trends of extreme precipitation, and local changes in the relationship between elevation and extreme precipitation through time. Annual gridded datasets of the flood indicator are produced from 1940 to 1999 on 800 m×800 m grids by using the space-time relationship between elevation and the index. Uncertainty evaluations of the proposed scenarios are also produced for each year. The results indicate that the relationship between elevation and extreme precipitation varies locally and has decreased through time over the study region. In wetter years the flood indicator exhibits the highest values in mountainous regions of the South, while in drier years the spatial pattern of extreme precipitation has much less variability over the study region. The uncertainty of extreme precipitation estimates also varies in time and space, and in earlier decades is strongly dependent on the density of the monitoring stations network. The produced maps will be useful in regional and local studies related to climate change, desertification, land and water resources management, hydrological modelling, and flood mitigation planning

Comparing different approaches - data mining, geostatistic, and deterministic pedology - to assess the frequency of WRB Reference Soil Groups in the Italian soil regions

Estimating frequency of soil classes in map unit is always affected by some degree of uncertainty, especially at small scales, with a larger generalization. The aim of this study was to compare different possible approaches - data mining, geostatistic, deterministic pedology - to assess the frequency of WRB Reference Soil Groups (RSG) in the major Italian soil regions. In the soil map of Italy (Costantini et al., 2012), a list of the first five RSG was reported in each major 10 soil regions. The soil map was produced using the national soil geodatabase, which stored 22,015 analyzed and classified pedons, 1,413 soil typological unit (STU) and a set of auxiliary variables (lithology, land-use, DEM). Other variables were added, to better consider the influence of soil forming factors (slope, soil aridity index, carbon stock, soil inorganic carbon content, clay, sand, geography of soil regions and soil systems) and a grid at 1 km mesh was set up. The traditional deterministic pedology assessed the STU frequency according to the expert judgment presence in every elementary landscape which formed the mapping unit. Different data mining techniques were firstly compared in their ability to predict RSG through auxiliary variables (neural networks, random forests, boosted tree, supported vector machine (SVM)). We selected SVM according to the result of a testing set. A SVM model is a representation of the examples as points in space, mapped so that examples of separate categories are divided by a clear gap that is as wide as possible. The geostatistic algorithm we used was an indicator collocated cokriging. The class values of the auxiliary variables, available at all the points of the grid, were transformed in indicator variables (values 0, 1). A principal component analysis allowed us to select the variables that were able to explain the largest variability, and to correlate each RSG with the first principal component, which explained the 51% of the total variability. The principal component was used as collocated variable. The results were as many probability maps as the estimated WRB classes. They were summed up in a unique map, with the most probable class at each pixel. The first five more frequent RSG resulting from the three methods were compared. The outcomes were validated with a subset of the 10% of the pedons, kept out before the elaborations. The error estimate was produced for each estimated RSG. The first results, obtained in one of the most widespread soil region (plains and low hills of central and southern Italy) showed that the first two frequency classes were the same for all the three methods. The deterministic method differed from the others at the third position, while the statistical methods inverted the third and fourth position. An advantage of the SVM was the possibility to use in the same elaboration numeric and categorical variable, without any previous transformation, which reduced the processing time. A Bayesian validation indicated that the SVM method was as reliable as the indicator collocated cokriging, and better than the deterministic pedological approach

Improving rainfall erosivity estimates using merged TRMM and gauge data

Soil erosion is a global issue that threatens food security and causes environmental degradation. Management of water erosion requires accurate estimates of the spatial and temporal variations in the erosive power of rainfall (erosivity). Rainfall erosivity can be estimated from rain gauge stations and satellites. However, the time series rainfall data that has a high temporal resolution are often unavailable in many areas of the world. Satellite remote sensing allows provision of the continuous gridded estimates of rainfall, yet it is generally characterized by significant bias. Here we present a methodology that merges daily rain gauge measurements and the Tropical Rainfall Measuring Mission (TRMM) 3B42 data using collocated cokriging (ColCOK) to quantify the spatial distribution of rainfall and thereby to estimate rainfall erosivity across China. This study also used block kriging (BK) and TRMM to estimate rainfall and rainfall erosivity. The methodologies are evaluated based on the individual rain gauge stations. The results from the present study generally indicate that the ColCOK technique, in combination with TRMM and gauge data, provides merged rainfall fields with good agreement with rain gauges and with the best accuracy with rainfall erosivity estimates, when compared with BK gauges and TRMM alone

Comparing Different Approaches - Data Mining, Geostatistic, and Deterministic Pedology - to Assess the Frequency of WRB Reference Soil Groups in the Italian Soil Regions

The assessment of class frequency in soil map legends is affected by uncertainty, especially at small scales, where generalization is larger. The aim of this study was to test the hypothesis that data mining or geostatistic techniques provide better estimation of class frequency than traditional deterministic pedology in a national soil map. In the map of Italian soil regions compiled at 1:5,000,000 reference scale, soil classes were the WRB Reference Soil Groups (RSGs). Different data mining techniques, namely neural networks, random forests, boosted tree, classification and regression tree, supported vector machine (SVM), were tested and the last one gave the best RSGs predictions, using selected auxiliary variables and 22,015 classified soil profiles. Given the categorical target variable, the multi-collocated indicator cokriging was the algorithm chosen for the geostatistic approach. The first five more frequent RSGs resulting from the three methods were compared. The outcomes were validated with a Bayesian approach on a subset of 10% of geographically representative profiles, kept out before the elaborations. The most frequent classes were uniformly predicted by the three methods, which instead differentiated notably for the classes with a lower occurrence. The Bayesian validation indicated that the SVM method was as reliable as the multi-collocated indicator cokriging, and both more than the deterministic pedological approach. An advantage of the SVM was the possibility to use numeric and categorical variable in the same elaboration, without any previous transformation, which notably reduced the processing time

Estimating rainfall and water balance over the Okavango River Basin for hydrological applications

A historical database for use in rainfall-runoff modeling of the Okavango River Basin in Southwest Africa is presented. The work has relevance for similar data-sparse regions. The parameters of main concern are rainfall and catchment water balance which are key variables for subsequent studies of the hydrological impacts of development and climate change. Rainfall estimates are based on a combination of in-situ gauges and satellite sources. Rain gauge measurements are most extensive from 1955 to 1972, after which they are drastically reduced due to the Angolan civil war. The sensitivity of the rainfall fields to spatial interpolation techniques and the density of gauges was evaluated. Satellite based rainfall estimates for the basin are developed for the period from 1991 onwards, based on the Tropical Rainfall Measuring Mission (TRMM) and Special Sensor Microwave Imager (SSM/I) data sets. The consistency between the gauges and satellite estimates was considered. A methodology was developed to allow calibration of the rainfall-runoff hydrological model against rain gauge data from 1960-1972, with the prerequisite that the model should be driven by satellite derived rainfall products for the 1990s onwards. With the rain gauge data, addition of a single rainfall station (Longa) in regions where stations earlier were lacking was more important than the chosen interpolation method. Comparison of satellite and gauge rainfall outside the basin indicated that the satellite overestimates rainfall by 20%. A non-linear correction was derived used by fitting the rainfall frequency characteristics to those of the historical rainfall data. This satellite rainfall dataset was found satisfactory when using the Pitman rainfall-runoff model (Hughes et al., this issue). Intensive monitoring in the region is recommended to increase accuracy of the comprehensive satellite rainfall estimate calibration procedur