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- Editorial & advisory board
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ACP | Articles | Volume 18, issue 2

Atmos. Chem. Phys., 18, 535–554, 2018

https://doi.org/10.5194/acp-18-535-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/acp-18-535-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Quadrennial Ozone Symposium 2016 – Status and trends...

**Research article**
18 Jan 2018

**Research article** | 18 Jan 2018

The impact of nonuniform sampling on stratospheric ozone trends derived from occultation instruments

^{1}NASA Langley Research Center, Hampton, VA, USA^{2}University of Toronto, Department of Physics, Toronto, Ontario, Canada

^{1}NASA Langley Research Center, Hampton, VA, USA^{2}University of Toronto, Department of Physics, Toronto, Ontario, Canada

**Correspondence**: Robert P. Damadeo (robert.damadeo@nasa.gov)

**Correspondence**: Robert P. Damadeo (robert.damadeo@nasa.gov)

Abstract

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This paper applies a recently developed technique for deriving long-term
trends in ozone from sparsely sampled data sets to multiple occultation
instruments simultaneously without the need for homogenization. The technique
can compensate for the nonuniform temporal, spatial, and diurnal sampling of
the different instruments and can also be used to account for biases and
drifts between instruments. These problems have been noted in recent
international assessments as being a primary source of uncertainty that
clouds the significance of derived trends. Results show potential
“recovery” trends
of ∼ 2–3 % decade^{−1} in the upper stratosphere at
midlatitudes, which are similar to other studies, and also how sampling
biases present in these data sets can create differences in derived recovery
trends of up to ∼ 1 % decade^{−1} if not properly accounted for.
Limitations inherent to all techniques (e.g., relative instrument drifts) and
their impacts (e.g., trend differences up to ∼ 2 % decade^{−1})
are also described and a potential path forward towards resolution is
presented.

How to cite

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How to cite.

Damadeo, R. P., Zawodny, J. M., Remsberg, E. E., and Walker, K. A.: The impact of nonuniform sampling on stratospheric ozone trends derived from occultation instruments, Atmos. Chem. Phys., 18, 535–554, https://doi.org/10.5194/acp-18-535-2018, 2018.

1 Introduction

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Ever since the Montreal Protocol came into effect, the global
scientific community has been monitoring the state of stratospheric ozone in
an effort to determine at first if the loss rate was decreasing and later if
ozone had begun to recover. Consequently, there has been an ongoing body of
work to use single (at first) or multiple (later) sources of data, spanning
the satellite record starting around 1979, for various multiple linear
regression (MLR) analyses to determine the long-term trends in stratospheric
ozone. A simple literature search would reveal the various techniques and
studies ranging from the earlier works
(e.g., Wang et al., 1996; Bodeker et al., 1998; Newchurch et al., 2003) revealing the loss
slowdown to a recent surge in efforts to determine potential ozone recovery
(e.g., Randel and Wu, 2007; Remsberg and Lingenfelser, 2010; Bodeker et al., 2013; Kyrölä et al., 2013; Bourassa et al., 2014; Gebhardt et al., 2014; Tummon et al., 2015; Harris et al., 2015; Steinbrecht et al., 2017).
These works have culminated in the most recent Scientific Assessment of Ozone
Depletion (WMO, 2014) that showed statistically significant “recovery”
trends of ∼ 2 % decade^{−1} in the upper stratosphere at
midlatitudes but identified three factors with a potential major impact that were
not readily accounted for in those analyses: diurnal variability of ozone,
biases between data sets, and long-term drifts between data sets. There is an
additional complication that is intricately tied to these three factors in
this kind of analysis, namely the nonuniform temporal, spatial, and diurnal
sampling of the different instruments used for these analyses. This
nonuniform sampling can have a detrimental impact not only on the regression
techniques used to derive long-term trends in ozone but also on other
analyses performed to determine diurnal variability or the magnitude of
potential biases and drifts between data sets. Herein, we discuss a recently
developed technique that not only accounts for the potential sampling issues,
but also the perceived diurnal variability, as well as any potential bias
and/or drift between instruments in a single analysis.

2 Data sets

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There have been several remote sensing instruments over the past several decades that have observed stratospheric ozone using the method of solar occultation, including but not limited to the Atmospheric Chemistry Experiment Fourier transform spectrometer (ACE-FTS); the Halogen Occultation Experiment (HALOE); the Polar Ozone and Aerosol Measurement (POAM) II and III; and the Stratospheric Aerosol and Gas Experiment (SAGE) I, II, and III. For the purpose of this study, however, SAGE I was ignored because it does not have any overlap with the other missions.

The Atmospheric Chemistry Experiment Fourier transform spectrometer (ACE-FTS)
was launched onboard the SCISAT-1 spacecraft in August 2003
(Bernath, 2017). The spacecraft occupies a 74^{∘} inclined orbit at an
altitude of ∼ 650 km that allows for observations
from 85^{∘} S to 85^{∘} N. The primary ACE instrument is a high
spectral resolution (0.02 cm^{−1}) Fourier transform
spectrometer operating in the spectral range
of ∼ 2.2–13.3 µm
(750–4400 cm^{−1}) that measures many trace gas species
and isotopologues (Bernath et al., 2005). Ozone is retrieved using the
spectral features near 10 µm (Boone et al., 2005). The
version of the ACE-FTS data product used here is version 3.5
(Boone et al., 2013), which produces vertical profiles of volume mixing
ratio (VMR) interpolated to a 1 km grid with a vertical resolution
of 3–4 km. The ACE-FTS instrument is still operating.

The Halogen Occultation Experiment (HALOE) was launched onboard the Upper
Atmosphere Research Satellite (UARS) in September 1991. The spacecraft
occupied a 57^{∘} inclined orbit at an altitude
of ∼ 585 km that allowed for observations
from 80^{∘} S to 80^{∘} N. The HALOE instrument used a combination of
broadband radiometry and gas filter correlation techniques to observe several
trace gas species in the spectral range
of ∼ 2.4–10.4 µm
(∼ 950–4150 cm^{−1}) and measured
ozone using the spectral band near 9.6 µm
(Russell et al., 1993). The version of the HALOE data product used here is
version 19.0 (Thompson and Gordley, 2009), which produces vertical profiles
of VMR interpolated to a 0.3 km grid with a vertical resolution of
2–3 km (Bhatt et al., 1999). The UARS mission was
decommissioned in December 2005.

The Polar Ozone and Aerosol Measurement II (POAM II) was launched onboard the
SPOT-3 spacecraft in September 1993. The spacecraft occupied a
sun-synchronous orbit, crossing the descending node at 10:30 LT, that allowed
for observations in two latitude bands at 88 to 62^{∘} S
and 65 to 71^{∘} N. The POAM II instrument used broadband
radiometry to observe aerosol and trace gases in the spectral range
of ∼ 350–1070 nm and measured ozone using the
spectral band near 600 nm (Glaccum et al., 1996). The version of
the POAM II data product used here is version 6.0, which produces vertical
profiles of number density interpolated to a 1 km grid with a
vertical resolution of 1 km (Lumpe et al., 1997). The SPOT-3
spacecraft ceased functioning in November 1997.

The Polar Ozone and Aerosol Measurement III (POAM III) was launched onboard
the SPOT-4 spacecraft in March 1998. The spacecraft occupied a
sun-synchronous orbit, crossing the descending node at 10:30 LT, that allowed
for observations in two latitude bands at 88 to 62^{∘} S
and 65 to 71^{∘} N. The POAM III instrument used broadband
radiometry to observe aerosol and trace gases in the spectral range
of ∼ 345–1030 nm and measured ozone using the
spectral band near 600 nm (Lucke et al., 1999). The version of
the POAM III data product used here is version 4.0
(Lumpe et al., 2002; Naval Research Laboratory, 2006), which produces vertical
profiles of number density interpolated to a 1 km grid with a
vertical resolution of 1 km. The POAM III instrument ceased
functioning in December 2005.

The Stratospheric Aerosol and Gas Experiment II (SAGE II) was launched
onboard the Earth Radiation Budget Satellite (ERBS) in October 1984. The
spacecraft occupied a 57^{∘} inclined orbit at an altitude
of ∼ 610 km that allowed for observations
from 80^{∘} S to 80^{∘} N. The SAGE II instrument was a broadband
spectrometer that operated in the spectral range
of ∼ 375–1030 nm for aerosol and trace gas
observations and measured ozone using the spectral band near 600 nm
(Mauldin III et al., 1985). The version of the SAGE II data product used here
is version 7.00 (Damadeo et al., 2013), which produces vertical profiles of
number density interpolated to a 0.5 km grid with a vertical
resolution of 1 km. The ERBS mission was decommissioned in
October 2005.

The Stratospheric Aerosol and Gas Experiment III (SAGE III) was launched
onboard the Russian Meteor-3M (M3M) spacecraft in December 2001. The
spacecraft occupied a sun-synchronous orbit, crossing the ascending node
at 09:00 LT, that allowed for observations in two latitude bands at 60
to 30^{∘} S and 45 to 80^{∘} N. The SAGE III instrument was
a grating spectrometer that operated in the spectral range
of ∼ 295–1025 nm for aerosol and trace gas
observations and measured ozone using the spectral features
near 600 nm (Mauldin et al., 1998). The version of the SAGE III
data product used here is version 4.00
(Cunnold and McCormick, 2002; Wofsy et al., 2002), which produces
vertical profiles of number density interpolated to a 0.5 km grid
with a vertical resolution of 1 km. The M3M spacecraft ceased
functioning in January 2006.

When making use of any data set, it is important to apply the proper filtering to ensure that bad data (e.g., fill values or data contaminated by clouds) are excluded. Since this analysis is constrained to the stratosphere, all data below the tropopause are ignored. If a data set provides a tropopause height, that value is used for filtering purposes, otherwise the World Meteorological Organization (WMO) definition is used (WMO, 1992). Beyond this, the data screening procedures recommended for each data set are performed. ACE-FTS data are screened as outlined in Sheese et al. (2015). HALOE data are screened for potential problematic “constant lockdown angle” and “trip angle” events as detailed by the data producers (http://haloe.gats-inc.com/user_docs/index.php). POAM II data could be screened for interference from polar stratospheric clouds (PSCs) by looking for outliers in the 1 µm data, though this is not performed. POAM III data are screened for potential sunspot interference and heavy aerosol interference through the use of the quality flags. SAGE II data are screened for this analysis in the same way as was done in Damadeo et al. (2014). Since SAGE III data were screened prior to release, no additional screening is performed.

3 Analysis technique

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In principle, this work is a continuation of the work first performed in Damadeo et al. (2014) and so the same techniques and methodologies are used. Each data set is filtered according to the stated filtering techniques and converted to the unit system of interest (i.e., number density or mixing ratio versus altitude or pressure) using the pressures, temperatures, and altitudes provided with the respective data sets. While we did apply the analysis to both combinations of unit systems, for the sake of brevity all results shown here are for regressions to data in number density on altitude (some mixing ratio on pressure results are shown in the Supplement). Additionally, the data for each instrument were interpolated to 0.5 km increments. These data are then consolidated into daily zonal means for each instrument separated by both satellite and local event types. A generalized least-squares regression technique that accounts for autocorrelation, heteroscedasticity, and data gaps is then performed on all data sets simultaneously, with the autocorrelation and heteroscedasticity corrections being applied separately for each instrument. In principle, this technique is applicable to data sets with higher sampling (e.g., the Microwave Limb Sounder, MLS, on the Aura satellite) but is demonstrated here on occultation data sets only to illustrate the impact of their sparse sampling patterns on derived trends.

The same simultaneous temporal and spatial (STS) MLR model as was used in Damadeo et al. (2014) is applied using the same proxy terms, albeit with nine spatial terms instead of seven and some additional changes to account for the incorporation of multiple instruments (see Appendix A). Terms accounting for the quasi-biennial oscillation (QBO), El Niño–Southern Oscillation (ENSO), solar variability, and long-term trends (two orthogonal equivalent effective stratospheric chlorine, EESC, functions) are applied to all data sets simultaneously. Terms accounting for volcanic eruptions (primarily the Mount Pinatubo eruption in 1991) are applied to the SAGE II and HALOE data sets only (and separately) to avoid potential overfitting of minor eruptions in data sets that do not cover the Mt. Pinatubo eruption. Diurnal variability (applied as a binary conditional term) is fit separately for each data set. While the sun-synchronous instruments (i.e., SAGE III, POAM II, and POAM III) sample both satellite event types, they do not adequately sample both local event types (Fig. 1) and so all local sunrises from these instruments are ignored for this analysis. The seasonal cycle is applied to all data sets simultaneously as a single seasonal cycle for all instruments. Lastly, a bias offset term and a linear drift term are applied separately for each instrument using SAGE II as the reference instrument.

Since the STS regression model uses a two-dimensional regression, it is best
utilized on data that adequately cover the full range of temporal and
spatial sampling to constrain the temporal and spatial variability present in
the data. Occultation instruments in mid-inclination orbits tend to deliver
near-global coverage at somewhat reduced seasonal sampling while occultation
instruments in sun-synchronous orbits tend to deliver highly localized
spatial coverage at nearly full seasonal sampling. The primary focus of this
work is the impact of sampling biases on long-term trends in ozone, which is
typically analyzed in the stratosphere between about 60^{∘} S
and 60^{∘} N. Since this work focuses on that latitude range and the
sun-synchronous instruments exhibit little to no coverage within that region
and thus very little influence on resulting trends there, the results
presented herein derive from an STS regression using only the SAGE II, HALOE,
and ACE-FTS data sets. We also applied an STS regression using all six data
sets and found that the long-term trends were not significantly affected (see
the Supplement) but did notice that the lack of spatial coverage in the
POAM II, POAM III, and SAGE III data sets detrimentally impacted the results
in the seasonal cycle and diurnal variability derived from a two-dimensional
regression. In the interest of brevity and to maintain the legibility of
certain figures in this paper, individualized results from the six-instrument
regression are not shown here.

4 Non-trend results

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Similarly to Damadeo et al. (2014), we investigate the residuals of the regression. 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. While the mean of the residuals is zero (as it should be), a clear pattern in the spread of the residuals emerges as a function of latitude at each altitude. The total residuals of the regression (i.e., the residuals from the ordinary least-squares regression) are a combination of the correlated residuals (i.e., those removed during the autocorrelation correction) and the uncorrelated residuals (i.e., the residuals from the generalized least-squares regression). The correlated residuals represent geophysical variability that is well sampled but not well modeled by the regression as well as any systematic instrumental variability (e.g., biased meteorological or ephemeris input data). The uncorrelated residuals represent both measurement noise and geophysical variability that is not well sampled (e.g., geophysical variability present within each daily mean).

Figure 2 shows the spread of the correlated and uncorrelated residuals for each instrument. All of the instruments exhibit increased residuals in the lower stratosphere, owing both to the increased uncertainty of measurements in that region as well as increased variability that is not adequately captured by the proxies used for this regression. Similarly, residuals are higher at higher latitudes where measurements can routinely dip into and out of the vortex both over multiple days and within a single day itself. SAGE II has greatly increased uncorrelated residuals at the highest altitudes compared to HALOE and ACE-FTS. While the influence of measurement noise and daily zonal variability in the uncorrelated residuals cannot be separated, the fact that SAGE II and HALOE (and to a lesser extent ACE-FTS) exhibit similar sampling patterns means that the increased uncorrelated residuals in the upper stratosphere and lower mesosphere in SAGE II compared to HALOE must be a result of increased measurement noise in SAGE II. Similarly, SAGE II and ACE-FTS display slightly lower uncorrelated residuals in the lower stratosphere while HALOE and ACE-FTS display lower uncorrelated residuals in the upper stratosphere. All three instruments show comparable uncorrelated residuals in the middle stratosphere.

The correlated residuals show an increased spread in the stratosphere at high latitudes, which is expected as variability within the polar vortex is not modeled in this regression. Similarly, increases can be seen in the tropical middle stratosphere near a local peak in QBO amplitude. This is a result of a two-dimensional fit using a proxy derived only at the Equator. While modulating the QBO with the seasonal cycle better represents the QBO at higher latitudes, the inability to accurately model the QBO at higher latitudes detracts from the ability to accurately model the QBO at lower latitudes (Damadeo et al., 2014). Another interesting feature is an apparent vertical banding structure in the correlated residuals present in each data set. The locations of this banding correlate to the turnover latitudes in each instrument's orbit (i.e., the latitudes at which measurements go from progressively closer to the poles to progressively further away). The autocorrelation correction accounts for the degree of correlation of data from day to day. However, the locations of daily means change in latitude from day to day, with rates of motion greater at the Equator and smaller near the poles, and so the degree of correlation is dependent upon both the temporal variability and the meridional variability, with the meridional variability being the primary driver. At the orbit turnover point, the meridional variability between each successive daily mean essentially disappears. While not explored in this study, it is possible that this additional source of correlated noise stems from the nature of how wave one action is sampled from day to day over the course of about 1 week until the instrument moves away from the turnover latitude. Because the measurements systematically shift in longitude over the day while the wave itself also rotates, the zonal variability is not evenly sampled and so these day to day differences will be highly correlated as the wave one action rotates and changes, thus revealing a potential additional source of sampling bias albeit more localized and on a shorter timescale.

Occultation instruments sample one sunrise (SR) and one sunset (SS) per orbit as seen by the spacecraft, which typically correlates to one sunrise and one sunset as seen by an observer on the ground at the measurement location. This means that occultation measurements of ozone sample its diurnal variability present in the mesosphere and upper stratosphere. Diurnal variability of ozone in the mesosphere has been investigated before and is well understood to be a result of rapid photochemistry across the terminator (Chapman, 1930; Herman, 1979; Pallister and Tuck, 1983). While the full attribution of sources is still not completely understood, diurnal variability in the stratosphere is well represented in various data sets. Analysis of the diurnal variability from occultation instruments is typically performed by looking for periods where the instrument's diurnal sampling “crosses itself” (i.e., local sunrise and sunset measurements occur at roughly the same latitude at roughly the same time). Sakazaki et al. (2015) used this method to analyze the diurnal variability present in SAGE II, HALOE, and ACE-FTS and found that not all data sets agree and the differences between SR and SS values differ typically by up to ∼ 5 %. The STS regression can extract the mean diurnal variability present in each data set and the results shown in Fig. 3 compare quite well with those in Fig. 5 of Sakazaki et al. (2015).

Volcanic eruptions periodically inject sulfur dioxide into the stratosphere where it goes on to form sulfate aerosols that can impact ozone either via chemical effects (Rodriguez et al., 1991; Solomon, 1999) or through changes in dynamics via changes in radiative forcing (McCormick et al., 1995; Robock, 2000). In either case, it is possible for volcanic aerosols to have a significant impact on stratospheric ozone levels such that their presence can complicate these regression analyses. Since ozone trend analyses utilize data from the past ∼ 30 years, usually only the Mt. Pinatubo eruption in mid-1991 is considered for special treatment. If the analysis goes back further, sometimes the El Chichón eruption in early 1982 is also considered. The punctuated nature of the eruptions and not completely characterized impacts on data quality often leads to many works simply excluding data from 1 to several years after these eruptions (e.g., Wang et al., 1996; Randel and Wu, 2007; Harris et al., 2015), while some works attempt to include a term in the regression to model the impact (e.g., Bodeker et al., 2001; Stolarski et al., 2006; Bodeker et al., 2013; Tummon et al., 2015), although the nature of these terms tends to be different between different analyses.

For this work we include an aerosol proxy that was derived in Damadeo et al. (2014). The proxy is a volcanic one, meaning that eruptions occur and the proxy rises, peaks, and subsequently decays back to zero. The proxy only covers the SAGE II mission time period and thus is zero throughout most of the ACE-FTS mission period. However, given that it takes a relatively large eruption (e.g., Mt. Pinatubo) to register any noticeable changes in stratospheric ozone in these regression analyses and the fact that only minor eruptions have occurred since (Vernier et al., 2011), this is assumed to be sufficient. Given that occultation instruments can (depending upon their spectral channels) have reduced measurement sensitivity in the presence of heavy aerosol loading (Wang et al., 2002; Bhatt et al., 1999), the volcanic proxy is applied separately for SAGE II and HALOE. The regression was applied under two conditions with regard to aerosol: one in which no filtering of events for the influence of aerosols was performed and another in which SAGE II was filtered under the recommendations in Wang et al. (2002) and HALOE was filtered under the recommendations in Bhatt et al. (1999).

Figure 4 shows the peak of the volcanic regression term surrounding the Mt. Pinatubo eruption for both the aerosol filtered and unfiltered cases. In the unfiltered case, both SAGE II and HALOE show similar responses of ozone to the eruption in the tropics between ∼ 24 and 35 km. Both instruments show a large region of negative correlation between ozone and aerosol in the lower stratosphere surrounding the aerosol layer itself and another large region of positive correlation in the middle stratosphere above the aerosol layer (the anomalously large responses in the lowermost stratosphere are a result of overfitting due to missing data). These results are in reasonably good agreement with Aquila et al. (2013) and Bodeker et al. (2013), which show results of the impact of the eruption on ozone levels from modeling and data respectively, and in surprisingly good agreement between the two separate instruments. The effect of the eruption on ozone derived from HALOE data is typically more difficult to quantify since HALOE did not begin to take measurements until shortly after the eruption, which has a tendency to negatively impact studies of long-term variation using only the HALOE data set (Remsberg, 2008).

When the regression was run with the stated aerosol filtering criteria applied to the data, the results of the volcanic regression term from HALOE remain unchanged (not shown). However, compared to the unfiltered case (middle plot in Fig. 4) the SAGE II responses with aerosol filtering applied (left plot in Fig. 4) remain unchanged above 28 km but significantly reduced in amplitude in the tropics below that. The aerosol filtering has no effect in the middle stratosphere in the region of positive correlation because the aerosol loading levels were not so high as to detrimentally affect the retrievals of these occultation instruments there. Between the middle and the lowermost stratosphere the data quality declined until measurements were no longer possible. The Wang et al. (2002) filtering criteria were meant to exclude anomalous ozone values based on aerosol extinction and aerosol extinction ratio values in the regions where data quality declines. However, the apparent agreement of the unfiltered results suggest that the Wang et al. (2002) filtering criteria are overly conservative and need to be revisited. Either that, or the SAGE II filtering criteria and results are reasonable and perhaps the HALOE data require a better aerosol correction in the retrieval algorithm than what is already applied (Hervig et al., 1995).

The impact of the ∼ 11-year solar cycle on stratospheric ozone has been an ongoing topic of study (e.g., Wang et al., 2002; Soukharev and Hood, 2006; Randel and Wu, 2007; Remsberg, 2008, 2014; Maycock et al., 2016; Dhomse et al., 2016). As such, it is worthwhile to show the results of the solar response in this work as well as to point out a few things about data usage and the determination of the solar cycle response to ozone when using MLR-based studies on SAGE II and HALOE data. The cited works show different solar cycles when using SAGE II data as well as different solar cycles between using SAGE II and HALOE data, with the latter exhibiting the greatest difficulty in determining the solar cycle from only HALOE data (Soukharev and Hood, 2006). Figure 5 shows the latitude- and altitude-dependent amplitude of the solar cycle response derived from this work, which is similar to other recent works based on the usage of the SAGE II data set (e.g., Maycock et al., 2016; Dhomse et al., 2016) and naturally very similar to those from Damadeo et al. (2014). One important distinction between the previous work and this one is the impact of the use of one or two solar terms. Previously, when applied to only SAGE II data, using two solar terms shifted the solar cycle response by about 2 years in the presence of the Mt. Pinatubo eruption in agreement with Remsberg (2014), though this was believed to be the regression algorithm simply trying to attribute some of the aerosol response to the solar cycle (Solomon et al., 1996). The inclusion of HALOE and ACE-FTS in this study, however, seems to better constrain the solar cycle such that using one or two solar cycle terms no longer creates temporal shifting in the presence of the eruption (not shown) and thus only a single solar cycle term is required for the regression. While also not shown here, we attempted to apply the STS regression to only HALOE data and found that no combination of proxies exhibited realistic-looking solar cycle responses, most likely due to the data having insufficient duration capable of constraining the solar cycle, aerosol, and trend terms simultaneously. This could potentially explain the often different solar cycle responses derived when using the instruments separately, while using them simultaneously creates SAGE II-like responses as well as very similar aerosol responses, though this requires further study. Lastly, it is worth noting that the large amplitude tropical response below ∼ 23 km is a result of the previously discussed anomalous aerosol response in that area.

5 Sampling biases

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Traditionally, data sets are reduced to monthly zonal mean (MZM) values for regression analyses to determine long-term trends. Practically speaking, these MZM values are utilized as though they are representative of the center of the month and the center of the latitude bin. Though this assumption holds mostly true for highly sampled data sets (e.g., nadir and limb sounders), it generally fails when applied to occultation data sets. This fact is well known and has been studied before. Toohey et al. (2013) and Sofieva et al. (2014) both investigated nonuniform temporal sampling as an added source of noise and uncertainty that could be characterized and included in trend analyses. Using deseasonalized anomalies for trend analysis can mitigate the impacts of sampling bias if the bias is constant with time. However, owing to the observational geometry of occultation instruments and orbital parameters (i.e., altitude, inclination, and precession rates) the sampling patterns often tend to systematically drift over time as shown in the top row of Fig. 6. Millán et al. (2016) investigated the impacts of nonuniform sampling biases on resulting trends from different instruments by using a “representative year” of sampling for each data set and repeating it over 30 years to analyze the effect on trends. While illustrative, this did not account for the actual sampling bias as it changed from year to year.

The systematic drift in sampling combined with the presence of sampling biases precludes the use of the MZM method to accurately determine the seasonal cycle that is represented by an occultation data set. The STS regression, however, is less sensitive to sampling biases and can thus be used to quantitatively assess the sampling biases that would be present in the MZM method. It is relatively straight forward to compute the temporal and spatial offset between the average time and location of sampling within a given month and latitude band and the center of that month and band that is considered the representative location for the MZM method. The spatially varying seasonal ozone cycle from the STS regression can then be used to compute the difference in fitted ozone values between the actual center of sampling and the representative center of sampling to compute a seasonal sampling bias for each month, latitude band, and altitude bin. Some typical results of these biases are shown in the bottom row of Fig. 6. It is evident from a simple visual inspection of these results that drifting sampling patterns create patterned monthly biases.

While it is clear that these sampling biases will create problems attempting to use the MZM method to assess the seasonal cycle or how it changes over time, this investigation is more focused towards the effects on long-duration variability. For each year, latitude band, and altitude bin, an average of the monthly sampling biases can be computed to produce a yearly averaged bias shown in the top row of Fig. 7. While the magnitudes of yearly averaged biases are smaller than those of monthly biases, systematic patterns are still evident. To illustrate the potential impact these sampling biases can have when incorporated into regression analyses, we can look at an individual sampling bias time series by extracting data from the top row of Fig. 7 and plotting it along with the low frequency variability from the STS regression. The bottom row of Fig. 7 shows this data in black with the solar cycle (red) and long-term trend (blue) overplotted to demonstrate how easily the drifting sampling patterns create patterned biases that alias into interannual and long-term geophysical variability. This will ultimately interfere with the ability of any analysis to accurately determine the “true” long-term trends. While it may appear that these results are being cherry-picked (and, since only so many figures can be shown, they are), in actuality it is a “fruitful tree” and results shown here are common (see the Supplement for more plots).

It should be noted that while the presence of seasonal sampling biases that alias into longer-duration terms is pervasive (in altitude and latitude for each data set), the actual degree of correlation with terms such as the solar cycle or trend is somewhat more random as it is dependent upon the chance combination of drifting sampling patterns, spatially varying seasonal gradients, and frequency of interannual variability. Additionally, the seasonal sampling biases will correlate with multiple terms simultaneously, making a simple and concise quantitative evaluation of their impact on the analysis results almost impossible. Ultimately, however, it is readily apparent that the use of an MZM analysis method on data with obvious seasonal sampling biases will produce biased results in derived long-term variability.

With a few exceptions (e.g., Kyrölä et al., 2013; Remsberg, 2014; Damadeo et al., 2014), most analyses of ozone trends make use of MZM values where SR and SS measurements are treated equally. This has been done with the assumption that the mean value will fall between the SR and SS means but that any sampling biases are a source of random noise and do not affect the trend. As a result, should the distribution of diurnal sampling not be evenly distributed, the risk of a diurnal sampling bias becomes apparent.

In a similar way as the seasonal sampling, the nature of the orbit of the spacecraft dictates how the instrument will sample local sunrises and sunsets as a function of time of year and latitude over the mission lifetime. An example of the diurnal sampling of the SAGE II instrument over its lifetime is shown in Fig. 8. The most apparent features are the increased rate of sampling at midlatitudes versus high and low latitudes and the presence of instrument problems during the mission that caused asymmetric diurnal sampling between mid-1993 and mid-1994 and after 2000. However, it is by taking the difference between the sunrise and sunset sampling that the true diurnal sampling differences become apparent. A close investigation of the bottom panel of Fig. 8 for any given latitude reveals a rapid oscillation of monthly biases between SR- and SS-dominant months. In the presence of significant (i.e., a few percent) diurnal variability such as in the upper stratosphere, this sampling bias will interfere with the derivation of the seasonal cycle for an MZM analysis.

To get a better idea of the systematic long-term nature of the diurnal
sampling bias, we have looked at larger latitude bands (i.e., 35–45^{∘} N–S and 15^{∘} S–15^{∘} N) using the data in
Fig. 8. By smoothing the data over a year, we can dampen
some of the seasonal effects and more easily investigate the long-term
changes. Also, to intercompare different latitude bands, it is preferable to
look at the differences between SR and SS sampling as a percentage of the
total events rather than the absolute number of events. This actually raises
the question of whether to convert the differences as a number of events to a
percentage of the total number of events (for each month) and then smooth or
the other way around. It is interesting because this question draws a
corollary with the concept of computing unweighted or weighted monthly mean
values. If MZM values were computed by first calculating a mean value for
each month and then computing a mean July (for example) by simply taking the
mean of all Julys, that would be unweighted (i.e., all Julys are treated
equally in the overall mean regardless of how many events are in each July)
and would be analogous to our converting to a percent first and then
smoothing. If, however, one were to compute the overall July mean by
factoring in how many events went into each month, that would be weighted and
would be analogous to our smoothing in number of events and then converting
to a percentage.

Figure 9 illustrates this approach for the three latitude bands for each of the three main data sets (i.e., SAGE II, HALOE, and ACE-FTS) where the “unweighted” approach is shown in the left column and the “weighted” approach is shown in the right column. As can be seen, the unweighted approach is more susceptible to creating a diurnal sampling bias that aliases into longer-duration variability than the weighted approach. However, even the weighted approach reveals that diurnal sampling biases cannot be avoided. The previously noted SAGE II instrument problem periods create large diurnal sampling biases with the net effect of creating large discrepancies in derived potential recovery trends (i.e., post-1997–1998) between the MZM and STS approaches. However, the diurnal sampling bias for HALOE appears to have some QBO-like periodicity that is hemispherically anticorrelated and that for ACE-FTS appears to have an overall trend in the tropics. For each case, it becomes apparent that even an attempt to account for the diurnal sampling of these instruments in an MZM analysis (i.e., the weighted case) will still introduce biases unless the diurnal variability is specifically modeled or corrected for beforehand.

6 Trends

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The nonuniform temporal, spatial, and diurnal sampling patterns present in
occultation instruments detrimentally impact trend results derived from the
MZM method. To illustrate this, we also employ an MZM regression to compare
with the STS regression. The MZM method employed is a one-dimensional (i.e.,
time only) regression that utilizes monthly means with a minimum of 5 events
in 10^{∘} wide latitude bins without differentiating between sunrise and
sunset events, but otherwise uses the same proxies and statistical analysis
as the STS method. Since the MZM method cannot compensate for the various
sampling biases but is the de facto methodology for data product usage (e.g.,
trend analyses or incorporation into models), we also used the results of the
STS method to create corrected versions of the different data sets for
incorporation into the MZM method. The first is a diurnally corrected data
set that simply applies the derived diurnal variability to bring all
individual sunrise events into the sunset regime. The second applies the
diurnal correction and also uses the spatially varying seasonal ozone
gradient to compute a correction based on the difference between the location
and time an event occurred versus the center of that month and the latitude
bin it would fall within for a particular MZM averaging scheme. It is
important to note that this “seasonal correction” retains variability between
events within a month and bin (i.e., it does not make all values the same)
and is specific to the latitude bin (i.e., width of the bin and center of the
bin). The MZM regression is then applied to each of these three data versions
(i.e., uncorrected, or “Raw”; diurnally corrected, or “DCorr”; and diurnally
and seasonally corrected, or “DSCorr”).

We compute trends and uncertainties using the resulting two orthogonal EESC-proxy functions and the method described in Appendix B. This uses a simple linear fit to the EESC component of the regression results evaluated over a desired time period to derive the trend and makes a correlation between the EESC-fit uncertainties and the functional form of the linear fit to derive the associated trend uncertainty. The derived trends from the MZM and STS methods for a typical decline period (1985–1995) are shown in Fig. 10. As expected, the difference between the MZM and STS methods during this time is small. The diurnal correction (i.e., comparing MZM DCorr with MZM Raw) has some limited impact in the upper stratosphere at midlatitudes while the seasonal correction (i.e., comparing MZM DSCorr with MZM DCorr) has larger influence at higher latitudes (at all altitudes) as well as some minor influence in the tropical middle stratosphere, though trends in this area are not significant. Overall, however, the resulting trends are typical of other studies, though we would like to note that the positive trends in the tropical lower stratosphere below ∼ 23 km are, similar to the solar cycle response, detrimentally affected by the anomalous aerosol response.

The derived trends from the MZM and STS methods for a potential recovery
period (2000–2012) are shown in Fig. 11. There are significant
differences between the raw MZM results and the STS results most noticeably
from the diurnal sampling biases. Trends in the upper stratosphere at
mid-southern latitudes decrease by ∼ 1 % decade^{−1} while trends
in the upper stratosphere at mid-northern latitudes increase
by ∼ 1 % decade^{−1}, which is consistent with the expectations
from diurnal sampling biases in the SAGE II data set. The seasonal
correction, as in the decline period, influences the trends at higher
latitudes as well as some minor influence in the tropical middle
stratosphere. It is worth noting that, generally, the fully corrected MZM data
results agree much better, though expectedly not identically given the
different data resolutions and techniques, with the STS results when compared
to the raw MZM results. Overall the results show statistically significant
trends of about 2–3 % decade^{−1} in isolated parts of the upper stratosphere at
midlatitudes as well as in the tropical middle stratosphere. However, as
discussed in the next section, there are other factors that affect these
results that may indicate these trends are not only statistically
insignificant but potentially biased as well.

7 Limitations and orthogonality

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One of the biggest issues in every regression technique is the combination of multicollinearity and orthogonality. Multicollinearity refers to the fact that the proxies used in the regression are not orthogonal to every other proxy used and that individual proxies or linear combinations of proxies are correlated with other proxies. The larger the collinearity between two or more proxies, the more difficult it is to separate their influences on the data. Sometimes proxies are sufficiently independent as to be useable, but when sampled in a particular way (e.g., to match the sampling of a particular data set) the resulting subsampled proxies exhibit larger collinearity. A clear example of this is seen in the diurnal and seasonal sampling patterns of the three instruments. Over their mission lifetimes, the diurnal and seasonal sampling patterns in SAGE II and HALOE are sufficiently orthogonal such that the regression can extract both the diurnal variability and seasonal cycles in each instrument separately. However, this is not the case for ACE-FTS as its diurnal and seasonal sampling patterns are highly correlated. Figure 12 illustrates the diurnal variability for each instrument when the regression allows each instrument to have its own seasonal cycle. When compared with Fig. 3, the results for SAGE II and HALOE are the same, illustrating sufficient orthogonality in their sampling patterns and the fact that their seasonal cycles are essentially the same as well. However, the results for ACE-FTS lose coherence and agreement with other studies. It is for this reason that this work made use of a single seasonal cycle as it allowed SAGE II and HALOE to constrain the seasonal cycle and thus make the extraction of the diurnal variability in the ACE-FTS data set possible. Furthermore, the fact that using a single seasonal cycle allows the independent extraction of diurnal variability that agrees well with other studies suggests that all three instrument do, in fact, observe the same seasonal cycle.

The most recent Scientific Assessment of Ozone Depletion (WMO, 2014) noted that a primary problem when attempting to derive long-term trends in ozone when incorporating multiple data sets is that of instrument offsets and drifts. Given any overlap between two instruments, the offset between instruments is easily characterizable though many trend analyses are performed on anomalies and so these offsets are inherently removed. Drifts between instruments, however, are much more difficult to characterize. Hubert et al. (2016) performed an extensive analysis of ground and satellite data sets in an attempt to assess the average drifts present in each satellite data set relative to the ground network. The results showed that some instruments were more stable than others (e.g., SAGE II, HALOE, ACE-FTS, and MLS), though the degree of overlap between the satellite sampling patterns and the available ground stations did preclude the ability to determine the full spatial extent of drifts for every instrument (e.g., ACE-FTS).

This work incorporates an offset and a drift term for HALOE and ACE-FTS
relative to SAGE II. The offset terms (not shown) are similar to those found
in other studies comparing these instruments and are not a focus here.
Figure 13 shows the linear drifts relative to SAGE II.
Throughout most of the stratosphere, HALOE shows a negative drift
of ∼ 2–3 % decade^{−1} relative to SAGE II, which is in good
agreement with other studies
(e.g., Morris et al., 2002; Nazaryan et al., 2005; Hubert et al., 2016).
The drift results from ACE-FTS, however, require a different interpretation.
A quick comparison of the ACE-FTS drifts in Fig. 13 and the STS
recovery trend results in Fig. 11 shows that the patterns in
the drifts somewhat match the patterns in the trends. This suggests that
trends in the ACE-FTS data set are different from those in the SAGE II data
set and highlights another example of the orthogonality problem. Over the
course of the ACE-FTS mission period, the long-term trend terms and the drift
terms are highly correlated, which is not the case for HALOE because HALOE
spans the ozone turnaround time in the late 1990s, creating sufficient
orthogonality between the long-term trend terms and its drift term. This
means that the long-term trends are constrained by SAGE II and HALOE (and an
independent HALOE drift can also be determined), but any difference in what
the ACE-FTS data may suggest the trend is goes entirely into the drift term.
This is further complicated by the fact that ACE-FTS data only have 2 years
of overlap with SAGE II and HALOE. When the regression is run without any
drift term (Fig. 14), the recovery trend results can be
changed by up to ∼ 2 % decade^{−1}, indicating a potential
additional uncertainty originating from possible drift between this
particular combination of data sets (similar to what was shown in
Harris et al., 2015) and that derived recovery trends are sensitive to how
potential drifts are incorporated or accounted for. Overall, the issue of
orthogonality highlights the limitations of regression techniques and
illustrates how it is actually impossible to simultaneously determine both
potential recovery trends and relative instrument drifts using data from only
after the ozone turnaround.

8 Conclusions and future work

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A simultaneous temporal and spatial regression applied to multiple
occultation data sets simultaneously without homogenization has been
presented. The technique allows for a stratospheric ozone trend analysis that
natively compensates for the nonuniform temporal, spatial, and diurnal
sampling patterns of the data sets, and results on data quality, diurnal
variability, response to aerosol, and the solar cycle were shown. The STS
regression shows the natural derivation of the diurnal variability captured
in each instrument and highlights the impact of how the seasonal cycle is
incorporated, revealing that only a single uniform seasonal cycle should be
used for regression analyses. Comparison of the aerosol responses in SAGE II
and HALOE suggests the need to potentially revisit suggested data usage
filtering criteria and the increased temporal extent of data used in the
study helps to separate apparent aerosol and solar cycle responses to reveal
how only a single solar cycle term should be used. Additionally, a detailed
discussion of the nature of the sampling biases reveals how they impact the
retrieval of long-term trends when performing regressions on MZMs causing
differences in potential recovery trends up to
∼ 1 % decade^{−1}, though we also introduce corrected versions
of the data sets for use with MZM methods that apply a first-order sampling
bias correction for use with trend analyses. While these corrected MZM data
sets naturally do not produce identical results as the STS, they are in
better agreement. This study also highlights the limitations inherent in
regression techniques and details how problems with multicollinearity and
lack of orthogonality can impede accurate determination of long-term trends
in ozone.

For future work, we would like to continue to address the topic of drifts and
orthogonality as this study has shown impacts of the drifts on derived trends
of up to ∼ 2 % decade^{−1}. It is currently impossible to
simultaneously determine both potential recovery trends and relative
instrument drifts but it is also impossible to ascertain a global picture of
drifts for every satellite instrument due to lack of necessary coverage
overlaps. That being said, an analysis could be performed where a relatively
stable and long-lived dense sampler (e.g., MLS) is used as the reference
instrument while incorporating all other desired instruments as well. With
sufficient overlap with all other data sets, an STS regression could
determine the globally resolved drifts between the reference instrument and
all other instruments. The derived trends, however, would come only from the
reference instrument but a follow-up analysis where the reference instrument
is compared to the ground network could ascertain its drift and use it as a
transfer standard. In this way, all instruments could be drift-corrected and
then fed into a final STS regression (without a drift term) so that all of
the data are used to constrain the trend.

Data availability

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Data availability.

All data from individual instruments are publicly accessible either from NASA at https://eosweb.larc.nasa.gov/ (i.e., for SAGE II and III, HALOE, and POAM II and III) or the SCISAT-1 team at https://databace.scisat.ca/level2/ (i.e., for ACE-FTS).

Appendix A

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This work is primarily a continuation and expansion of Damadeo et al. (2014). That work discusses the application of a simultaneous temporal and spatial (STS) multiple linear regression (MLR) analysis applied to SAGE II stratospheric ozone data. This work uses the techniques described in Damadeo et al. (2014) and expands them to include multiple occultation data sets. For the sake of brevity and to assist the reader, this appendix will summarize the methodology and detail how it was expanded to incorporate multiple data sets.

Occultation instruments provide observations at two distinct latitude bands
each day separated by spacecraft event type (i.e., sunrise or sunset as seen
by the spacecraft). These observations are evenly distributed in longitude
and span about 3^{∘} in latitude at the highest latitudes to
about 10^{∘} in latitude in the tropics. The locations of these bands
gradually move from day to day according to the spacecraft's orbit and can
occasionally cross each other. The data for each instrument are averaged
according to these daily zonal bands and are separated by both the local and
spacecraft event types so that each day can produce up to four data points
for a single instrument. When multiple instruments are used, this process is
done separately for each instrument, meaning that, on a given day, it is
possible to have multiple data points at the same latitude from different
instruments feeding into the regression simultaneously.

The regression model applied to all of this averaged data has the following form:

$$\begin{array}{}\text{(A1)}& \mathit{\eta}(\mathit{\theta},t)=\sum _{i}\sum _{j}{\mathit{\beta}}_{i,j}\phantom{\rule{0.25em}{0ex}}{\mathrm{\Theta}}_{i}\left(\mathit{\theta}\right)\phantom{\rule{0.25em}{0ex}}{T}_{j}\left(t\right),\end{array}$$

where *η* is the concentration of O_{3}, Θ_{i}(*θ*) is the
functional form of the latitude dependence (Legendre polynomials in spherical
harmonics), *T*_{j}(*t*) is the functional form of the temporal dependence, and
*β*_{i,j} are the coefficients of the regression. The *T*_{j}(*t*)
represent all of the typical proxies used in MLR analyses (e.g., QBO, ENSO,
solar) as well as several conditional proxies. Conditional proxies are
simply 0 or some value (typically 1 to make it a binary conditional term)
depending upon whether a condition is met or not for each data point. For
example, the diurnal variability proxy is 0 for every data point that is a
sunset and 1 for every data point that is a sunrise. In this way, the diurnal
coefficient (or rather set of coefficients because there are multiple “*i*”
values for each “*j*”) represents the difference between sunrise and sunset
events. Additionally, some proxies are applied separately by adding another
condition. Continuing the diurnal example, the diurnal variability is
actually applied separately for each of the instruments so instead of a
single *T*_{j}(*t*) there are three (one for each instrument). The condition is
a simple logical “AND” between the diurnal condition just described and a
test to see if the data point of interest comes from the instrument to which
the proxy applies. Similarly, there are two mean offset binary conditional
terms (i.e., one for HALOE relative to SAGE II and one for ACE-FTS relative
to SAGE II) and there are two drift conditional terms with forms
${T}_{j}\left(t\right)=t-{t}_{\mathrm{0},j}$, where *t*_{0,j} is chosen at the middle of each
instrument's mission period for HALOE and ACE-FTS. This process of creating
conditional proxies can be repeated to apply certain temporal proxies
separately to data points from different data sets (e.g., having a single
seasonal cycle applied to all data sets or having each data set have its
own).

Once the regression is applied, autocorrelation and heteroscedasticity corrections are applied as detailed in Damadeo et al. (2014). These corrections use the total and uncorrelated residuals from the regression to improve the uncertainties in the coefficients that would otherwise be underestimated. When using multiple data sets, these corrections are applied separately by first subsetting the residuals to only those from a single instrument, applying the corrections, and then repeating the process for each instrument. Applying these corrections separately ignores correlations between the data sets and their impacts on the uncertainties. However, we believe this to be a second-order effect as the occurrence of global perturbations is negligible and the number of coincidences between the occultation instruments is small when compared to the ensemble.

Appendix B

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The goal of this work is determine ozone trends and their uncertainties from the proxies used in the regression. In the case of a piecewise linear trend (PWLT) proxy, the trend is simply the coefficient corresponding to that particular time period (or, in the case of the STS regression, an aggregate coefficient evaluated at a particular latitude). Unlike a PWLT term, the EESC-proxy terms (from Damadeo et al., 2014) are comprised of two separate temporal coefficients and uncertainties with functional shapes that are nonlinear, making a simple determination of the resulting overall trends and uncertainties impossible. Instead, we begin by taking the EESC-proxy component of the fit and its associated uncertainties that have the following forms:

$$\begin{array}{ll}\text{(B1)}& {\displaystyle}y({\mathit{\theta}}_{\mathrm{0}},t)=& {\displaystyle}{C}_{{\mathrm{EESC}}_{\mathrm{1}}}\left({\mathit{\theta}}_{\mathrm{0}}\right)\phantom{\rule{0.25em}{0ex}}{T}_{j={\mathrm{EESC}}_{\mathrm{1}}}\left(t\right){\displaystyle}+& {\displaystyle}{C}_{{\mathrm{EESC}}_{\mathrm{2}}}\left({\mathit{\theta}}_{\mathrm{0}}\right)\phantom{\rule{0.25em}{0ex}}{T}_{j={\mathrm{EESC}}_{\mathrm{2}}}\left(t\right)\end{array}$$

and

$$\begin{array}{ll}\text{(B2)}& {\displaystyle}{\mathit{\sigma}}_{y}^{\mathrm{2}}({\mathit{\theta}}_{\mathrm{0}},t)=& {\displaystyle}{\mathit{\sigma}}_{{\mathrm{EESC}}_{\mathrm{1}}}^{\mathrm{2}}\left({\mathit{\theta}}_{\mathrm{0}}\right)\phantom{\rule{0.25em}{0ex}}{T}_{{\mathrm{EESC}}_{\mathrm{1}}}^{\mathrm{2}}\left(t\right){\displaystyle}+& {\displaystyle}{\mathit{\sigma}}_{{\mathrm{EESC}}_{\mathrm{2}}}^{\mathrm{2}}\left({\mathit{\theta}}_{\mathrm{0}}\right)\phantom{\rule{0.25em}{0ex}}{T}_{{\mathrm{EESC}}_{\mathrm{2}}}^{\mathrm{2}}\left(t\right),\end{array}$$

where ${C}_{{\mathrm{EESC}}_{\mathrm{1},\mathrm{2}}}\left({\mathit{\theta}}_{\mathrm{0}}\right)$ are the aggregate coefficients from
the regression evaluated at a particular latitude *θ*_{0} computed as

$$\begin{array}{}\text{(B3)}& {C}_{{\mathrm{EESC}}_{\mathrm{1},\mathrm{2}}}\left({\mathit{\theta}}_{\mathrm{0}}\right)=\sum _{i}{\mathit{\beta}}_{i,j={\mathrm{EESC}}_{\mathrm{1},\mathrm{2}}}\phantom{\rule{0.25em}{0ex}}{\mathrm{\Theta}}_{i}\left({\mathit{\theta}}_{\mathrm{0}}\right),\end{array}$$

with

$$\begin{array}{}\text{(B4)}& {\mathit{\sigma}}_{{\mathrm{EESC}}_{\mathrm{1},\mathrm{2}}}^{\mathrm{2}}\left({\mathit{\theta}}_{\mathrm{0}}\right)=\sum _{i}{\mathit{\sigma}}_{{\mathit{\beta}}_{i,j={\mathrm{EESC}}_{\mathrm{1},\mathrm{2}}}}^{\mathrm{2}}\phantom{\rule{0.25em}{0ex}}{\mathrm{\Theta}}_{i}^{\mathrm{2}}\left({\mathit{\theta}}_{\mathrm{0}}\right),\end{array}$$

and ${T}_{{\mathrm{EESC}}_{\mathrm{1},\mathrm{2}}}$ are the EESC proxies. The equivalent trend is then computed by performing a simple linear fit to these data over a desired time period (e.g., 2000–2012) and using the resulting slope as the trend. The resulting uncertainty in this slope, however, is more complicated because the uncertainty from the linear fit can vary with the arbitrary number of points used to create the EESC fit. We have concluded that the best way to relate a linear fit to the EESC fit was to draw a corollary to the uncertainties associated with a straight line fit. A linear fit to the EESC-fit data and their uncertainty have the following forms:

$$\begin{array}{}\text{(B5)}& y{}^{\prime}\left(t\right)={c}_{\mathrm{0}}+{c}_{\mathrm{1}}\phantom{\rule{0.25em}{0ex}}(t-{t}_{\mathrm{0}})\end{array}$$

and

$$\begin{array}{}\text{(B6)}& {\mathit{\sigma}}_{y{}^{\prime}}\left(t\right)=\sqrt{{\mathit{\sigma}}_{{c}_{\mathrm{0}}}^{\mathrm{2}}+{\mathit{\sigma}}_{{c}_{\mathrm{1}}}^{\mathrm{2}}\phantom{\rule{0.25em}{0ex}}(t-{t}_{\mathrm{0}}{)}^{\mathrm{2}}},\end{array}$$

where *y*^{′}(*t*) is the best fit to *y*(*θ*_{0},*t*) and *c*_{0} and *c*_{1}
come from the linear fit but there is no straightforward calculation to compute ${\mathit{\sigma}}_{{c}_{\mathrm{0}}}$
and ${\mathit{\sigma}}_{{c}_{\mathrm{1}}}$. It is worth noting that, for the linear fit to the EESC
fit, the choice of *t*_{0} is arbitrary when we only care about *c*_{1}. From
these equations, the correlation is made between the linear equation and its
functional uncertainties (i.e., ${\mathit{\sigma}}_{y{}^{\prime}}$ that are unknown) and the
actual uncertainties from the EESC fit (i.e., *σ*_{y}). From the above
we have

$$\begin{array}{}\text{(B7)}& {\mathit{\sigma}}_{{c}_{\mathrm{0}}}={\mathit{\sigma}}_{y{}^{\prime}}\left({t}_{\mathrm{0}}\right),\end{array}$$

to which we draw the corollary

$$\begin{array}{}\text{(B8)}& {\mathit{\sigma}}_{{c}_{\mathrm{0}}}=\mathrm{MINIMUM}\left\{{\mathit{\sigma}}_{y}({\mathit{\theta}}_{\mathrm{0}},t)\right\}={\mathit{\sigma}}_{y}({\mathit{\theta}}_{\mathrm{0}},{t}_{\mathrm{0}})\end{array}$$

that yields ${\mathit{\sigma}}_{{c}_{\mathrm{0}}}$ and *t*_{0}. From there, it is simple to look at ${\mathit{\sigma}}_{{c}_{\mathrm{1}}}$:

$$\begin{array}{}\text{(B9)}& {\mathit{\sigma}}_{{c}_{\mathrm{1}}}=\sqrt{{\displaystyle \frac{{\mathit{\sigma}}_{y{}^{\prime}}^{\mathrm{2}}\left(t\right)-{\mathit{\sigma}}_{{c}_{\mathrm{0}}}^{\mathrm{2}}}{(t-{t}_{\mathrm{0}}{)}^{\mathrm{2}}}}},\end{array}$$

to which we draw the corollary

$$\begin{array}{}\text{(B10)}& {\mathit{\sigma}}_{{c}_{\mathrm{1}}}=\mathrm{MEAN}\left\{\sqrt{{\displaystyle \frac{{\mathit{\sigma}}_{y}^{\mathrm{2}}({\mathit{\theta}}_{\mathrm{0}},t)-{\mathit{\sigma}}_{{c}_{\mathrm{0}}}^{\mathrm{2}}}{(t-{t}_{\mathrm{0}}{)}^{\mathrm{2}}}}}\right\}.\end{array}$$

Thus, using a direct correlation between the EESC fit and the functional form of the linear fit, the uncertainties in the EESC fit can be used to derive a reasonable estimate for the uncertainty in the fitted slope.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-18-535-2018-supplement.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Quadrennial Ozone Symposium 2016 – Status and trends of atmospheric ozone (ACP/AMT inter-journal SI)”. It is a result of the Quadrennial Ozone Symposium 2016, Edinburgh, United Kingdom, 4–9 September 2016.

Acknowledgements

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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. The Atmospheric Chemistry Experiment (ACE), also known as SCISAT,
is a Canadian-led mission mainly supported by the Canadian Space Agency and
the Natural Sciences and Engineering Research Council of Canada.

Edited by: Rolf Müller

Reviewed by: two anonymous referees

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Special issue

Short summary

An ozone trend analysis that compensates for sampling biases is applied to sparsely sampled occultation data sets. International assessments have noted deficiencies in past trend analyses and this work addresses those sources of uncertainty. The nonuniform sampling patterns in data sets and drifts between data sets can affect derived recovery trends by up to 2 % decade^{−1}. The limitations inherent to all techniques are also described and a potential path forward towards resolution is presented.

An ozone trend analysis that compensates for sampling biases is applied to sparsely sampled...

Atmospheric Chemistry and Physics

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