EP A/600/A-96/073
A ROTATED PRINCIPAL COMPONENT ANALYSIS OF
TOTAL COLUMN OZONE OBTAINED FROM TOMS FOR 1984 - 198V
Brian K. Eder* and Sharon K, LeDuc*
Atmospheric Sciences Modeling Division
Air Resources Laboratory
National Oceanic and Atmospheric Administration
Research Triangle Park, NC USA
1. INTRODUCTION
The global distribution of total column ozone,
arguably the most important trace gas in the atmosphere,
is attracting great international attention as concerns over
reduced ozone abundances escalate (WMO, 1990). In
addition to its biological importance, ozone also plays a
critical role in the chemical and meteorological dynamics
of the atmosphere. Ozone is a precursor of the hydroxyl
radical, the major cleansing agent in the troposphere and
its absorption of ultraviolet radiation is the major heat
source in the stratosphere, ultimately driving the global
circulation of the stratosphere.
Despite its great importance, the spatial and
temporal distribution of ozone is poorly understood. To
assess anthropogenic changes to date and to better
understand how ozone abundance may respond to future
perturbations requires a better understanding of its natural
intra- and interannual variability and the processes that
contribute to this variability. Unfortunately, many current
studies (Herman et. al., 1991; Stolarski et. al., 1991) use
either globally, hemispherically or zonally averaged (i.e
data from 30"N to 40"N) data to try and detect cycles in
the TOMS data. Although these studies provide a simple,
conceptually appealing picture of ozone abundance, much
of its variability is attributable to more complex
phenomena that are neither globally or zonally
symmetrical. As a result, such approaches often mask the
true strength of temporal cycles and do not allow
identification of the exact spatial extent of such cycles.
Accordingly, the purpose of this analysis is to
develop a better understanding of these natural variations
across various spatial and temporal scales. This will be
achieved through the application of a multivariate
statistical technique called rotated Principal Component
Analysis (PCA) to the total column ozone data derived
from Version 6.0 TOMS (Total Ozone Mapping
Spectrometer) for the period 1984 through 1989. The
main objective of principal component analysis (PCA) is
'On assignment to the National Exposure Research
Laboratory, U.S. Environmental Protection Agency.
to identify, through a reduction in data, the characteristic,
recurring and independent modes of variation across all
potential spatial and temporal scales (Eder et al., 1993).
The analysis sorts initially correlated data into a hierarchy
of statistically independent modes of variation which
explain successively less and less of the total variation;
thereby summarizing the essential information of that set
so that meaningful and descriptive conclusions can be
achieved. This technique is ideal for application to the
TOMS data set where the total number of observations
exceeds 3 million. Utilization of Kaiser's (1958) varimax
orthogonal rotation will allow delineation of homogeneous
subregions - that is, areas of the globe that experience
unique total ozone characteristics. Examination of the
time series associated with each unique subregion will be
based on spectral density analysis. This will allow further
elucidation (across many wavelengths) of the physical
phenomena (i.e. annual and semi-annual cycles, Quasi-
Biennial Oscillation (QBO), El-Nino-Southern Oscillation
(ENSO)) responsible for the natural variability of total
column ozone.
2. TOMS DATA
The daily TOMS (Version 6.0) data were
obtained from the archived data sets available on CD-
ROM at the U.S. National Space Science Data Center
(NSSDC) located at NASA's Goddard Space Flight
Center in Greenbelt, MD. TOMS is aboard the Nimbus
7 satellite, which is in a sun-synchronous nearly polar
orbit, launched in October, 1979. The instrument consists
of a single Ebert-Fastie monochromator that measures the
ultraviolet sunlight backscattered from the Earth's
atmosphere and surface at six wavelengths. Four of these
wavelengths (312.5, 317.5, 331.2 and 339.8 nm) are used
in pairs to infer, from the differential absorption of
scattered sunlight, the total column ozone. For instance,
ozone is calculated from the ratio of two wavelengths,
312.5 nm and 331.2, where one wavelength (312.5 nm) is
strongly absorbed by ozone, while the other is only
weakly absorbed. The two remaining wavelengths (360
and 380 nm) are used to measure surface reflectivity.
-------
TOMS collects 35 measurements every 8 s as it
rapidly scans from right to left in a plane perpendicular to
the orbital plane, producing roughly 200,000 daily ozone
measurements with a resolution of between 50 and 150
km. The total ozone column is then expressed as the
depth that the ozone alone would occupy at standard
temperature and pressure. This depth is measured in
thousands of centimeters or Dobson Units [1 DU = 2.69
x 101A molecules cm'2], such that typical column
abundances of between 250 and 400 DU would
correspond to pure ozone column depths of between 0.25
and 0.40 cm. The TOMS data used in this analysis were
regridded from the original resolution of 1" lat. by 1.25"
long, using an averaging technique which yielded a total
of 1440, 5" by 5" daily fields extending from 5Q"S to
50"N. Six years of data were examined, from January 1,
1984, through December 31, 1989. All missing data (<
5 %) were eliminated through a temporal interpolation
scheme resulting in 3,156,480 observations.
3. METHODOLOGY
3.1 Spatial Analysis
Mathematically, this analysis began with the
calculation of a square, symmetric correlation 1440Rlil4(>
from the original data matrix, which had dimensions of
1440 (grid cells) x 2192 (days). Selection of a correlation
matrix (as opposed to a covariance matrix) has two
advantages in PCA. First, use of a correlation is much
more suitable for resolving spatial patterns; second, use of
a correlation matrix allows maps of component loadings
(the correlation coefficient between the grid cell and
component) to be drawn (Overland and Preisendorfer,
1982). By using R and the identity matrix I of the same
dimensions, 1440 characteristic roots or eigenvalues (A.)
were derived, which satisfied the following polynomial
equation:
det [1440^1440 - ^!44(/i440 1 = 0 •
For each root (A,), a non-zero vector e can be derived:
1440®1440ei = ^1440*1 »
where the vector e is called the characteristic vector, or
eigenvector of the correlation matrix R, associated with its
corresponding eigenvalue (X). The eigenvectors derived
from the correlation matrix represent the mutually
orthogonal linear combinations (or modes of variation) of
the matrix. Their associated eigenvalues, which are
scalars, represent the amounts of total variance that are
explained by each of the eigenvectors. By retaining only
the first few eigenvector-eigenvalue pairs, or principal
components, a substantial amount of the total variance can
be explained while ignoring higher order principal
components which explain minimal amounts of the total
variance and can be viewed as noise. The exact number
of components that should be retained was determined by
the Scree Test (Eder et al., 1993), and indicated that
eleven components should be retained. Therefore, the
original data set, which contained 1440 intercorrelated and
noisy variables (the 5" by 5" grid cells), has been reduced
to one containing only eleven orthogonal and thereby
independent variables (the principal components), yet still
explains nearly three-fourths of the total variance.
Since one of the major goals of this research was
to define areas of homogeneous total column ozone, a
rotation was performed on the principal components in
order to better segregate the areas that have similar ozone
characteristics. Of the many types of rotation available,
an orthogonal method developed by Kaiser (1958) was
selected because it rigidly rotates the predetermined
principal components while retaining the constraint that
the individual components remain orthogonal or
uncorrelated. This method increases the segregation
between loadings, allowing better definition of distinct
groupings of intercorrelated data, thereby making spatial
interpretation easier (Horcl, 1981).
The elements of each eigenvector were then
multiplied by the square root of the associated eigenvalue
to obtain the component loadings (L,y), for grid cell j on
principal component i. These loadings represent the
correlations between the component and the grid cell. The
square of the component loading indicates the proportion
of variance in the individual grid cell that can be
attributed to that component. Maps of the loadings
associated with the first seven retained components can be
found at the tops of Figures 1 through 4. Space
constraints prohibit inclusion of all eleven components.
3,2 Temporal Analysis
Another useful parameter that can be derived
from PCA is the component score (PCS). Initially, the
rotated PCA replaced the 1440 grid cells, which were
measured over 2192 days, with 11 rotated principal
components having no temporal measure. By introducing
the PCS, a derivation of similar temporal measurements
for the rotated principal components over the same 2192
days can be achieved. The rotated principal components
are identified in terms of the original grid cells, the larger
the loading the more important the grid cell is in the
interpretation of the component. Therefore, if a day has
high values for grid cells with large loadings, it should
have a large value on the component. The PCS for day
-------
i on component k is designed to meet this requirement are
defined as follows:
(PCS)* = E OtjL]k ;
J
where O0 is the observation for day i on grid cell j, Ljk is
the loading of grid cell j on component k, and the PCS is
the component score for day i on component k and is
summed over all 1440 grid cells. As seen from the
equation, the PCSs are simply weighted summed values
for the days over the grid cells, the weights being the
component loadings. The larger the value that a day has
on a grid cell with large loadings, the larger the PCS.
When plotted as a time series (bottoms of Figures 1
through 4), the PCS, which are standardized, provide
excellent insight into the spectrum of temporal variance
experienced by each of the subregions.
Spectral density analysis (SDA) using the finite
Fourier transformation was then employed to examine
each of the 11 time series associated with the subregions.
Such analysis yields a measure of the distribution of
variance of the time series across all possible
wavelengths, each arbitrarily close to the next. It is used
to look for non-random, physically generated cyclical
patterns or periodicities in the time series data, which
would be represented by a peak in the spectrum at a
particular frequency. SDA decomposes the time series
into a sum of sine and cosine waves of varying
amplitudes and wavelengths as defined by:
(PCS)j = (PCS) + E [atcos«t(i-l)+fcjtsin(Dit(i-l)] ;
k
where the aks are the cosine coefficients, the bt's are the
sine coefficients and = 2nk/2l9l. From this
decomposition, the periodogram can be obtained and is
defined as follows:
h =
2191(4*
BD
where Ak and Bk are estimates of ak and bk and are given
by:
B,,
S [((PCS),. - PCS) cosut(i-l)] ;
2 [((PCS),. - PCS) sina>k(i-1)],
The spectral density is then estimated by smoothing the
periodogram using a triangular weight (often referred to
as the kernel or spectral window) as follows:
Spectral density estimates
E w,
J"P
where the W's are smoothing weights normalized to rc/4.
At the risk of decreasing the statistical stability of the
spectral estimates, a relatively narrow window (1 2 3 4 5
4 3 2 1) was used in order to prevent any bias that might
occur through the over smoothing of "real" peaks.
4. RESULTS
The first Rotated Principal Component (RPC) (X
= 409.48), which explains 28.44% ((409.48/1440) x 100]
of the total variance, defines an area encompassing much
of the Southern Hemisphere (SH). A large area of this
subregion contains grid cells with loadings in excess of
0.70 (meaning more than half of the variance of these
cells can be attributed to this component). The time
series and spectra analysis of this component reveal a
strong annual cycle (maximum spectral power at / =
0.01720, corresponding to a periodicity of (2jiff) = 365.33
days) that peaks during the austral Spring (September and
October) of each year and falls to a broader minimum
during the period of February through May. This cycle
is likely responding to the changing solar cycle which
alters the chemical and dynamical process in the
atmosphere. In terms of magnitude, the annual cycle of
this SH subregion exhibits similar strength during the
years: 1984, 1986, 1988 and 1989; however during 1987
and especially 1985 its magnitude and range is strikingly
reduced.
The second RPC (X = 287.49), which explains
19.96% of the total variance defines a comparable area
comprising much of the Northern Hemisphere (NH);
however, the poleward extent of this NH subregion is
limited to roughly 35"N, resulting in a more narrow
subregion. This difference is likely attributable to the
extensive landmasses found in the NH, which play a large
role in perturbing the Arctic circumpolar vortex and hence
the hemispheric scale circulation patterns. These
perturbations, called planetary waves, occur much more
frequently in the NH than in the SH. As a result, the
Antarctic circumpolar vortex is much less disturbed,
resulting in less variability in the ozone abundance of the
SH. The NH land masses, which coincide fairly well
with loadings on RPC2 of less than 0.30 induce additional
variability into the total column ozone which are not
represented well by this RPC. The time series and
spectral of this subregion likewise indicate a strong annual
-------
cycle, however its peak is much more broad and occurs
during the period of March through July. The minimum
tends to occur during December and January. It is
interesting that like the SH, the annual cycle associated
with the NH is weakest during 1985 and 1987.
Generally, the relative annual maximum is greater in the
SH than the NH. However, during 1985 and 1987, the
maxima are nearly equal with the NH slightly stronger.
The third RPC (X = 207.85, 14.43%) defines an
area that is fairly symmetrical about the equator,
dominating between 10"N and 10"S, This subregion
coincides well with a region associated with the Quasi-
Biennial Oscillation (QBO) of the tropical winds in the
lower stratosphere, where dominance of the NH and SH
annual variability is replaced with QBO dominated
variance (Ziemke and Stanford, 1994). This association
is confirmed by the SDA which reaches a maximum
power at / = 0.00860, corresponding to a periodicity of
731 days (roughly two-years). This component is slightly
more significant in the southeastern Pacific Ocean near
Ecuador and Peru. Its times series peaks in 1985 and
1987, the two years corresponding with the weak annual
cycles observed in both the NH and the SH.
The fourth RPC (X = 33.50, 2.33%) defines an
area in the SH from New Guinea in the Western Pacific
to central South America. This area coincides well with
that impacted by the El Nino - Southern Oscillation
(ENSO), which is the likely driving force. Note that
during the El Nino of 1986-1987, the PCSs are lower,
while in the non-El Nino years, especially, 1988 and
1989, the scores are high.
The fifth and sixth RPCs are both associated with
subregions where a strong semi-annual variance
dominates. The fifth subregion (X = 29.43, 2.04%)
defines a region from the western Indian Ocean into the
western Pacific Ocean. It is strongest north of the
Australian Continent over Indonesia and appears to be
associated with the Equatorial Semi-Annual Oscillation,
which is physically driven by the negative temperature
effect of ozone photochemistry associated with the semi-
annual modulation of the temperature field (Varotsos et.
ai., 1992). This component has a very strong semi-annual
cycle, with maximum spectral power at / = 0.03440,
corresponding to a periodicity of 182.67 days. It peaks
during the transitional seasons, (Spring and Autumn).
Similarly, the sixth subregion (K = 25.87, 1.80%), which
defines an area in central Asia may be related to the
Polar Semi-Annual Oscillation, which is driven by two
processes: the photochemical production of ozone and
dynamical transport from the equatorial regions of
maximum ozone production. This subregion also has a
strong periodicity of roughly 182 days. Reasons for the
areas in South America that load highly on this
component and any potential teleconnection implications
are not understood at this time.
The seventh RPC (X = 18.74, 1.30%) is
somewhat different from the first six RPCs in that it is
not contiguous, but rather a compilation of three separate
subregions, that are none-the-less driven by the same
physical process - namely Baroclinic Waves associated
with the polar jet stream. Each of these three areas are
associated with favored areas of tropospheric jet stream
cores, principally, the Aleutian coast, the Canadian North
Atlantic Coastal region and eastern Siberia/Asia. The
time series and spectra support this contention in that this
component generally reaches a maximum twice a year,
during transitional seasons which are periods of maximum
baroclinicity.
The remaining RPCs (8 through 11), like RPC7,
define various, small areas in the higher latitudes of both
hemispheres, and are likewise thought to be associated
with baroclinic waves.
5. REFERENCES
Eder, B.K., J.M. Davis and P. Bloomfield (1993) "A
characterization of the spatiotemporal variability
of non-urban ozone concentrations over the
Eastern United States", Atmos. Environ,, 27A,
2645-2668.
Herman, J.R., R McPeters and R. Stolarski (1991)
"Global average ozone change from November
1978 to May 1990", J. Geophys. Res., 96, D9,
17297-17305.
Horel, J.D. (1981) "A rotated principal component
analysis of the interannual variability of the
Northern Hemisphere 500 mb height field", Mn
Wea. Rev., 109, 2080-2092,
Kaiser, H.F. (1958) "The varimax criterion for analytical
rotation in factor analysis", Psychometrika, 23.
Overland, J.E. and R.W. Preisendorfer (1982) "A
significance test for principal components applied
to a cyclone climatology", Mon. Wea. Rev., 110,
1-4.
Stolarski, R.S., P. Bloomfield and R.D. McPeters (1991)
"Total ozone trends deduced from Nimbus 7
TOMS Data", Geophys. Res. Lett, 18, No. 6,
1015-1018.
-------
Varotsos, C., C. Helmis and C. Cartalis (1992) "Annual
and semi-annual waves in ozone as derived from
SBUV vertical global ozone profiles", Geophys.
Res. Lett., 19, No. 9, 925-928.
World Meteorological Organization (1990) Scientific
Assessment of Stratospheric Ozone: 1989,
Global Ozone Research and Monitoring Project,
Report No. 20, Geneva, Switzerland.
Ziemke, J.R and J. L. Stanford (1994) "Quasi-biennial
oscillation and tropical waves in total ozone", J.
Geophys. Res., 99, Dll, 23041-23056.
The information in this document has been funded by the
United States Environmental Protection Agency. It has
been subjected to Agency review and approved for
publication. Mention of trade names or commercial
products does not constitute endorsement or
recommendation for use.
-------
Varotsos, C., C. Helmis and C. Cartalis (1992) 'Annual
and semi-annual waves in ozone as derived from
SBUV vertical global ozone profiles", Geophys.
Res. Lett., 19, No. 9, 925-928.
World Meteorological Organization (1990) Scientific
Assessment of Stratospheric Ozone: 1989,
Global Ozone Research and Monitoring Project,
Report No. 20, Geneva, Switzerland,
Ziemke, J.R and J. L. Stanford (1994) "Quasi-biennial
oscillation and tropical waves in total ozone", J.
Geophys. Res., 99, Dll, 23041-23056.
The information in this document has been funded by the
United States Environmental Protection Agency. It has
been subjected to Agency renew and approved for
publication. Mention of trade names or commercial
products does not constitute endorsement or
recommendation for use.
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1984
1985
1986
1987
1988
1989
Figure 1. Rotated principal component loadings and scores associated with the Southern
Hemisphere Subregion.
-------
LOADINGS
~ LE 0300
¦ 0.301 TO 0300
¦ 0501 TO 0.10
¦ GT 0.700
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1984
1985
1986
1987
1988
1989
~ UE 0.300
0.301 TO 0.500
0,501 TO 0.700
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1984 1985 1986 1987 1988 1989
Figure 3. Rotated principal component loadings and scores associated with the El Nino -
Southern Oscillation Subregion (top figures) and the Equatorial Semi-Annual Oscillation
Subregion (bottom figures).
-------
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LOADINGS
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¦ 0301 TO 0J00
¦ 0301 TO 0.700
¦ GT 0.700
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1984 1985 1986 1987 1988
1989
Figure 4. Rotated principal component loadings and scores associated with the Polar Semi-
Annual Oscillation Subregion (top figures) and the Baroclinic Waves Subregion (bottom
figures).
-------
TECHNICAL REPORT DATA
1. REPORT NO.
EPA/600/A-96/073
2.
3.REC3
4. TITLE AND SUBTITLE
A Rotated Principal Component Analysis of Total
Column Ozone Obtained from TOMS for 1984-1989
5.REPORT DATE
i. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Brian K. Eder and Sharon K. LeDuc
». PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Same as Block 12
10.PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO,
12. 'SPONSORING AGENCY NAME AND ADDRESS
National Exposure Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
13.TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
IS. SUPPLEMENTARY BOXES
16. ABSTRACT
The purpose of this analysis is to develop a better understanding of these natural variations across
various spatial and temporal scales. This will be achieved through the application of a multivariate
statistical technique called rotated Principal Component Analysis (PCA) to the total column ozone data
derived from Version 6.0 TOMS (Total Ozone Mapping Spectrometer! for the period 1984 through 1989. The
main objective of principal component analysis (PCA! is to identify, through a reduction in data, the
characteristic, recurring and independent modes of variation across all potential spatial and temporal
scales (Eder et al., 1993S . The analysis sorts Initially correlated data into a hierarchy of
statistically independent modes of variation which explain successively less and less of the total
variation; thereby summarizing the essential information of that set ao that meaningful and descriptive
conclusions can be achieved. This technique is ideal for application to the TOMS data set where the
total number of observations exceeds 3 million. Utilization of Kaiser's (1958) varimax orthogonal
rotation will allow delineation of homogeneous subregions - that is, areas of the globe that experience
unique total ozone characteristics. Examination of the time series associated with each unique subregion
will be based on spectral density analysis. This will allow further elucidation (across many
wavelengths) of the physical phenomena (i.e. annual and semi-annual cycles. Quasi-Biennial Oscillation
(QBO), El-Nino-Southern Oscillation (ENSO)) responsible for the natural variability of total column
ozone.
17. KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
b.IDENTIFIERS/ OPEN ENDED
TERMS
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