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Single-Point Analytical Formulations 203

COMMENT 10.1 : Th e result suggested b y Proposition 10.3 can easily b e

extended to incorporate cross-variograms ^ x(Pi, Pj, Pk) between more than

one random field.
Equations 10.14, 10.16, and 10.21 tell us how to combine physical knowl-
edge about the space/time variation of the natural variable, hard, and soft data
to arrive at the most probable estimate. The BME estimate obtained from the
analytical formulations above is, in general, a nonlinear function of the data.
This is in contrast to the classical kriging estimators, which are restricted to
linear combinations of hard data and do not possess a systematic mechanism
to incorporate the variety of knowledge bases available in physical applications
(probabilistic data, scientific laws, etc.). As already mentioned, in addition to
the BMEmode estimates provided by the above analytical expressions (Corollary
10.1 and Propositions 10.2 and 10.3), other BME estimates can be obtained
from the general BME posterior pdf derived in the fundamental Proposition
10.1. A common case is the BMEmean estimate (see Chapter 7, "Other BME
Estimates," p. 147) which is used in the particulate matter study discussed next.

Participate Matter Distributions in

North Carolina

Particulate matter (PM) is the general term used for a mixture of solid particles
and liquid droplets found in the air (e.g., see USEPA, 1997). Particles may
come in a wide range of sizes; "fine" particles have a diameter d < 2.5/zm
(e.g., PM2.s) and "coarser" particles are such that d > 2.5/im (e.g., PMio).
Particles originate from many different stationary and mobile sources as well as
from natural sources. PMio is generally emitted from sources such as vehicles
traveling on unpaved roads, materials handling, and crushing and grinding
operations, as well as windblown dust. Some particles are emitted directly
from their sources (e.g., smokestacks and cars). Also, gases such as SOa,
NO X, and VOC interact with other compounds in the air to form fine particles.
Their chemical and physical compositions vary depending on location, time,
weather, etc. PM has been linked to public health risks by a number of authors
(Milne et al, 1982; Anderson et al, 1992; Dockery and Pope, 1994; USEPA,
1996; Janssen et al., 1999). These risks are associated mainly with PM of
aerodynamic particle sizes of 10 /zm or smaller. PM particles can accumulate
in the respiratory system and are associated with numerous health effects.
Exposure to coarse particles is primarily associated with the aggravation of
respiratory conditions, such as asthma; fine particles are most closely associated
with increased hospital admissions and emergency room visits for heart and lung
disease, increased respiratory symptoms and disease, decreased lung function,
and even premature death. Sensitive groups that appear to be at greatest risk
to such effects include the elderly, individuals with cardiopulmonary disease,
and children. In addition to health problems, PM is a major cause of reduced