Abstract A statistical method based on adjustments of spatial source oriented structure functions to observed concentration values is presented.

Transcription

1 A stochastic receptor model based on source oriented structure functions K.E. Gr0nskei, S.-E. Walker Norwegian Institute for Air Research, P.O. Box 100, n Airway Abstract A statistical method based on adjustments of spatial source oriented structure functions to observed concentration values is presented. Hourly spatial structure functions are determined by hourly source oriented model calculations based on a time dependent numerical algorithm on a km scale. Stationary Gaussian models are used to describe subgrid gradients in concentrations as a result of point and line sources in the area Parameters adjusting location and amplitude of source oriented structure functions and background concentrations to observed hourly concentration values are estimated on an hourly basis. The method is based on Bayesian statistical analysis and uses tentative apriori information about the uncertainties in source oriented model calculations in the area. The improvements in spatial concentration description are evaluated by using measurements of background values and by leaving out measurements from one station in the determination of correction parameters. This is repeated five times using measurements from each station for control. 1 Introduction Hourly calculated and observed concentrations may deviate considerably at receptor points in urban areas (Venkatram (1981), Hanna (1987)). Results of dispersion calculations give at best only an expected spatial concentration distribution. Measurements show actual concentration values at receptor points. Results of dispersion calculations are often inaccurate as a result of inaccurate input data and improper model formulations. In many situations,

2 240 Air Pollution Theory and Simulation however, the observed concentration fluctuations are simulated by the dispersion calculations as a result of varying input data such as emission variations or changes in the wind and dispersion conditions (Gr0nskei, Walker and Gram (1993)). Data on hourly deviations between observed and calculated concentrations may be used to acquire new information about the pollution conditions. Information from one single receptor may be misleading, and better data may be obtained when measurements from several stations and/or of several chemical components are taken into account. A description of the statistical model is given in section two. In section three the model is used to correct hourly calculated concentration values in accordance with observed values from five stations. The intention of the procedure is to combine concentration measurements with calculated spatial concentration distribution to be used for estimating people's or individual's exposure to air pollution (Clench-Aas et al. (1991)). 2 Description of the model Pollution concentrations (C0 are considered to be linearly dependent on the contribution from three different types of sources (Ak,i, Pk,i and P) for a number of receptor points (i = l,2...n). A%,i : The concentration contribution in a receptor point (i) from a group of small area sources such as home heating in an urban area (k). Pk,i : The concentration contribution in a receptor point (i) from one or several point sources (k). (3 : a spatially constant background concentration contribution. In the neighbourhood of point sources the gradients of pollution concentrations are large and a measure of the corresponding uncertainties at the receptor locations (AQ,) may be accounted for by using existing data on emissions from point sources, wind and dispersion conditions, i.e. V Ar = A^,..., A/; Spatial gradient operator Possible local spatial relocation vector for the concentration patterns at receptor points

3 Air Pollution Theory and Simulation 241 Factors of proportionality for the concentration contribution from area sources (cp^) and from point sources (cp^) take into account errors resulting from inaccurate data on emission and dispersion conditions or improper model formulation. A spatial homogeneous concentration value p take into account concentrations from other areas (background concentrations). Tentatively the following expression is used to describe uncertainties in the calculated concentration values. (Eq ' Wp,k ' correction factors for amplitude of calculated concentration distributions. The values A^ and P^ are calculated by the dispersion model. In our study: ~Ak (l,m,q) ' Spatial average concentrations given in three dimensional boxes (l,m,q). The average concentrations ~Pk (/,m,g) are given in a 1 km horizontal grid system (l,m). The vertical resolution (q) is 50 m, 50 m and 100 m. Al (x,,y,,z,) : Subgrid variation in concentration, as a result of line sources. In this study increased concentrations at the main roads were taken into account. P^ (jc,,>>,,z,) : Subgrid variation in concentration, as a result of point sources. In this study this contribution to concentration at receptor points were calculated by a puff/trajectory model. %,,)/,, z, : The co-ordinates of the measuring stations inside the grid boxes. z, = 2.5 m The height of the air inlet at the measuring station. A time-dependent finite difference model in three levels combined with a puff model to account for subgrid effects close to single sources was used to calculate hour-to-hour spatial concentration distributions.

4 242 Air Pollution Theory and Simulation The procedure for dispersion calculations is given by Gr0nskei, Walker, and Gram in The gradients are based on the results of dispersion calculations determining the spatial concentration distributions. ox oy oz, =!,..., Eq. 2.2 v ; ' ^ j ' // ~> ; ~ s i «; ox oy oz. S*.: concentration contribution from subgrid model. local sources calculated by the To account for spatial uncertainty the calculated concentration values within» distances given by Ar, from the receptor point are used. Let Ar=(An,...,AO be the vector of all spatial uncertainty parameters. Bayesian statistical techniques (Berger (1985)), are now applied in order to find optimal correction factors given the observed values Oi, i = l,...,n. Prior density distribution for the correction factors According to the principles of Bayesian analysis (Berger (1985)) a prior density distribution function must be defined for the correction factors accounting for existing information about the uncertainty in the model formulation and of the input data.» The correction factors (p^,cp^,ar,,p defined above are reasonably assumed to be statistically independent, and the prior density function is therefore written as a product of the individual prior density functions for each correction factor:»» ^ (^P A if? ^Pp it 5 ^ ^"> P/ ~ ^i \^P A k j' ^7 \^Pp k /' ^^(^ ^") * ^4 (p) (Eq. 2.3) Likelihood function for observed concentration values given information on emission and dispersion In Bayesian analysis this function describes how "likely" a set of observations is, given results of dispersion calculations and the parameter values.

5 Air Pollution Theory and Simulation 243 For each value of the parameters 9^,cp^,Ar and$, a deviation e, between the observed values 0, and the adjusted result of calculated values C, is defined: 8, =0, -C, (cp^,(p^,ar,p) / =!,...,«(Eq. 2.4) The deviation at each station is caused by an error of measurements and also by lack of model resolution and inaccurate data on emission, wind and dispersion. The observed values 0, have been carefully checked, and the values are believed to be correct, except for minor measurement errors which are assumed to be normally distributed with mean zero. Measurement errors on the different stations are assumed to be statistically independent. It is further assumed tentatively, that the remaining errors in the model, after adjusting the model calculations using the correction factors, are normally distributed with mean zero. Such errors on different stations are also assumed to be statistically independent. The deviations E, defined in Eq. (2.4) can then be viewed as normally distributed variables with mean zero and with standard deviations a^, for i = l,...,n. The following likelihood function for the observed concentrations, given the correction factors is therefore proposed: Posterior density distribution (Eq. 2.5) Given the prior density distribution and the likelihood function, a posterior density distribution function for the correction factors can be developed. The posterior density function describes how probable different combinations of correction factors are, given the apriori information and the actually observed concentrations. According to Bayesian analysis (Berger (1985)), the posterior distribution function is given by: density

6 244 Air Pollution Theory and Simulation Eq. 2.6 where n is the prior density distribution (Eq. 2.3) and L is the likelihood function (Eq. 2.5), and where d is a constant factor so that the PDF function integrates to 1 over the parameter space. -» Optimal values of the parameters <p^,(p^,ar,p, can now be calculated by maximizing the posterior density function. Such estimates of the correction factors are called generalized maximum likelihood estimates and can be viewed as the "most probable" values of the parameters given the apriori information and the actual observations. Using hourly observations O,, i = l,...,n, optimal correction factors are calculated for each hour. The NAG numerical algorithm library is used to determine the optimum values (NAG (1993)). In this paper all parameters are determined simultaneously as an alternative to an earlier method where optimal background concentrations were determined separately on an hourly basis. In the earlier approach (Grenskei, Walker and Gram (1993)) determining weights were given to stations not influenced by local sources. 3 Improvements in estimated spatial concentration distributions In order to evaluate the improvements in modified spatial concentration distributions, data from each of five stations in the Grenland area in Norway are reserved for control. Data from the other stations are used for prescribing the correction parameters cp ^ (p /> and p. The control data were considered in two different ways: 1. Data for situations when the dispersion model was rejected on apriori information (about 15% of the data). 2. The evaluation of the correction method was based on all hourly data. Table 1 shows the following parameters for each station for the period :

7 Air Pollution Theory and Simulation 245 ro,i : Correlation between observed concentrations (OO and results of corrected dispersion calculations (Q). Measurements from all stations have been used to specify the correction parameters rmse o,i ' Root mean square error corresponding to O;. ri j rmsei i*2,i : Correlation between observed concentrations (OO and results of corrected dispersion calculations (C0. For each station the correction parameters have been determined using data from the other four stations. All hourly data have been used. : Root mean square error corresponding to r%j. : Correlation between observed concentrations (OO and results of corrected dispersion calculations (Co,i). For each station the correction parameters have been determined using data from the other four stations. Data for situations when the dispersion model was rejected based on apriori information were taken out of the evaluation of the correction method i.e. eq >30/ig/m3, p >80 ^ o r p p smaller than 0.5 or larger than 1.5. ; Root mean square error corresponding to The values in Table 1 show that the correction procedure improves the correlation between observed and calculated concentrations and decreases the rmse also at stations not included in the definition of correction factors (see roj and r2,i remsegj and rmseo,i). Table 1: Correlation (r) and root mean square errors (rmse) between observed and calculated concentration values. Calculated time series are given for corrected and uncorrected results of dispersion calculations. Klyve G. Stangs gt. Nenset Frednes Uncorr. Corr. Uncorr. Corr. Uncorr. Corr. Uncorr. Con. ro,i mnsenj (ug/m3) '2,i rmse2,j (ug/m3) M,i rmse2,i (ug/m3) Subscript 0: All stations have been used to specify the correction parameters. Subscript 1: All values are used in the evaluation. Subscript 2. When the results of dispersion calculations were rejected based on observed and calculated data were left out of the evaluations. As Uncorr. Coir apriori information

8 246 Air Pollution Theory and Simulation When rejected results of dispersion calculations are excluded from the evaluation, the root mean square error (rmse^i) decreases on all stations except at Aas when this station is used for control. Table 2 shows that average values of corrected time series are closer to average observed values at Nenset and Frednes but not at the other stations. This indicates that maximizing the posterior density function does not render the best correction parameters in all situations. Table 2: «.., M), Average values and standard deviations of observed and calculated concentrations. Parameters are given for each station used as control of the statistical correction procedure. Klyve 25 (30) (23) G. Slangs gt. 35 (43) 35 ] (30) 24,' Nenset 54(81) 37 I (51) Frednes 71 (88) 42 ; (55) 30 j j 37 As 17 (15) (31) 24,' (obs) : Average value of observed concentrations. CT L. : Standard deviation of observed values. Average value of calculated concentrations. CT ^1 Standard deviation of calculated concentrations. Subscript 1: All values are used in the evaluation. Subscript 2. When the results of dispersion calculations were rejected based on apriori information observed calculated data were left out of the evaluations. To avoid misinterpretations it could be an advantage to require that the posterior density function is maximized under the condition that all ei values decreases after the correction of hourly calculated concentration function. It seems that the underestimations of observed concentrations at Nenset and Klyve is of local importance as well as the overestimation of observed concentrations at Aas. A closer study of the data indicates that roads close to the measuring stations cause local concentrations to be higher than the average value in the grid square. This should be accounted for in the analyses before using the data for statistical correction of pollution distributions. A better subgrid model to interpret these observations are important before using the measuring results to characterize air quality in the area. At Aas the observed concentrations in episodes are overestimated as a result of contribution from point sources and from area sources.

9 Air Pollution Theory and Simulation Discussion of results In this analyses systematic modelling errors resulting from errors in the subgrid model have not been accounted for. The correction procedure operating on an urban scale increases hour to hour correlations and reduces rmse. However, when average s-values are larger after the correction than before the correction, it may be a result of misinterpretation of signals as a result of the statistical probability functions. The model should be further developed, and source oriented reasons for systematic deviations between observed and calculated concentrations may be accounted for by improving the subgrid model before the procedure is applied on the urban scale. When the receptor model is validated, the intension is to use it as a diagnostic tool in an hour to hour surveillance system. It may also be used to combine observed concentration with apriori knowledge to improve the spatial concentration distributions for exposure estimations. References Gr0nskei, K.E., Walker, S.-E., and Gram, F. (1993): Evaluation of a model for hourly spatial concentration distributions. Atmos. Environ., Vol. 27B. Grenskei, K.E., Walker, S.-E., and Gram, F. (1990) Short term cohort study of the relationship between health and air pollution in Grenland, Norway. Calculation of spatial concentration distribution based on measurements. The data for evaluation of exposure. Lillestr0m (NILU OR 65/90) Hanna, S.R. (1987) A review of air quality model evaluation procedures. In: Proceedings of the WMO conference on air pollution modelling and its application. Leningrad Vol I. (WMO/TC No. 187) pp NAG (1993) Fortran library reference manual, Mark 16, Numerical Algorithms Group. Venkatram, A. (1981) Model predictability with reference to concentrations associated with point sources. Atmos. Environ., 15,