The Atmospheric Chemistry Experiment Fourier transform spectrometer (ACE-FTS) was launched onboard the SCISAT-1 spacecraft in August 2003 . 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 . Ozone is retrieved using the spectral features near 10 µm . The version of the ACE-FTS data product used here is version 3.5 , 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 . The version of the HALOE data product used here is version 19.0 , which produces vertical profiles of VMR interpolated to a 0.3 km grid with a vertical resolution of 2–3 km . 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 . 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 . 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 . The version of the POAM III data product used here is version 4.0 , 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 . The version of the SAGE II data product used here is version 7.00 , 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 . The version of the SAGE III data product used here is version 4.00 , 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 . Beyond this, the data screening procedures recommended for each data set are performed. ACE-FTS data are screened as outlined in . 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 . Since SAGE III data were screened prior to release, no additional screening is performed.

Similarly to , 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  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 . 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 . 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). 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.  compare quite well with those in Fig. 5 of .

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 or through changes in dynamics via changes in radiative forcing . 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.,, while some works attempt to include a term in the regression to model the impact e.g.,, 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 . 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 , 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 , 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 and HALOE was filtered under the recommendations in .

Figure  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 and , 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 .

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. ) the SAGE II responses with aerosol filtering applied (left plot in Fig. ) 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 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 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 .

The impact of the ∼ 11-year solar cycle on stratospheric ozone has been an ongoing topic of study e.g.,. 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 . Figure  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., and naturally very similar to those from . 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 , though this was believed to be the regression algorithm simply trying to attribute some of the aerosol response to the solar cycle . 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.

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. and 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. . 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. . 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. . 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.  and plotting it along with the low frequency variability from the STS regression. The bottom row of Fig.  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.,, 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. . 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.  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. . 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  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.