Cokriging and multivariate kriging methods based on data of a functional random field

Kriging and cokriging and their several related versions are techniques widely known and used in spatial data analysis. However, when the spatial data are functions a bridge between functional data analysis and geostatistics has to be built. I give an overview to cokriging analysis and multivariable spatial prediction to the case where the observations at each sampling location consist of samples of random functions. I extend multivariable geostatistical methods to the functional context. Our cokriging method predicts one variable at a time as in a classical multivariable sense, but considering as auxiliary information curves instead of vectors. I also give an extension of multivariable kriging to the functional context where is defined a predictor of a whole curve based on samples of curves located at a neighborhood of the prediction site. In both cases a non-parametric approach based on basis function expansion is used to estimate the parameters, and I prove that both proposals coincide when using such an approach. A linear model of coregionalization is used to define the spatial dependence among the coefficients of the basis functions, and therefore for estimating the functional parameters. As an illustration the methodological proposals are applied to analyze two real data sets corresponding to average daily temperatures measured at 35 weather stations located in the Canadian Maritime Provinces, and penetration resistance data collected at 32 sampling sites of an experimental plot.Peer ReviewedPostprint (published version

Extending Functional kriging to a multivariate context

Environmental data usually have a spatio-temporal structure; pollutant concentrations, for example, are recorded along time and space. Generalized Additive Models (GAMs) represent a suitable tool to model spatial and/or temporal trends of this kind of data, that can be treated as functional, although they are collected as discrete observations. Frequently, the attention is focused on the prediction of a single pollutant at an unmonitored site and, at this aim, we extend kriging for functional data to a multivariate context by exploiting the correlation with the other pollutants. In particular, we propose two procedures: the first one (FKED) combines the regression of a variable (pollutant), of primary interest on the other variables, with functional kriging of the regression residuals; the second one (FCK) is based on linear unbiased prediction of spatially correlated multivariate random processes. The performance of the two proposed procedures is assessed by cross validation; data recorded during a year (2011) from the monitoring network of the state of California (USA) are considered

Issues of scale for environmental indicators

The value of environmental indicators largely depends upon the spatial and temporal scale that they represent. Environmental indicators are dependent upon data availability and also upon the scale for which statements are required. As these may not match, changes in scales may be necessary. In this paper a geostatistical approach to analyse quantitative environmental indicators has been used. Scales, defined in terms of resolution and procedures, are presented to translate data from one scale to another: upscaling to change from high resolution data towards a low resolution, and downscaling for the inverse process. The study is illustrated with three environmental indicators. The first concerns heavy metals in the environment, where the zinc content is used as the indicator. Initially, data were present at a 1km2 resolution, and were downscaled to 1m2 resolution. High resolution data collected later showed a reasonable correspondence with the downscaled data. Available covariates were also used. The second example is from the Rothamsted’s long-term experiments. Changes in scale are illustrated by simulating reduced data sets from the full data set on grass cuts. A simple regression model related the yield from these condcut to that of the first cut in the cropping season. Reducing data availability (upscaling) resulted in poor estimates of the regression coefficients. The final example is on nitrate surpluses on Danish farms. Data at the field level are upscaled to the farm level, and the dispersion variance indicates differences between different farms. Geostatistical methods were useful to define, change and determine the most appropriate scales for environmental variables in space and in time

Climatic variations in Macerata province (Central Italy)

The province of Macerata, Italy, is a topographically complex region which has been little studied in terms of its temperature and precipitation climatology. Temperature data from 81 weather stations and precipitation data from 55 rain gauges were obtained, and, following quality control procedures, were investigated on the basis of 3 standard periods: 1931-1960, 1961-1990 and 1991-2014. Spatial and temporal variations in precipitation and temperature were analysed on the basis of six topographic variable (altitude, distance from the sea, latitude, distance from the closest river, aspect, and distance from the crest line). Of these, the relationship with altitude showed the strongest correlation. Use of GIS software allowed investigation of the most accurate way to present interpolations of these data and assessment of the differences between the 3 investigated periods. The results of the analyses permit a thorough evaluation of climate change spatially over the last 60 years. Generally, the amount of precipitation is diminished while the temperature is increased across the whole study area, but with significant variations within it. Temperature increased by 2 to 3 °C in the central part of the study area, while near the coast and in the mountains the change is between about 0 and 1 °C, with small decreases focused in the Appennine and foothill belt (-1 to 0 °C). For precipitation, the decrease is fairly uniform across the study area (between about 0-200 mm), but with some isolated areas of strong increase (200-300 mm) and only few parts of territory in which there is an increase of 0-200 mm, mainly in the southern part of the coast, to the south-west and inland immediately behind the coast. The monthly temperature trend is characterized by a constant growth, while for precipitation there is a strong decrease in the amount measured in January, February and October (between 25 and 35 mm on average)

Stochastic estimation of hydraulic transmissivity fields using flow connectivity indicator data

This is the peer reviewed version of the following article: [Freixas, G., D. Fernàndez-Garcia, and X. Sanchez-Vila (2017), Stochastic estimation of hydraulic transmissivity fields using flow connectivity indicator data, Water Resour. Res., 53, 602–618, doi:10.1002/2015WR018507], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/2015WR018507/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.Most methods for hydraulic test interpretation rely on a number of simplified assumptions regarding the homogeneity and isotropy of the underlying porous media. This way, the actual heterogeneity of any natural parameter, such as transmissivity ( math formula), is transferred to the corresponding estimates in a way heavily dependent on the interpretation method used. An example is a long-term pumping test interpreted by means of the Cooper-Jacob method, which implicitly assumes a homogeneous isotropic confined aquifer. The estimates obtained from this method are not local values, but still have a clear physical meaning; the estimated math formula represents a regional-scale effective value, while the log-ratio of the normalized estimated storage coefficient, indicated by math formula, is an indicator of flow connectivity, representative of the scale given by the distance between the pumping and the observation wells. In this work we propose a methodology to use math formula, together with sampled local measurements of transmissivity at selected points, to map the expected value of local math formula values using a technique based on cokriging. Since the interpolation involves two variables measured at different support scales, a critical point is the estimation of the covariance and crosscovariance matrices. The method is applied to a synthetic field displaying statistical anisotropy, showing that the inclusion of connectivity indicators in the estimation method provide maps that effectively display preferential flow pathways, with direct consequences in solute transport.Peer ReviewedPostprint (published version

Building an Environmental Quality Index for a big city: a spatial interpolation approach with DP2

The elaboration of Environmental Quality Indexes (EQI) for big cities is one of the main topics in regional and environmental economics. One of the usual methodological paths consists of generating a single measure as a linear combination of several air contaminants applying Principal Component Analysis (PCA). Then, as a final step, a spatial interpolation is carried out to determine the level of contamination across the city in order to point out the so-called ‘hot points’. In this article, we propose an alternative approach to build an EQI introducing some methodological and practical novelties. From the point of view of the selection of the variables, first we will consider noise -joint to air pollution- as a relevant environmental variable. We also propose to add ‘subjective’ data -available at the census tracts level- to the group of ‘objective’ environmental variables, which are only available at a number of environmental monitoring stations. This combination leads to a mixed environmental index (MEQI), which is more complete and adequate in a socioeconomic context. From the point of view of the computation process, we use kriging to match the monitoring stations registers to the Census data. We follow an inverse process as usual, since it leads to better estimates. In a first step, we krige the environmental variables to the complete surface and finally, we elaborate the environmental index. At last, in order to build the final synthetic index, we do not use Principal Components Analysis -as it is usual in this kind of exercises- but a better one, the Pena Distance method (DP2).Environmental index, Air pollution, Noise, Subjecive expectations, Kriging, Distance indicators