BME representation of particulate matter distributions in the

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. D9, PAGES , MAY 16, 2001 BME representation of particulate matter distributions in the state of California on the' basis of uncertain measurements George Christakos, Marc L. Serre, and Jordan L. Kovitz Environmental Modeling Program, Department of Environmental Sciences and Engineering School of Public Health and Center for the Advanced Study of the Environment (CASE) University of North Carolina at Chapel Hill, Chapel Hill, North Carolina Abstract. Maps of temporal and spatial values of annual averages of daily particulate matter (PMm) concentrations were generated throughouthe state of California using uncertain forms of physical data. The PMm estimates were derived in an integrated space/ time domain using the Bayesian maximum entropy (BME) mapping approach of modern spatiotemporal geostatistics. The approach possessesome interesting features which allow an insightful analysis of the PM m space/time distribution. A complete stochastic characterization of the pollutant involves the probability density function of the PM m map, which is the result of a rigorous knowledge-integration process. This process is considerably flexible, it can account for several physical knowledge bases and sources of uncertainty, and it may involve Bayesian or material conditionalization rules. Taking advantage of BME's flexibility, PMm estimates were chosen which offered an appropriate representation of the real distribution in space/time, and a meaningful assessment of the representation accuracy was derived. Depending on the space scales/timescales considered, the PM 0 distributions depicted considerable levels of variability, which may be associated with topographic features, climatic changes, seasonal patterns, and random fluctuations. The importance of integrating soft information available at surrounding sites as well as at the estimation points themselves was discussed. Comparisons were designed which demonstrated the usefulness of the BME-based maps to represent PMm distributions in space/time. Areas were identified where the annual PMm geometric mean reached or exceeded the California standard, which is valuable information for regulatory purposes. 1. Introduction tistics. General knowledge that can be processed by BME Particulate matter (PM) is a mixture of solid particles and includes scientific laws, multiple-point statistics, and empirical relationships, which are knowledge bases that cannot be taken liquid droplets, which poses hazard to humans if it deposits into consideration by most kriging techniques [Olea, 1999]. within the respiratory system [U.S. Environmental Protection Soft data at neighboring points as well as at the estimation Agency (EPA), 1996, also National ambient air quality standards for particulate matter, final draft, Federal Register 40 CFR Part 50, Washington, D.C., 1997; Janssen et al., 1999; Tsai et al., 2000]. In particular, PM o refers to PM in the air that has an aerodynamic diameter d < 10 /am. In practice, reference methods determine PM o by inertial samplers designed to achieve dso (50% collection efficiency) cutpoints of 10 /am aerodynamic diameter. In this work, spatiotemporal PM o concentrations in the state of California are studied using the Bayesian maximum entropy (BME) approach of modern spatiotemporal geostatistics [Christakos, 1990, 1992, 2000]. BME points are incorporated, and both single- and multipoint estimators are allowed. Kriging techniques are based on the minimum mean squared error criterion that may fail in the case of heavy-tailed random fields with large variances [Painter, 1998]. In contrast, BME permits more flexible estimation criteria (e.g., physical knowledge-based probability maximization) that are well-defined even for heavy-tailed fields. In general, BME is a nonlinear estimator which does not impose any constraints on the estimator sought, non-gaussian laws are automatically incorporated, and by taking into account physical models, BME possesses global estimation features. These are signifidistinguishes between a formal part (which involves stochastic cant improvements of the modern geostatistics approach since, concepts, logical axioms, and mathematical techniques) and an as is emphasized by Stein [1999], linear estimators commonly interpretive part (which focuses on real-world interpretations used in spatial statistics can be highly inefficient compared to nonlinear estimators associated with non-gaussian random of the formal concepts and techniques). The BME approach has certain advantages over other methods of mapping space/ fields. Furthermore, modern geostatistics methods can pertime pollutant distributions [see, e.g., Christakos and Setre, form in situations in which the physical conditions considered 2000]. It rigorously takes into consideration many forms of suggest different approaches to knowledge integration (Bayesphysical knowledge, which improve the accuracy and scientific ian conditionalization, material conditionalization, etc.). The space/time distribution of PM o concentrations consticontent of space/time mapping and provide the means to avoid tutes an important indicator of the air quality for which the the circular problem of empirical geostatistics and spatial stastate of California has set some stringent air quality standards Copyright 2001 by the American Geophysical Union. (see, e.g., California Code of Regulations Web site: Paper number 2000JD /01/2000JD ccr.oal.ca.gov/). These air quality standards include a maximum allowable limit of 50/xg/m 3 for the 24-hour average of

2 9718 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS PM o, and a maximum allowable limit of 30 /xg/m 3 for the annual geometric mean of PM o (see California Code of Regulations title 17õ70200). While the 24-hour average standard is aimed at protecting the population against exposure of acute PM o levels, the annual geometric mean standard is useful when one seeks to assess chronic exposure to particulate matter. The annual PM o geometric mean is defined as the natural antilog of the average of a set of natural log-transforme daily PM o measurements [e.g., Hinds, 1982, p. 75; Jacobson, 1999, p. 410]. One aspect that is particular to the annual geometric mean is that the set of measurements used to calculate this 2.1. Spatiotemporal Random Field Representation The natural variable (in this case, a PM m space/time distribution) is represented in terms of the spatiotemporal random field (S/TRF) X(p): p - {X}, fkb. (1) Let us clarify the meaning of the symbols in (1); the vector p = (s, t) defines a point in the space/time domain (s is the spatial position vector, and t denotes time); and the {X} is a set of possible X(p) values at point p. Furthermore, the vector X = (X1, X2,..., Xd) will be used to represent a space/time realization of X(p) values at a set of points Pi (i = 1, 2,..., d). To each one of these realizations, the S/TRF model assigns a probability value by means of the probability density function (pdf) rim(x) defined by fkb(x)dx = prob t=l X, Xt Xi 'q- dx, ; (2) the subscript "KB" denotes that the pdf model has been constructed on the basis of the physical knowledge bases available regarding the PM o distribution (see, also, below). One of the advantages of the S/TRF representation is that it integrates in a mathematically rigorous and physically meaningful way the spatiotemporal structure of the PM o distribution (which is the result of the underlying physicalaws) with the random features of its space/time fluctuations (which are due to local uncertainties, imperfect measurements, etc.). The X(p) representation considers a multiple processing of all physically possible realizations X, which may be diverse yet necessary for a complete description of the phenomenon in the stochastic sense (a "physically possible realization" is a PM o realization which has the same physical characteristics as the actual realization). Air quality studies are generally concerned with the space/ time estimation of the PM o distribution at points p, = (p,,..., p,,), given certain knowledge abouthe entire par- average changes from year to year and from monitoring station to monitoring station (the number of measurements may vary because of missing values, inaccurate readings, shutdown of a monitoring station, etc.). As a result, the calculated values of the annual PM o geometric mean may have varying levels of reliability and should be considered as uncertain (soft) information rather than as accurate (hard) data. In order to address this important issue we used the BME approach in which the annual PM o geometric mean is described by means of probabilistic (soft) data. Space/time correlations of PM o values are modeled in terms of nonseparable covariance functions. By rigorously accounting for the soft data pattern and its consid- ticulate matter field and a set of specific data at points Pdata = erable variability in both space and time, BME yields maps of (P,..., Pro), which includes but is not restricted to observathe annual PM o geometric mean which are more accurate and tions obtained at these points. At the points Pk either we have informative than the maps obtained using classical methods. no observations at all, or the available data are considerably These kind of predictive maps could be essential tools for uncertain (soft) and cannot be used as reliable estimates of the regulatory agencies and environmental decision makers. actual PM o values at these points. In terms of the S/TRF theory, the estimation problem is stated as follows: Let the S/TRF X(p) represent the PM o distribution. One seeks to 2. A Review of the BME Approach derive the pdf (2) that characterizes X(p) in light of the physical knowledge considered. Then, the PM o estimates at points BME is a stochastic approach that belongs to the field of modern spatiotemporal geostatistics. Broadly speaking, BME p,, which are denoted by ih, -- (i:,,-.., i:k,), are derived from the pdf (2) by means of a suitable criterion, that is, space/time analysis and mapping has four major components, criterion as follows: (1) the spatiotemporal random field representation fi B(Xk) ' ilk. of the pollutant under consideration; (2) the physical knowl- (3) edge bases available regarding the pollutant; (3) the formal spatiotemporal analysis and mapping process, which deals with concepts and mathematical formalisms; and (4) the interpretive process, in which the formal steps receive an interpretation or content contributed by the world of experience. Below, we will briefly review these four components of the BME approach in the context of PM1o distribution analysis. The inter- The choice of the criterion is not unique, but it depends on the goals of the study. Also, the criterion must be technically sound and should obey certain logical rules. In some situations, for example, the criterion may seek the most probable estimates; in some other cases, estimates may be sought which optimize some cost function, etc. Whatever the criterion selected, the ested reader is referred to the relevant literature for a more final outcome is a spatiotemporal map which provides a dedetailed exposition of the approach. tailed representation of the PM o distribution in space/time. Single-point analysis focuses on a single scalar estimate Y0, of X, at one point Pk at a time, whereas multipoint analysis deals with the vector estimate i:k = (i:,,"', i:, ) of Xk = (X,..., X, ) at several points p j (j = 1,..., t ) simulta- neously Physical Knowledge Bases A knowledge base (KB) is a systematically organized collection of physical knowledge relevant to the problem at hand, to be invoked by a reasoning process aiming at the solution of the problem. Usually, a logical distinction is made between general KB and specificatory (or case-specific) KB. The general KB, denoted as q3, includes information that is general enough to apply in a variety of situations (e.g., physical laws, scientific theories, statistical moments of any order, multiple-point statistics, and empirical relationships). The specificatory KB, denoted as 9, involves information about the specific situation

3 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS 9719 Table 1. Examples of Soft Data Xsoft Table 2. Examples of #a and ha Functions Equation Xsoft G I = (Itoh +1..., Ira) Prob [Xsoft <- / ] = Fv(/ ) Prob [Xsoft <-/, x k <- / k] = Fv(/, / k) Prob [Xsoft G I] : pv(i) Xhard ' --(,)('mh+l, (,)('1,, Xmh), Xm) t ' f>: Xsoft Equation q3 # h No. (14) (15) (16) (17) BME provides a systematic and rigorous framework for integrating various physical KBs into spatiotemporal analysis. As a model of physical knowledge integration, BME has two distinct parts: a formal part (which is conceptual and mathematically oriented), and an interpretive part (which deals with the content contributed by empirical observations). In this section we concentrate on the mathematical formulation of the BME space/time approach (interpretive issues will be the focus of a following section). The six main steps involved in the formal BME approach are as follows (an epistemic justification of these steps is given by Christakos [2000]): In step 1, in view of the physical KB, q3, available, formulate Mean functions Xi xi Covariance functions (Xi -- Xi)(Xj -- Xj) Cx(Pi' Pi) the corresponding stochastic equations. In many applications these equations are as follows: ha = f dxma p goz(xmap) f 3(Xmap), (6) (e.g., in situ hard data, uncertain observations, topographic contours, and case-specific details). The total physical knowledge available f is the union of q3 and 9. For example, two a=l,...,n major KB commonly used in air pollution studies are as folwhere Xmap = (Xdata, Xk) and, in this case, KB = q3. The ga are lows. properly chosen functions so that they express the q3 base 1. The first is the general KB, q3, consisting of the mean considered, and the total number N of (6) is such that stochasfunctions mx(p) and the covariance functions Cx(p, p') over tic moments are included which involve all the space/time the space/time domain of interest, that is, points Pi Pmap = (Pdata, Pk) of the map. In the special case of (4) above, the g a functions are selected so that the expec- (4) tations ha provide the known space/time mean and covariance q3: { mx(p) Cx(p, p') = [X(p) - f (p)][x(p') - f (p')] functions throughout the space/time domain of interest (see, (the bars denote stochastic expectation). The mean character- e.g., Table 2). Note that by convention, go -- 0o = 1 (norizes trends and systematic structures in space/time, whereas the malization constraint). covariance expresses spatiotemporal PM1o correlations and In step 2, assume that the qg-based pdf has the general form dependencies. 2. The second is the specificatory KB, 9, which consists of f (Xmap) = e ø+ Tg, (7) hard PM1o data (e.g., accurate measurements) at a set of points where g = {g a; a = 1,..., N} is the vector of the g a Pi (i = 1,..., mh) and soft (uncertain) data at another set of points Pi (i = mh -Jr- 1,..., m); that is, functions above, and = {/a a; a = 1,..., N } is a vector of coefficients associated with the g a functions. The coefficients are functions of the space/time coordinates and will be determined in the following step, whereas /Xo is a coefficient that accounts for the normalization constraint oo- 1. such that Xdata: (Xhard, Xsoft). In Table 1, for example, the soft In step 3, substitute (7) into (6) and solve for the coefficients data Xsoft may be expressed in terms of intervals of varying la.. Insert these multipliers back into (7) to find the exact form lengths and probabilistic functions of arbitrary shapes. Note of the qg-based pdf f of the map. that equation (16) in Table 1 corresponds to the case that soft In step 4, in light of the specificatory KB, 9, available, the (probabilistic) data are also available at the estimation points pdff is updated by means of Bayesian conditionalization (bc) themselves. As we shall see below, (16) represents a desirable (see, also, equation (9) below) leading to an integration (or situation that can improve considerably the predictiveness of posterior) pdf for the map as follows: the PM1o maps obtained from BME analysis. The adequate processing of the relevant physical KB, also known as KB conditionalization, is a vital aspect of PMlo space/time modeling and mapping. The KB conditionalization approaches can be divided into two major groups: approaches where KB = f = q3 U 9; A = J'z> de (Xsoft)f õ(xdata) is a based on the Bayesian conditionalization concept (this is, e.g., normalization coefficient that is independent of Xk, and the the case of BME), and these using a non-bayesian condition- forms of the integrand and the domain D depend on the alization rule (e.g., material conditionalization). corresponding hard and soft data, 9, considered (see, e.g., Table 3) Formal Spatiotemporal Analysis and Mapping Process In step 5, from (8), select the appropriate space/time estimates k, depending on the goals of the study (Table 4). The (8) fb [(Xk) = A-1 ID de (Xsøft) f (Xmap), BME mode estimate represents the most probable PM1o realization, whereas the BME mean minimizes the mean square Table 3. Examples of Ese and D Es D Equation (14) a X søft I Equation (15) F (Xsoft) I Equation (16) Fse(Xsoft, Xk) I U I k a These are equations of Table 1.

4 9720 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS Table 4. Examples of Space/Time Estimates ; k where Q = 2Pcg[Xdata = Xdata] and Q2 = 1 - Q /2 - P [Xk --< Xk]' Example Equation (10) is notably differenthan (9), which is the result of Equation No. the fact that, while (10) presupposes some dependence (physical or otherwise) between Xdatand Xk, (9) does not involve any such BME mode k, mode: max ff (X ) (18) dependence (see, also, next section). The shapes of the probabil- BME mean :... = f dxkxk f b xk) (19) ity functions in the right-hand side of (10) depend on the KB available. Assume, for example, that the case-specific data are of the soft form considered in the previou sections. Then, Q = 2A, and Q2 -' l - A - Fm(Xk), where A is a normalization coefficient as before. The different mathematical expressions of the probaestimation error. Several other estimates could be derived so bilities (8) and (10) may lead, in general, to different space/time that an expected objective function (I)(x,) is optimized with estimates of the pollutant distribution. respecto these estimates, where the form of the function (I) as well as the optimization conditions depend on the characteristics of the situation considered. The estimated PM o values 2.5. Interpretive Spatiotemporal Analysis and Mapping Process are used to create spatiotemporal maps, which can be scien- The problem with applying sophisticated space/time maptifically interpreted to provide a useful picture of reality. ping techniques in air pollution situations is often not in the In step 6, because of the inherent randomness of the PM o formal part itself, but in the appropriateness of the application distribution and the inaccuracies of the physical data, one can and the validity of the interpretation. The interpretive part use the BME pdf (8) to obtain an uncertainty assessment goes beyond mathematics into the realms of common sense, associated with :k' For example, a popular measure of estimaphysical knowledge, and empirical observation. Interpretation tion accuracy is the variance rr,lx 2 of [ calculated at each issues are relevant when one needs (1) to establish relationestimation point of the space/time domain. Several other acships between the natural pollutants and the formal mathecuracy parameters, including confidence intervals and confimatics which describe them, (2) to measure and test the formal dence sets, can be also calculated. structure, or (3) to justify certain methodological steps of the Unlike the classical kriging variance that is independent of space/time mapping procedure. For example, the assumptions the data values (and, as a consequence, it has been the subject of the theoretical and computational models need to be tested of some criticism among geostatisticians [see, e.g., Goovaerts, 2 versus the real-world data before the models are used in a case 1997]), the BME variance rr,l depends on the specific data 2 study. When this is not possible at the initial stage of the study, set considered. The rr,lx can provide an adequatestimation one should carefully keep track of the untested assumptions. error assessment when the shape of the pdf is not very com- Also, site-specificontrol of these models is necessary to make plicated. For a Gaussian pdf, for example, the probability that sure that the relevant environmental influences are taken into x, lies in the interval k,rnean q- 1.96rr,lx is 95%. In some account by the model solutions and predictions. other cases where the fix c has a complicated shape, a realistic assessment of the mapping error is achieved using BME con- An interesting interpretation of equations (9) and (10) is fidence sets, etc. obtained in a predictability framework involving some sense of physical causality. Given that the Xdata and Xk realizations 2.4. Conditionalization Rules of Knowledge Integration belong to the same natural field X(p), one could reasonably postulate that underlying the probability equation of the map Equation (8) is based on the Bayesian conditionalization there is a physical connection between Xdata and Xk in an (bc) rule, in which case the bc-based integration probability of integrated space/time domain. In other words, here we are not the map is given by merely talking about separate events without any causal conp[cexk --< Xk] = P [Xk -< XklXdata], (9) nection, but about realizations of the same natural field, which implies that the probability equation should assert a physical where P, [Xl, -< XklXaata] is the classical conditional probabilconnection between Xdata and Xk that is above and beyond the ity. Ideally, the applicability of definition (15) in a particular mere fact that if Xdata occurs, the Xk occurs too and vice versa. situation should be verified either by comparing it to another This physical connection is expressed by means of the general way of evaluating conditional probability or by examining its and specificatory KB relevant to the phenomenon. Then, the consistency with physical and logical arguments that apply in question arises whether a Bayesian or a non-bayesian condithe situation. When this is not possible in practice, the validity tionalization provides the most meaningful characterization of of (15) is often established by means of the meaningful conthe spatiotemporal framework provided by the laws of nature. clusions and useful results it leads to. Non-Bayesian condition- One could argue, for example, that in certain cases in which alizations of the KB in step 4 above are also possible [see the KB available support a strong physical connection, equa- Christakos, 2000]. An example of non-bayesian conditionalization is material biconditionalization (mb), denoted as the truth tion (10) may be able to offer a more meaningful representafunctional Xdata <--> X k. The latter focuses on the cases that the tion of this connection than equation (9) since the latter merely field values Xdata = Xdata and Xk --< Xk are either both true or expresses the probability of the prediction Xk given the data both false. This "both-way dependence" makes sense, given the Xdata, whether or not there is any physical connection between fact that Xdat and Xk belong to the same pollutant field X(p), the two. Nevertheless, the present PM o study focuses on the which implies a physical space/time connection. Then, the mb- derivation of BME estimates of the PM o space/time distribubased integration probability of the map may be defined as tion which use equations (8) and (9). In this case, it is assumed follows: that the physical connection has been taken into consideration P t'[xk--< Xk] = OlP[C[Xk -< Xk] + 02, (10) at the Cg-based steps leading to the formulation of the probability function P, which is subsequently used in equation (9).

5 CHRISTAKOS ET AL.' BME REPRESENTATION OF PM DISTRIBUTIONS N * County Monitoring boundaries Sites 41 N 4O N 39 N 38 N 37 N 36 N 35 N 34 N 33 N 124W 123W 122W 121W 120W 119W 118W 117W 116W 115W 114W Figure 1. Map of the state of California showing the county boundaries and the locations of the monitoring stations. Non-Bayesian maps which are based on equation (10) will be presented in a future publication. In the remaining sections of this manuscripthe interpretive part of BME analysis, in which the formal steps 1-6 above receive a meaningful physical interpretation and lead to informative maps and probabilistic assessments of the real phenomenon, are examined by means of the PM o California data set. 3. Spatiotemporal PM o Distribution Over California 3.1. Data Set The PM o data set used in this analysis is based on daily PMm measurements for an 11-year period ( ), obtained from the California Air Resources Board on CD num- ber PTSD CD (California Ambient Air Quality Data ). The daily PM m measurements in our data set were generally collected every 6 days from 191 monitoring stations distributed throughouthe state of California (Figure 1). Exploratory data analysis revealed that the time series of log-transformed daily PM o data show annual oscillations. This is apparent in Figure 2a, which represents a monitoring station in Imperial County, California, during the time period as typical. The oscillation in Figure 2a suggests that there is a seasonal pattern in the level of PM m. Note that similar oscillations have been observed in a study of PM m distribution over Egypt [Sarra at al., 2000]. The investigation of the California data set also revealed that some monitoring stations had several missing consecutive values (due to, e.g., a shutdown period). Because these missing values may span the entire period of low or high seasonal PMm levels, using the arithmetic mean of the log-transformed PM m measurements may introduce a bias in the calculated geometric mean. Furthermore, the variation in the number of missing values from site to site and year to year leads to calculated averages that have varying levels of reliability. As a consequence, we proposed and applied an alternative approach to calculate the geometric mean, which uses a sinusoidal function to capture the seasonal fluctuation of log-transformed PMxo measurements, and represents the calculated geometric mean in terms of probabilistic (soft) data. This kind of soft data provides a meaningful quantitative representation of the case-specific KB available regarding the California data set Obtaining Soft PM o Data Let Z(p) = Z(s, t) = log PM o(s, t) be the S/TRF representing the log transform of daily PM1o concentrations, where s = (s, s2) denotes the spatial coordinates, t is the time, and log denotes the natural logarithm. Then, the annual Z(p) mean is expressed as Y(p) = Y(s, t)= T - a. Z(s..). where T = days. Note that the Y(p) is also a S/TRF and the exp (Y) is the annual geometric mean of PM m. Soft data for Y at given monitoring stations and calendar years are obtained by modeling the seasonal fluctuations of daily log PMm values in terms of a sinusoid curve, namely, Y + /3 cos [2rr(t - to)/t ]. For illustration, one such curve is plotted in Figure 2a for the calendar year 1995 at a site located in Imperial County. To ensure meaningful regression, monitoring stations and calendar years were selected which meet the com- (ll)

6 9722 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS 6 - (a) I 5200 I I I I I I I I I Time t (days, day 1 = January 1, 1980) (b) _. I I I I I I I I I I $inusoid (c) ' Y Figure 2. Plots showing (a) the log PM o versus time t (dots) and the sinusoid fit Y + /3 cos [2 r(t - to)/ ] during the year 1995 (plain line); (b) the log PM o versus the sinusoid cos [2 r(t - to)/ ] (dots) and the linear fit that yielded the Y estimate (plain line with a slope; the 95% confidence interval for Y is defined by the two horizontal lines); (c) the generated pdffse(y) expressing a probabilistic (soft) datum for Y. The PM o is in/ g/m 3. pleteness criterion of having at least one PM o sample during each one tenth of the year. For each sampling location and calendar year, linear regression of the logarithm of daily average PMm againsthe computed sinusoid cos [2½r(t- to)/t] for all phase shifts to corresponding to midnight, 0600 LT, noon, and 1800 LT throughout the year were computed (where the time t was measured in days so that the annual period T = days). From the resulting linear regressions the one

7 _ CHRISTAKOS ET AL.' BME REPRESENTATION OF PM DISTRIBUTIONS 9723 with greatest r 2 value was chosen. Figure 2b presents the chosen regression result for the same typical site in Imperial cy(r, r) = [Y(s, t) - (s, t)][y(s', t') - (s', t')], (12) County for the year where r = Is - s'l and r = It - t'l. The covariance function The regressed intercept for the chosen linear regression (12) is a second-order statistical moment describing the Y corresponds to the annual log PM m mean, Y, for which we variability in space and time. As its dependence on spatial and seek to establish soft probability densities fv(y). Conventional temporal lags (r and r) reveals, (12) is a spatially isotropic and linear regression theory [Biswas, 1991] holds that confidence temporally stationary covariance. Using the experimental data, intervals on the regressed intercept Y may be derived from a it is possible to calculate covariance values at different spatial Student's t distribution with (n - 2) degrees of freedom and temporal lags, and then to fit a suitable theoretical model (where n is the number of daily log PM m values used in the to the experimental values obtained. The theoretical model regression). Figure 2b depicts the intercept of the linear reselected to fit to the experimental covariance values for the gression and its 95% confidence interval. Figure 2a shows that annual log PM m mean field Y is a nonseparable model that the regressed intercept represents adequately the annual log consists of the superposition of two exponential models with PM o mean and that the associated 95% confidence interval is different spatial and temporal scales, as is shown in the fola reasonable indicator of the uncertainty with which the annual lowing equation: log PM m mean may be regressed from the data. Within the context of BME analysis, we used confidence intervals for the regressed Y in order to build the pdf's fv(y), which quantify the uncertain (soft) data in the particulate matter mapping situation under consideration. An illustration of the fv(y) is Cy(r, r) = Cox exp (r/asx) exp (r/ato + c02 exp (r/as2) exp (7'/at2). (13) given in Figure 2c for the calendar year 1995 at the same The Co and Co2 denote the corresponding variances which typical monitoring site. Similar soft pdf's were developed for offer an assessment of local variabilities; a s and as2 are the all sites and years that included sets of daily PMm measure- spatial ranges of the fluctuations; and a t and at2 are the ments that satisfied the completeness criterion mentioned temporal ranges of the fluctuations. The first covariance comabove. This completeness criterion ensures that monitoring ponent (with parameters Co, as, and ate) serves to model the stations with too many consecutive missing values in a given small-scale structure of the air pollutant fluctuations in space year (which may lead to a biased estimate of the year's aver- and time, whereas the second component (with parameters age) are removed from the list of valid sites for that year. Co2, as2, and at2 ) is used to model the large-scale structure of Following application of the completeness criterion, the num- the fluctuations. The covariance parameters obtained by adber of valid sites used to generate soft pdf's increased from 65 justing the theoretical model to the experimental covariance sites in 1987 to 141 sites in values are as follows: Co , as = 2 km, at = 1 year, Co2 = , as2 = 90 km, and at2 = 15 years. The 3.3. Space/Time Correlation Structure Co and Co2 values correspond, respectively, to small-scale and of PM o Distribution large-scale structure variances of the annual log PM m means The regression procedure described above yielded Y values (with PM m expressed in / g/m3). The large-scale structures (annual log PM o means) at each site and every year with may be linked to large meteorological patterns (such as largesufficient PM o samples. Geostatistical investigation of the re- scale wind patterns, temperature inversions, etc.), whereas the gressed values revealed the existence of a large-scale trend in small-scale structures are probably related to anthropogenic the space/time distribution of the random field Y(s, t). In activities and point source pollution. The covariance model order to account for this trend we used the decomposition plotted in Figure 3a properly illustrates the existing spatial and model, Y(s, t) = (s, t) + X(s, t), where X(s, t) denotes a temporal interdependencies. Figure 3b presents covariance residual field. The latter is a homogenous in space and station- profiles along the spatial and temporal axes. Together with the ary in time field with a zero mean. The mean function (s, t), mean values above, these covariance models are the inputs in also called the mean trend, is modeled in terms of a determin- (6) of the BME approach. Then, by inserting (7) into (6) we istic (nonrandom) function and then removed from the data in can solve for the coefficients /x, (a = 0, 1,..., N) at all order to work with the homogenous/stationary residual X(s, space/time mapping points. It is apparent from these figures t). An investigation of the data showed that a mean trend that the nonseparable covariance model (13) provides a satismodel with additive spatial and temporal components, that is, factory fit to the experimental data over the lags considered. Y(s, t) = ms(s ) + mr(t), is a reasonable choice. The purely Moreover, the experimental covariance values decrease toward spatial component ms(s) was obtained by averaging the re- zero as the spatial and temporal lags increase, which indicates gressed Y values at each monitoring station and then applying that the rather simple mean trend model assumed was adea spatial exponential smoothing function to remove local fluc- quate. Finally, we note that for space/time estimation purposes tuations. Similarly, the purely temporal component mr(t) was the generated maps correspond to calendar years only. That is, obtained by applying a temporal exponential smoothing func- the temporal lags r considered were equal to exactly 0 years, 1 tion to Y averages calculated during each monitoring year. This year, 2 years, etc. (e.g., covariance values at a temporal lag 0 < is, essentially, a moving average approach that has the advan- r < 1 year were not considered in the analysis). tage of being very simple to implement. The approach appeared to yield a satisfactory model for the mean trend, so that 3.4. other more sophisticated approaches were not implemented Space/Time BME Estimation and Confidence Intervals for this analysis. On the basis of the mean and covariance models and the set Next, the space/time variability of the annual log PM o mean of probabilistic (soft) data, we were able to estimate the coefwas described in terms of the (centered) covariance function ficients /. in (6) and (7), and then use (8) to calculate the defined by BME pdf's of the annual log PM o mean at any desired space/

8 9724 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS (a) Time lag, ' (year) 5 ><" Spatialag, r (Km) t i 0.01 (b) spatial lag, r (Km) I -- 0 Experimental Exponential model values o i i i i I Experimental values Exponential model Q (3 O Time lag, ' (year) Figure 3. Plots showing (a) the spatiotemporal covariance model of Y as a function of the spatialag r and the temporal lag r; (b) the covariance model of Y (plain lines) along the spatial axis (r = 0) and along the temporal axis (r = 0). The experimental covariance values are shown in circles. time point in the state of California. For illustration purposes, consider the BME pdf [ calculateduring the year 1997 at three estimation points P1, P2, and P3 (Figure 4a). These points are located on a straight line connecting two monitoring stations near San Diego. The monitoring stations are about 13 km apart and in this study are denoted as MS82 and MS181. Also shown in Figure 4a are the (soft) probability densities fse at these monitoring stations. Note that as the estimation point moves from MS82 to MS181, the BME pdf's [ take on a shape that gradually changes from the (soft) probability density fse of MS82 to the density of MS181. The set of soft data points used to calculate each BME pdf of the map constitutes a local space/time neighborhood; that is, the set includes soft data points obtained during the same calendar year (but at different spatial locations than the estimation point) as well as soft data points during different years. A numerical investigation showed that the number of soft data needed to obtain an accurate BME pdf was rather small. Because of the shielding

9 CHRISTAKOS ET AL.' BME REPRESENTATION OF PM DISTRIBUTIONS 9725 (a) MS82 -- Soft BME probabilistic posterior pdf data MS181 '",,,, P2 o Y (b) 3.5 P2 2.5 / \ / \ // \\ / \\ 68% CI \ \ \ \ \ \ 95% CI "m 99% CI ' 3'.5' -- 4'.5 Figure 4. Plots showing (a) the probabilistic (soft) data at the monitoring stations MS82 and MS181 (plain lines), and the BME pdf at the estimation points P1, P2 and P3 (dashed lines); (b) the BME pdf at point P2 together with the 68, 95, and 99% confidence intervals (CI). effect (resulting from data points located nearby the estima- ance functions. As a consequence, the BME estimator used in tion point [see Olea, 1999]), using the three soft data points this case was the BME mean estimator. exhibiting the highest correlation with the estimation point (as determined by the space/time covariance function) yielded In Figure 5 we give temporal plots of the annual log PM 0 mean Y at a few selected monitoring stations. In the same pdf's [ for the final map that essentially stayed unchanged figure the (soft) probability curves are shown at each calendar when the number of soft data was increased. As a safety factor, in most cases up to five soft data were used (e.g., this is the case of the pdf shown in Figure 4). An advantage of the BME method is that at each estimation point it generates the f-based pdf of the annual log PM 0 mean, which provides a more informative and accurate deyear where soft data were available. Since the air quality standard for the annual PM o geometric mean in California is 30 /xg/m 3, the maximum allowable limit of the annualog PM 0 mean is log (30) units, which is also depicted in Figure 5. On the basis of the soft data, we calculated the BME estimates of the annual log PM 0 mean during each calendar year. scription of the air pollutant concentrations than most classical These estimates are shown in Figure 5 together with the corgeostatistics methods (which merely provide an estimated value of the pollutant). By giving a complete probabilistic description of the pollutant distribution, the BME pdf offers the responding lower and upper bounds of the 68% confidence intervals. In particular, the Y temporal profile of Figure 5a corresponds to the monitoring station numbered 73 in our flexibility of obtaining any estimator desired (such as the mode, the mean, the median, or any quartile), as well as an assessment of the associated estimation accuracy. This flexibility is especially useful when evaluating health risks, which is an issue of significant concern when studying PM 0 concentrations throughout the state of California. In Figure 4b we show the BME pdf at the estimation point P2 as well as the associated 68, 95, and 99% confidence intervals. A reasonable measure of the estimation uncertainty is given by the size of the 68% confidence interval. As is apparent from Figure 4b, in this particular case the pdf may be approximated by a Gaussian distribution, which means that in order to characterize this pdf it is sufficiento calculate the corresponding mean and vari- study. This monitoring station was chosen because it is representative of the characteristi configuration of the soft data in California. While Figure 5a gives a close-up view of the Y temporal profile at the monitoring station 73, in Figure 5b we plot temporal profiles at other representative monitoring stations (numbered 3, 8, and 11) for comparison. The temporal plots above highlight some interesting fea- tures of the BME methodology. As is apparent in Figure 5a, the shapes of the probabilistic (soft) data may vary from one data point to another. This observation illustrates the fact that the annual PM o geometric means can have varying reliability levels during different calendar years. The BME method provides estimates during calendar years where no data are avail-

10 9726 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS (a) Monitoring Station Soft probability data... - Max allowed by CA Law '.. Estimated profile -, x [... 68% Confidence interval i( '"'"..:" 3..:..,...: I I, I I! / Time (years) (b) [ i ii..... Monitoring Station I! I : ', 4..'--' I i. i' ' i. "' ) i[ [.. Monitoring Station I I I I "",i.. Monitoring Station '--: Time (years) Figure 5. Temporal plots of the estimated annualog PMzo mean versus time t at (a) the monitoring station 73 and (b) the monitoring stations 3, 8, and 11. The PMzo is in g/m 3. able by using soft data obtained during previous and preceding years, as well as soft data obtained during the same calendar year at nearby monitoring stations (the totality of these soft data constitutes a local space/time neighborhood). Whenever available, BME can also use soft data at the estimation points. Combining the information available in the space/time neigh- borhood with the soft datum available at the estimation point itself is a unique feature of BME, which can lead to more accurate space/timestimates [Christakos and Setre, 2000]. The size of the 68% confidence interval indicates the uncertainty associated with each estimated value. As was expected, the size of the confidence interval is smaller during calendar years for

11 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS 9727 which soft data were available, and it increases for the calendar years where no soft data are available. It is worth noticing, however, that the size of the confidence intervals did not increase significantly at points where soft data were not available, which implies that at the monitoring sites the space/time estimation uncertainty is small compared to the uncertainty associated with the soft data. This result emphasizes the usefulness of rigorously accounting for soft data in terms of the BME method, which is not possible for most classical geostatistics methods that account only for hard data. Finally, it is interesting to note that the PM o levels have been, generally, decreasing during the period (one may notice a slight increase in 1997, but this is an issue that deserves further investigation) Space/Time Maps of the Annual PM o Geometric Mean tween a minimum of (at the monitoring stations) to a Numerical implementations of the BME method produced accurate estimates for use in spatiotemporal mapping applications, as well as realistic measures of the relevant uncertainties for use in decision making processes and risk assessment. We calculated the BME mean estimates of the annual log PM o mean Y at the space/time nodes of a regular grid covering the entire state of California. Then, by taking the antilog transform maximum of (away from these stations). Thus the distribution of the normalized confidence interval is affected by the space/time correlation pattern between the estimation points and the information points (e.g., geometric configuration of data points within each local neighborhood and relative to the estimation point), as well as by the quality of information (i.e., the uncertainty level of the soft data). Uncertainty maps such of the estimated Y, that is, exp (Y), we were able to construct as that of Figure 7 are useful in risk analysis and decision contour maps of the annual geometric PM o means for each calendar year during the period. The contour maps of the annual geometric PMm means obtained during the years 1988, 1991, 1994, and 1997 are shown in Figures 6a-6d. When calculating the BME estimates for these maps, the local neighborhood around each estimation point included soft data points that were highly correlated to the estimation point. making. For example, on the basis of the uncertainty map we could reconsider the interpretation of the nonattainment areas depicted in Figures 6a-6d. Instead of interpreting them merely as areas of nonattainment of the air quality standard, these plots may be viewed as areas where the probability of not attaining the standard was greater than 0.5, which may acquire a different meaning in terms of risk analysis [e.g., Maynard and These soft data points belonged to a composite space/time Howard, 1999]. neighborhood (rather than to a purely spatial or a purely temporal neighborhood). In other words, estimation at each calendar year used a local space/time neighborhood that included soft data obtained at monitoring stations during the same calendar year as well as during the preceding and following calendar years. As has been demonstrated in previous studies Finally, to obtain a numerical assessment of the mapping improvements gained by BME space/time mapping versus spatial statistics and time series analysis, we compared the annual PM o geometric mean estimates derived from each one of these three different techniques at a set of monitoring stations (where the actual geometric means were available but assumed [e.g., Vyas and Christakos, 1997], using a composite space/time unknown when calculating the estimated values). The estimaneighborhood is an additional factor contributing to the accuracy of the BME maps. Other PM m studies [Christakos and Serre, 2000] have shown that by incorporating soft data in space/time, BME can provide considerably more accurate predictions at control points compared to the predictions obtained tion errors were expressed in terms of the absolute value of the difference between the actual value and the corresponding estimate of the geometric mean at each point. In particular, the estimation errors at 116 monitoring sites during the year 1996 were calculated using: (1) spatially distributed data during the by the classical geostatistics techniques. This issue is also dis- same year (purely spatial analysis); (2) data collected during cussed belo with the help of some numericai comparisons. the previous and following years at the monitoring station Furthermore, in Figures 6a-6d one can identify those areas in the state where the annual PM o geometric mean reached or under consideration (purely temporal analysis); and (3) data obtained by means of the composite space/time analysis (difexceeded the California standard (30/xg/m3). This identifica- ferent space and time neighborhoods were used at each station is useful for regulatory purposes, as it allows to observe how the nonattainment spatial areas have evolved in time. Figure 6a shows that in 1988 the nonattainment areas formed tion). In this way, the following estimation errors were obtained at each site: es (spatial analysis), e r (temporal analysis), and esr (BME space/time analysis). In Table 5 we present the a rather continuous domain that covered most of Southern improvement in estimation error (normalized) gained by California. Then, during the following years 1991, 1994, and space/time mapping over: (1) the purely spatial analysis, that is, 1997 the nonattainment domain reduced in size to become a set of three large disconnected areas (Figure 6d). The first of these areas is located in the central valley area which generally extends from Fresno to Bakersville, the second one encompasses the city of Los Angeles, and the third one is located along the southern borderline with Mexico and encompasses part of San Diego. Together with the BME mean estimates of the annual PM o geometric mean we also computed the size of the associated 68% confidence interval; then, by dividing this size by the BME mean estimate we obtained the normalized size of the 68% confidence interval. As an example, in Figure 7 we plot the spatial distribution of the normalized size for the year This plot is interesting for it describes the uncertainty associated with the map of annual PM o geometric mean of Figure 6d. Figure 7 accounts for the uncertainty due to the soft data (i.e., the error associated with the regression of an annual mean from the set of PM o values at a monitoring station), as well as the uncertainty associated with space/time estimation (i.e., the error introduced when estimation uses data at nearby monitoring stations). Figure 7 shows that, as was expected, the mapping uncertainty is smallest at the monitoring sites and increases as one moves away from these sites. We note that the value of the normalized 68% confidence interval varies be- Ae = (es - esr)/esr, and (2) the purely temporal analysis, that is, Ae 2 = (er - esr)/esr, at six representative monitoring stations. As can be seen from this table, in all monitoring stations considered, the BME space/time analysis led to more accurate estimates of the annual PM o geometric mean than either the spatial statistics or the time series analysis. The same calculations were carried out at all 116 monitoring stations, and the average improvement was found to be 0.32 for Ae and

12 9728 CHRISTAKOS ET AL.' BME REPRESENTATION OF PM DISTRIBUTIONS 42 N (a) [, Monitoring Sites 41 N 40 N 39 N 38 N 37 N 36 N 35 N /') 34 N 33 N 124W 123W 122W 121W 120W 119W 118W 117W 116W 115W 114W 42 N (b) [ * Monitoring Sites] 41 N 40 N 39 N 38 N '\, i-.x\ 37 N 36 N 35 N 34 N 33 N 124W 123W 122W 121W 120W 119W 118W 117W 116W 115W 114W Figure 6. Map of the BME mean estimates of the annual PM o geometric mean for (a) 1988; (b) 1991; (c) 1994; and (d) Plain contour lines indicate areas where the California standard of 30/xg/m has been reached or exceeded; dashed contour lines correspond to concentrations smaller than the standard. The PM o is in/xg/m 3.

13 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS N 41 N / *'x / /'* " t..._,' /'%/ I[' -- ' /.1'\ (c) I * Monitoring Sites I 40 N / 8_. '-. 39 N.5','. ' 38 N \.' N 36 N / 35 N 34 N / / / -' / / /',,? 33 N 124W 123W 122W 121W 120W 119W 118W 117W 116W 115W 114W 42 N. \ ' *./ r' '5 (d) I * Monitoring Sites I 41 N./! '%..._.._./ 4O N 39 N 38 N 37 N 36 N 35 N / / / / / 34 N 33 N 124w 123w 122w 121w 120w 119w 118w 117w 116w 115w 114w Figure 6. (continued) 1.02 for Ae 2. In other words, the estimation error in the case of purely spatial estimation was on the average 32% larger than that of the space/time analysis, and the estimation error for the purely temporal analysis was on the average 102% larger than that of the space/time analysis. These findings demonstrate that the systematic application of the BME approach can lead to significant improvements in the prediction of the annual PM o geometric mean in the state of California.

14 9730 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS 42 N Monitoring Sites I 41 N 4O N 39 N 38 N 37 N 36 N 35 N 34 N 33 N 124W 123W 122W 121W 120W 119W 118W 117W 116W 115W 114W Figure 7. Map of the normalized size of the 68% confidence interval for the BME mean estimates of the annual PM m geometric mean for Conclusions Modern spatiotemporal geostatistics was used to analyze a set of uncertain PM m data available throughout the state of California and during a series of years. In particular, a BME approach was able to account for the considerable variability of the PM m data in both space and time, and to generate accurate and informative PM o maps. On the basis of these maps, areas were identified where the annual PM o geometric mean reached or exceeded the California standard. These predictive maps may prove to be useful tools for regulatory agencies, environmental decision makers, etc. The knowledge-based pdf's of the annual log PM o means generated by BME at estimation points throughouthe state of California provided a more meaningful description of air pollutant concentrations than most classical methods (which Table 5. Effect of Composite Space/Time Analysis in the Estimation of the Annual PM o Geometric Mean Improvements in Estimation Error (Normalized) Obtained by Space/Time Analysis Versus Monitoring Purely Spatial Purely Temporal Station Analysis, Ae Analysis, Ae O merely calculate an estimation value at each point). Moreover, this description could improve as the knowledge base expands and becomes more accurate. Depending on the form of the physical knowledge available, knowledge processing may involve Bayesian as well as material conditional formulations of the probability distribution of the pollutant. By offering a complete probabilistic description of the pollutant distribution, BME allows considerable flexibility in selecting the appropriate estimator form (including the mode, the mean, or any quartile), as well as a meaningful assessment of estimation accuracy. This flexibility is particularly valuable when assessing health risks, which is an issue of significant concern in the state of California. On the basis of the analysis it appears that annual PMm geometric means may have varying reliability levels during different calendar years. This is consistent with the varying shapes of the probabilistic (soft) data deduced for the monitoring stations for respective years. Combining the information available in the space/time neighborhood with soft data available at the estimation point itself is a unique feature of BME which leads to more accurate estimation. On the basis of space/ time confidence intervals it was found that the estimation un- certainty at the monitoring sites was small compared to the soft data uncertainty, which leads toward greater confidence in the results obtained. The composite space/time mapping provided by BME led to improved estimates of the annual PM o geometric mean in the state of California, compared to traditional techniques of spatial kriging and time series. The improvement in estimation accuracy is due to several factors, including (1) the use of a space/time neighborhood in the estimation, (2) the rigorous

15 CHRISTAKOS ET AL.: BME REPRESENTATION OF PM DISTRIBUTIONS 9731 incorporation of soft data into the analysis, and (3) the processing of soft data at the estimation points themselves. The reader is referred to previous works for additional demonstrations (both theoretical and by means of numerical studies) of the significant advantages of space/time BME compared to the methods of spatial statistics and classical geostatistics [Christakos and Serre, 2000; Christakos, 2000, and references therein]. Finally, this study emphasized the need of establishing rigorous procedures for converting uncertain data into forms that can be processed by the BME method. To the best of our knowledge, this is the first particulate matter case study to apply BME where all space/time data are considered as soft. This may be viewed as an important step in demonstrating the capability of the BME method in practical situations of space/ time pollution assessment and mapping. Acknowledgments. The comments and suggestions of the two referees are greatly appreciated. This work has been supported by grants from the National Institute of Environmental Health Sciences (grant P42 ES ), the Army Research Office (grant DAAG ), the Department of Energy (grant DE-FC09-93SR18262), and the U.S. Civilian Research and Development Foundation (project RG2-2236). References Biswas, S., Topics in Statistical Methodology, John Wiley, New York, Christakos, G., A Bayesian/maximum-entropy view to the spatial estimation problem, Math. Geol., 22(7), , Christakos, G., Random Field Models in Earth Sciences, Academic, San Diego, Calif., Christakos, G., Modem Spatiotemporal Geostatistics, Oxford Univ. Press, New York, Christakos, G., and M. L. Serre, BME analysis of spatiotemporal particulate matter distributions in North Carolina, Atmos. Environ., 34, , Goovaerts, P., Geostatistics for Natural Resources Evaluation, Oxford Univ. Press, New York, Hinds, W. C., Aerosol Technology, John Wiley, New York, Jacobson, M. Z., Fundamentals of Atmospheric Modeling, Cambridge Univ. Press, New York, Janssen, L. H. J. M., E. Buringh, A. van der Meulen, and K. D. van den Hout, A method to estimate the distribution of various fractions of PMw in ambient air in the Netherlands, Atmos. Environ., 33, , Maynard, R. L., and C. V. Howard (Eds.), Particulate Matter: Properties and Effects Upon Health, BIOS Sci. Publ., Springer-Verlag, New York, Olea, R. A., Geostatistics for Engineers and Earth Scientists, Kluwer Acad., Norwell, Mass., Painter, S., Numerical method for conditional simulation of Levy random fields, Math. Geol., 30(2), , Serre, M. L., G. Christakos, and J. Howes, Powering an Egyptian air quality information system with the BME space/time analysis toolbox, paper presented at GeoEnv2000 (Third European Conference on Geostatistics for Environmental Applications), Avignon, France, Nov , Stein, M. L., Interpolation of Spatial Data: Some Theory for Kriging, Springer-Verlag, New York, Tsai, F. C., K. R. Smith, N. Vichit-Vadakan, B. D. Ostro, L. G. Chestnut, and N. Kungskulniti, Indoor/outdoor PM w and PM2.5 in Bankok, Thailand, J. Exposure Anal. Environ. Epidemiol., 10, 15-26, U.S. Environmental Protection Agency (EPA), Air quality criteria for particulate matter, 3 vols., Rep. EPA/600/P-95/OOlaF, Washington, D.C., Vyas, V., and G. Christakos, Spatiotemporal analysis and mapping of sulfate deposition data over the conterminous USA, Atmos. Environ., 31, , G. Christakos, J. L. Kovitz, and M. L. Serre, CASE, University of North Carolina at Chapel Hill, Chapel Hill, NC (george_christakos@unc.edu) (Received June 29, 2000; revised September 6, 2000; accepted November 7, 2000.)