Page 223 - Modern Spatiotemporal Geostatistics

204 Modern Spatiotemporal Geostatistics — Chapter 10

visibility in many parts of the United States. Airborne particles also can cause
damage to paints and building materials.
In the case of the State of North Carolina, the specificatory knowledge base
S includes measurements of ambient PMi 0 which are available from the EPA
(Environmental Protection Agency) Aerometric Information Retrieval System
(AIRS). Averaged (24-hr) air ambient concentrations of PMio were measured
at 47 monitoring stations distributed throughout North Carolina during the
years 1995 and 1996. The PMio data were collected at these monitoring sta-
tions every 6 days (thus, while these two years had 729 calendar days, there
were only 122 measurement days, starting January 3, 1995). The PMio mea-
surements of the EPA data base that were considered accurate were classified
as hard data Xhard- Furthermore, the classification of uncertain measurements
as soft data xSOft was based on expert opinion (e.g., expertise gained in simi-
lar situations, intuition, understanding of the atmospheric processes, and error
biases). This implies that, (a) the missing PMio values were replaced by soft
intervals (Eq. 3.32, p. 85) and (b) at some other points, the shape of the
soft probabilistic data (Eq. 3.33) that were used represented the distribution of
measurement errors around the reported uncertain values of PMio concentra-
tion. At each missing data point, the lower bound of the interval was assumed
equal to zero and the upper bound was assumed equal to the maximum PMio
concentration measured within the local neighborhood surrounding the missing
data point. Using local neighborhoods leads to physically meaningful bounds,
as well as smaller (and, thus, more informative) interval data.
For illustration, in Figure 10.1 we show the PMio measurements collected
every six days at monitoring station no. 13 during the year 1995 and part
of 1996 (Christakos and Serre, 2000b). The time axis is labeled in calendar
days, with day 1 corresponding to January 1, 1995. The first measurement
day was January 3, 1995; hard (exact) data, as well as soft (uncertain) data
of the interval and the probabilistic types are shown in Figure 10.1. The
general knowledge base included the space/time covariance of the particulate
matter distribution. The experimental covariance was first calculated from the
available PMio data for the years 1995 and 1996. A theoretical model was
then obtained by fitting the following model to the experimental values

3 2
where GI = 45, c 2 = 50 [both in (jug/m ) ], a r ,i = 20, a r, 2 = 1,000 (in km),
and a T]i = 1, a T]2 = 5 (in days). The covariance model (Eq. 10.27) is plotted
in Figure 10.2. Equation 10.27 is the sum of two distinct exponential covariance
functions which are separable with respect to space and time. While the first
exponential term addresses short-range interactions with respect to both space
and time, the second term addresses long-range interactions (this behavior is in
agreement with the pattern of curvature shown in Fig. 10.2). The short-range
interaction term has parameters consistent with a metropolitan scale of 20 km