This paper is based on a study performed within the model world using the Chemical Lagrangian Model of the Stratosphere, CLaMS . The model set-up is described in detail elsewhere e.g.. Briefly, CLaMS is a modular Lagrangian chemistry transport model based on 3D-forward trajectories with parameterization of small-scale mixing. CLaMS consists of several modules, such as Lagrangian advection (TRAJ), stratospheric mixing (MIX), sinks of H2O (CIRRUS), stratospheric chemistry, and several other modules responsible for simulation of various physical and chemical processes. The modules act successively at each time step of 24 h. The boundary conditions at the surface are prescribed based on ground-based measurements in the lowest model level (below ≈ 4 km). CH4 mixing ratios are taken from the zonally symmetric NOAA/ESRL dataset e.g. from 1990–2011 and from zonally resolved AIRS data e.g. for 2011–2017. The CH4 boundary condition was switched in 2011 to take advantage of the better sampling of AIRS data in comparison to NOAA, although accepting the apparent discontinuity. Because the results of CLaMS are internally consistent, the discontinuity has only negligible effects; see Fig.  .

The CLaMS trajectory module TRAJ performs fully Lagrangian three-dimensional advection of an ensemble of approximately 2 million air parcels. The position of each air parcel is defined in hybrid isentropic coordinates and longitude–latitude space. Horizontal resolution is about 100 km, while the vertical resolution is defined via a critical aspect ratio, of about 250 . The CLaMS simulations cover the atmosphere from the surface to the stratopause (2500 K or ≈ 60 km). The advection of forward trajectories in CLaMS is calculated using 6-hourly wind fields and total diabatic heating rates from meteorological ERA-Interim reanalysis . Wind fields are linearly interpolated from the adjacent grid points to the locations of the air parcels. The trajectory calculation advects air parcels to new positions within one model time step.

The parameterization of small-scale mixing in the CLaMS mixing module is based on the deformation rate in the large-scale flow: air parcels may be merged, or new air parcels may be inserted at each time step, depending on the critical distances between the air parcels . Note that the mixing parameterization affects both horizontal (associated with deformation in the horizontal flow) and vertical (related to the vertical shear) diffusivity .

Dehydration in CLaMS is performed with the CIRRUS module e.g.. The calculation includes freeze-drying in regions of cold temperatures, mainly occurring around the tropical tropopause and in the southern polar vortex. These cold temperatures cause formation and sedimentation of ice particles. If saturation along a CLaMS air parcel trajectory exceeds a critical saturation, then the H2O amount in excess is instantaneously transformed to the ice phase and sediments out. The parameterization of sedimentation is based on a mean ice particle radius, the characteristic sedimentation length, and the corresponding fall speed. The fall distance of the ice particles is calculated from the fall speed and the computation time step. After comparison of the fallen path with a characteristic sedimentation length of the vertical grid size, a respective fraction of ice is removed. If the parcel is sub-saturated and ice exists, the ice will be instantaneously evaporated to maintain saturation. CH4 oxidation is included in CLaMS as a source of H2O in the middle and upper stratosphere for details see. Note that due to the simple parameterization of ice microphysics and the omission of a parameterization of convective processes, the simulated H2O results are meaningful only above the tropopause.

In general, the mixing ratio of any long-lived trace gas χ(r,t) at a specific time and specific location in the stratosphere, with assumed absence of integrated loss, can be expressed as the following integral over all past times : 2χ(r,t)=∫0∞χr0,t-t′Gr,t|r0,t-t′dt′, where t is the field time when the volume is sampled, t′ is the transit time, and G(r,t|r0,t-t′) is the boundary propagator or Green function of the transport operator. Here, G is interpreted as a transit time distribution (the “age spectrum”) and is the probability that the transit time of the air parcel travelling from the source r0 to the sample point r is in the range between t′ and t′+dt′. The first momentum of the age spectrum is the mean age of air (AoA). In this way, the stratospheric tracer distribution can be described through the contributions of the tropospheric evolution and transport.

In our study, the age spectrum is computed with CLaMS driven by the ERA-Interim reanalysis, using the “boundary impulse (time-)evolving” (BIER) method based on multiple tracer pulses . For the inert tracer χ with a pulse at the location r0, the field time t0, and the source time t0*, the time evolution of the source can be described with a δ-distribution as χ(r0,t0)=δ(t0-t0*). Thus, Eq. () can be transformed to 3χ(r,t′+t0*)=G(r,t′+t0*|r0,t0*), where t′=t0-t0* is a transit time, and G(r,t′+t0*|r0,t0*) is the boundary impulse response at location r to a δ-boundary condition at the location r0 at source time t0*. Having N different tracers χi (i=1,…,N) with pulses at the source location r0 at times t0[i] provides the field time dependence of the propagator G. The age spectrum may be constructed at each field time t and location r as G(r,t|r0,t-t[i]′)=χi(r,t). Hence, the N different tracers provide N pieces of information for the age spectrum at the discrete transit times t[i]′=t-t0[i].

Here, N=60 different boundary pulse tracers were used. These pulses were released directly at the tropical tropopause between 30∘ S–30∘ N. Precisely, the source region covers the potential temperature layer from 10 K below to 10 K above the WMO (lapse rate) tropopause. As a remark, this specific choice of the source region causes the uncertainty in our analyses as it does not exactly correspond to the region that determines the H2O and CH4 entry mixing ratios; the tropically controlled transition region bounds between approximately 380 and 450 K . However, this mismatch impacts the results only close to the tropopause, so the reconstruction of H2O and CH4 by the modelled age spectrum ensures the reliability of the method in most of the stratosphere. The particular tracer mixing ratio is set to 1 for each pulse for a period of 30 d at the location r0, and it is set to 0 in r0 at other times. Pulses are launched every month. Consequently, to build the age spectrum for January 1990, the most recent tracer pulse has source times in January 1990, the second tracer pulse in December 1989, and so on. In our study, the original length of each age spectrum is 10 years (threshold transit time). The tail of the age spectrum is approximated with an exponential function when transit time exceeds 10 years e.g.. We used the exponential correction for the tail back to January 1979 for each age spectrum.

Changes in stratospheric H2O are determined by the stratospheric H2O entry mixing ratio through troposphere–stratosphere exchange and by chemical sources, mainly oxidation of CH4 and molecular hydrogen (H2) in the middle and high stratosphere . H2O in the troposphere is continuously supplied from the Earth's surface. CH4 is largely emitted at the Earth's surface because of anaerobic reactions, and H2 originates from biomass burning and other natural sources; CH4 and H2 are transported from the troposphere into the stratosphere. Based on satellite and balloon observations, the sum of the principal components of the hydrogen budget (H2O, 2 × CH4, and H2) is constant with altitude over most of the stratosphere e.g..

In this paper, we investigate the methodology to estimate AoA trends within a closed model environment of CLaMS, in which the mean stratospheric H2O and CH4 and their trends are known. Trends in AoA can be calculated from trends in stratospheric H2O mixing ratios by using the conservation property of total hydrogen in the stratosphere and assuming that H2 production from CH4 oxidation is balanced by H2 oxidation, namely that the sum of H2O and 2 times CH4 mixing ratios is approximately constant e.g.. This conservation property implies, for the H2O mixing ratio at a given location r and time t in the stratosphere, 4H2O(r,t)=H2O[entry](r,t)+2α(r,t)CH4[entry](r,t), where H2O[entry](r,t) and CH4[entry](r,t) are H2O and CH4 mixing ratios respectively at the specific location and time in the stratosphere, being transported from their entry location at the tropical tropopause without any chemical effects. Note that the usage of the simple parameterization (see Eq. ) for the ratio between oxidized CH4 and produced H2O has its limitations; e.g. a ratio of 2 overestimates the production of H2O in the lower stratosphere and somewhat underestimates it in the upper stratosphere . It is questionable whether this parameterization can be used for future climate projections, when the BDC is expected to accelerate e.g., and, as a result, the transport of H2 molecules becomes an important factor for the vertical profile of the H2O in the stratosphere. We also note that an increase in tropospheric H2 might gain importance in a future hydrogen economy e.g..

The time series of CLaMS H2O and CH4 at the entry to the stratosphere, averaged over the tropics at 30∘ S–30∘ N and in the potential temperature layer 390–400 K (approximately 80 hPa), chosen just above the cold tropical tropopause region, are shown in Fig. . The H2O time series is highly variable (Fig. a), while the CH4 time series shows a clear positive trend (Fig. b). Besides freeze-drying at the tropical tropopause and in the Antarctic polar vortex, stratospheric H2O is controlled mainly by CH4 oxidation in the middle and high stratosphere (Fig. c, d). Consequently, CH4 mixing ratios generally decrease with increasing altitude, as it is gradually chemically transformed into H2O.

The fractional release factor (FRF), α, describes the fraction of CH4 which has been dissociated in the stratosphere , and it is determined as 5α(r,t)=1-CH4(r,t)/CH4[entry](r,t). The FRF is strongly affected by the vertical transport of the BDC. Hence, information on circulation trends (in particular on AoA) can be deduced from trends in FRF . It should be noted that the change in FRF is due to transit time (age spectra) and circulation pathway (path spectra) changes, but AoA is a measure for only transit time and not the transit pathway dependency.

Assuming long-term trends as small perturbations to the basic state, Eq. () can be rewritten as an equation for the linear trend in stratospheric H2O over a given time period 6ΔH2O(r,t)=ΔH2O[entry](r,t)+2α(r,t)ΔCH4[entry](r,t)+2CH4[entry](r,t)Δα(r,t). Here, ΔH2O[entry](r,t) and ΔCH4[entry](r,t) are the trends in water vapour and methane transported from their stratospheric entry location at the tropical tropopause layer (neglecting chemistry), and Δα(r,t) is the trend in FRF. Note that ΔH2O[entry](r,t) and ΔCH4[entry](r,t) depend only on changes in transit times, while Δα(r,t) generally also depends on the changes in transport pathways. In addition, the sum of H2O and 2 times CH4 mixing ratios is not conserved around and below the tropopause and in the polar vortex, where dehydration causes loss of H2O. Hence, the presented analysis does not apply at the lowermost stratosphere, below the tropopause, and in the polar vortex or near its edges.

Using the above approach, it is possible to investigate the different contributions to stratospheric H2O changes related to changes in stratospheric H2O entry mixing ratio, changes in CH4 entry mixing ratio, and changes in the FRF, which are affected by stratospheric circulation changes and can be converted to AoA trend. In addition, the dependency of FRF changes on circulation pathways is implicitly taken into account by the AoA–FRF correlation functions in all used methods (Sect. ).

There are four methods of AoA trend calculation used in this study, depending on the assumed approximations of (i) an instantaneous propagation of stratospheric entry mixing ratios and (ii) stationarity of the correlation between AoA and the FRF. The summary of the different terms needed for AoA estimation with respect to the used method is shown in Table .

The full reconstruction method (FULL) includes the most detailed representation of the true atmospheric processes. In the FULL method, H2O and CH4 entry mixing ratios are propagated through the convolution of the tropical tropopause layer mixing ratios with the modelled age spectrum. The monthly varying AoA–FRF correlations are used for translating FRF into AoA changes, to include effects of the non-stationarity of the correlation. Neither of the approximations (i) or (ii) is used in the FULL method. Note that for estimating AoA trend with the FULL method, the propagated entry H2O mixing ratios are not needed; the AoA trend is deduced from the FRF, which requires only propagated entry CH4 mixing ratios.

The constant correlation method (C-CORR) includes the propagation of entry H2O and CH4 mixing ratios by the modelled age spectrum (same as for FULL), but the stationary relationship between FRF and modelled AoA is used (ii approximation). The difference between C-CORR and FULL is in the correlation between AoA–FRF and the method of calculating FRF and its trend.

The method based on both approximations (i) and (ii) is named “approximation method” in the following and is abbreviated APPROX. In fact, the APPROX method is used in the literature, and we evaluate it in this paper in detail. Finally, we introduce an improvement to the approximation method: instead of using approximation (i), stratospheric entry H2O and CH4 mixing ratios are propagated with the parameterized idealized age spectrum.

AoA trends can be estimated from the separation of H2O changes into three different contributions (see Eq. ). In a first step, we estimate here each term of Eq. () using the full reconstruction method FULL through propagation of H2O and CH4 entry mixing ratios by the modelled age spectrum (for details see Table ).

We calculate the first term of Eq. () by the convolution of H2O mixing ratio at the tropical tropopause with the transport operator's Green function using Eq. (). The Green function, or stratospheric age spectrum, has been simulated by CLaMS and is known over the considered period (for details see Sect. ). Hence, the propagation of boundary mixing ratios to each grid point in the stratosphere provides the full reconstructed stratospheric tracer field. The analogous calculation is applied to derive CH4 mixing ratios in the stratosphere, transported without including chemical effects.

In our study, the entry time series of H2O and CH4 are taken from zonally averaged monthly mean data simulated with CLaMS driven by ERA-Interim reanalysis (see Fig. a, b). The location of entry to the stratosphere is approximated as the 390–400 K layer between 30∘ S–30∘ N, which is located just above the cold point tropopause to avoid complications related to H2O dehydration processes at the tropopause. The small difference between the age spectrum source region (tropopause ± 10 K) and the trace gas entry region has only a negligible impact on our results from above ≈420 K due to the small difference in the transit time between the two regions. The propagation procedure yields zonally averaged stratospheric entry H2O and CH4 distributions with a monthly resolution on the latitude–potential temperature grid.

For deducing FRF, the relation from Eq. () is used, where CH4(r,t) is monthly mean CH4 simulated with CLaMS driven by ERA-Interim reanalysis. The CH4[entry](r,t) is calculated as described above, through the convolution of the tropical entry mixing ratio time series with the age spectrum. Consequently, the resulting FRF has a monthly resolution.

Thus, we can estimate each contribution to the stratospheric H2O mixing ratio. All trends are calculated through a linear regression, which minimizes the standard deviation at each latitude–potential temperature grid. The different contributions to the total stratospheric H2O change for 1990–2017 are shown in Fig. . Note that in this study we applied the propagation procedure to the period 1990–2017 because of the availability of the CLaMS data and the necessary age spectrum length. Figure a, b represent the first two terms of Eq. (), related to the entry H2O and CH4 mixing ratio trends. Figure c shows the impact from circulation changes, in terms of the change in FRF. In general, the different contributions affect the stratospheric H2O changes differently in different regions, consistent with the findings of . The strongest regional pattern is apparent in the contribution related to the stratospheric circulation change (Fig. c).

For assessing the reliability of the method applied to estimate the different contributions, we compare the stratospheric H2O trend reconstructed as the sum of the calculated contributions (in Fig. ) with the actual trend of CLaMS-simulated H2O. Figure a shows the true H2O trend from the CLaMS simulation, while Fig. b shows the sum of the three terms from Fig. . Figure c shows the absolute difference between the true and reconstructed trends, indicating clear quantitative differences. Particularly large differences occur in the Antarctic region. This is expected due to the strong local dehydration occurring in that region and the related failure of the total hydrogen conservation. Hence, the results of the reconstruction method should be interpreted with caution in the southern polar region. Further disagreement between Fig. a and b is partly related to inaccuracies in the modelled age spectrum (monthly pulsing, limited spectrum length) and to inaccuracies in the boundary time series (averaging in the layer of 390–400 K potential temperature and 30∘ S–30∘ N). Note that through the convolution of the age spectrum and the stratospheric entry time series it is possible to reconstruct mixing ratios only above the tropopause (or the level of the boundary time series, if chosen differently). Outside of the southern high-latitude regions and in the proximity of the extratropical tropopause (> 30∘ N/S), the overall differences shown in Fig. c are small, and the propagation procedure provides a good estimate of stratospheric H2O change and its contributions, at least regarding the large-scale patterns. Note that neither stratospheric entry H2O (Fig. a) nor CH4 changes (Fig. b) explain the pattern of the true H2O trend (Fig. a). Instead, the circulation change term (Fig. c) includes the regional characteristics of the actual stratospheric H2O trend.

The changes in FRF can be translated into changes in AoA using the correlation between estimated FRF and known AoA, following the procedure described by . In the full reconstruction method FULL (for details see Table ), a monthly varying correlation between estimated FRF and CLaMS AoA is used, where AoA = f(α), and f is an empirically determined correlation function (Fig. ).

In an example for January, April, and July of 1995 shown in Fig. a, the AoA–FRF correlation functions are unique for each month, because the differences in magnitude of the coefficients are greater than the standard error's range. Moreover, monthly AoA–FRF correlation functions have a very small difference for relatively young air (< 4 years) and low FRF (< 0.4), but there are visible differences towards older AoA.

It is worth mentioning that the monthly AoA-FRF correlation functions are still a simplification and might introduce some bias in the reconstruction. In general, an accurate AoA-FRF correlation function depends not only on the considered time but also on longitude, latitude, and altitude. An example of the relationship between AoA and FRF with regard to different latitude ranges (30–40∘ S, 10∘ S–10∘ N, 30–40∘ N) is shown in Fig. b for January 2000. The AoA–FRF correlation functions are unique for each latitude range, because the differences in the magnitude of coefficients are out of the standard error's range. And, for instance, at the same FRF level of 0.3 air at the northern tropics (30–40∘ N) is younger than at the southern tropics (30–40∘ S) by almost half a year. It is likely due to stronger and deeper BDC in the Northern Hemisphere during boreal winter e.g., causing air parcels of the same age to travel deeper pathways through the stratosphere and experience more chemical depletion compared to the Southern Hemisphere.

AoA trend is calculated from the resulting AoA applying a linear fit (minimizing the standard deviation) at each latitude–potential temperature grid point. The reference true AoA trend calculated directly from CLaMS-simulated AoA is shown in Fig. a, whereas the resulting AoA trend calculated with the FULL method is shown in Fig. b. The estimated AoA trend with FULL is qualitatively and quantitatively highly reliable, when comparing to the reference CLaMS AoA trend. Note that for estimating AoA trends with the FULL method, the propagated entry H2O mixing ratios are not used; the AoA trend is deduced from the FRF, which requires only propagated entry CH4 mixing ratios. Visible differences between Fig. a and Fig. b are related to approximations in the AoA–FRF correlation: monthly AoA–FRF correlation functions are still a simplification and might introduce some bias in the reconstruction. The averaged (between 390–400 K potential temperature, from 30∘ S–30∘ N) CH4 boundary mixing ratios, which are used for the reconstruction, could induce biases as well.

Accurate estimation of AoA from observed trace gas distributions is a complicated task. Even though it is desirable to have a complete age spectrum for AoA calculations, it is very difficult to obtain it from measurements. Consequently, different approximations are often applied when deriving AoA from trace gases observation with a non-linear increase, as well as assumptions about the age spectrum and its shape e.g.. In the following, we further investigate the method of AoA trend estimation from a combination of H2O and CH4 changes, applying the two major approximations introduced above: (i) instantaneous propagation of stratospheric entry mixing ratios and (ii) constant correlation (stationary relationship) between FRF and modelled AoA.

In the approximation method APPROX, each term of Eq. () is approximated. The actual trend of stratospheric H2O (left side of the Eq. ) is assumed to be known beforehand. In our case it is the CLaMS-simulated H2O change over the considered period; if the method is applied to observations it would be the observed H2O change. The needed variables for the first and the second terms of Eq. () are obtained as a linear trend of the model entry mixing ratio time series, averaged over the 390–400 K potential temperature layer and 30∘ S–30∘ N, from the considered period (see Fig. ). The FRF required for the second term is derived from Eq. (). Therefore, following we use zonal mean CLaMS-simulated CH4(r,t) mixing ratios averaged over 2005–2006, and we calculate CH4[entry] as a mean mixing ratio between 390–400 K potential temperatures and 30∘ S–30∘ N during 2002–2006. The third term is calculated as a residual between the actual CLaMS H2O trend over the considered period (left side of Eq. ) and the other two terms. Dividing the third term by 2CH4[entry] yields the FRF changes.

In order to estimate the AoA trend induced by the changes in stratospheric H2O, we define the relationship between FRF from APPROX and CLaMS-modelled AoA (taken as 2005–2006 climatology in the APPROX method). The correlation function is derived by fitting a third-order polynomial, as suggested by . In our study, the empirical relationship between APPROX FRF and CLaMS AoA is described by the function f(α)=0.85+16.49α-25.30α2+13.77α3 (see Appendix ). The approximation method assumes that the AoA–FRF relationship is stable in time (stationary). Note that by applying a constant AoA–FRF correlation function some atmospheric variability can be lost. Using the correlation function f(α) we obtain the AoA trends from previously estimated FRF and its changes, as ΔAoA=f(α+Δα)-f(α). A summary of all terms defined within the APPROX method is provided in Table .

The resulting decadal AoA trend for 1990–2017 estimated with the approximation method is shown in Fig. c and can be compared to the true AoA trend from the reference CLaMS simulation, shown in Fig. a. There are visible quantitative differences between the true and the estimated trends of the AoA, especially in the Antarctic region. These differences are related to the dehydration processes occurring in that region. Also, the approximation method overestimates the AoA trend in the northern hemispheric subtropical middle stratosphere. In the extratropical lowermost stratosphere (below about 380 K), AoA trends calculated with the approximation method are even opposite compared to the true trends, likely related to significant transport into this region across the subtropical tropopause e.g., which is not represented in the APPROX. But overall, both APPROX and the true AoA trends show good agreement: decreasing AoA in the lower stratosphere and increasing AoA in the northern hemispheric middle stratosphere. Interestingly, for the 1990–2017 period the AoA trend shows clear differences between the Northern Hemisphere and Southern Hemisphere. The hemispheric differences might be related to the effect of mixing e.g. and shifting stratospheric circulation patterns , previously found in the long-term AoA trend derived from the observed stratospheric CH4 . Overall, APPROX provides a good estimate of the AoA trend for 1990–2017, corroborating the validity of the applicability of this method to H2O observations over similar time periods e.g..

To assess the general applicability of the approximation method APPROX, we consider another period, 1990–2006. The true AoA trend for this period from the CLaMS simulation is presented in Fig. a, and the result from APPROX is shown in Fig. c. In this case, the APPROX AoA trend disagrees substantially when compared with the true CLaMS trend. Differences occur even in the sign of the AoA trend. Particularly clear differences occur in the strength of the AoA trend and its detailed pattern. Thus, the accuracy of the estimated AoA changes from APPROX largely depends on the considered period, which should be long enough to ensure that the effects of variability are small. In the following section, we further investigate the effects of the applied approximations and discuss their impact on the quality of the estimated AoA trend.

Firstly, we evaluate the effect of approximation (i) of the instantaneous entry mixing ratio propagation. For this purpose, we perform an additional sensitivity study, where the stationary relationship between FRF and modelled AoA is kept, but entry H2O and CH4 mixing ratios are propagated through the convolution of the CLaMS mixing ratios with the modelled age spectrum. This method is termed the constant correlation method, C-CORR, in the following (see Table  for details). Note that in both methods, C-CORR and APPROX, the same FRF distribution is used, but the changes in circulation (namely Δα) are different.

The estimated AoA trend from C-CORR for 1990–2006 is shown in Fig. d. The approximation of the instantaneous entry propagation largely affects the AoA trend, as evident from comparison of the resulting AoA trends from C-CORR, APPROX, and the CLaMS reference AoA trend (Fig. a, c, d). Including the entry H2O and CH4 propagation by the age spectrum in the method clearly improves the estimated AoA trend. When comparing C-CORR to APPROX the general trend patterns stay similar, but improvements are visible in the extratropical lower stratosphere and above about 600 K.

For evaluating the effect of the approximation (ii) of a constant correlation between FRF and AoA, we compare the resulting AoA trends from the C-CORR and FULL methods. The difference between the AoA trend results from the C-CORR and FULL methods stems from the differences in the AoA–FRF correlations used in each method and the explicit FRF trend calculation (see Table ).

In C-CORR, the AoA trend is estimated from the residual circulation contribution (see Table ). Consequently, the monthly varying AoA–FRF correlation improves the accuracy of the estimated AoA trend both qualitatively and quantitatively, when compared to the stationary AoA–FRF correlation (Fig. a, b, d). It was mentioned earlier in the paper that stratospheric H2O is a highly variable tracer and can lead to difficulties in estimating AoA trends. Hence, the good performance of the FULL method can be related to the fact that stratospheric entry H2O mixing ratios do not influence the calculation, provided that the polar regions are excluded (as explained in Sect. ).

For a more precise assessment of the effects of the two approximations, the differences among AoA trends estimated with different methods (APPROX, C-CORR, FULL) and for two different periods (1990–2017 and 1990–2006) are analysed. The difference between AoA trends from APPROX and C-CORR gives an estimate of the effect of the approximation (i) assuming instantaneous entry mixing ratio propagation, whereas the difference between C-CORR and FULL gives an estimate of the effect of the approximation (ii) using a constant AoA–FRF correlation (Fig. ).

Figure a, b show the differences in AoA trends for 1990–2017 between C-CORR and APPROX as well as FULL and C-CORR. The differences are less than 5 % per decade above 600 K. Below 600 K, the differences in AoA trends are higher, with the maximum at approximately 480 K. The larger differences in the lowest stratosphere directly above the tropopause should not be over-interpreted as the transit time resolution of used age spectra of 1 month is too coarse for a reliable reconstruction there. Interestingly, the effects of the first and second approximations are opposite in sign for 1990–2017 (Fig. a, b). Consequently, the effects from both approximations cancel out to some extent, such that the APPROX method yields results remarkably close to the CLaMS reverence AoA trend (see Fig. ). However, in general such a cancellation can not be expected.

Figure c, d show the difference in AoA trends for 1990–2006. The difference above 600 K is around 5 % per decade or less. A more complex structure is found below 600 K. Generally, the instantaneous entry mixing ratio propagation causes an error in the estimated stratospheric entry H2O and CH4 contributions (see Sect. ). This, in turn, causes an error in the derived residual circulation impact in C-CORR, which is further translated into the estimated AoA trend error.

The above analysis shows that the effects of both approximations (instantaneous propagation of stratospheric H2O and CH4 entry mixing ratios and a constant correlation between FRF and AoA) on the estimated AoA trend are comparable in magnitude; however they depend on the exact period considered. Interestingly, the effects of the approximations can be opposite in sign, cancelling each other out to some extent. Consequently, the approximation method can lead to a reliable estimation of AoA trends for certain periods, but this should not always be expected. A further improvement of the approximation method is proposed in the following Sect. .

For estimating AoA trends from H2O and CH4 observations, the approximations (i) instantaneous entry mixing ratio propagation and (ii) constant AoA–FRF correlation are necessary. Nevertheless, the higher reliability of AoA trends can be achieved if the used approximations are adjusted. As a simple and practical improvement, we propose using an analytical, parameterized age spectrum for propagating stratospheric entry H2O and CH4 mixing ratios. Note that for a further improvement of AoA trend estimates, a non-stationary AoA–FRF relationship in principle would be needed as well. But due to the sparseness of available stratospheric CH4 measurements, deducing such a relationship from observations is challenging, and we refrain from including it in the methodological improvement.

In the following, we discuss the results of an additional sensitivity study with CLaMS stratospheric entry H2O and CH4 mixing ratios propagated by the parameterized idealized age spectrum and using a constant AoA–FRF correlation. This method is hereinafter referred to as the “improved approximation method” (see Table  for details). In this method we use an inverse Gaussian distribution e.g. as a parameterized age spectrum: 7G(t,Γ)=Γ34πΔ2t3⋅exp⁡Γ⋅(t-Γ)24Δ2t, where Γ is the mean AoA and Δ is the width of the age spectrum. Here, we parameterize AoA in different zones or “regions” depending on the considered latitude, longitude, and height. The finer the separation into different regions, the less pronounced the discontinuities at the edges of the regions are. For a simple and practical method without assuming a priori knowledge of model age of air, we propose dividing the stratosphere into seven regions, prescribing one mean value of AoA for each region (see Appendix ). We apply the empirical relation between the age spectrum width and AoA proposed by and used in several other studies e.g.: 8Δ2Γ=C, with the constant C=0.7 years, although we note that recent work of suggests a larger value (2.0 years) for the lower stratosphere.

The resulting AoA trend estimated with the improved approximation method for 1990–2006 is shown in Fig. e. There is a clear improvement in the AoA trend estimation when compared to the pure approximation method APPROX (Fig. c). Note that in both methods the same FRF distribution is used, but the FRF changes are different depending on the propagation method of stratospheric entry H2O and CH4. The discrepancies in estimated AoA trends between APPROX and the improved approximation method stem from the residual calculation of the third term of Eq. (), CH4[entry](r,t)Δα(r,t), with the major impact of the stratospheric entry H2O trend (see Sect. ). The propagation of stratospheric entry H2O and CH4 by the proposed parameterized idealized age spectrum results in AoA trends close to those from the C-CORR method (Fig. d, e), which is the best estimate possible for the improved approximation method due to the usage of constant AoA–FRF correlation.

Hence, we encourage the usage of the improved approximation method when estimating AoA from the combination of H2O and CH4 observational data. Stratospheric entry mixing ratio time series for H2O and CH4 can be deduced from satellite measurements, such as ACE-FTS, HALOE, MIPAS, or SCIAMACHY e.g.. Due to the limited stratospheric CH4 observations, the stratospheric entry CH4 mixing ratio contribution can be kept as in APPROX (instantaneous propagation), because this term has only little effect on the resulting AoA trend (see Sect. ). Our study shows that the usage of the parameterized idealized age spectrum clearly improves the representation of the stratospheric entry H2O term and hence the final AoA trend estimate.