Spatial mapping of ground-based observations of total ozone

Total column ozone variations estimated using ground-based stations provide important independent source of information in addition to satellite-based estimates. This estimation has been vigorously challenged by data inhomogeneity in time and by the irregularity of the spatial distribution of stations, as well as by interruptions in observation records. Furthermore, some stations have calibration issues and thus observations may drift. In this paper we compare the spatial interpolation of ozone levels using the novel stochastic partial differential equation (SPDE) approach with the covariance-based kriging. We show how these new spatial predictions are more accurate, less uncertain and more robust. We construct long-term zonal means to investigate the robustness against the absence of measurements at some stations as well as instruments drifts. We conclude that time series analyzes can benefit from the SPDE approach compared to the covariance-based kriging when stations are missing, but the positive impact of the technique is less pronounced in the case of drifts.


Introduction
The ground-based total column ozone data set is based on Dobson and Brewer spectrophotometer and filter ozonometer observations available from the World Ozone and UV Data Centre (WOUDC) (http://www.woudc.org/).Large longitudinal inhomogeneities in the global ozone distribution and limited spatial coverage of the ground-based network make it difficult to estimate zonal and global total ozone values from station observations directly (Fioletov et al., 2002).The Total Column Ozone (TCO) data set is comprised of the ozone observations from the set of ground-based stations worldwide.Most of those stations are located on land in the Northern Hemisphere, and relatively few stations are over the Southern Hemisphere and oceans.Therefore the spatial distribution of ground-based stations is highly irregular.In addition, durations of operations for each station are different.One of the major difficulties in assessing long-term global total ozone variations is thus data inhomogeneity.Indeed recalibration of ground-based instruments, or interruptions in observation records result in data sets which may have systematic errors that change with time (Fioletov et al., 2002(Fioletov et al., , 2008)).
The TCO data set and corresponding satellite measurements have also been widely discussed in the statistics literature.Some authors have noticed space-time asymmetry in ozone data (Stein, 2005(Stein, , 2007;;Jun andStein, 2007, 2008).The other important feature of TCO data is that the spatial dependence of ozone distributions varies strongly with latitude and weakly with longitude, so that homogeneous models (invariant to all rotations) are clearly unsuitable (Stein, 2007).This is why Jun and Stein (2007) assume that the spatial process driving the TCO data is an axially symmetric process whose first two moments are invariant to rotations about the Earth's axis, and constructed space-time covariance functions on the sphere × time that are weakly stationary with respect to longitude and time for fixed values of latitude.Jun and Stein (2008) further used linear combinations of Legendre polynomials to represent the coefficients of partial differential operators in the covariance functions.These covariance functions produce covariance matrices that are neither of low rank nor sparse for irregularly distributed observations, as it is the case with ground-based stations.Hence, likelihood calculations can thus be difficult in that situation, and we will not follow this approach.
The aim of this article is to apply a new technique, the stochastic partial differential equation (SPDE) approach in spatial statistics (Lindgren et al., 2011) in order to best evalu-Published by Copernicus Publications on behalf of the European Geosciences Union.ate total column ozone spatially from ground-based stations.The SPDE approach has already been applied to regularly spaced ozone satellite data by Bolin and Lindgren (2011) but not to ground-level stations, where gains in accuracy are potentially larger due to the gaps in coverage.Furthermore, we quantify the impact of the improvement of these spatial estimations on the computations of time series over various regions.Finally, the SPDE and covariance-based approaches are also used to calculate monthly zonal mean total ozone values and compare them with zonal means calculated from ground-based data and available from the WOUDC (Bojkov and Fioletov, 1995;Fioletov et al., 2002).
Section 2 gives a brief introduction to the theoretical framework of the SPDE technique, a basic description of the covariance-based kriging and related model selection and diagnostic techniques.Section 3 describes the spatial analysis using TCO data from WOUDC on a monthly, seasonal and annual basis.Furthermore, the estimated results of SPDE and covariance-based kriging are compared with the Total Ozone Mapping Spectrometer (TOMS) satellite data to examine which method yields approximations closer to satellite data.Finally, the long-term zonal mean trends enable us to conduct a sensitivity analysis by removing stations at random and by introducing long-term drifts at some groundbased stations.

Methods
Our main problem is to estimate ozone values at places where it is not observed.Models in spatial statistics that enable this task are usually specified through the covariance function of the latent field.Indeed, in order to assess uncertainties in the spatial interpolation with global coverage, we cannot build models only for the discretely located observations, we need to build an approximation of the entire underlying stochastic process defined on the sphere.We consider statistical models for which the unknown functions are assumed to be realizations of a Gaussian random spatial process.The standard fitting approach, covariance-based kriging, spatially interpolates values as linear combinations of the original observations, and this constitutes the spatial predictor.Not only large data sets can be computationally demanding for a kriging predictor but covariance-based models also struggle to take into consideration in general nonstationarity (i.e., when physical spatial correlations are different across regions) due to the fixed underlying covariance structure.Recently, a different computational approach (for identical underlying spatial covariance models) was introduced by Lindgren et al. (2011), in which random fields are expressed as a weak solution to an SPDE, with explicit links between the parameters of the SPDE and the covariance structure.This approach can deal with large spatial data sets and naturally account for nonstationarity.We review below some of the recent development on the covariance structure modeling on the sphere, with a particular focus on SPDE and covariancebased kriging.Computational implementations of SPDE and covariance-based kriging, with mathematical details, are relegated to Appendix A.
The Matérn covariance function is an advanced covariance structure used to model dependence of spatial data on the plane.On the sphere, Guinness and Fuentes (2013) show that the kriging prediction using the Matérn function with chordal distance outperforms many other types of isotropic covariance functions, both in terms of accuracy and quantification of uncertainty.The shape parameter ν, scale parameter κ and the marginal precision τ 2 parameterize it: where h ∈ R d denotes the difference between any two locations s and s : h = s − s , and K ν is a modified Bessel function of the second kind of order ν>0.

SPDE approach
Let X(s) be the latent field of ozone measurements Y (s) under observation errors (s).Lindgren et al. (2011) use the fact that a random process X(s) on R d with a Matérn covariance function (Eq. 1) is the stationary solution to the SPDE: where W(s) is Gaussian white noise, and is the Laplace operator.The regularity (or smoothness) parameter ν essentially determines the order of differentiability of the fields.
The link between the Matérn covariance (Eq. 1) and the SPDE formulation (Eq.2) is given by α = ν + d/2, which makes explicit the relationship between dimension and regularity for fixed α.Unlike the covariance-based model, the SPDE approach can be easily manipulated on manifolds.On more general manifolds than R d , the direct Matérn representation is not easy to implement, but the SPDE formulation provides a natural generalization, and the ν parameter will keep its meaning as the quantitative measure of regularity.Instead of defining Matérn fields by the covariance function on these manifolds, Lindgren et al. (2011) used the solution of the SPDE as a definition, and it is much easier and flexible to do so.This definition is valid not only on R d but also on general smooth manifolds, such as the sphere.The SPDE approach allows κ and τ to vary with location: where κ(s) and τ (s) are estimated by expanding them in a basis of a function space such as the spherical harmonic basis.We estimate the SPDE approach parameters and supply uncertainty information about the surfaces by using the integrated nested Laplacian approximations (INLA) framework, available as an R package (http://www.r-inla.org/).For latent Gaussian Markov random fields used to efficiently solve SPDEs on triangulations, INLA provides good approximations of posterior densities at a fraction of the cost of Markov chain Monte Carlo.Note that for models with Gaussian data, the calculated densities are for practical purposes exact.

Covariance-based approach
For locations on spherical domain, s i = (L i , l i ), let Y (s i ) denote the ozone measured at station i.Then the kriging representation can be assumed additive with a polynomial model for the spatial trend (universal kriging): where P is a polynomial which is the fixed part of the model, Z is a zero mean, Gaussian stochastic process with an unknown covariance function K and e i are i.i.d.normal errors.
The estimated latent field is then the best linear unbiased estimator (BLUE) of P (s)+Z(s) given the observed data.Note that the Gaussian process in the spatial model, Z(s), can be defined as a realization of the process X(s) from the previous section.
For the covariance-based approach, the hurdle that we are facing is that we have to define a valid (but flexible enough) covariance model and, furthermore, compared to data on the plane, we must employ a distance on the sphere.Two distances are commonly considered.The chordal distance between the two points (L 1 , l 1 ) and (L 2 , l 2 ) on the sphere is given by ch , where r denotes the Earth's radius.The more physically intuitive great circle distance between the two locations is gc Gneiting (2013) pointed out that using the great circle distance in the original Matérn covariance function (Eq. 1) would not work, as it may not yield a valid positive definite covariance function.Therefore in this study we use the chordal distance for covariance-based kriging.The main advantage of using the chordal distance is that it is well defined on spherical domains, as it restricts positive definite covariance functions on R 3 to S 2 (Jun and Stein, 2007).For the ozone data, we specify the Matérn covariance function defined in Eq. ( 1) in the covariance-based approach in order to compare the performance with the SPDE approach for exactly the same covariance function, whereas the model parameters are optimized according to different techniques.
The relevant model diagnostic and selection criteria are described in Appendix B. Note that the smoothness parameter ν is allowed to be selected in the covariance-based kriging, whereas we fix it at ν = 1 in the SPDE approach.

Mapping accuracy
In this section, we produce statistical estimates of monthly ozone maps, using TCO data from WOUDC.We consider TCO data in January 2000 as an illustration, which contain 150 ground-based ozone observations around the world.All ozone values in this article are in Dobson units (DU).We first choose the model setups for both SPDE and covariancebased approaches below.
With the SPDE approach, as the smoothness parameter ν is defined through the relationship α = ν + d/2 (see Appendix A1), we only need to choose the basis expansion order to represent κ and τ .To choose the best maximal order of the spherical harmonic basis, we fitted models with different maximal orders of spherical harmonics for the expansions of κ and τ in order to estimate them thereafter (the default formulation in the R-INLA package).The best fitted model is for a spherical harmonics basis with maximal order 3 since it yielded the lowest generalized cross-validation (GCV) standard deviation (SD) computed in Eq. ( B2), which yields a total of nine parameters to be estimated (four parameters for the expansions of κ and τ and one parameter for the variance).For the covariance-based approach, we need to choose the smoothness parameter in the Matérn covariance function before estimating the univariate quantities κ and τ 2 .The same σ GCV criterion is used to evaluate the model performance.We fit a wide range of values for the smoothing parameter, and ν = 20 minimizes the σ GCV for these ozone data.
To compare the performance of the SPDE approach with the covariance-based kriging, the same σ GCV criterion is used as it balances well predictive power vs. overfitting across methods.We first estimate the optimized model parameters, i.e., κ and τ in a SPDE and ν in a covariancebased model, by the same generalized cross-validation criterion (with different computational methods, the estimated method for the chordal distance covariance model is maximum likelihood estimates), then compare the uncertainty of spatial estimation over the surface.For the prior specifications of the SPDE approach, κ and τ follow log normal priors by default: precision (theta.prior.prec)= 0.1, median for τ (prior.variance.nominal)= 1 and median for κ (prior.range.nominal)depends on the mesh.We use a regression basis of spherical harmonics for both of these parameters; therefore by default the coefficients follow the log normal priors.We do not change the R-INLA default prior settings throughout the analysis.
In order to achieve a better estimation, the monthly mean "norms" (Bojkov and Fioletov, 1995) (or total ozone "climatology") are calculated for each station and each month of the year over the whole period and then subtracted from the data.The norms are used as first approximations to remove the general spatial trend.For each station and for each month, spatial interpolations through SPDE and covariancebased approaches were performed to these deviations.We now compare the results estimated by four strategies described in Table 1.We start an illustration to 6 years of monthly observations.The results of the analysis of ozone data averaged from 2000 to 2005 are shown in Table 2.The averaged number of available stations is denoted by n, and the residual sum of square (RSS) indicates residuals sum of square, which is defined in Eq. (B1).The SPDE approach (strategies C and D) provides a better fit than the covariancebased kriging (strategies A and B) for all months.The effective degree of freedom n eff is higher in the SPDE approach as it is more complex than covariance-based kriging.Higher effective degrees of freedom means smaller values in the σ GCV denominator (n−n eff ) and thus higher values for σ GCV to account for overparametrization.Nevertheless, σ GCV values for the SPDE approach are still all much lower than for covariance-based kriging in all cases.This means that the RSS in the SPDE approach is drastically smaller than the RSS for covariance-based kriging.Thus the SPDE approach supplies a much better fit to the true ozone observation.Note that the covariance-based model is computing a weighted average of the neighborhood values around the location, while the SPDE model is constructed through a triangular mesh.The mesh can be more adaptive and flexible to irregularly distributed observations.In addition, with spatially varying κ and τ , the table indicates that the results could be improved further by applying a nonstationary extension.
Table 2 also reports regional RSSs by dividing the Earth into the Northern Hemisphere (>30 • N), tropics (30 • S-30 • N) and Southern Hemisphere (<30 • S).In general, over half of the ground-based stations are located on the Northern Hemisphere, which gives rise to a higher RSS with respect to other regions as the RSSs are not normalized by the number of observations.RSSs estimated by SPDE and covariancebased kriging show similar patterns across months and across regions.Lower estimation errors can be found in August-October and higher errors occur in December-February.
Figure 1 shows the predicted mean and SD ozone maps by strategies B and D on January 2000.The spatial distributions of ozone means are similar for SPDE and covariancebased kriging in the Northern Hemisphere, but there are differences in the Southern Hemisphere.These differences arise from the asymmetry of available stations in the two hemispheres.The spatial distributions of the SD present similar general patterns for the two techniques.The uncertainties are higher where with fewer stations are available, see the large uncertainty distribution over the South Pacific Ocean.SDs of SPDE predictions are much smaller than the SDs of the covariance-based kriging predictions, especially where fewer observations are available (e.g., mid-Atlantic and South Pacific), but are larger near the North Pole; this may be due to covariance-based kriging underestimating its own uncertainty.

Seasonal and annual effects
Seasonal ozone data are obtained by averaging the corresponding monthly data (but all months of every season must be available to create such seasonal averages).Table 3 shows the seasonal results for different strategies over the years 2000-2005.Their respective highest errors are in December-January-February (DJF) for strategies A, B and C and in June-July-August (JJA) for strategy D. The results are hardly different in March-April-May (MAM) and September-October-November (SON) for strategies C and D, while a significant improvement can be achieved in DJF by applying a nonstationary SPDE model.Moreover, the values of σ GCV and RSS from this seasonal analysis are smaller than the corresponding analysis for each month of the associated season, both for SPDE and covariance-based kriging, and are also closer across the two competing techniques due to additional averaging smoothing out the gains in accuracy.Nevertheless the SPDE approach still provides a better fit than covariance-based kriging in all seasons.Figures 2 and 3 show the seasonal ozone maps by strategy D of means and   SD, respectively.The maps for SD again reveal the higher estimated error in regions without stations.
The annual ozone data are obtained by creating an annual average, which also means that stations with record in-terruptions are not used.Therefore fewer stations are available for this exercise.To see the improvement of the annualbased analysis over seasonally and monthly analyses, Table 4 shows the annual averaged results by month and seasonally, and the results directly obtained by doing the analysis on the annual mean.Although there are fewer stations in annual-based and seasonal-based data than in monthly data, the errors are lower than for monthly data over the years for all strategies and the results directly obtained from annual means yield even lower RSS and σ GCV than the results averaged by seasons due to smooth variation.

Comparison with satellite data
In this section, we assess the match between satellite observations and spatial predictions based on ground-level stations.The TOMS data on monthly averages are obtained from the NASA website (http://ozoneaq.gsfc.nasa.gov/), where we collected the Earth Probe (25 July 1996-31 December 2005) satellite data with grid 1 • × 1.25 • .We calculate the differences over all grid cells and summarize it by the root mean square error (RMSE).Let ŷi be the estimated result from the SPDE or covariance-based kriging on grid i, and let y s i denote the satellite value on grid i, then the  5. Ozone surfaces predicted by the SPDE approach are closer to the satellite data than the predictions from covariance-based kriging over all months.The highest improvement of SPDE over covariance-based kriging is 77.15 % in September and the lowest is 33.60 % in May.Also, in contrast with relatively unstable monthly predictions by covariance-based kriging, SPDE shows more consistency in predictions of monthly ozone variation.
Figures 4 and 5 map the differences of surface predictions of covariance-based and SPDE methods with respect to satellite data over 60 • S-60 • N on January, April, July and October 2000.The differences in October turn out to be much larger than in other months, and therefore a different scale is used.These maps indicate the overestimation (red) and underestimation (blue) with respect to satellite data.Similar patterns in deviations are revealed for both techniques, but SPDE displays less magnitude in the deviations than covariance-based kriging.One noticeable feature is that the pattern of deviation from satellite data is strongly related to the distribution of ground-based stations: for instance, the covariance-based kriging predicted surfaces tend to underestimate the values over the South Pacific Ocean, where few stations are available.The surface predictions by SPDE achieve a clear improvement in predictions compared to covariance-based kriging over areas with less stations, especially in January and October.
The seasonal predicted total ozone is obtained by averaging the corresponding monthly means.We excluded stations that have interruptions in their records.Therefore fewer observations are used to predict seasonal means.The RM-SEs between predicted surface and satellite data are presented in Table 6.In general, seasonal maps should agree better with satellite-based maps than monthly maps.However, fewer observations are used in seasonal predictions and that may trigger high RMSEs in the covariance-based kriging es-

Zonal mean time series analysis
To see how the ozone zonal means change over time over the same stations with different algorithms, we choose the stations which supplied data for at least 25 years between 1979 and 2010.Hence 67 stations are used to construct these zonal mean time series.There is a strong asymmetry between the Southern Hemisphere (6 stations) and the Northern Hemisphere (48 stations); there are 13 stations at the tropics (defined as 30 • S-30 • N).The zonal means were constructed by averaging the estimations obtained from either SPDE or covariance-based kriging over a grid of 1  In order to overcome the underestimation over the South Pacific (see Fig. 4) and achieve a better estimation of longterm global zonal means, the monthly mean norms for each station were subtracted from observations over all the period.Then for each month, spatial interpolation through SPDE and covariance-based kriging were performed to the deviations.
The ozone norms were added back to these deviations in order to compare zonal means over the corresponding belts.
In this study, we compare the zonal mean time series estimated by SPDE and covariance-based kriging with Solar Backscatter Ultraviolet (SBUV) satellite instrument merged ozone data described by Frith et al. (2014)   long-term ozone zonal mean change, we choose 57 stations (39 stations in the Northern Hemisphere, 10 stations in the tropics and 8 stations in the Southern Hemisphere) which provided data over the entire period from 1990 to 2010.We randomly remove 5, 10, 20 and 30 stations out of these set of stations by taking into account the relative weights of the respective regions and estimate the zonal mean trends in each case.The stations removed are randomly chosen by the design in Table 7.
Furthermore, to illustrate possible variations in the sensitivity analysis, we randomly draw 5 sets of stations which need to be removed, labeled cases 1-5.The time series for different zonal mean trends over the latitude band 30-60 • N and 30-60 • S are displayed in Figs. 9 and 10, respectively.The impact of randomly removing stations in the Northern Hemisphere is small even in the case of 30 stations removed (over half of the observations).The Southern Hemisphere is more sensitive to a loss of information because only few stations are located in there.The more stations are removed, the more fluctuations appears in the time series.The main finding is that the long-term effects estimated by SPDE are again more robust than the ones obtained by covariancebased kriging, especially for the case of 30 stations removed (with only two stations left in the Southern Hemisphere).The covariance-based kriging estimated trends can become chaotic for cases 2 and 4, and under this circumstance the total ozone annual cycle become unidentifiable.We use case 1 for further illustration, where both SPDE and covariance-based approaches estimated well with respect to other cases.Figure 11 shows deviations in time series in the annual mean total ozone estimated by SBUV data, WOUDC data set, SPDE and covariance-based kriging.We can see that both SPDE and covariance-based kriging estimate well in the Northern Hemisphere.Covariance-based kriging underestimates means significantly over the tropics and the Southern Hemisphere, while SPDE estimated means are close to SBUV trends.Note that SPDE estimated trends using all stations are closer to SBUV observations than the WOUDC data set at the tropics and Southern Hemisphere overall.
For the second part of sensitivity analysis, we add random long-term drifts into observations due to instrument-related problems.In reality, all observations from a ground-level station are often be biased by 5-10 DU (2-3 %) over a period of several years.For the setting of drifts, let y ij be the ozone ob- servations at station i and time j .We randomly select some stations i and set where a i ∼ N (1, 0.03 2 ) is the slope over every 5-year periods, i.e., one slope factor for 1990-1994, then different drifts for 1995-1999, 2000-2004 and 2005-2010.This setting means that stations were randomly selected and drift values were then randomly generated, but the drifts are fixed for each particular station over every 5 or 6 years.Using the same 57 stations which provided data consistently over 1990 to 2010, the zonal mean trends were estimated with these added drifts over subsets of randomly selected 5, 10, 20 and 30 stations.We consider five sets of random drifts as well to account for possible random vari- ations in the selection process.The time series in each case are shown in Figs. 12 and 13 for the Northern and Southern Hemisphere.Covariance-based kriging estimations hold in the case of over half of stations are drifted, SPDE also displays robustness to drift.We use case 1 as further illustration.Figure 14 shows the annual mean total ozone deviations in time series for SBUV, WOUDC data set, SPDE and covariance-based kriging estimated means when drifts are present.SPDE estimated trends turn out to be superior to covariance-based kriging over the tropics and Southern Hemisphere overall.  is always superior to covariance-based kriging.Furthermore, SPDE zonal means are closer to satellite zonal means than WOUDC zonal means for annual and seasonal averages in the Northern Hemisphere, despite using fewer stations (39) than WOUDC that uses all available stations each month.It shows the remarkable ability of SPDE to interpolate variations over the globe than WOUDC.However, for monthly zonal means, SPDE zonal means are further away from satellite zonal means than WOUDC zonal means.Indeed, there is much less averaging over 1 month, and the SPDE approach can suffer from the lack of stations at some locations to describe particular monthly features that can be more pronounced than seasonal or annual averages.SPDE zonal means over the tropics and Southern Hemisphere in monthly and seasonal analysis suffer greatly as only 10 stations are used in the tropics and 8 in the Southern Hemisphere (whereas WOUDC can use up to 20-30 in the Southern Hemisphere).We expect that for operational purposes, using all the available stations (usually around 130-150 as seen in Table 4, not 57 as done here for convenience) for each month would allow SPDE to clearly outperform WOUDC everywhere at all frequencies, as it does already with fewer stations in the Northern Hemisphere for annual and seasonal averages.Such a data set would constitute an improvement for the study of trends based on ground-level instruments.

Conclusions
In summary, the covariance-based kriging method may perform fairly well globally, but displays misfit locally.The misfits will be averaged out when zonal means are estimated, but they will reveal themselves as relatively higher errors in estimations compared to the SPDE spatial prediction method for mapping.Moreover, both the estimation uncertainty of SPDE and covariance-based methods considerably depend on the location of stations, but the SPDE approach outperforms covariance-based kriging in terms of the uncertainty quantification in areas with few stations.The estimation of trends in time series over the Northern Hemisphere are more accurate for both methods than over the Southern Hemisphere as there is a much denser network of stations than in the Southern Hemisphere.The sensitivity analysis also suggests that the ground-based network can provide a reliable source of data for estimation of the long-term ozone trends.
In the Northern Hemisphere, annual means can be successfully estimated even if half of the available sites is excluded from the analysis.This is not the case for the tropical belt and Southern Hemisphere where the number of sites is very limited.Additional 3 % biases over 5-year intervals at up to the half of the network have a relatively small impact on the estimated zonal means.This suggest that the network can tolerate some systematic errors as long as instruments are calibrated on a regular basis (5 years in our tests) in order to remove such biases.Overall, when stations are removed or have added drift, the SPDE approach shows more robustness than covariance-based kriging and thus, for current observations, should be a preferred method.

Figure 1 .
Figure 1.Surface predicted ozone (DU) mean and SD for SPDE (strategy D) and CBK (strategy B) from January 2000.The red points indicate the locations of stations.

Figure 3 .
Figure 3. Surface predicted ozone (DU) standard deviation from SPDE approach (strategy D) by season from 2000.

Figure 4 .
Figure 4. Total ozone (DU) difference mapping of SPDE and covariance-based kriging (CBK) estimated mean with respect to satellite data from January, April and July 2000.

Figure 5 .
Figure 5.Total ozone (DU) difference mapping of SPDE and covariance-based kriging (CBK) estimated mean with respect to satellite data from October 2000.Estimation in October shows worse prediction than other months; hence it used different scale from Fig. 4.

Figure 6 .Figure 7 .
Figure 6.Ozone mapping from TOMS data in (a) DJF and (b) MAM; global difference mapping of SPDE and covariance-based kriging (CBK) predicted mean with respect to TOMS data in DJF (c and e) and MAM (d and f) 2000.

Table 1 .
Specifications of the different strategies in the spatial estimations.

Table 2 .
Comparison of the generalized cross-validation error (σ GCV ) and the residual sum of square (RSS) for different strategies averaged of 2000-2005 by month.Statistical summaries for Northern Hemisphere (NH), tropics and Southern Hemisphere (SH).

Table 3 .
Generalized cross-validation error (σ GCV ) and residual sum of square (RSS) for different strategies averaged over 2000-2005 by season.

Table 4 .
Generalized cross-validation error (σ GCV ) and residual sum of square (RSS) for different strategies over2000-2005.Monthly and seasonally results are averaged over each year, and annual results are directly estimated using annual means from each station.

Table 7 .
Design of the sensitivity analysis: stations to be removed are randomly selected within each region.

Table 8 .
RMSEs of annual, seasonal and monthly total mean ozone from WOUDC data set, and SPDE and covariance-based kriging (CBK) estimated means (using 57 stations) against SBUV data over 1990-2010.