NCAR Technical Notes NCAR/TN-530+STR

Ozone and Foliar
Damage Analysis:
NCAR and St. Louis
Will Kaufman
Pulong Ma
Dorit Hammerling
Danica Lombardozzi
National Center for
Atmospheric Research
P. O. Box 3000
Boulder, Colorado
80307-3000
www.ucar.edu
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NCAR/TN-530+STR
NCAR Technical Note
______________________________________________
2016-08
Ozone and Foliar Damage Analysis:
NCAR and St. Louis
Will Kaufman
Student assistant,
National Center for Atmospheric Research, Boulder, CO
Pulong Ma
Department of Mathematical Sciences,
University of Cincinnati, Cincinnati, OH
Dorit Hammerling
Institute for Mathematics and Applied Geosciences,
National Center for Atmospheric Research, Boulder, CO
Danica Lombardozzi
Climate and Global Dynamics Laboratory,
National Center for Atmospheric Research, Boulder, CO
Computational and Information Systems Laboratory (CISL)
Institute for Mathematics Applied to the Geosciences (IMAGe)
Climate and Global Dynamics Laboratory (CGD)
Terrestrial Science Section (TSS)
______________________________________________________
NATIONAL CENTER FOR ATMOSPHERIC RESEARCH
P. O. Box 3000
BOULDER, COLORADO 80307-3000
ISSN Print Edition 2153-2397
ISSN Electronic Edition 2153-2400
Ozone and Foliar Damage Analysis: NCAR and St. Louis
Will Kaufman, Pulong Ma, Dorit Hammerling, Danica Lombardozzi
∗
August 23, 2016
Abstract
Surface level ozone, or tropospheric ozone, is a common pollutant and can cause harm
to a wide variety of plant species. Observing and analyzing ozone concentrations and foliar
damage jointly over time provides the possibility to predict plant damage as a function of ozone
concentration. This technical report is a first step towards this goal and presents an overview
and exploratory analysis of several datasets of surface level ozone concentrations and foliar
damage from the National Center for Atmospheric Research and the St. Louis Science Center.
Keywords: Surface Level Ozone, Ozone Garden, W126, AOT40
∗
Will Kaufman is a student assistant, National Center for Atmospheric Research, PO Box 3000, Boulder CO 30307–
3000 ([email protected]), Pulong Ma is a Graduate Student, Department of Mathematical Sciences, University of
Cincinnati, 2815 Commons Way, Cincinnati, OH 45221–0025 ([email protected]), Dorit Hammerling is a Project
Scientist II, National Center for Atmospheric Research, PO Box 3000, Boulder CO 30307–3000 ([email protected]),
Danica Lombardozzi is a Project Scientist I, National Center for Atmospheric Research, PO Box 3000, Boulder CO
30307–3000 ([email protected]).
1
Contents
1 Introduction
5
2 Description of Ozone Data
5
3 Accumulated Ozone
3.1 W126 Index . . .
3.2 AOTx Index . . .
3.3 Night Ozone . . .
Indices
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4 Plant Damage
4.1 NCAR Mesa Laboratory Ozone Garden
4.1.1 Damage Observations for 2015 .
4.1.2 Damage Observations for 2016 .
4.2 St. Louis Ozone Plant Damage . . . . .
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5 A Bayesian Model for Ozone Plant Damage
21
5.1 The Dirichlet-Multinomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Discussion of the Multinomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Ozone and Meteorological Data
27
7 Future Work
28
2
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.1
3.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.1
5.2
6.1
6.2
Minute-by-minute concentrations of ozone versus time, at the Mesa Lab from June
to September 2015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ozone concentration, recorded every 15 minutes, versus time at St. Louis in 2013. .
Ozone concentration, recorded every 15 minutes, versus time at St. Louis in 2014. .
Histogram of the minute-by-minute ozone concentrations at the Mesa Lab from June
to September 2015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histogram of the ozone concentrations every 15 minutes at St. Louis from 2013. . .
Histogram of the ozone concentrations every 15 minutes at St. Louis from 2014. . .
Ozone concentration from August 17, 2015 to August 20, 2015 at the Mesa Lab. .
Diurnal behavior of ozone concentration at the Mesa Lab, shown as the difference
between the hourly ozone concentration and their respective daily average. . . . . .
Diurnal behavior of ozone concentration at the Mesa Lab, shown as the percent
difference between the hourly values and their respective daily average. . . . . . . .
Daily W126 and nW126 indices for the Mesa Lab 2015 ozone data, calculated by
summing the sigmoidally-weighted 1-hour averages for each day. . . . . . . . . . .
Daily AOT40 and nAOT40 indices for the Mesa Lab 2015 ozone data, calculated by
summing all 1-hour ozone concentration averages over 40 ppb for the entire day. .
Examples of stipples (plant damage) caused by ozone in four different species. . . .
Concentration of ozone versus time at the Mesa Lab in 2015. The first signs of plant
damage occurred at the vertical yellow line. . . . . . . . . . . . . . . . . . . . . . .
Daily W126 indices at the Mesa Lab. The first signs of plant damage occurred at
the vertical yellow line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cumulative daily AOT40 indices at the Mesa Lab. The first signs of plant damage
occurred at the vertical yellow line. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Percent occurrence of damage scores for the cutleaf coneflower, surveyed by an expert
ecologist. The dates shown correspond to when samples were taken. The dashed
vertical line marks the date that the Bootcamp participants gathered plant damage
data (see section 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cumulative W126 indices at St. Louis in 2013, calculated by summing the daily
W126 index, and the damage scores reported for common milkweed during the same
time period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cumulative W126 indices at St. Louis in 2014, calculated by summing the daily
W126 index, and the damage scores reported for common milkweed during the same
time period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cumulative AOT40 indices at St. Louis in 2013, calculated by summing the daily
AOT40 index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cumulative AOT40 indices at St. Louis in 2014, calculated by summing the daily
AOT40 index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Posterior means for each damage score (θ0−4 ) using Jeffreys prior for the multinomial
distribution. The error bars denote three standard deviations. . . . . . . . . . . . .
Marginal prior and posterior density plots of the Dirichlet distribution under different
parameters from top left panel to bottom right panels corresponding to variables
θ1 , θ2 , θ3 , θ4 , and θ5 . The vertical lines denote the means of each marginal posterior
distribution for each θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hourly ozone concentration versus temperature at the Mesa Lab in 2015. . . . . .
Hourly ozone concentration versus relative humidity at the Mesa Lab in 2015. . . .
3
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List of Tables
2.1
4.1
4.2
4.3
4.4
5.1
5.2
Summary of the Mesa Lab and the St. Louis ozone datasets. . . . . . . . . . . . . .
Summary of dates of emergence and first signs of plant damage for three species of
plant in the Mesa Lab ozone garden in 2015. . . . . . . . . . . . . . . . . . . . . .
Plant damage score used by the Mesa Lab, related to the percent of leaf affected. .
Plant damage scores used by the St. Louis garden, related to the percent of leaf
affected. Note that these damage scores are different than those used at the Mesa
Lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cumulative ozone indices at the time of plant damage increase for milkweed. The
ozone indices were from St. Louis in 2013 and 2014. . . . . . . . . . . . . . . . . .
Number of leaves recorded in each damage score category for each bootcamp participant. Sums for each damage score are presented at the bottom. . . . . . . . . . .
Means and variances for θ given different priors: the uniform prior, Jeffreys prior,
and two informative priors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
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6
. 15
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1
Introduction
Ozone is created through a chemical reaction in which UV radiation breaks apart diatomic oxygen to
form two free oxygen atoms. These oxygen atoms then combine with a diatomic oxygen molecule
to form ozone (O3 ). Pollutants in the atmosphere, such as nitrogen oxides and volatile organic
compounds, catalyze ozone production by providing an extra oxygen molecule, thereby raising the
concentration of ozone.
Ozone is present both in the stratosphere and in the troposphere. While the ozone is chemically
equivalent in both places, stratospheric ozone is beneficial as it blocks UV radiation from reaching
the surface of the earth, while tropospheric, or surface-level, ozone can be harmful to humans,
plants, and animals. Ozone can cause rapid oxidation, thereby damaging plant tissue and negatively
affecting respiratory systems in humans and animals.
According to the US Environmental Protection Agency (EPA), ozone concentrations as low as 40
ppb can cause adverse reactions in plants. The European Union (EU) limits ozone concentrations
to a maximum of 60 ppb, while the EPA limits maximum concentrations at 70 ppb, averaged
over 8 hours. The EPA decided to reduce the maximum ozone concentration from 75 to 70 ppb
in 2015, despite the scientific community arguing for a lower concentration of 60 ppb to better
protect human and plant health. The World Health Organization (WHO) recommends an ozone
concentration below 51 ppb.
Plant damage from ozone concentrations is cumulative, and depends on ozone concentrations,
the length of time that the plant is exposed to ozone, and factors that govern plant sensitivity
to ozone like stomatal conductance and antioxidant capacity. As the growing season progresses,
certain plant species develop visual signs of ozone damage, known as stipples. Several gardens,
known as ozone gardens, have been established throughout the US to display the ozone damage
that plants sustain, and contain a variety of plants that are sensitive to high concentrations of
ozone.
This report explores several datasets of ozone concentration and plant damage from two different
ozone gardens in the US. The National Center for Atmospheric Research (NCAR) Mesa Laboratory
has ozone and plant damage data from the 2015 growing season and plant damage data from part
of the 2016 growing season. Ozone and plant damage data are available from the St. Louis Science
Center from the 2013 and 2014 growing seasons. The growing season is generally considered from
May 1 to September 1, though some of the datasets exceed this time frame.
2
Description of Ozone Data
The NCAR Mesa Laboratory (the Mesa Lab) ozone dataset contains minute-by-minute observations
of ozone concentration at the Mesa Lab in Boulder, Colorado, from June to September 2015. The
ozone concentrations are measured in parts per billion (ppb), and vary significantly over the course
of a single day and the observation period (as can be seen in figure 2.1). There were periods of
high ozone concentration, such as the end of June and the end of August, and periods of low
concentration, such as the middle of July and beginning of September.
The St. Louis Science Center has an ozone garden that gathered ozone concentration data at
15 minute intervals, in the 2013 and 2014 growing seasons (from May 1 to October 1). The ozone
concentrations exhibit similar variability as the Mesa Lab data, as can be seen in figures 2.2 and 2.3.
5
Result
Mesa Lab, 2015
St. Louis, 2013
St. Louis, 2014
Number of observations
Observation frequency
Time period
Minimum
1st quartile
Median
Mean
3rd quartile
Maximum
122,303
every minute
1 June-16 Sept. 2015
0.00
37.06
44.86
45.82
54.05
118.67
14,884
every 15 minutes
1 May-1 Oct. 2013
0.00
19.20
29.10
30.32
41.20
90.20
12,503
every 15 minutes
1 May-1 Oct. 2013
0.00
18.50
28.40
30.04
41.00
105.00
Table 2.1: Summary of the Mesa Lab and the St. Louis ozone datasets.
By comparing the values in table 2.1, St. Louis ozone concentrations were comparable between
2013 and 2014 (mean concentrations of 30.32 and 30.04 ppb for 2013 and 2014, respectively). The
Mesa Lab ozone concentrations from 2015 were significantly higher (45.82 ppb mean) than both
years of St. Louis data. The difference in ozone concentrations between these two locations can
be partially explained by the fact that Colorado tends to have more sunny days than St. Louis,
leading to more ultraviolet radiation that catalyzes ozone formation at the Mesa Lab.
Figure 2.1: Minute-by-minute concentrations of ozone versus time, at the Mesa Lab from June to
September 2015.
6
Figure 2.2: Ozone concentration, recorded every 15 minutes, versus time at St. Louis in 2013.
Figure 2.3: Ozone concentration, recorded every 15 minutes, versus time at St. Louis in 2014.
The breaks in the Mesa Lab data seen in figure 2.1, in the middle of June and the end of July,
were due to a required calibration of the sensor and malfunctions of the sensor. There are similar
periods of missing data in the St. Louis dataset from 2014, including a period from mid May to early
June. The 8-hour rolling averages are represented in green and 24-hour averages are represented
7
in brown, with 8-hour values above the EPA maximum concentration indicated by a red color.
The histogram of the 2015 Mesa Lab data (figure 2.4) presents a relatively symmetric distribution with some right-skewedness. In contrast, both years from the St. Louis data exhibit much
more pronounced right-skewed distribution (see figures 2.5 and 2.6).
0.035
Ozone concentration values
EPA Limit
0.020
0.015
0.000
0.005
0.010
Relative frequency
0.025
0.030
WHO Limit
0
20
40
60
80
100
Ozone concentration
Figure 2.4: Histogram of the minute-by-minute ozone concentrations at the Mesa Lab from June
to September 2015.
Ozone concentration values
0.025
EPA limit
0.015
0.010
0.000
0.005
Relative frequency
0.020
WHO limit
0
20
40
60
80
100
Ozone concentration (ppb)
Figure 2.5: Histogram of the ozone concentrations every 15 minutes at St. Louis from 2013.
8
Ozone concentration values
0.025
EPA limit
0.015
0.010
0.000
0.005
Relative frequency
0.020
WHO limit
0
20
40
60
80
100
Ozone concentration (ppb)
Figure 2.6: Histogram of the ozone concentrations every 15 minutes at St. Louis from 2014.
There is a feature found in the St. Louis data where a significant number of observations are
very close to or exactly zero. This feature is especially noticeable in figure 2.6. These low ozone
observations appear to be accurate observations, normally occuring with high relative humidity
and early in the morning, oftentimes between 00:00 and 8:00.
Because of the importance of UV radiation in the formation of ozone, ozone concentrations
exhibit a diurnal behavior as the UV radiation from the sun changes on a daily cycle. For example,
observing ozone concentrations from 17 to 20 August 2015 at the Mesa Lab (see figure 2.7), there
is a clear diurnal behavior, where ozone concentration increases during the day, peaks near sunset,
and decreases as the sun sets and the UV radiation decreases.
9
Figure 2.7: Ozone concentration from August 17, 2015 to August 20, 2015 at the Mesa Lab.
To get a more general understanding of the diurnal cycle, the deviation from the daily mean
concentration was calculated for every hour. As can be seen in figures 2.8 and 2.9, the maximum
ozone concentration occurs at 15:00 in the afternoon and minimum ozone concentration occurs at
8:00 in the morning, indicated by the red and blue colors.
10
20
0
−20
−40
Ozone concentration, absolute difference from daily average (ppb)
40
Diurnal Behavior of Ozone
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Hour of Day
Figure 2.8: Diurnal behavior of ozone concentration at the Mesa Lab, shown as the difference
between the hourly ozone concentration and their respective daily average.
50%
0%
−50%
−100%
Ozone concentration, percent difference from daily average
100%
Diurnal Behavior of Ozone
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
Hour of Day
Figure 2.9: Diurnal behavior of ozone concentration at the Mesa Lab, shown as the percent difference between the hourly values and their respective daily average.
11
3
Accumulated Ozone Indices
Since ozone damage accumulates through time, it is important to calculate ozone metrics that
account for the instantaneous concentrations, as well as the duration of the ozone exposure. Several
ozone indices have been previously developed to quantify cumulative ozone damage. To enforce
environmental regulations in the US, the EPA records ozone concentrations in a given area and
calculates 8-hour rolling averages. Over the course of three years, the fourth highest 8-hour average
is taken for every year, and the average of these values over the three years is calculated. The
EPA then considers this value with respect to the maximum allowable concentration of 70 ppb. In
figure 2.1, the 8-hour averages above the current EPA maximum concentration are marked in red.
Some common indices are the W126 and the AOTx. Many common ozone indices (including
the W126 and AOTx) only consider ozone during “daylight hours” and during the “growing season.” However, these definitions vary among the different indices, leading to inconsistencies in the
resulting indices. When considering existing metrics and developing new metrics, it is important
to consider what “daylight hours” and the “growing season” are, because those vary by location
and species of plant.
3.1
W126 Index
The W126 index is “a weighted index designed to reflect the cumulative exposures that can damage
plants and trees during the consecutive three months in the growing season when daytime ozone
concentrations are the highest and plant growth is most likely to be affected.” 1 W126 stands for
“weighted” concentration, where one of the constants used in the calculation, A, is 126.
The weight for each concentration is calculated using the following equation:
Wi =
1
1 + M · e−A·ci
Where Wi is the weight corresponding to the concentration ci in ppm, M and A are constants equal
to 4403 and 126 ppm-1 , respectively.
The use of the equation above weights higher concentrations of ozone more than lower concentrations. If the concentration is close to zero, then the weight for that concentration is also close
to zero. As the concentration increases the weight also increases, eventually approaching an upper
bound of 1.
Daily indices are calculated by taking one-hour averages of ozone concentration from 8:00 to
19:00, multiplying the concentrations by their respective weights (ci ·Wi ), and summing the weighted
concentrations.
3.2
AOTx Index
The AOTx index stands for “accumulated dose of ozone over a threshold of x ppb”. AOT40,
a specific implementation of AOTx commonly used as an ozone index, can be calculated by the
following equation:
X
AOT 40 =
max(0, ci − 40 ppb)
i
Where each hourly concentration ci is taken and, if greater than 40ppb, added to the daily AOT40
index. The AOT40 index only considers observations from 8:00 to 20:00 each day, and from May
1 to July 31.
1
https://www.epa.gov/air-quality-analysis/ozone-w126-index
12
The AOT40 index is finally determined by summing all the daily indices (from 8:00 to 20:00)
from May 1 to July 31 (the three month growing season). For the Mesa Lab ozone data, the total
sum of the AOT40 index is 5130 ppm hr. To account for missing observations (such as the two-week
period in June), AOT40 is adjusted according to the following equation:
AOT 40estimate = AOT 40measured ·
Ntotal
Nobserved
Where Ntotal is the total possible observations that could have been made, and Nobserved is the total
number of observations actually made. For AOT40, Ntotal is the number of hourly observations per
day (12) multiplied by the number of days between May 1 and July 31 (91), for a total of 1,092
total possible observations.
So the AOT40 estimate from the Mesa Lab ozone data is 5130 · 91·12
505 = 11, 220 ppb hr.
3.3
Night Ozone
New research has pointed to the possibility that some plants keep their stomata open at night,
allowing for further ozone damage. The above indices only consider daytime concentrations (from
8:00 to 19:00 for the W126 and 8:00 to 20:00 for the AOT40) with the previous understanding that
plants would not sustain as much or any ozone damage at night.
In order to account for the nighttime ozone exposure, the W126 and AOT40 indices are recalculated, considering all 24 hours of the day. The indices adjusted for nighttime ozone exposure are
denoted as nW126 and nAOT40, respectively.
Summing the nAOT40 daily values and adjusting for missing data yields the final nAOT40
index of 15,018 ppb hr for the Mesa Lab ozone data.
Below is a plot of the daily W126 and nW126 values, calculated using both the standard method
and the new method, considering all ozone concentrations including those during night time.
500
W126 Daily Indices vs. Time
W126
300
200
0
100
W126 Daily Index (ppb hr)
400
nW126
Jun
Jul
Aug
Sep
Time
Figure 3.1: Daily W126 and nW126 indices for the Mesa Lab 2015 ozone data, calculated by
summing the sigmoidally-weighted 1-hour averages for each day.
13
Similarly, below is a plot of the daily AOT40 values and nAOT40 values.
500
AOT40 Daily Indices vs. Time
AOT40
300
200
0
100
AOT40 Daily Index (ppb hr)
400
nAOT40
Jun
Jul
Aug
Time
Figure 3.2: Daily AOT40 and nAOT40 indices for the Mesa Lab 2015 ozone data, calculated by
summing all 1-hour ozone concentration averages over 40 ppb for the entire day.
4
Plant Damage
4.1
NCAR Mesa Laboratory Ozone Garden
The Mesa Lab has two ozone gardens (at the main entrance and in the cafeteria patio) that contain
ozone-sensitive plants.2 During the growing season, these plants develop visible signs of damage
caused by high concentrations of surface-level ozone.
4.1.1
Damage Observations for 2015
In addition to observing ozone concentrations, three different species of plant were observed for
signs of ozone damage in the form of stipples. Examples of stipples can be seen for four different
species of plant in figure 4.1. When the first signs of damage were observable, the date was recorded.
Table 4.1 summarizes the plant species, the approximate date of emergence, and the date of first
observable signs of damage.
These dates of first signs of ozone damage are marked by a yellow line in the following plots of
ozone concentration, the W126 index, and the AOT40 index.
2
More information can be found at Danica Lombardozzi’s blog post here.
14
relatively high for extended periods of time. Visible ozone
injury on broadleaf plants starts as stipple.
They are typically separate and uniform in
size, but may merge and cover much of the
Additional symptoms such as leaf yellowing or patches
of tissue death can occur as ozone damage accumulates
and becomes more severe.
leaf surface as ozone exposure continues.
Stipples: Here are examples from our garden in past years
Snap bean
Coneflower
Potato
Milkweed
What’s the difference between ozone damage and other leaf damage?
Be
careful4.1:
in your
identification
of ozone
injury!damage)
Insects and
other by
diseases
symptoms
that are
Figure
Examples
of stipples
(plant
caused
ozonecan
in cause
four different
species.
often mistaken for ozone injury. To distinguish between ozone damage and other kinds of leaf damage,
keep the following points in mind:
•
Date
Date of First
Ozone injury only occurs between thePlant
leaf veins,
not Emerged
on the veins themselves.
•
Most ozone injury occurs
only on
the top of the leaf.
Cutleaf
Coneflower
Late
•
Older leaves
sensitive
plants will
show the mostMid
damage.
SnapofBean
(sensitive
genotype)
June
•
Damage
May
22 July 2015
23 July 2015
var. LaWith
Chipper
Mid June
24 July
2015
Ozone damage Potato,
starts as stipple.
extended exposure
to ozone, stipple
can mix
with leaf
yellowing and patches of tissue death, making markings less distinct and more difficult to diagnose.
Table 4.1: Summary of dates of emergence and first signs of plant damage for three species of plant
in the Mesa Lab ozone garden in 2015.
Figure 4.2: Concentration of ozone versus time at the Mesa Lab in 2015. The first signs of plant
damage occurred at the vertical yellow line.
15
8000
6000
4000
2000
0
Cumulative W126 Index (ppb hours)
10000
Cumulative W126 Indices
Jun
Jul
Aug
Sep
Time
Figure 4.3: Daily W126 indices at the Mesa Lab. The first signs of plant damage occurred at the
vertical yellow line.
The W126 daily indices are summed from the plants’ first emergence up to the first sign of plant
damage, and scaled to account for missing data, to calculate the cumulative W126 index of 4456
ppb hours. For the W126 index, the units are in weighted ppb hours according to the weighted
distribution mentioned in Section 3.1.
16
4000
3000
2000
0
1000
Cumulative AOT Index (ppb hours)
5000
6000
Cumulative AOT40 Indices
Jun
Jul
Aug
Time
Figure 4.4: Cumulative daily AOT40 indices at the Mesa Lab. The first signs of plant damage
occurred at the vertical yellow line.
Similar to the W126 cumulative index, the AOT40 daily indices are summed and scaled to
account for missing data. This calculation yields the cumulative AOT40 index of 9000 ppb hours
above the 40 ppb threshold at the first signs of damage (22–24 July 2015).
4.1.2
Damage Observations for 2016
In 2016, plant damage was surveyed more frequently and systematically at the Mesa Lab. To survey
plant damage, an observer samples 10 random leaves from a single plant and observes the amount
of leaf showing signs of ozone damage (these visible signs of damage are presented in figure 4.1).
The observer then records the damage score for each plant using the following plant damage score
scheme in table 4.2.
Score
Percent Affected
0
1
2
3
4
0
0–30
30–60
60–90
> 90
Table 4.2: Plant damage score used by the Mesa Lab, related to the percent of leaf affected.
An ecologist recorded plant damage data for the cutleaf coneflower from 1 June to 18 July
2016. The percentage of observed leaves in each damage score category is presented in figure 4.5.
At the beginning of the growing season, 100% of the leaves surveyed show no signs of damage
17
(damage score 0). As the growing season progresses, a portion of the leaves begin to show more
signs of damage, and the highest damage score of four was observed most recently on 18 July 2016.
Figure 4.5 also shows that there is some random sampling variability as there was less damage
observed on June 30 than earlier.
Plant damage scores vs. time
100
90
Percent damage score
80
70
Damage score
0
60
1
50
2
3
40
4
30
20
10
0
June 1
June 10 June 16
June 24 June 30
July 8
July 18
Date
Figure 4.5: Percent occurrence of damage scores for the cutleaf coneflower, surveyed by an expert
ecologist. The dates shown correspond to when samples were taken. The dashed vertical line marks
the date that the Bootcamp participants gathered plant damage data (see section 5).
4.2
St. Louis Ozone Plant Damage
The St. Louis ozone garden observed damage to milkweed plants over the 2013 and 2014 growing
season. The milkweed damage scores were recorded by several different observers multiple times
during the growing season, then averaged across all observers for each time. The damage scores
followed a scale from 1 to 6 described in Table 4.3.
18
Score
Percent Affected
1
2
3
4
5
6
0
1–6
7–25
26–50
51–75
76–100
Table 4.3: Plant damage scores used by the St. Louis garden, related to the percent of leaf affected.
Note that these damage scores are different than those used at the Mesa Lab.
The following plots overlay the cumulative ozone indices for both the W126 and the AOT40,
and the plant damage scores over time. Note the missing data in the plots of ozone data from 2014
(Figures 4.7 and 4.9), visualized by a flat line in mid-May and early June. The missing data may
be accounted for by scaling the cumulative ozone index at a given time by the ratio of total possible
ozone observations that could have been recorded to the total ozone observations actually taken.
This method of accounting for the missing data is not included in the plots, but is included in the
subsequent calculations.
5
4
3
1
0
2
Damage Score (1−6)
6000
4000
2000
Cumulative W126 Index (ppb hours)
8000
6
10000
Cumulative W126 Indices
May
Jun
Jul
Aug
Sep
Oct
Time
Figure 4.6: Cumulative W126 indices at St. Louis in 2013, calculated by summing the daily W126
index, and the damage scores reported for common milkweed during the same time period.
19
5
4
3
1
0
2
Damage Score (1−6)
6000
4000
2000
Cumulative W126 Index (ppb hours)
8000
6
10000
Cumulative W126 Indices
May
Jun
Jul
Aug
Sep
Oct
Time
Figure 4.7: Cumulative W126 indices at St. Louis in 2014, calculated by summing the daily W126
index, and the damage scores reported for common milkweed during the same time period.
5
4
3
1
0
2
Damage Score (1−6)
6000
4000
2000
Cumulative AOT Index (ppb hours)
8000
6
10000
Cumulative AOT40 Indices
May
Jun
Jul
Aug
Sep
Oct
Time
Figure 4.8: Cumulative AOT40 indices at St. Louis in 2013, calculated by summing the daily
AOT40 index.
20
5
4
3
1
0
2
Damage Score (1−6)
6000
4000
2000
Cumulative AOT Index (ppb hours)
8000
6
10000
Cumulative AOT40 Indices
May
Jun
Jul
Aug
Sep
Oct
Time
Figure 4.9: Cumulative AOT40 indices at St. Louis in 2014, calculated by summing the daily
AOT40 index.
The milkweed plant damage scores increased rapidly in a short time frame, around 15 August
in 2013 and 1 August in 2014. The cumulative indices for W126 and AOT40 at these times are
summarized below.
Plant and year
Cumulative W126 Index
Cumulative AOT40 Index
Milkweed, 2013
Milkweed, 2014
4300
4500
5900
5400
Table 4.4: Cumulative ozone indices at the time of plant damage increase for milkweed. The ozone
indices were from St. Louis in 2013 and 2014.
The cumulative ozone indices are similar between 2013 and 2014, both for the W126 and AOT40
indices.
5
A Bayesian Model for Ozone Plant Damage
As part of the second data analytics bootcamp for high school students held at the Mesa Lab in
June 2016, the bootcamp participants were introduced to the problem of ground level ozone. The
m = 24 participants were instructed how to gather data on plant damage, and randomly collected
n = 10 samples, using the damage score scheme shown in table 4.2 (categorized into c = 5 levels
from 0 to 4). Table 5.1 displays the number of leaves observed for each damage score, for all 24
participants.
21
Participant
Damage 0
Damage 1
Damage 2
Damage 3
Damage 4
Sum
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
2
4
4
7
4
6
1
2
2
3
4
4
2
0
4
5
5
6
6
3
2
3
5
3
4
3
4
3
5
4
8
7
6
5
6
5
6
7
5
2
5
3
3
6
4
5
3
6
3
2
2
0
1
0
1
1
1
2
0
1
1
2
1
1
0
1
1
0
1
2
2
1
1
1
0
0
0
0
0
0
1
0
0
0
1
1
0
1
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Sum
87
115
27
8
3
240
Table 5.1: Number of leaves recorded in each damage score category for each bootcamp participant.
Sums for each damage score are presented at the bottom.
As an additional part of the instruction, the participants were taught how to analyze these data
using a multinomial Bayesian model.
5.1
The Dirichlet-Multinomial Model
Let xk = (xk1 , . . . , xkc )0 be a random vector where each of xki is the number of leaves corresponding
to a plant damage score from 0 to c − 1 with k = 1, . . . , m and i = 1, . . . , c. Suppose that
each xk identically and independently follows the multinomial distribution (where the multinomial
distribution can be viewed as an extension of the binomial distribution to more than two categories)
with parameters n = 10 and θ = (θ1 , . . . , θc )0 , denoted by M ult(θ). Its probability mass function
is given by
p(xk |θ) =
n!
θxk1 . . . θcxkc ,
xk1 ! . . . xkc ! 1
(1)
P
P
where ci=1 xki = n and ci=1 θi = 1 with 0 ≤ θi ≤ 1. The parameters θi describe the probability
of belonging to category i.
To carry out Bayesian inference, we need to choose prior distributions for parameters θ,
which describe information about the unknown parameters before obtaining the data. The most
22
widely used prior for multinomial distribution is the Dirichlet distribution with parameters α =
(α1 , . . . , αc )0 , denoted by Dir(α). Its probability density function is given by
P
c
Γ( ci=1 αi ) Y αi −1
f (θ|α1 , . . . , αc ) = Qc
θi
,
i=1 Γ(αi )
(2)
i=1
P
where ci=1 θi = 1 and αi > 0. Notice that αi = 1 corresponds to the uniform prior in Rc−1 ,
and αi = 1/2 corresponds to the Jefferys prior in Rc−1 . The other priors used ((1,5,2,4,3) and
(2,6,1,3,1)) are informative. Multiple priors are used to investigate how sensitive the results are to
the choice of prior.
After specifying prior distributions for parameters θ, the posterior distribution of θ, given
x1 , . . . , xm , can be derived according to Bayes’ rule as:
! c
m Y
c
Y
Y α −1
p(θ|x1 , . . . , xm ) ∝
θixki
θi i
(3)
∝
k=1 i=1
Pm
c
Y
θi
k=1
i=1
xki +αi −1
,
i=1
P
which is also a Dirichlet distribution with parameters α̃ = m
k=1 xk +α. Due to the explicit form of
the posterior distribution, we can directly calculate quantities such as the posterior mean, posterior
variance, and credible intervals. Table 5.2 shows the posterior mean and standard deviations given
the four different priors.
Prior
(1,1,1,1,1)
(0.5,0.5,0.5,0.5,0.5)
(1,5,2,4,3)
(2,6,1,3,1)
Posterior θ1
Posterior θ2
Posterior θ3
Posterior θ4
0.359,
0.361,
0.345,
0.352,
0.473,
0.476,
0.471,
0.478,
0.114,
0.113,
0.114,
0.111,
0.037,
0.035,
0.047,
0.043,
0.0306
0.0308
0.0296
0.0300
0.0317
0.0319
0.0311
0.0313
0.0202
0.0202
0.0197
0.0197
0.0118
0.0118
0.0134
0.0126
Posterior θ5
0.016,
0.014,
0.024,
0.016,
0.00836
0.00774
0.00949
0.00774
Table 5.2: Means and variances for θ given different priors: the uniform prior, Jeffreys prior, and
two informative priors.
23
Posterior Bayesian Bootcamp Results, June 30
0.6
●
0.4
Damage score
●
● 0
Mean
● 1
● 2
● 3
0.2
● 4
●
●
●
0.0
0
1
2
3
4
Damage score
Figure 5.1: Posterior means for each damage score (θ0−4 ) using Jeffreys prior for the multinomial
distribution. The error bars denote three standard deviations.
By looking at table 5.2, it is clear that the results are not very sensitive to the choice of prior.
Even between informative and uninformative priors, the mean values of θ vary only slightly. In
figure 5.2, the marginal prior and posterior distributions are presented for each θ, and it can be
seen that the posterior distributions are consistent across all the choices for the prior distribution.
5.2
Model Diagnostics
The Dirichlet-Multinomial model allows for fast and easy computation of the posterior distribution.
However, other Bayesian models do not allow for easy computation of the posterior. It is therefore
necessary to sample the posterior distribution to better understand the posterior distribution. Even
though we can easily calculate the posterior distribution, we can use posterior predictive samples
to check the adequacy of the model and report the posterior predictive p-value. For any new
24
observation xP = (xP1 , . . . , xPc )0 , the posterior predictive distribution is given by
Z
Z
P
P
f (x |x) =
f (x , θ|x)dθ = f (xP |θ)f (θ|x)dθ
!
!
Pc
Z
c
c
Y
Y
α̃
)
Γ(
Γ(n + 1)
xP
i
Qc i=1
Qc
θiα̃i −1 dθ
=
θi i
P + 1)
Γ(α̃
)
Γ(x
i
i=1
i=1
i
i=1
i=1
Pc
Z Y
c
Γ( i=1 α̃i )
Γ(n + 1)
xP +α̃ −1
Qc
= Qc
θi i i dθ
P
i=1 Γ(α̃i )
i=1 Γ(xi + 1)
i=1
Pc
Qc
Γ( i=1 α̃i ) i=1 Γ(xPi + α̃i )
Γ(n + 1)
Qc
Pc
= Qc
.
P
i=1 Γ(α̃i ) Γ(n +
i=1 α̃i )
i=1 Γ(xi + 1)
(4)
Since this distribution is a non-standard distribution, Monte Carlo methods can be used to generate
samples from this posterior predictive distribution, and the detailed procedure is given as follows:
• For j = 1, . . . J, sample
1. θ [j] from posterior distribution p(θ|x), and
2. x∗[j] from sampling distribution p(xP |θ [j] ).
The quantities {θ [j] , x∗[j] } are joint samples from the distribution f (xP , θ|x). Therefore, x∗[1] , . . . , x∗[J]
are samples from f (xP |x). We can compute P (T (x∗ ) ≥ T (x)|x) based on the posterior predictive
samples for any test statistic T (·).
As a first step, we would typically plot the Dirichlet distribution Dir(α) of θ. Since the Dirichlet
distribution is defined in Rc−1 with c = 5, we cannot plot this distribution
P directly. Instead, we
can use the fact that if θ ∼ Dir(α̃), θi |x ∼ Beta(a, b) with a = α̃i , b = ( ci=1 α̃i ) − α̃i . Figure 5.2
shows the marginal plots of θ1 , . . . , θ4 under different values for α, which shows how the marginal
posterior distributions of θ given x vary with different prior distributions. We check the model by
drawing posterior predictive samples and the predictive p-values with the identity function as the
test statistic T (·) for each variable are 0.4964, 0.4565, 0.5323, 0.5900, 0.6525, respectively, which
indicate that there is not enough evidence to show that the model does not fit the data very well.
25
12
p(θ2)
0.2
0.4
0.6
0.8
0 2 4 6 8
12
0 2 4 6 8
p(θ1)
0.0
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
θ2
10
20
p(θ4)
10
0
0
5
p(θ3)
15
30
20
θ1
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
θ3
θ4
30
α = (2,6,1,3,1)'
marginal prior
marginal posterior
0 10
p(θ5)
50
α = (1,1,1,1,1)'
α = (.5,.5,.5,.5,.5)'
α = (1,5,2,4,3)'
0.0
0.2
0.4
0.6
0.8
1.0
θ5
Figure 5.2: Marginal prior and posterior density plots of the Dirichlet distribution under different
parameters from top left panel to bottom right panels corresponding to variables θ1 , θ2 , θ3 , θ4 , and
θ5 . The vertical lines denote the means of each marginal posterior distribution for each θ.
5.3
Discussion of the Multinomial Model
We also tried to model the datasets collected by each person individually, and found that if x
contains lots of zeros, the model does not fit the data very well as assessed by the predictive p-value.
When considering m = 24 data points together, there are numerical issues when computing the
gamma function in the Dirichlet distribution, which can be alleviated by computing the logarithm
of gamma function. Currently, the model assumes independence for the data collected by each
person, and is used to analyze datasets collected at a single time point. For future analyses, this
model can be generalized to incorporate the dependence assumption among data points collected by
26
different people and the temporal dependence structure for data collected at multiple time points.
6
Ozone and Meteorological Data
The concentration of ground level ozone is highly dependent on many meteorological variables.
This dependency can be seen as a correlation between several variables. As a very brief exploration
into the relationships between ozone and certain variables, the following scatter plots of ozone
concentration versus surface temperature and relative humidity are presented below.
60
40
20
Ozone concentration (ppb)
80
Ozone concentration vs. temperautre
10
15
20
25
30
Temperature (C)
Figure 6.1: Hourly ozone concentration versus temperature at the Mesa Lab in 2015.
27
60
40
20
Ozone concentration (ppb)
80
Ozone concentration vs. relative humidity
20
40
60
80
Relative humidity (%)
Figure 6.2: Hourly ozone concentration versus relative humidity at the Mesa Lab in 2015.
There is a positive correlation between ozone and surface temperature, and a negative correlation
between ozone and relative humidity. This may be explained by the fact that sun exposure increases
both surface temperature and ozone concentration. Further analysis of ground-level ozone related
to meteorological variables may also lead to better understanding of its impact on foliar damage.
7
Future Work
The ultimate goal of this project is to model foliar damage as a function of ozone concentration
observed by ozone monitors and potentially satellites. The work presented here is an exploratory
analysis serving as initial groundwork towards this goal.
References
[1] D.
Lombardozzi,
“Boulder’s
ozone
boulder-s-ozone-gardens, July 2015.
gardens.”
https://scied.ucar.edu/blog/
[2] E. T. C. on Air Pollution and C. C. Mitigation, “Data aggregation, calculation of statistics and
nox values in airbase.” http://acm.eionet.europa.eu/databases/airbase/aggregation_
statistics.html.
[3] E. F. Commission, “Ozone exposure indices: Aot40 exposure index.” http://www.forestry.
gov.uk/fr/infd-622kb5.
[4] E. P. Agency, “Ozone w126 index.” https://www.epa.gov/air-quality-analysis/
ozone-w126-index.
28