Introduction
The ozone layer in the stratosphere protects the Earth's biosphere from
harmful solar ultraviolet (UV) radiation. The use of ozone-depleting substances
(ODSs), including chlorofluorocarbons (CFCs), led to a decline in ozone
globally over the latter half of the 20th century , particularly in the polar regions
. The implementation of the Montreal Protocol (MP),
which banned the use of most ODSs, has halted this decline, and in some
regions there has been a recovery in total column ozone
. However, there is large uncertainty in the sign and
magnitude of recent trends depending on altitude and latitude, and a clear
signal is difficult to determine .
Ozone responds to forcings from below, e.g. injections of aerosols from
volcanoes or wave activity from the troposphere
, and from above, e.g. from solar sources such as UV
radiation and particles . In order to quantify and understand the variability
forced by a particular driver, and long-term trends in ozone – not just in
terms of the total column ozone (TCO) but also resolved vertical profiles –
observations spanning multiple decades are needed. Such a dataset can only be
provided by combining data from multiple sources . The method used to combine the data needs to consider
different inherent attributes, the most important of which include temporal
resolution, vertical and horizontal spatial resolution
, time of day and geolocation of observations
, absolute calibration
, and stability estimates and instrument
uncertainty . All of these factors, if not well
accounted for, can introduce additional (artificial) trends, uncertainties,
and errors, which may leak into statistical analyses of decadal trends
and estimates of the magnitude
of the response to drivers such as the Sun . This
can lead to conflicting results from different datasets .
Observational records of atmospheric ozone began with ground-based
observations in 1921 and were joined by satellite
observations in the 1960s . These records are an
invaluable tool to understand not only the long-term trends in ozone but
also how the middle atmosphere operates. Ground-based observations have the
advantage of being longer records and can be recalibrated on a continuous
basis, but they are point-source observations and thus cannot account for
large differences in ozone concentration and variability with latitude and
longitude. The introduction of satellite observations has allowed for
near-global, continuous observations over many decades but has the
disadvantages of typically only operating for a limited number of years and
being subject to space-based degradation.
Creating an accurate record of stratospheric ozone profiles is a non-trivial
task and much work has been done at every stage, from design, construction,
and operation during flight, to post-processing and combining datasets into composites
. Recently,
several composites were published by multiple groups in connection with the
SI2N initiative (SPARC (Stratosphere-troposphere Processes And their Role in
Climate)/IO3C (International Ozone Commission)/IGACO-O3 (Integrated Global
Atmospheric Chemistry Observations – Ozone)/NDACC (Network for the
Detection of Atmospheric Composition Change)) .
Nevertheless, even when problems are flagged and uncertainties are
minimized, the fact that different composites can lead to trend estimates that
differ by more than their uncertainties (e.g. Fig. 6 of
, and Fig. 8 of
, ) means that at
least one, if not all, are insufficiently stable during some periods to
provide a robust estimate of changes in ozone throughout the stratosphere.
further notes that the choice of instruments to
merge has more impact on trends than the merging technique used, that the
construction approach needs careful consideration of the method used to avoid
contaminating trends with artefacts, and that so far it has not been possible
to remove biases from any individual, vertically resolved dataset.
Despite these difficulties, it is possible to account for many of these
problems. There is common information within all the composites, e.g. the
annual variability is similar in most composites ,
and the differences between composite datasets due to the issues listed above
should, in principle, point to where potential artefacts such as steps and
drifts are located in time and by latitude and altitude. This can be
especially effective in the case of an unexpected or erroneous change
occurring in one dataset, which is absent in all the others. Once the
instrument or composite at fault is identified, there is the possibility of
flagging, removing, or rectifying an error, and confidence in applying a
correction increases if the deviation or fault can be linked to a known
issue. Thus, together with this prior knowledge and an unbiased uncertainty
estimate, one can evaluate the likelihood of an observation being correct or,
indeed, estimate the most likely value.
Our goal here is to provide a technique whereby the most likely ozone
variability throughout the stratosphere can be identified by using the
information embedded within multiple datasets simultaneously. The natural
approach with which to tackle such a problem is using Bayesian inference
. In adopting a Bayesian
approach, we develop a detailed probabilistic model for the (multiple)
datasets, carefully allowing for outliers and accounting for all knowledge
(and ignorance) of measurement uncertainties and any known problems during
instrument operation. Additionally, by incorporating (data-driven) prior
information about the underlying ozone variability, we are able to identify
– using only the data and knowledge of the instruments – where some datasets
are systematically biased due to measurement artefacts whilst others are
consistent with the anticipated month-to-month variability. In this way, our
approach combines the multiple datasets in such a way that they
“self-calibrate” each other, resulting in a single ozone time series that is
cleaned of many of the artefacts affecting any individual dataset (although
if a problem is common to all datasets, it cannot be identified).
This paper has three main parts. In the first part (Sect. ), we
introduce the composite datasets we use (Sect. ) by
explicitly presenting the problems we will later attempt to fix. Ozone
composites have been updated since important intercomparison papers by
and , so our results cannot
be directly compared with theirs; we briefly present some of these
differences (Sect. ). The ozone composites, described in
Sect. , form a good starting point from which to combine
information and account for differences, since the effort put into producing
them already considers and accounts for many instrument and observational
issues. However, some remaining problems are clear in the composites. In the
second part, we present the Bayesian method to self-correct the ozone
composites (Sect. ), construct uncertainty estimates
(Sect. ), form the Gaussian-mixture likelihood
(Sect. ), develop transition priors to estimate how ozone
is expected to vary on monthly timescales (Sect. ), and
discuss how we include additional prior information that we have available
(Sect. ). We call this combined set of steps and
algorithms the BAyeSian Integrated and Consolidated (BASIC) approach. The
resulting BASIC composite time series are presented and compared with the
composites in Sect. and . In the final part
(Sect. ), we primarily use dynamical linear modelling (DLM) to
evaluate long-term trends (Sect. ), although we compare our
results with multiple linear regression (MLR) analysis, and present our
results for ozone changes over the 1985–2012 period in Sect. .
We conclude in Sect. .
Bayesian inference of the underlying ozone time series
We want to combine the information from the various
composites and correctly account for uncertainties, artefacts, and drifts. To
this end, we adopt a Bayesian approach to infer constraints on the
(unknown) true time series, y, given the full set of data,
d. The data consist of nc composites, indexed by c;
each composite is made up of nt measurements, indexed by t. A
single measurement is hence dt,c, where the index ordering is chosen to
match that required for the matrix manipulations used in
Sect. . The underlying time series that is to be
inferred, y, hence has individual elements yt.
Bayesian inference necessarily involves conditioning on our knowledge about
uncertainties and potential artefacts and drifts, and any prior assumptions
about the month-to-month variability, through our model which we denote as
M. Bayes's theorem allows us to combine this information in the form of
the posterior distribution of the true time series given the data, model, and
any prior information P(y|d,M):
P(y|d,M)=P(y|M)P(d|y)P(d|M),
where P(y|M) encodes our prior information and assumptions about
the month-to-month variability of the underlying true time series, the
likelihood P(d|y) summarizes our probabilistic model
for the data given the associated measurement uncertainties (including our
knowledge and assumptions about the possibility of instrumental artefacts
systematically biasing the observations at certain times), and the marginal
likelihood P(d|M) in this situation just plays the role of a
normalizing constant.
In order to form the desired posterior distribution, we require a
probabilistic model for the data (Sect. ) that incorporates
our knowledge and assumptions about the observational uncertainties
(Sect. ), and a clear statement of our prior assumptions
(Sect. ). The resulting posterior density is a
high-dimensional probability density over y, where the length of
the vector y (i.e. the number of time points in the time series)
is typically of order ∼102. Whilst direct evaluation of such
high-dimensional probability densities on a grid is computationally
unfeasible, they can be effectively reconstructed through sampling algorithms
such as Markov chain Monte Carlo (MCMC), discussed in Sect. .
Uncertainty estimation
Our method requires uncertainties for each composite
that reflect the actual differences between the reported values and the true
state of ozone at the time of each measurement, as encoded in the likelihood
(Eq. ). We cannot use the uncertainties published by the
composite teams as they are (in general) not derived in the same way and so
they potentially encode information differently. The quoted uncertainties can
include (i) uncertainties propagated at each step of the data and composite
processing, e.g. in regression analysis used to combine individual
instruments; (ii) uncertainties in the absolute offsets; (iii) the total
number of observations in each dataset; and (iv) precision and calibration
errors. A natural choice might be to scale the uncertainty with the inverse
square root of the number of observations used to form the monthly ozone
value from each instrument, but this would not correctly deal with
systematics such as slow instrument drift (as experienced by the SBUV
instruments during the 1995–2000 period). Using the number of data points to
weight the monthly mean in each composite would lead to the most likely value
simply following the SBUV data almost exclusively until 2005 (see
Fig. ), and drifts would remain in the final
product (see Sect. ).
Instead, we seek to estimate the noise level from the data and in particular
from the discrepancies between the different composites. Estimating the
uncertainties is not the main focus of this paper, so a simple heuristic
method is used here, but this is clearly an aspect of this overall data
analysis problem which should be investigated further. Our approach is based
on a principal components analysis (PCA) of the composites to model the
differences between them, with the time-dependent noise level of each
composite then estimated from the variance of the higher-order components.
The starting point of this approach is to treat the full dataset
d as an nt×nc matrix with elements
dt,c as defined above. We then use this to construct the mean-subtracted
data matrix d′ with elements given by
dt,c′=dt,c-1nt∑t′=1ntdt′,c,
where each composite is treated separately.
The PCA is implemented via singular value decomposition (SVD) in which the
mean-subtracted data matrix is factorized as
d′=UWVT,
where U is an nt×nc matrix in which the
columns are the orthogonal component time series, W is an nc×nc matrix giving the weights of the components, and
V is an nc×nc matrix that encodes the
contributions of the components to the composites. A standard PCA
reconstruction of the (mean-subtracted) composites would then have the form
dt,c′=∑c′=1ncUt,c′Wc′,c′Vc′,c,
where the sum has to go from c′=1 but is often truncated to include
only the first few terms with the highest weights.
Our method of estimating the uncertainties in the composites is based on the
above reconstruction formula but is only heuristic in the sense that it does
not follow a rigorous calculation. We start by ignoring the leading,
i.e. the highest weighted, mode in U as it is common to all
composites, and so it provides no extra information. The various noise artefacts
are separated across the other nc-1 components, which must be
combined somehow to reconstruct the noise. We make the natural choice to
weight the modes by their respective contributions to each composite and then
sum the resultant variances to obtain uncertainty estimates as
σt,c2=∑c′=2nc(Ut,c′Wc′,c′Vc′,c)2.
Visualization of the components of the
SVD algorithm within the BASIC approach used to estimate the uncertainty on
each ozone composite for two examples at (a) 10 hPa and (b) 2.2 hPa
at 0–10∘ N. The left column of the first four rows show the
determined singular value decomposition (SVD) unitless modes (black time
series); the mode weighting (%) is given in the bottom right; the right
column is the mode weighting for each composite. All colours represent
information related to GOZCARDS (blue), SWOOSH (light blue), SBUV-MER
(yellow), and SBUV-MOD (red). Vertical lines represent dates an instrument
change in the composite occurred. The grey time series is the arbitrarily
rescaled difference between SBUV-MER–GOZCARDS, SWOOSH–GOZCARDS, and
SBUV-MER–SBUV-MOD in panel (a) in rows 2–4, and SBUV-MER–SBUV-MOD in panel (b)
in row 4. The bottom panel (row 5) in panels (a) and (b) represents the uncertainty derived
from the root sum of the squares of the modes (rows) 2–4, weighted by the
mode and composite weight, in units of ppm. Grey vertical lines represent
dates when data in any composite are missing and filled with the median
uncertainty for the sub-period in which they lie (i.e. between the vertical
lines in rows 2–4).
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The steps of this method are illustrated in Fig. . The
left set of panels shows the SVD applied to ozone at 10 hPa 0–10∘ N:
the SVD modes (i.e. from matrix U) (black lines; first
four panels), each have a different weight (percentage value in the lower
right of each plot, from matrix W). The first mode contains most of
the variance (84 %) with the remainder split between the other three (13,
1, and 2 %). The first mode is common to all four datasets, and its relative
weight within each dataset is represented by the coloured dots (from
matrix V) to the right of each mode ranging from -1 to +1; the
weight of the first mode is similar in all four datasets. The second mode is split
roughly equally between the two pairs of composites as indicated by the dots
on the right, suggesting that it is the difference between the pairs, and for
which the rescaled difference of SBUV-MER and GOZCARDS confirms, plotted in
grey and with an almost identical variance to the SVD mode. The
SBUV composites have almost zero weight in the third mode, indicating that
the mode represents artefacts only within the SAGE composites, again
confirmed by the difference between SWOOSH and GOZCARDS (grey). With almost
zero weight for the SAGE pair in the fourth mode, the rescaled difference
between SBUV pairs confirms the mode represents artefacts in SBUV.
From this, we form the uncertainty estimate for each of the composites in the
bottom panel, σt,c. Unfortunately, the SVD can only be formed when
there are data available in each composite, which leads to gaps, represented
by the grey shading in the bottom panel. Because composite sub-periods have
different uncertainty characteristics, we fill gaps using the median of the
period between instrument changes in the composites (vertical lines in the
four modes; colours relate to each composite).
In principle, the time series at each latitude–altitude location in the four
composites should be the same, and any deviations from the true value should
be a result of one or more of the potential reasons listed in
Sect. . By this assertion, the composites each contain the
real time series and an additional set of artefacts. The problem is that we do
not know for sure in which dataset a problem might be, especially if the true
trend is only apparent in (or missing from) one composite or one composite
pair (i.e. SAGE or SBUV based). Thus, the SVD approach allows us to
separate the common signal (the leading mode in U corresponding to
the highest weight in W, from those that form the differences
between the composites (with lower weights) and the real ozone. This
leads to an attribution of higher uncertainty for single datasets that
exhibit variance not present in the other three, and allows us to assign
higher uncertainties in all the composites when one pair (e.g. SAGE pair)
acts differently to the other pair (e.g. SBUV pair). In this way, it is a
relatively conservative estimate.
The example at 10 hPa was ideal since modes were easy to associate with
artefacts within and between the composite pairs. Another example of the
usefulness of applying the SVD approach to estimate the uncertainty is shown
for 2.2 hPa and 0–10∘ N in the right-hand panels of
Fig. b. The first mode is ubiquitous to the composites, and
the fourth mode shows a clear attribution to the SBUV composites (the
rescaled difference is shown in grey). However, it is not possible to
attribute modes two and three as confidently, though the artefacts are more
likely from GOZCARDS and SWOOSH, respectively. Since complete separation of
this mode from the other composites is not possible (e.g. that SWOOSH is
definitely the reason for the third mode), some uncertainty is given to the
other composites. This is an intuitive approach to assigning uncertainty to
each of the composites.
Satisfyingly, the error estimates display higher uncertainty to individual
composites during periods already known to have anomalous behaviour
(Sect ). For example, in the lower panel of
Fig. at 2.2 hPa (right), GOZCARDS is assigned a
particularly high uncertainty during the first 5 years, as expected
(Sect. ). At 10 hPa (left), the SBUV composites generally have a
higher assigned uncertainty, especially around mid-1995, and until 2000, when
we know there are instrument drifts in the SBUV composites
(Sect. ). In summary, the SVDs allow us to independently and
fairly assign an uncertainty to each of the composites.
The expected monthly ozone changes (or
“transitions”) between month n and the next month, n+1, i.e. index 1
represents a change between January and February. We show two examples at
0–10∘ N: (a) 2.2 and (b) 10 hPa. The box-and-whisker plots
are for all observations when no change in the underlying instrument of the
composites occurred and represent the interquartile range (IQR) covering the
25th to 75th percentiles (box) and 1.5 times the IQR or the maximum,
whichever is smallest (whisker); outliers are plotted as dots. Plotted to the
left of the vertical lines at each index are the changes between months for
each composite (represented by the different colours); Gaussian distributions
to the right of the vertical lines represent those formed from the mean and
standard deviation of all the composite transitions from 1000 bootstraps.
These Gaussians are used as transition prior estimates and are calculated for
all pressures and latitudes.
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As the SVD approach is not always able to assign a known artefact explicitly
to a specific composite, it is necessary for us to provide additional
information regarding the composite uncertainties, whereby in three cases we
increase the estimated uncertainty by a factor of 2. These are (i) when an
instrument changes in a composite, which is appropriate since there are many
examples of jumps in a composite on, or immediately after, these dates (e.g.
Fig. c in 1994 and 1995); (ii) during known and significant
instrument drifts in SBUV – the SBUV drift from the SVD uncertainty
estimate is typically assigned equally to both pairs of composites and so
additional information is needed, and tests show that it is only partially
accounted for when this additional information is not included –
specifically 1995–2000 for both SBUV composites and additionally
1994–1995 in SBUV-MER (these periods are marked by black shading and white
text in Fig. ); and (iii) following the eruption of Mount
Pinatubo in SBUV-MER only (see Fig. and
Sect. ).
The likelihood
With estimates of the uncertainties on each composite, we can construct the
joint-likelihood function for the set of composites as a product over the
individual likelihoods at each time step (indicated by t) and composite
(indicated by c), so that
P(d|y)=∏t∏cP(dt,c|yt),
which implicitly assumes that the measurement errors at different time steps
and between different composites are uncorrelated.
A common assumption would be that, ordinarily, the likelihood for a single
measurement would be taken to be a normal distribution with a mean given by
the true value, yt, and a standard deviation of σt,c, the
measurement uncertainty in composite c at this time step. However, it is
clear from even a quick inspection of the data that there are significant
disagreements between the different composites, implying several of them –
and possibly all – are far more prone to extreme errors (i.e. outliers)
than would be predicted by a simple Gaussian likelihood. We hence adopt the
model of Box and Tiao in which there is a probability 0≤β≤1 that any given measurement has an uncertainty inflated by
a factor of γ≥1, such that the likelihood for a single
measurement is
P(dt,c|yt)=12πσt,c{βγexp[-(dt,c-yt)22γ2σt,c2]+(1-β)exp[-(dt,c-yt)22σt,c2]}.
Smaller values of β encode more faith that uncertainties,
σt,c, are correct; higher values of γ correspond to more
catastrophic outliers. The standard normal distribution is recovered if
either β=0 or γ=1. Both β and γ must either be
fixed by hand or kept as hyperparameters to be inferred. We fix β=0.1 and γ=100, which implies that we consider outliers reasonably
rare but extreme should they occur; this choice leads to multi-modal
behaviour as desired (see Sect. and Fig. ).
When the multiple measurements of the different composites are combined in
the product over c, the resultant likelihood can be multi-modal when
considered as function of yt. In cases where the composites disagree, the
implication is that it is most likely that one of the measurements is good
but not necessarily that which is to be preferred. By contrast, simply multiplying
Gaussian likelihoods together in such a situation would result in a joint
likelihood that sits between the two (or more) peaks and does not represent
likely values according to any of the composites (left column of
Fig ). However, under the model prescribed by
Eq. (), the joint likelihood is multi-modal where
subsequent application of the prior may elicit which of the peaks is
representative of the truth and which observations were likely dominated by
artefacts (or indeed if all composites might be systematically biased
simultaneously but in different ways, in which case the resulting posterior
for that point will have an inflated uncertainty as desired).
Transition prior
We factorize the prior into a product of transition priors
for each month-to-month transition, i.e.
P(y)=P(y0)∏t=1N-1P(yt+1|yt).
The transition prior provides a way to estimate if measurements of ozone
values from the composites in the month being evaluated are more likely or
not and hence provide a way of assessing anomalous behaviour. The annual, or
semi-annual, variability that makes up the seasonal cycle, is the largest
mode of ozone variability. It is also a relatively consistent mode, so
together with information from the observations, it can provide a way to help
differentiate between artefacts and real anomalous behaviour.
We form the transition prior from all four composites together. Two examples
are given in Fig. at 2.2 and 10 hPa at 0–10∘, where
the expected change between month n and n+1 for the whole year is shown,
with, e.g. n=1 being the transition between January and February. The monthly
changes for all composites are shown with the box-and-whisker plots, which
show the mean (white horizontal line), interquartile range (IQR, 25–75th
percentiles; thick stem), and full range or 1.5 times the IQR (thin line),
with any outliers given as dots; data in a composite where instruments change
are not included in the estimates. The grey Gaussian distributions are formed
from all the changes between 2 months treated independently and then
performing 1000 bootstraps. We note that in the examples shown in
Fig. , the SAGE-based composites typically have a larger range
of month-to-month variance, which we suggest may be due to the higher
resolution of the SAGE composite instruments, but we cannot exclude the
possibility that this is also related to the low sampling and higher scatter
of, e.g. the earlier observations from SAGE-II.
Posterior sampling
With the likelihood (Sect. ) and prior in hand, we can
construct the posterior density for the true time series given the data and
our prior knowledge and assumptions, i.e. Eq. (). The product
of the prior (Eq. ) and the likelihood (Eq. )
over all the observations gives the numerator of the posterior density
defined in Eq. (). The normalizing denominator cannot be
calculated analytically, but fortunately the numerator is sufficient to
obtain samples from the posterior distribution. We sample the posterior using
Hamiltonian Monte Carlo (HMC) sampling implemented in
stan ; HMC is an MCMC method
that is particularly effective at sampling high-dimensional densities
. The resulting inferred ozone time series forms the BASIC
composite.
BASIC approach as an approximation to a Bayesian hierarchical state–space model
When constructing the month-to-month transition prior as described above, we
use the data to estimate and fix the prior's hyperparameters, i.e. the means
and variances of each month-to-month transition (January–February, February–March, etc.). This
is using the data twice – once to construct the transition prior and once
in the main posterior inference. However, we note that estimating and fixing
the hyperparameters from the data is an approximation, similar to “empirical
Bayes” methods, to a full Bayesian hierarchical treatment where the
parameters of the prior would be kept as free unknown parameters and inferred
jointly with the true ozone time series. In cases where the hyperparameters
are tightly constrained by the data and do not strongly co-vary with the
parameters of interest (here the underlying ozone time series), estimating
and fixing the hyperparameters from the data before the main analysis is an
excellent approximation to the full hierarchical model.
Testing BASIC with synthetic data
We designed synthetic tests to evaluate whether the BASIC approach was
effective in retrieving the “true” ozone time series given a set of four ozone
composites that had jumps, drifts, and noise, similar to those we encounter in
the existing datasets. Overall, we found the BASIC approach to be successful
at estimating ozone and, in particular, better than any individual composite
that contains artefacts. These synthetic tests are presented in
Sect. .
The BASIC composite result for the 0–10∘ N 2.2 hPa time series is
given in Fig. a, with all four composites, and the BASIC
composite with uncertainties at 2 standard deviations (dotted lines) and
68, 95, and 99 % credible intervals (dark, medium, and light grey shading); the
differences in Fig. a relative to the BASIC composite are
shown in Fig. b. It is clear that the BASIC approach has
successfully accounted for (i) the early drift prior to 1991 in GOZCARDS
resulting from the use of NCEP reanalysis temperatures and (ii) the high
scatter in both the SAGE composites prior to 1991 and mainly in SWOOSH prior
to 2004 resulting from the low sampling of the occultation instruments used.
When disagreement between composites increases, or the priors inflate the
uncertainties, the BASIC composite uncertainty estimate naturally inflates to
allow for the higher uncertainty during that period; on the other hand, the
BASIC composite uncertainties reject most of GOZCARDS prior to 1989 by being
outside the 99 % credible interval.
Another example, at the higher pressure of 10 hPa, is given in
Fig. c and d. Here, we see that the BASIC approach has
accounted for (i) the SBUV-MER problem following the Mt. Pinatubo eruption,
during which SBUV-MOD measurements are not provided, (ii) rapid steps in the
SBUV composites between 1995 and 2001, and (iii) some of the drifts in the
SBUV composites during the same period. What is clear here, especially in the
period after 2002, is that while the BASIC composite reproduces most of the
variance, it cannot determine whether the higher amplitude variance of the
QBO signal in the SAGE composites is more likely to be correct than the
SBUV composites, though we know the reason is due to the lower vertical
resolution of the SBUV-type instruments and that the QBO represented by the
SAGE composites is more likely to be correct (see Sect. ). We
do not currently have a solution for this particular issue, though the errors
do inflate naturally to accommodate this uncertainty, and so typically within
the uncertainties this issue is captured by the BASIC approach.
Finally, to show how the BASIC approach operates in a completely different
regime to that near the Equator, in Fig. e and f we give an
example at 6.8 hPa and 50–40∘ S. Here, ozone lacks a semi-annual
component of variability. Except for between 1993 and 2001, all four
composites show broadly similar variability. The SAGE composites again appear
to show spikes that are not present in the SBUV composites, and indeed on many
occasions do not occur in both SAGE composites. Therefore, many of these are
rejected by the BASIC composite. We cannot discount that some of these
artefacts are a result of the better resolution in the SAGE composites and
may be real, for example, unexplained artefacts after 2008, but these are
generally found to remain at or within the 99 % credible interval. Following
the instrument change in SBUV-MER in 1994, and until 2001, we see anomalous
behaviour in SBUV-MER that is rejected by the BASIC composite at the 99 %
level throughout this period; between 1995 and 1997, SBUV-MOD also displays
behaviour quite different to the other composites, and this is also generally
rejected.
Ozone time series at three stratospheric
locations from 1985 to 2012, all bias shifted to the mean of SWOOSH after
August 2005. (a) Absolute ozone at 2.2 hPa over 0–10∘ N
from SWOOSH (light blue), GOZCARDS (blue), SBUV-MER (yellow), and SBUV-MOD
(red). The BASIC composite mean estimation (black) is plotted with shading
representing 68 % (dark grey), 95 % (grey), and 99 % (light grey) credible
intervals (CIs); these CIs are not Gaussian, so 2 times the standard
deviation is also plotted with thin dotted lines. Panel (b) is the same as (a),
but now for the difference relative to the BASIC composite. Panels (c) and
(d) are as the same as (a) and (b) at 10 hPa and 0–10∘ N, and
(e) and (f) are the same as (a) and (b) at 6.8 hPa and
50–40∘ S. Vertical dashed and solid lines in panels (b), (d), and
(f) identify changes in the instruments used in the composites.
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Further examples of problems resolved by the BASIC approach
Ozone time series at two stratospheric
locations from 1984 to 2014, all bias shifted to the mean of SWOOSH after June
2005. (a) Absolute ozone at 2.2 hPa over 0–10∘ N from
SWOOSH (light blue), GOZCARDS (blue), SBUV-MER (yellow), and SBUV-MOD (red).
(b–g) The difference between each pairing of the four composites and
with the BASIC composite (see legends). Panels (h–n) are the same as (a–g) but at
4.6 hPa and 0–10∘ N. Solid and dashed vertical lines represent
months with a change in the instrument used to construct the composite
(colours are with respect to the composite colour in panels a and h).
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In Sect. and , we showed examples of differences
between composites based upon the same, or similar, instrument data. It is
not always clear by looking at the pairs of composites, however, which is
more likely to be correct: drifts and rapid changes occurring over a few
months cannot be immediately attributed to a specific composite. However, as
we will now demonstrate, additional information from the literature,
knowledge of when instruments are added or removed within the composites, and
looking at the differences of all four composites at the same time, helps to
build confidence in attributing the source and reason for the deviation, and
then correcting it – these are encoded in the uncertainties of each
composite as discussed in Sect. . We also show the
effectiveness of the BASIC approach in accounting for most of these
artefacts. The final BASIC ozone composite product that integrates
information from all four composites is denoted “BASIC”. However, it is also
possible to only use information from either the SBUV pair (“BASIC(SBUV)”) or
SAGE pair (“BASIC(SAGE)”) of composites (with SVD uncertainties and
transition priors constructed using only the respective pairs of data), which
elucidates how the prior information applied in the BASIC algorithm is able
to perform if information is missing from the other pair of composites. In
other words, a correction of artefacts (e.g. drifts in SBUV composites) that
do not appear in differences of just one composite pair strengthens our claim
that the BASIC approach is correctly accounting for artefacts in the
composites. For clarity in the figures introduced here, we do not provide
uncertainties on the BASIC results presented.
In Fig. , we show two examples of the four ozone
composites at 0–10∘ N, at 2.2 hPa (Fig. a–g) and 4.6 hPa (Fig. h–n). Below
the absolute time series (Fig. a and h) are six plots (Fig. b–g and i–n), which are
the differences between each pairing of composites (black); the absolute
BASIC composite ozone is shown with a dotted line, and differences of the
BASIC, BASIC(SAGE), and BASIC(SBUV) compared to the composites are given in
red, blue, and orange, respectively. Once again, the early drift (e.g. Fig. b SWOOSH – GOZCARDS)
and the steps (e.g. Fig. n SBUV-MER–SBUV-MOD) are clearer
in these restricted latitude bands than in the broader equatorial band
presented in Figs. c and c. However, considering
these different pressures and latitudes, and the SBUV–SAGE differences (Fig. c–f),
additional anomalous behaviour is revealed, which we list and discuss
in the following.
The most significant problem in creating a unified calibration for all SBUV
instruments is the orbital drift . Ideally, the local
time at Equator crossings should be the same each orbit, and the orbit should be
near polar to attain near-global coverage. However, NOAA satellites slowly drifted
over time, changing from near 14:00 LT (local time)
(10:00 LT, NOAA-17) Equator crossings to late afternoon
(early morning, NOAA-17) Equator crossings. NOAA-9, -11, -14, and -16 drifted through
the terminator and began making early morning measurements. The Equator-crossing
time for each of the SBUV satellites is shown in Fig. between
the SBUV-MOD and -MER composite information. Any instrument or calibration errors
may be significantly enhanced for observations taken as the orbit approaches the
terminator, such that the orbit drift can lead to an apparent time-dependent
trend in ozone that could be misinterpreted as real; ,
, and do not recommend the use
of near-terminator data for this reason. Accordingly, SBUV-MOD, with the exception
of NOAA-11, does not include any observations taken outside the 08:00–16:00 LT equatorial
crossing time range (marked as dotted horizontal lines in Fig. )
and similarly SBUV-MER prioritizes measurements made while instruments are in their
optimum orbits. The clearest example of this drift-related trend can be seen in
Fig. k, m, and n in all differences with respect to SBUV-MOD
between 1995 and 1998 (until 2000 with respect to SBUV-MER in Fig. n);
there is then a reversed drift until after 2000. The differences with the SAGE composites
indicate that a 1994–1995 drift is likely in SBUV-MER from the exclusive use of NOAA-9;
for 1995–1997, the drift is probably in both but more prominent in SBUV-MER differences;
the 1997–2000/2001 drift is more likely in SBUV-MER with the exclusive use of NOAA-11
(SBUV-MOD merges NOAA-11 with NOAA-14). Other smaller drifts between the SBUV composites
are visible in Fig. , e.g. in 2001 and 2002. While BASIC(SBUV) and BASIC were able to account
for the large discontinuity present in Fig. n, BASIC(SBUV) is unable to
account for the 1997–2000 drift in SBUV-MOD. We do inform the BASIC approach that the
uncertainties should be increased in the SBUV composites during the drift period 1995–2000
(from 1994 in SBUV-MER), so uncertainties are equal for this period in BASIC(SBUV).
Nevertheless, with the inclusion of the SAGE composites this drift can be accounted
for (red line in Fig. j–n), which further reinforces the need
for information from all composites to resolve problems. Confirmation of drift problems
during the periods mentioned
justifies using it as prior information to down-weight these data for this time (see Sect. for more information).
The apparent high scatter at 2.2 hPa in all differences involving SAGE composites
(i.e. Fig. b–f) during the periods of 1985–1991 and 1997–2004 coincides with
periods when only occultation instruments were active (SAGE-II, UARS/HALOE, and ACE-FTS).
and convincingly demonstrated that
insufficient and/or inhomogeneous sampling can result in inaccurate monthly estimates
and even induce spurious spikes in ozone time series; coarse-sampling occultation-type
instruments such as GOMOS and ACE-FTS can lead to differences of up to 20 %. This can
especially affect seasonal cycle representation, especially at high altitudes where
ozone undergoes rapid variations with latitude and time of day. This is why spurious
variability from occultation instruments is clearly evident in Fig.
during the aforementioned periods. Even though satellite measurements from limb viewers
have a lower vertical resolution than occultation, these are still sufficient to reduce
the monthly zonal-mean scatter in the SAGE-based composites when overlaps with occultation
instruments occur (e.g. 1992–1997 in GOZCARDS). The BASIC–GOZCARDS difference in
Fig. e agrees closely with the month-to-month artefacts that are
highlighted in the SBUV-MER–GOZCARDS difference. This is not because of the information
provided in the SBUV composites, which do not display this behaviour, but because the
deviation from the natural seasonal cycle is so high that the month-to-month seasonal
variability is more informative. This is confirmed by the high agreement between
BASIC–GOZCARDS with BASIC(SAGE)–GOZCARDS on these short timescales in Fig. b,
the latter of which contains no knowledge from the SBUV composites.
The drift between the SAGE composites prior to 1991 (Fig. b and i; see
Sect. ) is largely absent in the SWOOSH composite compared to SBUV composites
(Fig. c and d), confirming it as a feature of GOZCARDS only. It is clear
from Fig. e that the artificial trend in GOZCARDS prior to 1991 is fully
accounted for by BASIC, and once again the agreement of BASIC–GOZCARDS with BASIC(SAGE)–GOZCARDS
in Fig. b shows that the information in the SAGE composites alone is sufficient
to eliminate most, though not all, of this problem. No prior information about the drift being in
GOZCARDS is provided to the BASIC approach – the ability for the BASIC approach to account for
the drift is most likely because SWOOSH agrees with the prior information from the seasonal
variability (in the transition prior) much better than GOZCARDS.
A small downward step in the SAGE composite difference in Fig. b and
i in 2004 occurs around the time both SAGE composites have an instrument change. This feature
is more evident in the differences between GOZCARDS and the SBUV composites than for SWOOSH,
at both altitudes. At the lower altitude of 4.6 hPa in Fig. i, it appears
that BASIC(SAGE) could not account for the jump in GOZCARDS and ends up slightly offset
from the black difference line. The BASIC approach performs better with the additional
information provided by the SBUV composites and fully accounts for this jump.
A prominent feature in Fig. j–m is the approximately 2- to 3-year
oscillation. This is the result of lower vertical resolution in the SBUV observations, which
leads to a damping of the quasi-biennial oscillation (QBO) signal in SBUV relative to the
higher resolution instruments of the SAGE-based composites; at 3 hPa SBUV has a vertical
resolution of approximately 6–7 km, while the SAGE-based instruments are usually better
than 3.5 km – the vertical resolution only gets larger for SBUV with lower altitude,
reaching a maximum of ∼ 15 km below the tropopause .
After 2003, the resolution effect is more clearly visible in Fig. h,
since many of the other instrument-data/composite artefacts are absent.
showed that by applying the SBUV resolution kernel
to higher vertical-resolution Aura/MLS data led to good agreement with SBUV data.
Focusing on the period after 2005 in Fig. h–n, it is evident
that BASIC is unable to distinguish between the QBO represented in the
SAGE and SBUV composites; this is because uncertainties are similar during
this period and composite issues are generally absent. We discuss this further in Sect. .
Following the eruption of Mt. Pinatubo in June 1991, there is a large drop
in SBUV-MER at 10 and 16 hPa due to interference in viewing from volcanic aerosols
(not shown here, but see Fig. c and d), which is absent in the
SAGE composites; SBUV-MOD does not include data during this period. Ozone is
usually depleted by sulfate aerosols following a volcanic eruption but at lower
altitudes. Due to the rapid departure of SBUV-MER from the SAGE composites, the
BASIC composite predicts that the SAGE composites are more likely to be correct
during this period. To be clear, the BASIC approach can adapt to rapid,
unexpected changes in ozone: if all the datasets had shown a sudden and
similarly large change that was significantly different from the prior
expectation for that month, it would tend towards a tighter cluster of
observations as more likely than the broader prior estimate. We discuss
this period further in Sect. .
For completeness, steps in the SBUV composites in Fig. k,
m, and n, discussed in Sect. , occur in 1995 and in 2003, 2004, and 2007 in Fig. n;
though these are not the only times that steps occur; prominence of steps depends on
altitude and latitude. The BASIC approach accounts for these discontinuities, which
is most clear for the large jump in the SBUV–MOD composite in Fig. k,
m, and n; absence of a jump in Fig. i confirms the success of the BASIC
approach. For the BASIC(SBUV)–SBUV-MOD case in Fig. n (orange), which
relies exclusively on the SBUV composites, the large step in 1994/1995, and drift that follows, is mostly accounted for.
Results
Now that we have established the validity of the BASIC approach and
constructed an ozone composite from GOZCARDS, SWOOSH, SBUV-MOD, and SBUV-MER,
we turn to analysing trends and modes of variability. This is often performed
using MLR
.
However, the use of DLM, first applied to ozone
data by , appears to be more robust at estimating
the background trend, especially if it is non-linear.
noted this when comparing their DLM results with
the MLR results of where linear trends were sometimes
found to be inverse to those estimated using DLM. We performed tests upon the
artificial time series used to evaluate the performance of both methods with
the BASIC approach (Sects. and ).
We briefly introduce both methods below. We compare their performance on the
artificial time series and the BASIC correction, introduced in
Sect. , in Sect. . We found
that in every test case the DLM did equally well, or better, at estimating
the true background “trend” than the linear estimate from MLR (see
Figs. and ) both for non-linear background trends
and for time series with large artefacts.
MLR analysis
We perform MLR analysis on deseasonalized time series (i.e. by
subtracting monthly means) using five regressors: the F30 radio flux (solar),
which is superior to the F10.7 cm radio flux for representing solar UV
variability ; the stratospheric aerosol optical
depth (SAOD; , for volcanic eruptions); the El Niño–Southern Oscillation (ENSO); and two orthogonal modes of the dynamical
quasi-biennial oscillation (QBO). These regressors are displayed in the upper
part of Fig. . When we analyse decadal trends between
1985–1997 and 1998–2012, we use a linear trend to estimate the long-term
trend. We use prewhitening and a first-order autoregressive process (AR1) to
account for autocorrelation in the residuals . Statistical
significance of the regression coefficients was evaluated with a Student's
t test.
DLM analysis
We perform a DLM analysis following
very closely the model and formalism of . We use
the same five regression components as in the MLR. We allow for two modes of
seasonal variability in the fit (with 6- and 12-month periods), where
additional (Gaussian-process) variability of the sinusoidal seasonal modes is
also allowed for (following ), and variance of
the (Gaussian) seasonal model variability σseas2 is
kept as a free parameter in the fit. We include an AR1 process, where the
variance σAR2 and correlation coefficient
ρAR of the AR process are also kept as free parameters in
the fitting process. In contrast to MLR, the DLM approach allows for a fully
non-linear “trend”, where the degree of non-linearity
σtrend is also kept as a free parameter in the fit (see
for details). In further contrast to MLR, the
Bayesian DLM approach jointly fits for the non-linear time-varying trend, the
regression coefficients of the five proxies and seasonal modes, as well as
the nuisance parameters σseas, σAR,
ρAR, and σtrend; uncertainties in the
nuisance parameters and regression coefficients are formally marginalized
over when stating inference of the trend, leading to a principled propagation
of uncertainties. Similarly, uncertainties in the nuisance parameters and
trend can be marginalized over when we are interested in the regression
coefficients.
Our DLM analysis follows except for some small
differences in the prior choices. For σtrend, we use a
positive half-Gaussian prior with zero mean and dispersion 0.0005. For
σseas and σAR, we take positive uniform
priors over [0,∞], and for the correlation coefficient of
the AR process we take a uniform prior over [0,1], assuming that
negative correlations are unphysical in this context. We also do not impose
an external prior on the initial value of the AR process, as is done in
, but draw the initial value of the AR process
from its stationary distribution, i.e.
N(0,σAR/(1-ρAR2)).
Recovery of the
DLM parameters
{σtrend,σseas,σAR,ρAR}
under the chosen priors is shown in a set of figures in
Sect. . As in ,
we use MCMC to sample the joint posterior of the DLM parameters, regression
coefficients of the proxies, seasonal cycle, and non-linear trend.
The percentage change in ozone from DLM between 1985 and 1997 (a–c), and 1998
and 2012 (d–f), over 60–35∘ S (a, d),
20∘ S–20∘ N, and 35–60∘ N (c, f).
GOZCARDS, SWOOSH, SBUV-MER, and SBUV-MOD are shown with error bars
representing 95 % credible intervals; for the BASIC composite (black),
shading represents uncertainties. The mean linear trend estimate from
MLR for the BASIC composite is given as a black
dashed line (no uncertainties) and is the scaled version of the MLR-BASIC
decadal trend shown in Fig. .
![]()
Multi-decadal changes in ozone
Here, we present estimates of changes in ozone between 1985 and 1997, and
between 1998 and 2012 (Fig. ). This is the first time that DLM
has been applied to these composite datasets, including recently updated
SWOOSH and SBUV-MER. While we focus on the DLM results, we also refer to
results using MLR given in Fig. .
Typically, ozone trends are reported as linear decadal percentage changes in
three latitude bands in the Southern Hemisphere
(60–35∘ S), over the Equator
(20∘ N–20∘ S), and in the Northern Hemisphere
(35–60∘ N) with sub-periods ending and starting in
December 1997 and January 1998, respectively, as shown in Fig.
(Fig. for MLR) . These integrated latitude bands were formed by averaging
the area/latitude-weighted 10∘, with the 30–40∘ band
receiving half the weight of the equivalent full band; the resultant
time series were then analysed.
It does not make sense to provide a linear trend estimate for the non-linear
DLM background trend. Instead, in Fig. we give the percentage
change of ozone between the first and last months of the sub-periods, i.e.
between January 1985 and December 1997 (top row), and January 1998 and
December 2012 (lower panels). Uncertainties represent the 95 % credible
intervals of the change for all 100 000 samples estimated with the DLM
algorithm (shading for BASIC, bars for all others). Since we do not show
decadal trends for the DLM (but do for MLR in the Appendix), we also show as
dashed black lines in Fig. the mean MLR-BASIC linear trend
profiles from Fig. , scaled from decadal changes to
the longer 13- and 15-year sub-periods.
In the earlier period (1985–1997), the DLM and MLR profiles agree well
(within the DLM uncertainty). The DLM-BASIC typically displays better
agreement with the GOZCARDS profiles than the others in the northern and
southern midlatitudes, but the mean profile is generally closer to that of
SBUV-MOD over the Equator. Indeed, above 4 hPa, SWOOSH is typically at or
outside the BASIC composite 95 % credible interval in northern and equatorial
bands (this is also the case with MLR). Interestingly, the SBUV composites
are often outside the MLR-BASIC uncertainty range above 7 hPa at
midlatitudes in both hemispheres; DLM uncertainties are larger and the four
composites are in closer agreement when trends are analysed using DLM. This might
hint that MLR is being biased by residual variance and/or underestimating
error bars, in contrast to DLM, as was observed in the test cases (see
Sect. ). Overall, the 1985–1997 DLM results are
consistent with previous studies and MLR, with a significant decline in ozone
above 7 hPa at all latitudes, especially at midlatitudes, and negative but
usually insignificant trend at lower altitudes.
The results for the latter period, 1998–2012, show a significant positive
trend in the upper stratosphere above 7 hPa, as expected to occur following
the implementation of the Montreal Protocol. The result is significant in
every dataset analysed with DLM in both the northern and southern
midlatitudes for at least one pressure level; for the BASIC composite, the
result is clear at multiple altitudes. We note that the MLR results are only
statistically significant at northern midlatitudes for both SBUV composites
and for all composites in the southern midlatitudes at 3.2 and 4.6 hPa.
There are also statistically significant differences between the mean
MLR-BASIC and the DLM-BASIC profiles over the Equator and at northern
midlatitudes; in the southern region, DLM profiles for composites are less
consistent than when using MLR, but the DLM-BASIC results are in good
agreement. The DLM profile shapes in the Northern Hemisphere are consistent
with each other, with a negative trend in the lower stratosphere, though
usually insignificant at the 95 % level, and a positive response in the upper
stratosphere, confirming the result of .
Interestingly, with the exception of SBUV-MOD, the large and significant
negative MLR equatorial trends seen in most of composites at 7 hPa disappear when using
DLM, except in GOZCARDS. This anomaly was found in an integrated set of seven
composites by , though not in the multi-model mean of
the same composites in . These results suggest that
it may be an artefact of the analysis approach rather than a real feature and
further investigation is required.
The percentage change in ozone (left
axis) relative to 1998 (vertical dashed line; horizontal zero line) for the
integrated latitude bands 60–35∘ S (a),
20∘ S–20∘ N (b), and 35–60∘ N (c)
and pressure levels from 1 hPa (top; right axis) to 46 hPa (bottom).
Only the mean trend lines are shown for GOZCARDS, SWOOSH, SBUV-MER, and
SBUV-MOD; the BASIC composite is shown in black with shading representing the
95 % credible interval. The MLR trend estimates for the period before and
after January 1998 are given as dashed black lines.
![]()
In Fig. , we plot the DLM moving trends as a percentage
change in ozone relative to 1998; only the BASIC composite uncertainty is
presented, and the MLR-BASIC linear trends pre-1998 and
post-1997 are given as dashed lines; as a guide the MLR uncertainties are
typically smaller than the DLM (see Fig. ). From
Fig. , significant disagreement at 5–10 hPa at the
Equator and 15–22 hPa in the Southern Hemisphere is very much apparent at
the altitudes where DLM and MLR trend estimates disagree on the sign of the
trend; this instability of MLR was also noted by
and requires investigation in a future publication to understand.
Figure also allows us to observe how the background
evolves with time; from this we can see that, while SBUV-MER often displays
large deviations from the group (e.g. especially at 5 and 7 hPa in all
latitude bands), the BASIC composite results are almost always smoothly
varying and generally monotonic to/from the years 1998–2002, meaning that a
comparison between MLR trends and a change between fixed dates from the DLM
are indeed valid (the exception possibly being at 1.5 and 1 hPa over the
Equator where all datasets display relatively rapid variations in the sign of
the DLM gradient, though we note this is where data are more sparse, and
temporal sampling can easily be biased by the large diurnal variability; even
so, this altitude region appears to be where MLR and DLM are most consistent).
Figure is entirely consistent with, and explains why,
was able to show that the choice of pivot date from
the piecewise linear trend using MLR on GOZCARDS led to larger positive
trends the later the date of pivot was chosen, i.e. from 1998 to 2002, most
prominently above 10 hPa in both midlatitude bands (see Fig. 7 of
). We see from the DLM trends in
Fig. that at many locations above 10 hPa the gradient
is typically zero in 2002, not 1998, especially at 3, 2, and 1.5 hPa, the
exact region where the biggest increase in the trend was found by
; northern midlatitude ozone at 1.5 hPa actually
appears to start increasing a little later, perhaps in 2004. These results
are consistent between all the composites analysed, including the BASIC
composite.
It is interesting to note that the two 1998–2012 midlatitude BASIC
composite profiles in Fig. , while determined independently of
each other, display remarkably similar shapes in the DLM analysis, suggesting
a symmetry in the stratospheric driving of ozone changes over this period
and, indeed, a similar hemispheric recovery following the Montreal Protocol.
In contrast, the lower stratospheric mean-profile changes from MLR (dashed
black lines in Fig. ) are not similar, with a generally (and
sometimes statistically significant) positive trend in the Southern
Hemisphere and (an almost significant) negative trend in the northern
midlatitudes.
We propose that the profiles determined by DLM-BASIC are likely to be a
better representation of the change in stratospheric ozone than previous
estimates. We base this conclusion upon the knowledge that (i) the BASIC approach was
successful in identifying and correcting most known artefacts in the ozone
composites, (ii) the DLM performed better than the MLR in the artificial
ozone time series test cases, and (iii) the DLM-BASIC outperformed both
MLR-BASIC and DLM of all the “artefact-damaged” artificial time series. The
consistency of independent northern and southern midlatitude DLM profiles
for both periods would suggest that additional explanation for why the
different hemispheres should evolve in different ways is not required
. However, this also means that further investigation into why
MLR and DLM trend estimates can differ so substantially is needed.
Conclusions
We have presented a novel approach to identify and account for data artefacts
that remain in multiple ozone composites of satellite observations. These
artefacts are one of largest remaining causes of disagreement between decadal
trend estimates made from the many composites available. Our approach
includes estimates of uncertainties using singular value decomposition, a
Gaussian-mixture outlier model for the likelihood, and prior information in
the form of expected monthly transitions and knowledge of problems in ozone
observations; these are combined via Bayesian inference. The main output of
this process we term the BAyeSian Integrated and Consolidated (BASIC)
composite, which has been designed to account for differences in ozone
composites that are constructed in different ways and with observations from
different sources. The need for better approaches to combine ozone composites
has been raised in recent years as an issue needing resolution (e.g.
). stated
that it is not currently possible to make definite assumptions about the best
way to combine data and in what way, especially when considering multiple
composites that use similar, or identical, underlying datasets.
noted that the key to good estimates of
long-term trends is the combination of high-quality measurements and multiple
instruments. Our method both requires and benefits from the availability of
both. further state that the consideration
of uncertainties and artefacts is essential, especially when the trends are
small compared to the large natural variability (e.g. seasonal cycle), so
detailed information is needed about measurement uncertainties, data jumps
due to instrument changes, and drifts. Again, our method is specifically
designed to address these concerns.
The presence of data gaps, biases between instruments, and issues with
sampling, noise, and differences in resolution also enhance uncertainties in
trend estimates, which might lead to artificial trends being extracted in
multiple linear regression (MLR) analysis. To avoid this, we employed, with
refinements, dynamical linear modelling (DLM)
and found it to be more accurate than MLR when considering test cases where
all variance is understood. The combination of the BASIC approach with DLM
shows that the problems listed above can indeed be resolved to improve
estimates of ozone changes on decadal timescales.
The results presented here are a step forward, but we do not consider the
composite a definitive and final product; there are still issues to resolve,
which we extensively discuss (Sect. ). These
caveats include the concern of using the same instrument dataset more than
once, even though it may be used in separate composites with different
preprocessing . Our recommendation to resolve this
problem, and as the natural next step forward, is to apply the posterior
sampling approach as a method to combine as many independent datasets as
possible, integrating all the known caveats and uncertainties. This will
require an additional step to the methodology outlined here in order to
account for absolute bias between the datasets, but we do not consider that
this will cause significant difficulties.
From the DLM analysis, the estimated changes in ozone between 1985 and 1997,
and then between 1998 and 2012, show good agreement with the shape of the
ozone profiles presented by , where seven composite
datasets were combined with various approaches to estimate errors. The BASIC
composite results using DLM (and MLR) show remarkably similar profile shapes
and magnitudes for the earlier period. The implication for the latter period,
then, is that ozone is indeed clearly and significantly recovering in the
upper stratosphere as a result of the Montreal Protocol, which has not
previously been demonstrated universally with significance from observations,
though demonstrated that the recovery was indeed
underway by removing dynamics that interfere with calculating trends using a
model with specified dynamics. The largest uncertainty in the estimates of
came from considering instrument drift. Since the
BASIC composite has accounted for much of this uncertainty, we can be
confident that our smaller uncertainties represent an improvement. Further,
the BASIC composite typically rejects outliers inconsistent with other
composites, or otherwise inflates uncertainty estimates, leading to our
assertion that the estimated uncertainties are probably a reasonable
reflection of the uncertainty in the observations. Uncertainties on the
decadal trends can be further reduced with additional regressors, in addition
to a new composite based upon independent instrument datasets rather than the
four composites we considered here.
We will make the BASIC composite available and
provide supporting documentation should the composite be updated. The
composite is available for public use at
https://data.mendeley.com/datasets/2mgx2xzzpk/1 .
In future work, we will extend the latitude and altitude range and time period
covered, which should lead to more robust results and an improved assessment
of ozone trends in the stratosphere.