ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus GmbHGöttingen, Germany10.5194/acp-14-13455-2014Reevaluation of stratospheric ozone trends from SAGE II data using a simultaneous temporal and spatial analysisDamadeoR. P.robert.damadeo@nasa.govhttps://orcid.org/0000-0002-1466-839XZawodnyJ. M.ThomasonL. W.https://orcid.org/0000-0002-1902-0840NASA Langley Research Center, Hampton, VA, USAR. P. Damadeo (robert.damadeo@nasa.gov)17December2014142413455134702June20142July201421October201421November2014This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.atmos-chem-phys.net/14/13455/2014/acp-14-13455-2014.htmlThe full text article is available as a PDF file from https://www.atmos-chem-phys.net/14/13455/2014/acp-14-13455-2014.pdf
This paper details a new method of regression for sparsely sampled data sets
for use with time-series analysis, in particular the Stratospheric Aerosol
and Gas Experiment (SAGE) II ozone data set. Non-uniform spatial, temporal,
and diurnal sampling present in the data set result in biased values for the
long-term trend if not accounted for. This new method is performed close to
the native resolution of measurements and is a simultaneous temporal and
spatial analysis that accounts for potential diurnal ozone variation. Results
show biases, introduced by the way data are prepared for use with traditional
methods, can be as high as 10 %. Derived long-term changes show declines in
ozone similar to other studies but very different trends in the presumed
recovery period, with differences up to 2 % per decade. The regression model
allows for a variable turnaround time and reveals a hemispheric asymmetry in
derived trends in the middle to upper stratosphere. Similar methodology is
also applied to SAGE II aerosol optical depth data to create a new volcanic
proxy that covers the SAGE II mission period. Ultimately this technique may
be extensible towards the inclusion of multiple data sets without the need
for homogenization.
Introduction
The Stratospheric Aerosol and Gas Experiment (SAGE) II flew
onboard the Earth Radiation Budget Satellite (ERBS) for over 20 years
from its launch in October 1984 until its retirement in August 2005. It
employed the solar occultation technique to measure multi-wavelength
slant-path atmospheric transmission profiles at seven channels during each
sunrise and sunset encountered by the spacecraft. During its operation,
SAGE II produced high-precision vertical profiles of atmospheric ozone
(∼ 1 % 1σ uncertainty in the middle stratosphere)
with excellent vertical resolution (∼ 1 km)
. The combination of precise measurements and a long data
record has seen SAGE II data consistently used for long-term ozone trend
analysis e.g.,. Traditionally this is
performed via multiple linear regression of monthly zonal mean ozone data to
a set of predictor variables. However, given the sparse sampling of SAGE II
measurements (∼ 30 observations per day), biases can be
introduced if the data are not carefully treated. Herein we present a new way
to perform time-series analysis on a sparsely sampled data set, in particular
SAGE II, and compare the results and influence on trends with monthly zonal
mean methods. The method outlined in this paper is similar to that of
in that it utilizes a simultaneous temporal and spatial
regression (though the terms used and how the regression is applied are
different), but differs fundamentally in that it regresses to daily mean
values separated by event type.
Predictor variables
The choices of predictor variables are important and thus are chosen based on
atmospheric variability that ozone has historically shown to be responsive
to, namely seasonal cycle, quasi-biennial oscillation (QBO), El Niño–Southern Oscillation (ENSO), solar variability, volcanic eruption, and
long-term trend terms. Ideally, each predictor variable is orthogonal to each
other predictor variable. Since this is almost never the case, predictor
variables are pretreated to create normalized orthogonal functions from
multiple component data sets using empirical orthogonal function (EOF)
analysis (also known as principal component analysis). When the use of EOF
analysis becomes impossible due to having only one component data set, an
orthogonal function is created by shifting the original function in time
until the dot product between the shifted function and original function is
zero over the overlap period, hereafter referred to as the temporal shift
method. The use of orthogonal functions for predictors is better than the use
of a single term because it allows for the regression model to account for both
magnitude and phase changes in the response. Ultimately, however, each set of
orthogonal functions, while orthogonal internally within a set, is not
necessarily orthogonal between sets.
With these tools in mind, a full set of predictor variables is created from
component data sets in order to reduce the amount of multicollinearity. The
seasonal cycle is simply a Fourier series with periods of 12 (annual), 6, 4,
and 3 months (semiannual). To create a set of QBO predictor variables, EOF
analysis is performed on compiled monthly mean equatorial wind data
(http://www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo/)
at seven pressure levels (70, 50, 40, 30, 20, 15, and 10 hPa),
resulting in seven orthogonal basis functions. The leading four terms account
for over 99 % of the variance in the QBO data, so these are used as the QBO
predictor variables. To create a pair of ENSO predictor variables, the
temporal shift method is applied to multivariate ENSO index data
(http://www.esrl.noaa.gov/psd/enso/mei/). A pair of solar predictor
variables is created by applying the temporal shift method to solar
10.7 cm radio flux data
(ftp://ftp.geolab.nrcan.gc.ca/data/solar_flux/). Each of these
ancillary data sets is smoothed before the creation of orthogonal functions
(Fig. ) in order to minimize the effect of noise on the
creation of orthogonal functions.
Two separate terms are explored for use to represent long-term changes in
ozone. One is a simple piecewise linear term joined at the beginning of 1997
(i.e., both terms are linear, with one being zero everywhere before 1997 and
the other being zero everywhere during 1997 and after) like that in
. The other is the use of terms representing equivalent
effective stratospheric chlorine (EESC), which represents the total amount of
chlorine and bromine loading in the stratosphere that contributes to ozone
depletion. EOF analysis is performed on EESC data sets
for multiple mean ages of air (1, 2, 3, 4, 5, and 6 years) to
retrieve two primary orthogonal functions (Fig. ) that account
for over 99 % of the EESC data variance. A simple linear term in addition to
EESC terms could also be included, but we found that it results in
pathological behavior in the tropics in extrapolated data and is thus
omitted.
Orthogonal functions used for the regression. There are four QBO
EOFs (blue [1], red [2], green [3], and black [4]), two time-shifted ENSO
orthogonal functions (blue [1] and red [2]), two time-shifted solar orthogonal
functions (blue [1] and red [2]), and two EESC EOFs (blue [1] and red [2]).
The creation of a predictor variable to represent volcanic eruptions is
ideally performed with atmospheric aerosol data. Often data from periods of
heavy aerosol loading are omitted
e.g., or a simple
functional form for an eruption is used
e.g.,, but this does not take into
account the varying times of injection, the change in rate of accumulation
via transport, or varying decay rates, which are all functions of latitude.
Other aerosol databases exist, but these are representative of total aerosol
instead of just eruptive effects and seasonality cannot be trivially removed.
A seasonal cycle in a predictor variable would alias into the seasonal cycle
in ozone. Since the seasonal variation of ozone is related to but not
entirely dependent upon aerosol, it is preferential to have a purely eruptive
term. The same is true for QBO effects or spatially varying means in the
aerosol data set. SAGE II has aerosol measurements alongside ozone
measurements, so in theory these data could be used as a predictor variable.
However, these data have noise that is autocorrelated, making them a poor
choice for use as a predictor variable. In addition, the effects of aerosol
on ozone are not purely local (i.e., chemical reactions with local aerosol)
but also dependent upon radiative effects from aerosol layers above and below
the altitude layer in question. In the end, we chose to create our own
volcanic predictor variable based on stratospheric aerosol optical depth in
the 1020 nm channel. This procedure is described in
Appendix .
In addition to predictor variables that act as proxies for geophysical
variability, a number of cross-terms (products of terms) can also be
considered. In the end, only one is included. The data used for the QBO proxy
come from equatorial data up to an altitude of
∼ 32 km. However, it has been shown that the
frequencies at which the QBO oscillates are different at higher latitudes
than at the Equator and at higher altitudes
. While the use of a proxy is better than simply using
oscillating functions of different frequencies (since the QBO changes
frequencies over time) and the use of orthogonal functions allows for the
change in amplitude and phase of the response, it cannot account for the
change in frequencies. In other words, regressing a QBO proxy from the
Equator at higher latitudes will not capture all of the variation, nor will
regressing a QBO proxy from lower altitudes at higher altitudes even at the
Equator. It has been shown, however, that the annual cycle modulates the QBO
at higher latitudes , and thus the inclusion of this
cross-term would better fit the response of ozone to the QBO at higher
latitudes. Ideally a multi-dimensional QBO proxy would be used that captures
global variability, but to the authors' knowledge, no such proxy exists.
Pretreatment of dataSAGE II observations
SAGE II was launched into a 57∘ inclined orbit and took two
measurements per orbit. Each measurement was either a sunrise or sunset as
seen by the spacecraft (spacecraft event type) and also either a sunrise or
sunset as would be seen by an observer on the ground (local event type). Most
of the time the spacecraft and local event types are the same, except
occasionally at high latitudes. Given the nature of the orbit and observation
technique, the instrument took measurements in two ground-track swaths across
the surface of the Earth in a given day: one made of spacecraft sunrises and
one made of spacecraft sunsets (Fig. ). Each swath spans
about 3 to 10∘ in latitude for high to low latitudes,
respectively, and ∼ 360∘ in longitude.
Locations of SAGE II events for a single day. Ground-track swaths
are separated by satellite event type. While local event types are typically
uniform within a swath, they can occasionally be different at high
latitudes.
Data filtering
For the purpose of this work, the same basic methodology is applied to the
same source data, with the difference being how those data are pretreated. The
process begins by extracting SAGE II version 7.0 ozone (O3) number
density profiles for all events not flagged as “dropped” in the SAGE II
inversion algorithm and for all altitudes above the reported tropopause. A
modification of the filtering criteria is applied, which
includes the following: exclusion of any profile where the O3
uncertainty exceeds 10 % between 30 and 50 km, exclusion of all data
below where the O3 uncertainty exceeds 200 % below 30 km, and
exclusion of any data where the O3 uncertainty exceeds 100 % above
30 km. These filtering criteria also include aerosol filters to
remove data within and below clouds (typically within the troposphere) and
also to remove periods of heavy aerosol interference from the volcanic
eruption of Mount Pinatubo. However, since this regression includes a
volcanic term, data within the eruptive periods are not filtered out via
aerosol criteria.
Data binning
The binning of data is one of the primary differences between the two
methods. The first method, hereafter referred to as MZM (monthly zonal mean),
is simply to take all data within a latitude zone (in this case 10∘
wide) and within a particular month and compute the mean value at each
altitude. The time associated with this mean value is the center of the month
and the latitude is the center of the zone. Different event types are not
separated for these monthly zonal means. The second method, hereafter
referred to as STS (simultaneous temporal and spatial), utilizes the data on
a daily basis for each altitude. For each day, the events are separated into
four subsets governed by the combination of local and spacecraft sunrises and
sunsets (most of the time each day only contains two subsets of events). The
mean of each subset is taken and the time associated with each mean is at the
center of the day and the latitude is the mean of the latitudes of each event
in the subset. The time of day is actually irrelevant here, since no diurnal
model is used and each daily mean is already separated by event type, which
will be treated differently as shown later.
Data averaging
We can consider, for each subset of events that we are taking a mean, that we
have a collection of N data points (Y). The daily mean and standard
deviation of the mean are simply the following:
Y‾=1N∑i=1NYi,σY‾=1N(N-1)∑i=1N(Yi-Y‾)2.
Each data point also has some uncertainty value (δ) such that
δY‾=1N∑i=1Nδi2.
Therefore, for each subset, the mean value η is simply Y‾ and the uncertainty in η is
δη=σY‾2+δY‾2.
In this way, the uncertainty in the daily or monthly mean is representative
both of the uncertainty in the measurements and the overall variance.
This is important, particularly at higher latitudes, where it is possible that one
or two measurements of a subset dip into the vortex and produce abnormally
low values that are not representative of the entire zonal band.
Regression methodology
After filtering and binning the data for both the MZM and STS methods, the following functional form is regressed to the data:
η(θ,t)=∑i∑jβi,jΘi(θ)Tj(t),
where η is the concentration of O3, Θ(θ) is the
functional form of the latitude dependence, T(t) is the functional form of
the temporal dependence, and β are the coefficients of the regression.
The concept of a two-dimensional regression is similar to the work of
. The measure of time for the purposes of regression is
year fraction (e.g., 1994.655). This regression is done separately for each
altitude. For the MZM method, Θ(θ) is identically 1, since the
data are regressed separately within each latitude zone. For the STS method,
Θ(θ) is a series of seven Legendre polynomials in spherical
harmonics (no longitudinal dependence), which have the properties of a zero
derivative at the poles and mutual orthogonality. The temporal dependence is
simply the sum of the predictor variables and a constant. However, the STS
method also includes a conditional temporal term based on the local event
type. This accounts for any differences in the mean values between sunrise
and sunset events based on diurnal variation in the data, be it geophysical or
algorithmic in origin.
A generalized least-squares regression technique, outlined in
Appendix , is applied, which accounts for
autocorrelation, heteroscedasticity, and data gaps. Due to the nature of the
data, very careful consideration is given when calculating the
autocorrelation coefficient ϕ for the STS regression method. Only daily
means with the same satellite event type can be correlated, and even then the
temporal and spatial separation must be considered due to gaps in the data.
We make the assumption that the level of autocorrelation does not change
significantly over sufficiently small separations, because the amount of
autocorrelation is related to geophysical variability that is not well
modeled, which is nearly constant over sufficiently small time and spatial
scales. Thus, any consecutive data points that are within 5 days and
20∘ in latitude are included in the calculation of ϕ. No
dependence upon the temporal or spatial separation was found within these
limiting criteria, and so a simple lag-1 autocorrelation was still considered.
In this way, a set of pairs of points was created for each event type, which
were then all fed into the calculation of ϕ (Eq. )
to determine a single value for use in the autocorrelation correction. The
autocorrelation correction is then iterated until ϕ converges to within
0.05.
To account for the iterative correction of the heteroscedasticity, it was
assumed that values of the adjustment vector k in Eq. ()
had a simultaneous spatial and seasonal dependence (or purely seasonal for
the MZM method). Regression was thus performed to a combination of the
seasonal and spatial predictor variables. The fit to these data was used as
the values of k, and this process is iterated until k converges to
values sufficiently near 1.
Residual filtering is performed using the deweighted, uncorrelated residuals
(ϵ in Eq. (), which have zero mean and unit
variance) to remove outliers in the data. In order to create a filtering
criterion, the median absolute deviation (MAD) is computed
. Values of ϵ that deviate from the median by six
MADs are omitted in future processing. Because residual filtering is
performed on the deweighted, uncorrelated residuals, only data that both
disagree with the model and are highly uncorrelated with any other data are
omitted. This process is iterated and the number of additional data points
omitted decreases rapidly after each iteration. Since the filtering is
performed with the use of MADs instead of standard deviations, an iteration
can converge to exclude no additional data points, though in practice only a
few iterations are required.
When performing regressions to data of this type, a common problem can be the
excess of predictor terms. To overcome this, one can use a priori knowledge
of what terms are and are not significant, or one can use all terms and
manually determine which terms are statistically insignificant and omit those
terms. This can, however, prove to be a tedious process, particularly when
performing the same regression repeatedly for multiple latitudes and
altitudes and with potentially several hundred terms. We instead choose to
automate this process. A predictor term could be considered statistically
significant if it is statistically different from zero at the 2σ
(∼ 95 %) level. In other words, if the ratio of the
1σ uncertainty in a coefficient (σβi,j) to the
coefficient itself (βi,j) is less than 0.5, it can be considered
significant. After the residual filtering process is completed, an analysis
of this ratio is done. During the first iteration, all coefficients with a
ratio greater than 8 are omitted from future processing and the entire
process is repeated (from the beginning). This threshold is iteratively
lowered until only those coefficients that are statistically significant at
the 2σ level remain. In this way, all final coefficients are
statistically significant at the 2σ level. This does not, however,
mean that all resulting terms are non-negligible. However, excluding predictor
terms that are negligible (i.e., their contribution to the overall variance
is minimal) is, in practice, unnecessary. Additionally, while each
coefficient is statistically significant at the 2σ level, groups of
terms (e.g., piecewise linear slopes or EESC) can collectively become
statistically insignificant depending upon their interactions.
Regression quality and resultsResidual analysis
A quick look at some examples of the fit (Fig. ) shows that
the algorithm works well, with the exception of potential overfitting in the
regions of data gaps. However, as with any regression technique, great care
must be taken in interpreting the results. The process is ultimately a
numerical one, and just because the solution converges and yields a result
does not mean that result is accurate. As such, a proper investigation of the
total residuals is still required to ensure that the data and model
reasonably agree. If the data and model agree well, one would expect no
systematic biases to emerge in the residuals. A quick look at the total and
uncorrelated residuals (Fig. ) shows this to be the case. The
total and uncorrelated residuals show no systematic bias with respect to time
or latitude. However, a clear pattern in the spread of the residuals can be
seen as a function of latitude, though this is expected. The total residuals
are a combination of the correlated and uncorrelated residuals. Correlated
residuals (or those removed from the lag-1 autocorrelation correction)
represent geophysical variability that is well sampled but not well modeled.
Uncorrelated residuals (or those that remain after the lag-1 autocorrelation
correction) represent instrumental noise present in the data as well as
geophysical variability that is not well sampled, which mathematically
represent the heteroscedasticity in the data.
Some examples of the STS regression. Data within the stated latitude
bin are shown in blue, while a fit at the center of the bin is shown in red.
The data shown have autocorrelation and diurnal variation removed for the
purposes of plotting.
The residuals from the regression can be used to ascertain the quality of the
model and the data set itself, independent of any offset in the mean value.
Since the mean of the residuals is nearly zero (as it should be), the spread
of the residuals is used instead. To avoid the over-influence of outliers in
the data, a weighted running mean of the absolute value of the residuals as a
function of latitude at a particular altitude is taken. The result is shown
in Fig. . Analysis of the uncorrelated residuals reveals the
amount of uncertainty in the data that are being regressed (in this case,
daily means). The uncorrelated residuals will increase in the presence of
increased noise in the instrument data (e.g., at higher and lower altitudes)
or in the vicinity of increased geophysical variability within a daily mean
(e.g., in the tropics, where each daily mean spans a greater range in
latitude, or at high latitude in the local winter, where measurements may dip into and
out of the vortex), but the attribution of this uncertainty to each source
separately cannot be determined. However, the contribution to the uncertainty
from unresolved geophysical variability could be minimized by applying the
regression model to the data at their native resolution. The difference between
the total and uncorrelated residuals are the correlated residuals. These
correlated residuals are a measure of the discrepancy between the model and
the data. Namely, they are a result of the geophysical variability that is
well sampled but not well modeled as well as any instrumental variability
(e.g., biased meteorological or ephemeris input data). Residual analysis is
useful, because applying the same model to different data sets could be used
to independently assess the quality of the measurements via the uncorrelated
residuals, as well as to ascertain deficiencies in the model or the data
itself via analysis of the total residuals. A preliminary version of this
technique was applied in to both the previous (v6.2) and
current (v7.0) versions of the SAGE II data set to demonstrate and assess
improvements made to the new version.
Total and uncorrelated residuals (as percentages) as a function of both
time and latitude at 25 km from the STS regression (see
Eq. ). Results are similar at other altitudes, though
scales may change.
Predictor coefficient analysis
One of the problems with multiple linear regression is the issue of
multicollinearity, or the possibility that two or more predictors are highly
correlated. Multicollinearity, or even the presence of any correlation
between predictors, does not affect the regression results as a whole (short
of the possibility of poor inversion for numerical algorithms), but it does
affect the interpretation of individual predictors. For example, if aerosol
data were used (instead of just a volcanic proxy) that had an annual cycle in
addition to the fitting of an annual cycle term, the annual cycle term would
be biased because some of the annual variation in ozone would be attributable
to the aerosol term. Fortunately, this effect is captured in the
uncertainties in the predictor coefficients, but it still illustrates a
problem when attempting to interpret single-predictor coefficients. One could
analyze the covariance matrix that results from the regression to determine
the level of correlation between predictors, but care should nonetheless be
taken when interpreting results.
Due to these possible shortcomings, the analysis of any single predictor
requires the analysis of all of the predictors in order to ensure they are
reasonable. The annual cycle is fairly trivial with the exception of some
overfitting in regions of missing data (Fig. ) and ENSO lacks
any substantial contribution above 20 km (not shown). Since long-term
trends are discussed later, this section will take a brief look at the
remaining influential terms: QBO, solar cycle, and volcanic. It has been
shown that SAGE II data quality is best between 20 and 50 km and there are fewer gaps in sampling below 60∘ in
latitude. As such, the following analysis will focus on this region.
Spread of the total and uncorrelated residuals as a function of
latitude and altitude from the STS regression. White regions show areas where
insufficient data exist, generally due to being in the troposphere or early
profile termination during retrievals.
QBO
After the annual cycle, the QBO is the largest source of variation in ozone.
Figure shows the amplitude of the response of ozone to the
QBO as a function of latitude and altitude as a percentage of the local mean
value (i.e., the mean value is a function of latitude and altitude). The
amplitude is computed as the root-mean-square amplitude multiplied by
2, and is analogous to half of the peak-to-peak amplitude for a sine
wave. As can be seen, the influence of the QBO is largest in the tropics in
the lower and middle stratosphere as well as in the middle stratosphere at
midlatitudes. Figure shows examples of the QBO term at the
Equator as a function of time and altitude and at 23 km as a function
of time and latitude. The altitude-dependent and latitude-dependent phase
lags are easily noticeable in the two figures. However, it should be noted
that some deficiencies still remain due to the fact that the QBO proxy term
originated from data at the Equator. The frequencies at altitudes above where
the proxy is available do not change and the frequencies at midlatitudes are
slightly different only because of the inclusion of a cross-term with the
annual cycle. It should also be noted that the amplitude of the QBO is larger
around the time of the Pinatubo volcanic eruption, which may be a physical
response of the QBO itself to the Pinatubo eruption , or it
may simply be a byproduct of correlation with the volcanic term.
Additionally, the fact that the regression is both temporal and spatial means
that the inability to accurately model the QBO at higher latitudes will
detract from the ability to accurately model the QBO at lower latitudes.
Amplitude of oscillation of the QBO term as a percentage of the local mean.
Examples of the QBO term. Contour lines are plotted at intervals of 2 %.
Amplitude of oscillation of the solar term as a percentage of the
local mean for the use of one and two solar proxy terms while including a
volcanic term. Contour lines are plotted at intervals of 0.5 %.
Solar
To include a response of ozone to the solar cycle, the regression model can
include either one or two solar predictor variables. The biggest differences
between one or two terms are seen in the tropics between 25 and
35 km. The amplitudes of the oscillation are similar, as shown in
Fig. , with the exception of a very weak oscillation in this
region if only a single term is used. This is because, when two terms are
used, the solar term, if allowed to change phase, exhibits strong
correlations with the volcanic term in this region as shown in
Fig. . Here, the solar cycle is shifted later in phase by
about 2 years to coincide with the peak of volcanic increase surrounding
the Pinatubo eruption, which is similar to results shown in
.
The inclusion of a volcanic term reduces the overall residuals regardless of
whether one or two solar terms are included (not shown), but there is a
negligible difference in residuals and resulting trends between the use of
one or two solar terms if a volcanic term is also used. It is unclear whether the
response of ozone to the solar cycle really does lag by about 2 years in
the mid-stratosphere in the tropics or whether the algorithm is simply trying to
attribute some of the response of ozone to aerosol to the solar cycle instead
.
Examples of the solar term for the use of one and two solar proxy
terms while including a volcanic term. Contour lines are plotted at intervals
of 0.5 %.
Volcanic
The results of the volcanic term need to be interpreted very carefully. On
the one hand, it is clear from the data that ozone responds to changes in aerosol,
particularly after Pinatubo. On the other hand, the SAGE II inversion
algorithm can produce biases in ozone in the presence of high aerosol loading
, and so some of the response to aerosol, particularly at
lower altitudes in the tropics, can have algorithmic rather than physical
origins. However, omitting data based on aerosol extinction
e.g., and assuming that the influence of aerosol has
been removed would be incorrect.
A look at the response of ozone near the Pinatubo eruption reveals both
physical and algorithmic responses as well as regressive responses.
Figure shows the peak of the volcanic term in the few
years after the Pinatubo eruption as a percentage of the mean. Ozone shows a
positive response above 28 km with a corresponding negative response
just below that in the tropics, which are similar to results in
. These responses can be the result of local (i.e.,
chemical) effects of aerosol, radiative (e.g., thermal and/or photochemistry)
effects of aerosol at other altitudes, or algorithmic responses of ozone to
aerosol retrievals. The anomalously large responses at low altitudes are the
result of overfitting to data gaps (e.g., Fig. top).
However, given the results from the QBO and solar terms, some correlation
between these terms exists. Regardless of these correlations, however, it is
clear that ozone does respond to changes in aerosol in SAGE II data and that
the use of a volcanic term in these regressions is necessary.
SAGE II sampling and biases
As previously mentioned, SAGE II took ∼ 30 observations per
day in two ground-track swaths that each span 3 to 10∘ in
latitude and ∼ 360∘ in longitude
(Fig. ). This sparse sampling caused SAGE II measurements at
a particular latitude to occur at roughly the same times of the year,
resulting in full seasonal coverage at midlatitudes, and restricted (or
sparser) seasonal coverage at high (or low) latitudes.
Figure shows SAGE II sampling at both the beginning and
end of the mission. Sampling is sparser at the end of the mission due to
problems with the azimuth pointing system forcing the instrument to operate
at 50 % duty cycle starting in late 2000. The increased spread during the
later period is a result of an increased rate of precession of the orbit.
This demonstrates a form of potential bias due to sampling present throughout
the mission, though more pronounced in the later period, where the orbit
crosses a particular latitude but at progressively earlier times each
successive year. If the sampling were constant over the lifetime of the
instrument, it would only result in biases in the MZM seasonal cycle.
However, because the sampling drifts over time, this bias also aliases into
the MZM long-term trend.
Peak of volcanic term near the time of Pinatubo as a percentage of
mean. Anomalously high values at lower altitudes are the result of
overfitting to gaps in data as shown in Fig. .
Locations of daily means for each satellite event type (sunrises are
blue and sunsets are red) at the beginning and end of the mission.
Given the nature of this sampling, another potential problem could clearly
arise if any difference existed between the mean values of sunrise and sunset
events. Figure illustrates the differences in the means of
local sunrise and sunset event types. These differences can be the result of
geophysical variability and/or algorithmic biases .
Regardless of the source, however, these differences are present in the data
and must be accounted for in the regression.
Due to this nonuniform sampling, every monthly zonal mean value computed for
ozone is biased. The primary reason is that the data sampled within a given
month and zonal band has a mean sampling time and place that is not at the
exact center of the month and zone. The true spatial and temporal center of
each monthly zonal band can be computed and values can be extracted from the
results of the STS regression method. These values can then be compared to
the results of the MZM regression method to produce Fig. .
Figure shows the spatial and temporal dependence of the
bias (i.e., how the MZM method is biased compared to the STS method) at two
altitudes. Since the MZM method does not differentiate between event types,
the bias is computed against the STS method for each type, showing how the
MZM method is biased against each type, but only for where data of that type
exist. As can be seen, biases in individual monthly zonal bands exist as
large as 10 % due to nonuniform spatial and temporal sampling, and large
systematic biases exist at higher altitudes due to differences between event
types. Large gaps in sampling that are asymmetric in event type (both in
location and bias) are seen later in the mission, illustrating the problem
with not accounting for the differences in sampling and event type in the
regression.
Results of the local event type piecewise term from the STS
regression plotted as the percent difference between sunrise and sunset
events.
Long-term trends
The primary focus of time-series analysis of long-term ozone data sets is
typically the long-term trend of ozone. Most often this has been done using
two piecewise linear trend terms joined at some predetermined time. The
regression is performed four different ways utilizing the combinations of the
MZM and STS regression methods and two piecewise linear trend terms or two
orthogonal EESC trend-like terms. The resulting mean trends are computed both
for the traditional decrease in ozone (in this case between 1985 and 1995)
and for the traditional increase in ozone (in this case between 1998 and
2005) for all four analyses. The results for the earlier period are shown in
Fig. and the later period in Fig. .
At first glance, the four plots in Fig. seem very similar.
Each plot shows regions of significant decreases in ozone between 35 and
50 km at middle to high latitudes, as well as some slight positive
trends in the tropics at lower altitudes. However, there are some important
discrepancies to point out. The MZM method, regardless of which pair of trend
terms is used, shows a slight positive trend in the tropics between 30 and
35 km, which is consistent with other studies
e.g.,, though those
studies show this increase to be statistically insignificant. However, the
use of the STS method removes this feature, regardless of which trend terms
are used. In addition, the magnitude of the trends, when using the same trend
terms, is biased slightly negative for the MZM method compared to the STS
method. This is a result of the biases from nonuniform sampling, and is
explained in more detail in the next paragraph.
Biases for each monthly zonal mean for the MZM method compared to
the STS method computed as (MZM-STS)/STS⋅100. The MZM method does not
differentiate between event types, so it does not have different values for
each type. White areas show regions where data do not exist or where one or
both regression methods failed to converge to a solution.
Mean trend between 1985 and 1995 for four analysis scenarios
computed as (Tr(1995)-Tr(1985))/Tr(1985)⋅100, where Tr(t) represents the value
of the long-term trend term at time t. The analysis was run for both the MZM
and STS regression methods, each using either a piecewise linear term joined
at 1997.0 or two orthogonal EESC terms. Stippling shows regions where the
linear slope is not significant at the 2σ level. No similar
calculation can be done for multiple EESC terms.
The results shown in Fig. are very different. Whereas the
results of the MZM method are consistent with other studies
e.g.,which make use of multiple data sets extending to
2013, showing regions of large ozone recovery in
the Southern Hemisphere and smaller recovery in the Northern Hemisphere, the
results from the STS method show a significant increase only in the Northern
Hemisphere, which is slightly smaller than in the MZM method. To understand
this difference, one should take another look at Fig. and
Fig. . In the Southern Hemisphere at midlatitudes before
the pointing problems in late 2000, the concentration of sampling shows a
roughly equal mix of sunrise and sunset events. Given the difference in ozone
between sunrise and sunset events in this region at higher altitudes, the
mean of the data is expected to be somewhere in the middle of the sunrise and
sunset mean. However, later in the period, there is a significant decrease in
sunrise events in this region, which results in the mean of the data skewing
more towards the sunset mean. With the beginning of the potential recovery
period starting at the overall mean, and the end of the calculable recovery
period residing at the sunset mean, the computed trend is artificially biased
high. Proper treatment of the differences between sunrise and sunset events
accounts for this effect and results in smaller recovery trends in the STS
method.
Mean trend between 1998 and 2005 for four analysis scenarios
computed as (Tr(2005)-Tr(1998))/Tr(1998)⋅100, where Tr(t) represents the value
of the long-term trend term at time t. The analysis was run for both the MZM
and STS regression methods, each using either a piecewise linear term joined
at 1997.0 or two orthogonal EESC terms. Stippling shows regions where the
linear slope is not significant at the 2σ level. No similar
calculation can be done for multiple EESC terms.
Some examples of the long-term trend term computed using the STS
regression. The term in the top left comes from the use of a piecewise linear
term, while the other three come from the use of two orthogonal EESC terms.
Contour intervals are 1 %.
It is clear that the differences caused by the SAGE II nonuniform sampling
are important, and that the STS method is preferable to the MZM method
regardless of which pair of long-term trend terms is used. However, there are
still some differences in trends between the two pairs of trend terms as
shown in Figs. and . To understand this,
one needs to look at the time evolution of the long-term trend for the use of
each pair of trend terms. Figure illustrates the difference
between the two pairs of trend terms at 50∘ N. The piecewise linear
trend terms force any turnaround in ozone to occur at 1997, while the use of
the two orthogonal EESC terms allows this turnaround point to move in time.
As shown in Fig. (top right), the turnaround time is earlier
at lower altitudes. This is consistent with the fact that stratospheric ozone
is inversely related to stratospheric chlorine, and the EESC proxies from
show that the EESC peaks earlier for smaller mean ages of
air. Figure would thus suggest that the mean age of air in
the Northern Hemisphere decreases with decreasing altitude, which is
consistent with results shown in .
The study outlined in is performed only for the Northern
Hemisphere and the assumption is made that the hemispheres are symmetric.
However, the time evolution of the long-term trend at high southern latitudes
(Fig. , bottom right) shows no clear change in turnaround time
with altitude, and in some cases never turns around (i.e., is always
decreasing). It is, at present, unclear whether this hemispheric asymmetry is
geophysical or a result of correlation between the long-term trend and other
terms (e.g., solar or volcanic).
Conclusions and future work
A new method for performing
time-series analysis of sparsely sampled data, in particular SAGE II, has
been presented. The differences between the MZM method and the STS method
have been discussed and the impacts on the long-term trends in ozone
detailed. It has been shown that the nonuniform sampling in SAGE II data
will produce biased long-term trend values in ozone if not properly accounted
for. The STS method shows declines in ozone that are similar to those from
other studies in the upper stratosphere at middle to high latitudes but very
different results for the presumed recovery period, namely a noticeable
reduction in the magnitude of ozone increase in the Southern Hemisphere. The
use of two orthogonal EESC predictor variables instead of a piecewise linear
trend allows for a variable turnaround time in ozone due to differing mean
ages of air. Results show a hemispheric asymmetry in the middle to upper
stratosphere, with an earlier turnaround time with lower altitude and
latitude in the Northern Hemisphere but no coherent pattern in the Southern
Hemisphere. It has also been shown that the STS method can be used to assess
the quality of a data set's measurements independent of other data sets. In
addition to ozone, the STS method was applied to SAGE II aerosol optical
depth data to create a new volcanic proxy that covers the SAGE II mission
period.
For future work, we would like to extend this technique to other ozone data
sets and also include multiple data sets to better constrain the
long-term trends in the presumed recovery period. The benefit of this
technique for the creation of a single time series derived from multiple data
sets is that it does not require the homogenization of the different data
sets prior to regression. Instead, instrument-dependent conditional terms
representing mean offsets, different diurnal variation, time-dependent
drifts, or other terms could be included as necessary. Another consideration
is to expand upon the creation of a volcanic proxy term to one that is
altitude dependent, so that the response of ozone to both local (i.e., at the
same altitude) and total aerosol can be assessed. Lastly, it could be
beneficial to experiment with other coordinate systems in order to reduce
uncertainties in regions of larger variance. For example, regression could be
done on the data at their native resolution, using models for diurnal and
longitudinal variation, as this would reduce some of the variance in the
tropics, where each daily mean spans ∼ 10∘ in latitude,
as opposed to ∼ 3∘ in latitude at high latitudes.
Another coordinate transformation would be to perform this regression
methodology on equivalent latitude instead of latitude, as it would remove
much of the variance at high latitudes, where observations constantly dip into
and out of the polar vortex.
The creation of a volcanic proxy term is achieved via the simultaneous
temporal and spatial regression to SAGE II aerosol data. This is done using
the same STS regression technique as is done for ozone. The process begins by
extracting 1020 nm aerosol extinction coefficient profiles. During
periods of high aerosol loading, SAGE II profile retrievals stopped at
altitudes well within the stratosphere. To compensate, data below retrieval
termination are filled in via the process outlined in Chapter
4.3.1. Each profile is then integrated from the top
(40 km) down to 3 km above the reported tropopause. These
event-specific stratospheric aerosol optical depth values are then compiled
into daily means, except that no distinction is made between local event
types as no significant diurnal cycle in aerosol is seen. The regression uses
the same latitudinal dependence (albeit with 11 terms) and a temporal
dependence that includes annual, QBO, and eruptive terms for each significant
volcano during the SAGE II mission. The iterative regression technique
outlined in this paper is applied, though the data that is regressed is the
logarithm of optical depth. The logarithm is used instead of the raw data
because many physical effects, such as the annual cycle or QBO, are
inherently multiplicative effects (i.e., their magnitudes are related to the
magnitude of the instantaneous mean). The same is also true of ozone, but
ozone does not vary by several orders of magnitude over time at a particular
altitude. The primary reason for this appendix, however, is the difficulty
regarding the creation of the eruptive terms for each volcano used as
predictor variables in the regression.
The creation of a model term for use with linear regression that accurately
represents changes in aerosol as a result of a volcanic eruption is not
trivial. At any given location after a major volcanic eruption, changes in
aerosol are characterized by a delay after the eruption (i.e., time it takes
for aerosol to reach that location from the eruption), followed by an
increase in aerosol up to some peak value over time, followed lastly by a
long decay back to background levels (unless another eruption occurs). It
makes sense to create a piecewise function to model this rise and fall, but
the choice of these functions is important. Previous attempts
e.g.,Chapter 5.4.2 use a simple polynomial from eruption
to peak values followed by an exponential decay with some characteristic
decay constant. However, a simple exponential decay model would assume that
the data, when plotted in log space, are linear, which they clearly are not (see
Fig. ).
While analyzing the logarithm of the aerosol data, we choose a piecewise pair
of second-order polynomials in order to fit the eruptive effects on aerosol.
However, the two functions are restricted to maintain continuity of both the
functions and their derivatives where they join, as well as to assume the
eruptive term returns to zero (i.e., background) after some amount of time
has elapsed. In this way, a pair of piecewise second-order polynomials can be
defined by the declaration of three parameters: time of injection (tI or
time aerosol first arrives at a location after an eruption), peak time
(tP or time after injection at which aerosol values are at their peak),
and return time (tR or time after injection at which the eruptive term
and its derivative return to zero). The time at which these two functions
join is also constrained by these three parameters.
The downside to this methodology is that the functional form of the eruptive
term is not linearly dependent upon the parameter times (tI, tP,
and tR). Additionally, the times themselves are functions of latitude
and different for each eruption. Since we have no intrinsic knowledge of the
value of these times, or their spatial dependence, a nonlinear least-squares
fitting technique is applied to binned data. The process begins with data
taken in a 10∘ wide bin in latitude centered at a particular latitude.
Initial guesses are made for the three parameters for each of seven volcanic
eruptions: El Chichón (1982), Nevado del Ruiz (1985), Kelut (1990), Pinatubo
(1991), Cerro Hudson (1991), Ruang (2002), and Manam (2004). The MPFIT
algorithm is used, as it allows for restrictions on solvable
parameters to be placed, which greatly aids in convergence. Too few data are
available to constrain tI or tP for El Chichón, so these parameters
are tied to Pinatubo. Likewise, tR for Manam was set constant at 5 years.
Some examples of the nonlinear regressions can be seen in
Fig. .
Some examples of the nonlinear fit to aerosol at three different
center latitudes (35∘ N, 5∘ N, and 45∘ S). Data
within each band are shown in blue, while fits are shown in red. The vertical
lines denote the times of the seven eruptions used for the fit (El Chichón is
off scale). Note the difference in rates of rise and peak times at different
latitudes, most easily visible for Pinatubo.
The volcanic term resulting from the STS regression to stratospheric
aerosol optical depth in the 1020 nm channel. This is the term used
as a volcanic proxy term for the regression to ozone data. Relative optical
depth means relative to background values.
This nonlinear regression was performed for each latitude between 70∘ S and
70∘ N in increments of 1∘. The parameters (tI,
tP, and tR) were then smoothed, and any iterations that did not
converge properly were ignored. To create the eruptive terms for the STS
regression, the parameters were interpolated (or held constant at last value
for extrapolation) to all latitudes to create functional forms for each
volcano for the entire record. These eruptive terms are then easily linearly
regressed to, where the STS regression is allowed to determine spatial
dependence but not temporal variation for each volcano as a function of
latitude. The final product to be fed into the regression for ozone is a
single volcanic term that represents the eruptive changes in aerosol at all
times during the record for all latitudes (Fig. ).
Ultimately the creation of a volcanic proxy is an empirical result.
Theoretically, one could look at the parameters from the nonlinear least-squares regression individually, but most of the terms would have no real
meaning. For example, there are a large number of volcanic eruptions around
the time of the eruption of Nevado del Ruiz. While the parameters near the
eruption make sense, the parameters at higher latitudes merely represent the
algorithm's attempt to fit the overall increase in aerosol from a multitude
of eruptions in that time period (e.g., tP∼ 2 years and
tR∼ 8 years). The regression fits the overall data
well, but each individual term is not necessarily representative of that
eruption alone.
The principle of multiple linear regression is predicated upon the simple
assumption that a dependent variable (Y) is linearly dependent upon a set
of predictor variables (X) that produce a simple equation of the following
form:
Y=Xβ+R,
where Y is a vector of N data points (index i),
β is a vector of coefficients for M predictor variables
that include a constant (index j), X is an N by M matrix of
each predictor variable corresponding to each data point, and R is a
vector of N residual differences between the data and the fit. It should be
noted that generally Xi,j=0 is identically 1 for all i from 1 to N
and βj=0 is simply the overall constant of the fit. These
coefficients can be solved for using a simple ordinary least-squares (OLS)
regression technique, which can be found in any of a number of textbooks
related to statistics. In fact, the methods outlined in this appendix derive
from . The uncertainties in the coefficients
(σβ) can also easily be solved for provided the
following assumption holds:
var(Y)=σ02I,
where var(Y) denotes the variance–covariance matrix of
Y, σ0 is a constant, and I is the identity
matrix. If this assumption holds, then the residuals have the property of
being Gaussian with a constant conditional variance of σ02.
If the assumption of Eq. () does not hold, then Eq. () reverts to a general form:
var(Y)=Σ0.
This produces coefficients that are still unbiased, but the estimates of
their uncertainties are biased small. To overcome this problem, we turn to
generalized least squares (GLS), which applies a transformation matrix
G to Eq. () to obtain
Y*=X*β+R*.
If G=σ0Σ0-12 (where
Σ0-12Σ0-12′=Σ0-1),
then R* has the property of being Gaussian with conditional variance
σ02. The coefficients and their respective uncertainties can be
computed in the following way:
β=(X′Σ0-1X)-1X′Σ0-1Y=(X*′X*)-1X*′Y*,var(β)=(X′Σ0-1X)-1=σ02(X*′X*)-1.
It follows that σβj is the square root of the jth
diagonal element of Eq. (). It is worth noting that, when
solved explicitly using Σ0, the values of the
coefficients and their uncertainties are invariant to the value of
σ0, but when a transformation of variables is used, the equations
revert to the form of solutions from OLS regression.
In time-series analysis it is often the case that the assumption of
Eq. () does not hold. The residuals are, in fact, both
heteroscedastic (have a nonconstant variance) and serially correlated
(temporally autocorrelated). If we assume the residuals have the following
properties,
Ri=ϕRi-1+σiϵi,
where ϕ is the autocorrelation coefficient, Ri are the total
residuals, ϕRi-1 are the correlated residuals,
σiϵi are the uncorrelated residuals, and
ϵ is Gaussian with unit conditional variance, then
Eq. () reverts to Eq. (), where
Σ0 is an N by N symmetric matrix with components
of the following form:
Σ0i,j=σiσjϕ|i-j|1-ϕ2,
where, for this case, j goes from 1 to N. Computing G leads to the following transformation of variables:
Yi*=Yi-ϕYi-1σi,Xi,j*=Xi,j-ϕXi-1,jσi,Ri*=Ri-ϕRi-1σi=ϵi.
This is just the Cochrane–Orcutt transformation , which
ignores the first data point. The Prais–Winsten transformation
can be used to include the first data point and an
additional modification outlined in can be used to
account for data gaps. It should be noted that, when performing OLS
regression to the transformed variables, it will be necessary to force
regression about the origin if using a packaged algorithm that performs
regression and always assumes a constant exists in the regression.
If the heteroscedasticity and autocorrelation are known precisely and
everything about the regression is perfect, then theoretically
σ0=1. In reality this is never the case. A good estimate of
σ0, however, can be obtained from the weighted mean-square error
(also known as the reduced, weighted chi-squared error statistic):
σ02=ϵ′ϵN-M.
It is important to compute σ0 so as to not underestimate the
uncertainties in the coefficients in Eq. () when regressing to
transformed variables.
In theory, one would want to know a priori what the values for ϕ and
σ are. Instead, these parameters are solved for
iteratively towards convergence. The value for the autocorrelation
coefficient is solved for by first performing OLS regression and then
computing ϕ in the usual manner:
ϕ=∑i=1N-1(Ri-R‾)(Ri-1-R‾)∑i=1N(Ri-R‾)2,
which is itself a simple modification of the Pearson product-moment correlation coefficient:
ϕ(X,Y)=∑i=1N(Xi-X‾)(Yi-Y‾)∑i=1N(Xi-X‾)2∑i=1N(Yi-Y‾)2.
As can be seen, X and Y have been substituted with adjacent values of the
residuals as well as a slight modification of the limits of summation to
account for the number of pairs versus the number of total points.
The nature of the heteroscedasticity can be slightly more complicated. In
practice, one only has an estimate for the heteroscedasticity
(δ) such that
σi=δiki.
If the initial guess of the heteroscedasticity is correct, then k is
identically 1. However, generally k is more complex, having a
dependence on the predictor variables themselves. A practical way to solve
for k is to first assume that σ=δ
and solve for ϵ. If k=1, then the mean value of
ϵ2 should also be 1. As such, one can regress a
function f=log(ϵ2) to predictor variables and
obtain a fit value (fi) for each ϵi, then
ki=efi. In this way, σ can be
iteratively updated until k converges towards 1. However, the choice of
predictor variable dependence of k may or may not be straightforward.
From a practical standpoint, this regression methodology is applied by first
performing the regression (Eq. ) with the assumption that
there is no autocorrelation (ϕ=0). The resulting residuals are used to
compute the autocorrelation coefficient (Eq. ) and the
regression is repeated. The heteroscedasticity correction
(Eq. ) can then be applied. This process of applying the GLS
regression, applying the heteroscedasticity correction, and recomputing
ϕ can be iterated towards convergence of ϕ. Any residual filtering
to be performed would require iteration of everything performed thus far. If
filtering of regression coefficients is desired, it too would require an
additional level of iteration of all steps performed thus far (including
residual filtering).
Acknowledgements
The ongoing development, production, assessment, and analysis of SAGE data
sets at NASA Langley Research Center is supported by NASA's Earth Science
Division.
Edited by: D. Loyola
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