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# Stochastic modeling of atmospheric pollution a spatial time series framework. Part I methodology

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realizations of a series of 1D processes, the standardized residual process \ufffdRK (u, t), t3¹\ufffd at each loca- tion u, using the corresponding covariance model C \ufffdK (\ufffd; \ufffd(u)). Any stochastic simulation algorithm can be used for simulating these residual pro"les, e.g., sequential simulation or simulation via Cholesky decomposition of the covariance matrix (Deutsch and Journel, 1998). Simulated realizations of the spatiotemporal residual "eld RK (u, t) exhibit both temporal and spatial correla- tion. Temporal correlation is imposed via the (location- speci"c) temporal covariance model C \ufffdK (\ufffd; \ufffd*(u)). Spatial correlation is induced via the spatial correlation of the entries of the parameter vector \ufffd*(u), i.e., via the covariance models C \ufffd (h) and C \ufffd\ufffd (h) of the sill and range values of each component TS \ufffdRK \ufffd (u, t), t3¹\ufffd. A non- conditional realization \ufffdr( \ufffd\ufffd\ufffd(u, t), t3¹\ufffd of standardized residual pro"le at any location u, however, cannot be generated independently from a realization \ufffdr( \ufffd\ufffd\ufffd(u\ufffd, t), t3¹\ufffd at another location u\ufffd, even if the corresponding two parameter vectors \ufffdH(u) and \ufffdH(u\ufffd) are spatially corre- lated; the reason is that the random numbers used for simulating the two TS (e.g., Chat"eld, 1996) at locations u and u\ufffd should not be drawn independently at these locations. Additional spatial correlation is induced by an unobservable random "eld of innovations, which is spa- tially correlated and serially uncorrelated (Wikle and Cressie, 1997; Wilks, 1998). If sequential Gaussian simulation, i.e., autoregression, is used for simulating P.C. Kyriakidis, A.G. Journel / Atmospheric Environment 35 (2001) 2331}2337 2335 standardized residual r( -pro"les at each unmonitored lo- cation u, such innovation values are spatially colored and temporally white standard normal deviates; their spatial covariance could be identi"ed to that of the sample standardized r( -residuals after the latter are transformed to a Gaussian distribution. The above innovations "eld is also used for indirectly conditioning simulated r( -pro"les to sample residual r( - data measured at nearby monitoring stations. When simulating a non-conditional realization \ufffdr( \ufffd\ufffd\ufffd(u, t \ufffd ), i" 1,2,¹\ufffd of the standardized residual TS \ufffdRK (u, t), t3¹\ufffd at location u, sample residual r( -data available at nearby monitoring stations are not taken into account; only previously simulated r( -values at u. Consequently, a high simulated residual value r( \ufffd\ufffd\ufffd(u, t \ufffd ) can be generated next to a low sample residual r( (u\ufffd , t\ufffd ) measured at the same time instant t \ufffd at an adjacent station location u\ufffd . In the con- text of sequential Gaussian simulation, indirect condi- tioning to contemporaneous r( -data can be achieved by conditioning innovation realizations to the normal scores (quantiles of a Gaussian distribution) of the con- temporaneous sample r( -data. Such a conditioning step results into simulating high innovation values, and con- sequently high r( -values, nearby high sample r( -data (Kyriakidis, 1999). Simulated realizations \ufffdz\ufffd\ufffd\ufffd(u, t), t3¹\ufffd of concentra- tion pro"les are "nally built by adding the simulated trend \ufffdm\ufffd\ufffd\ufffd(u, t), t3¹\ufffd and residual \ufffdr\ufffd\ufffd\ufffd(u, t), t3¹\ufffd pro"les. Since the spatiotemporal trend and residual components are assumed uncorrelated, see de"nition (1), simulation of the trend "eld is performed independently from that of the residual "eld. Often, concentration re- cords \ufffdz(u\ufffd , t\ufffd ), i"1,2,¹\ufffd\ufffd at the n monitoring station locations u\ufffd , \ufffd3(n) are not complete. In this case, the missing values of the station pro"les are in-"lled by stochastic simulation. The set of S alternative, equally probable, realizations \ufffdz\ufffd\ufffd\ufffd(u, t), u3D, t3¹\ufffd provide a model of uncertainty for the unknown process levels in both space and time, which can be used for risk assess- ment studies or for deciding on additional sampling (Haas, 1998). 3. Discussion A framework for stochastic spatiotemporal modeling of atmospheric pollution has been presented in this pa- per. Observed concentration levels are viewed as a joint realization of a collection of spatially correlated time series, thus capitalizing on the typically better informed time domain. The stochastic spatiotemporal "eld is de- composed into a stochastic trend and a stochastic resid- ual component. Parametric temporal trend models are established at all monitoring stations, independently from one location to another, and their parameters are coregionalized in space. This amounts to mapping the task of simulating a space}time trend "eld to that of generating realizations of a set of cross-correlated RFs modeling the joint spatial distribution of the temporal trend parameters. Joint conditional simulation of such parameters ensures reproduction of important cross-cor- relations between them. Simulated realizations of the spatiotemporal residual component are "nally generated via a series of TS realizations, whose parameters are correlated in space. The proposed methodology is a (distributed para- meter) spatial time series framework merging modeling approaches typically applied independently in a spatial and temporal context; this comprehensive framework allows a consistent modeling of spatial and temporal variability through stochastic simulation. Concentration pro"les obtained from mechanistic model predictions at each location could alternatively be viewed as local trend components (Venkatram, 1988), and the spatiotemporal distribution of the resulting residual pro"les could then be modeled along the lines proposed in this work. The set of alternative concentration realizations provides a model of uncertainty regarding the unknown concen- tration levels in both space and time. Such an uncertainty model could be used in a risk analysis context for study- ing the impact of emission control policies or for design- ing monitoring networks. References Bilonick, R.A., 1985. 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