Predicting Air Quality
Exposures - Kriging
Because of a lack of air quality
data collected at rural and remote locations, it has been necessary
to use interpolation techniques to estimate O3 exposures in nonurban
areas. In the absence of actual O3 data, interpolation techniques
have been applied to the estimation of O3 exposures across the
United States (Reagan, 1984; Lefohn et al., 1987; Knudsen and
Lefohn, 1988). Kriging, a mathematical interpolation technique,
has been used to provide estimates of seasonal O3 values for
the National Crop Loss Assessment Network (NCLAN) for 1978 through
1982 (May-September for each year) (Reagan, 1984). These values,
along with updated values, coupled with exposure-response models,
were used to predict agriculturally related economic benefits
anticipated by lower O3 levels in the United States (Adams et
al., 1985; Adams et al., 1989).
Kriging is a statistical
tool developed by Matheron (1963) and named in honor of D.G.
Krige. Although originally developed specifically for ore reserve
estimation, kriging has been used for other spatial estimation
applications, such as analyzing and modeling air quality data
(Grivet, 1980; Faith and Sheshenski, 1979). At its simplest,
kriging can be thought of as a way to interpolate spatial data
much as an automatic contouring program would. In a more precise
manner, kriging can be defined as a best linear unbiased estimator
of a spatial variable at a particular site or geographic area.
Kriging assigns low weights to distant samples and vice versa,
but also takes into account the relative position of the samples
to each other and the site or area being estimated.
Because of the importance
of the higher hourly average concentrations in eliciting injury
and yield reduction for agricultural crops (U.S. EPA, 1986; 1992),
Lefohn et al. (1992) used the W126 index (see Lefohn and Runeckles,
1987) and the number of hours greater than or equal to 100 ppb
(N100) in its kriging to characterize the O3 exposures that occurred
during the period 1985-1989. Lefohn and co-workers used kriging,
coupled with the Palmer Drought Index, to identify areas of concern
for possible vegetation effects in the southern Appalachian mountains
(Lefohn et al., 1997). The figure above illustrates the use of
kriging to predict O3 exposures in the Southern Appalachian Mountains
Initiative (SAMI) region. In September 1998, A.S.L. &
Associates (Lefohn, 1998) completed a report identifying
"Areas of Concern" for vegetation in the region.
An important question
is: why not use only the W126 integrated exposure index instead
of both the W126 and the N100 indices? The answer is that there
are important concerns expressed in the literature with using
the various cumulative exposure indices in predicting yield loss
for agricultural crops or trees. For example, the same value
of an exposure index may relate to different vegetation responses
(Lefohn and Foley, 1992). Results reported by Yun and Laurence
(1999) showed that the same SUM06 value resulted in very different
foliar injury when exposure regimes with different numbers of
high concentrations were applied. Similarly, Hogsett et al. (1985)
showed that the same SUM07 value resulted in different yield
when exposure regimes, some containing peaks and some without
peaks, were used. Lefohn and Foley (1992) proposed an approach
to get around the problem. They combined the W126 cumulative
exposure index with the N100 index. The N100 index was selected
because the original experimental NCLAN data used to determine
the exposure-response relationships contained numerous occurrences
of hourly average concentrations greater than or equal to 100
ppb. Recently, Davis and Orendovici (2006) reported that they
had successfully developed a model that showed a statistically
significant relationship between indices of ozone symptoms and
the following parameters: plant species, Palmer Drought Severity
Index, and the interaction of the W126 exposure index and the
N100 index. The authors reported that the incidence of ozone
symptoms was most related to the ozone metrics W126 index coupled
to the N100 index.
In 2003, Lefohn
and co-workers used kriging to characterize the number of hours
greater than or equal to 100 ppb (N100) and the W126 exposures
for 2001 in the United States. The figure below illustrates the
N100 and 24-hour W126 values for the summer of 2001.
A brief discussion
follows on why ordinary kriging was selected for the analysis
above rather than another method.
is used when it is desired to estimate a distribution of values
within an area rather than just the mean value of an area. As
the purpose of the study was to estimate the mean values of the
N100 and W126 exposure indices within an area rather than the
distribution of values, Indicator Kriging was not selected.
was not chosen for this study for several reasons. Universal
kriging is used to estimate spatial means when the data have
a strong trend and the trend can be modeled by simple functions.
Trend is scale dependent. For example The University of Montana
Tech sits on the south side of a hill high above the valley Butte,
Montana is located in. A model of the elevations around Montana
Tech would show that a trend in the values exists when you look
north. If you want to model the elevation to the north of Montana
Tech, it can be accurately done with a simple straight line.
At the scale of 1/4 mile the local data has a trend. This trend
does not exist for far however. If you continue north for say
60 miles, you encounter Helena, Montana. Along the way the elevation
rises and falls many times as you cross mountains and valleys.
On the scale of 60 miles, there is no trend in the elevation.
Ozone data may display trends over small geographic areas but
at the scale of the entire United States, there is no trend that
can be modeled by simple functions.
Co-kriging is an
extension of kriging used when estimating a one variable from
two variables. The two co-variables must have a strong relationship
and this relationship must be defined. Use of Co-Kriging requires
the spatial covariance model of each variable and the cross-covariance
model of the variables. The method can be quite difficult to
do because developing the cross-covariance model is quite complicated.
Developing the relationship between the two variables can also
be complicated. Practice in the mining industry limits Co-Kriging
to the case when the variable being estimated is under sampled
with respect to the second variable. If all samples have both
variables, industry has found no benefit gained from the use
Co-Kriging was not
chosen for this study because the O3 indices N100 and W126 are
sampled at each location. Also, there has been no study that
has identified a secondary variable from more sampling sites
that is highly correlated to these exposure indices that can
be used to predict these indices.
was selected for this study based on how well it has performed
on prior years data and because the statistical characteristics
of the data in 2000 and 2003 make Ordinary Kriging the appropriate
choice of estimator. The data displayed no trend at the scale
of the modeling; thus universal kriging was not appropriate.
The covariance models (variogram) exhibited local stationary
and thus, Ordinary Kriging was the appropriate technique to use.
Our research team
has used ordinary kriging to make ground-level O3 models for
the years from 1982 to 2003. While the O3 values vary from year
to year, the statistical character of the data remains remarkably
constant from year to year. The covariance models are similar
in each year and the spatial anisotropy exhibited by the co-variance
models is similar in each year. If you have a need for our technical
expertise, please email
us. We would be pleased to discuss how your specific needs may
match with our capabilities to integrate spatial variability
quantification with unique mapping presentations.
It has been stated
that kriging has its shortcomings. In particular, the current
version of the EPA's Criteria Document (2006) states that kriging
variance underestimates the uncertainty in spatial prediction
and relies on the assumption of spatial covariance isotropy.
Kriging does not require an isotropic covariance model. For example,
the spatial covariance model used in our study was anisotropic.
If the spatial covariance were underestimated or there were an
additional source of variation that was not accounted for in
the model (e.g., temporal variation), then the kriging variance
could be underestimated. However, in our kriging effort, we did
not construct a model that modeled the temporal variation because
seasonal cumulative data were selected for the study. Also, in
many of our current studies, instead of using the Ordinary Kriging
variance, we use the Ordinary Interpolation variance developed
by Yamamoto (2005). Yamamoto derived an error variance for Ordinary
kriging that is conditional to the data values. He has shown
that the Ordinary interpolation variance is a better measure
of accuracy of the kriging estimate. In our current efforts,
the Ordinary kriging programs used are modified to calculate
the new error variance. We believe that the new method used in
our study to determine the interpolation variance is a better
estimate of the error variance than the kriging variance. In
particular, for skewed data, it is believed that the new variance
is a much better estimate of the error variance.
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The identification of ozone exposures that result in vegetation
injury and growth loss for specific species grown in the Southern
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