In the last two decades, increasing attention has been given to air pollution around the world, mainly because of its impact on human health and on the environment. In the Po valley (northern Italy), one of the most troublesome pollutant is PM 10 (particulate matter with an aerodynamic diameter of less than 10 m). In order to assess PM 10 concentration over an entire region, environmental agencies need models to predict PM 10 at unmonitored sites. To choose among possible predictive models and then meet the agencies' request, we focus on the class of Bayesian hierarchical models as they provide a flexible framework for incorporating relevant covariates as well as spatio-temporal interactions. We consider six alternative models for PM 10 concentration in Piemonte region (north-western Po Valley), during the winter season October 2005-March 2006. Our aim is to choose a model that is satisfactory in terms of goodness of fit, interpretability, parsimony, prediction capability and computational costs. In order to support this choice, we propose a comparison approach based on a set of criteria summarized in a table that can be easily communicated to non-statisticians. The comparison findings allow to provide Piemonte environmental agencies with an effective statistical model for building reliable PM 10 concentration maps, equipped with the corresponding uncertainty measure.

SUMMARYAir quality monitoring networks are important tools in management and evaluation of air quality. Classifying monitoring stations via homogeneous clusters allows identification of similarities in pollution, of representative sites, and of spatial patterns. Instead of summaries by statistical indicators, we propose to consider the air pollutant concentrations as functional data. We then classify using functional cluster analysis, where Partitioning Around Medoids (PAM) algorithm is embedded. The proposed data analysis approach is applied to the air quality monitoring network in Piemonte (Northern Italy); we consider the three more critical pollutants: NO 2 , PM 10 , and O 3 .

Functional data featured by a spatial dependence structure occur in many environmental sciences when curves are observed, for example, along time or along depth. Recently, some methods allowing for the prediction of a curve at an unmonitored site have been developed. However, the existing methods do not allow to include in a model exogenous variables that, for example, bring meteorology information in modeling air pollutant concentrations. In order to introduce exogenous variables, potentially observed as curves as well, we propose to extend the so-called kriging with external drift-or regression kriging-to the case of functional data by means of a three-step procedure involving functional modeling for the trend and spatial interpolation of functional residuals. A cross-validation analysis allows to choose smoothing parameters and a preferable kriging predictor for the functional residuals. Our case study considers daily PM 10 concentrations measured from October 2005 to March 2006 by the monitoring network of Piemonte region (Italy), with the trend defined by meteorological time-varying covariates and orographical constant-in-time variables. The performance of the proposed methodology is evaluated by predicting PM 10 concentration curves on 10 validation sites, even with simulated realistic datasets on a larger number of spatial sites. In this application the proposed methodology represents an alternative to spatio-temporal modeling but it can be applied more generally to spatially dependent functional data whose domain is not a time interval.

Abstract. The quantification of measurement uncertainty of atmospheric parameters is a key factor in assessing the uncertainty of global change estimates given by numerical prediction models. One of the critical contributions to the uncertainty budget is related to the collocation mismatch in space and time among observations made at different locations. This is particularly important for vertical atmospheric profiles obtained by radiosondes or lidar.In this paper we propose a statistical modelling approach capable of explaining the relationship between collocation uncertainty and a set of environmental factors, height and distance between imperfectly collocated trajectories. The new statistical approach is based on the heteroskedastic functional regression (HFR) model which extends the standard functional regression approach and allows a natural definition of uncertainty profiles. Along this line, a five-fold decomposition of the total collocation uncertainty is proposed, giving both a profile budget and an integrated column budget.HFR is a data-driven approach valid for any atmospheric parameter, which can be assumed smooth. It is illustrated here by means of the collocation uncertainty analysis of relative humidity from two stations involved in the GCOS reference upper-air network (GRUAN). In this case, 85 % of the total collocation uncertainty is ascribed to reducible environmental error, 11 % to irreducible environmental error, 3.4 % to adjustable bias, 0.1 % to sampling error and 0.2 % to measurement error.

Atmospheric thermodynamic data are gathered by high technology remote instruments such as radiosondes, giving rise to profiles that are usually modelled as functions depending only on height. The radiosonde balloons, however, drift away in the atmosphere resulting in not necessarily vertical but threedimensional (3D) trajectories. To model this kind of functional data, we introduce a "point based" formulation of an heteroskedastic functional regression model that includes a trivariate smooth function and results to be an extension of a previously introduced unidimensional model. Functional coefficients of both the conditional mean and variance are estimated by reformulating the model as a standard generalized additive model and subsequently as a mixed model. This reformulation leads to a double mixed model whose parameters are fitted by using an iterative algorithm that allows to adjust for heteroskedasticity. The proposed modelling approach is applied to describe collocation mismatch when we deal with couples of balloons launched at two