For the interpretation of atmospheric observations in the context of, for example, atmospheric pollution or in that of global warming, atmospheric models are often used. Atmospheric models vary in complexity from simple one-box models to state-of-the-art 3-D transport models. Different types of models are suitable for addressing different types of problems to different degrees of scrutiny. Therefore, there is no model category that fits all problems. Simple box models are easy to set up, computationally cheap, and transparent. For these and other reasons, their use in atmospheric studies is ubiquitous and has provided useful insights (e.g. ; ; ; ; ). However, simple box models also put limitations on the derived results, as they are by definition less comprehensive than complex models. For example, box models do not explicitly contain much information on a species' spatial distribution, which can be important if interacting quantities (e.g. loss processes) are distributed non-homogeneously in space. Where exactly these limitations lie and what the gain is from increasing model complexity can be difficult to diagnose and depends on the application.

A problem that has often been approached in box models is that of constraining the global atmospheric oxidizing capacity, which is largely determined by the tropospheric hydroxyl radical (OH) concentration . OH is dubbed the detergent of the atmosphere for its dominant role in the removal of a wide variety of pollutants, including urban pollutants (CO, NOx), greenhouse gases (CH4, HFCs), and HCFCs, which are greenhouse gases, and also contribute to stratospheric ozone depletion. The budgets of many of these pollutants have been strongly perturbed since pre-industrial times, and it is important to understand what consequences this has had in the past, and could have in the future, for the atmosphere's oxidizing capacity.

Due to its high reactivity, OH has a lifetime of seconds, which inhibits extrapolation of direct measurements. Moreover, OH abundance is the net result of many different reactions and reaction cycles, and thus modelling it process-based in full-chemistry models is complex and dependent on uncertain emission inventories of the many gases involved. Therefore, the most robust observational constraints on OH on the larger scales are thought to be derived indirectly from its effect on tracers: gases that are predominantly removed by OH. Depending on how well the tracer emissions are known, the time evolution of the global mixing ratio of such a tracer can serve to constrain OH. The most well-established tracer for this purpose is methyl chloroform (MCF; e.g. ; ). In part, this is because it was identified early on as a tracer with a well-defined production inventory that allowed emission estimates with small errors, relative to other gases (; ). Moreover, production of MCF was phased out in compliance with the Montreal Protocol, and the resulting rapid drop in emissions made loss against OH the dominant term in the MCF budget .

Research and debate surrounding OH (; ; ; ; ) has lead to considerable improvements in its constraints, for example, a likely upper bound on global interannual variability of OH of a few percent . Two recent studies derived OH variations in a tropospheric two-box model through an inversion of atmospheric MCF and CH4 observations (; ). In such an inversion, a range of parameters is optimized (most prominently emissions of MCF and CH4 and OH) so that the modelled mixing ratios best match atmospheric observations of the tracers involved.

Both studies found that constraints on OH in this set-up were weak enough that a wide range of OH concentration variations over time and, by extent, CH4 emission scenarios were possible as an explanation for the post-2007 increase in its measured global mole fraction. This is an important conclusion, because the CH4 growth rate, combined with the CH4 lifetime (in turn dominated by MCF-derived OH), is generally assumed to provide the strongest top-down constraints on global CH4 emissions and variations therein. We note that in the two tropospheric boxes were supplemented by a single stratospheric box, making it technically a three-box model. However, due to our focus on the troposphere, we hereafter treat this type of model, too, as a two-box model, and where relevant we discuss the implication of the addition of a stratospheric box.

There are two important reasons to approach the problem of constraining OH in a model of exactly two tropospheric boxes. Firstly, through the focus on annual timescales and hemispheric spatial scales, the result is only sensitive to interannual variability in large-scale transport of the modelled tracers. Moreover, by focusing on interannual variability as opposed to absolute OH or emission levels, remaining systematic offsets are not thought to significantly affect the outcome.

Secondly, a crucial part of the optimization consists of disentangling the influence of OH and that of emission variations on observed MCF mixing ratios. Ideally, MCF emission variations would be prior knowledge. However, though MCF production is well documented, the emission timing is much more uncertain . MCF was mainly used as a solvent in, for example, paint and degreasers of metals. In these applications, MCF is released only when used, rather than when produced, which results in uncertainty in the emission timing. Moreover, due to the continuing decline of the atmospheric MCF mixing ratios, small, ongoing MCF emissions could eventually become important. Observation-inferred emissions exceeding bottom-up emission inventories have been identified both from the US and from Europe as well as from other processes, such as MCF re-release from the ocean . Therefore, in the absence of other constraints, emission uncertainties would limit the use of MCF for deriving interannual variability of OH. However, in a two-box set-up, an additional constraint is provided by the IH mole fraction gradient of MCF. Emission inventories show that MCF emissions are predominantly located in the Northern Hemisphere (NH), whereas OH has a NH to SH ratio that is uncertain, but the ratio has a likely range of 0.80 to 1.10 (; ). This means that emission variations have a strong effect on the IH mole fraction gradient of MCF, whereas the effect of large-scale OH variations is much weaker. Thus, the IH gradient is an important piece of information that can help to disentangle the influence of emissions from the influence of OH on MCF growth rate variations. This use of the IH gradient for constraining global emissions of anthropogenically emitted gases has also been recognized in previous research (; ).

Despite the appealing degree of simplicity offered by the two-box model, its results still hinge on many simplifying assumptions, both explicit (e.g. interhemispheric transport) and implicit (e.g. intrahemispheric transport). In this context, the uncertain outcome of the two recent two-box model studies puts forward an important question: how do the simplifying assumptions inherent to the two-box set-up affect the conclusions drawn from it? Or, conversely, would these conclusions change when moving the analysis to a 3-D transport model? A recent study partly explored these questions. The study investigated how to incorporate information from 3-D transport models in a two-box model to increase the robustness of two-box model-derived constraints on OH. They found that there are key parameters in the two-box model that can be tuned to better represent the 3-D simulation results and thus ideally better represent atmospheric transport in general. For example, they found that IH transport rates can be species-dependent.

Here, we provide a different approach to the issue. In the first part of our study, we parametrized results from the 3-D global transport and chemistry model TM5 into a two-box model. Through this parametrization, we explored difficulties in the translation from the “reality” of a 3-D transport model to a two-box model and the assumptions made in the process. We focused on four aspects of the parametrization.

Firstly, we investigated the tracer-dependent nature of IH transport as reported by . Secondly, we analysed the IH OH ratio. Previous research has shown that because of tracer-specific source–sink distributions, different tracers can be exposed to different global mean OH concentrations . We extended this observation to a species-dependent IH OH ratio. Thirdly, we looked at the stratospheric loss for MCF specifically. This net loss to the stratosphere might be slowing after its emissions dropped (; ). Fourthly, we used the 3-D simulation to investigate differences between the burden seen by the surface measurement network of the National Oceanic and Atmospheric Administration Global Monitoring Division (NOAA-GMD) and the true tropospheric and hemispheric burden in our 3-D model, a bias that was also discussed in .

In the second part of this study, we assessed the impact of these four potential biases on derived OH variations in a two-box inversion set-up that is very similar to and . The objective was to provide a quantitative estimate of the impact of biases in a two-box inversion and to explore if and how these can be accounted for. Though this study is focused on the problem of OH, it also serves as a case study of potential pitfalls in two-box models in general, when applied to interpreting global-scale atmospheric observations.

A first point that deserves discussion is the low global CH4 emissions (1994–2015) we derived compared to those reported in literature. Our best estimate corresponds to 539 Tg yr-1 (Table ), which is significantly lower than the 580–600 Tg yr-1 estimates reported by the two-box inversions of and . Our estimate is also on the low end of 3-D modelling studies; derived CH4 emissions from 30 3-D model inversions and found emissions of 558 [540–570] Tg yr-1 over the 2000–2012 period. performed a full 3-D inversion of CH4, using OH fields that were optimized against MCF in a separate 3-D model inversion . In their study, CH4 emissions of 525±8 Tg yr-1 were found over the 1984–2003 period, so their estimate does not include the renewed CH4 growth.

We found that several factors contribute to the differences. Firstly, in the model used by the atmospheric mass was taken as the global atmospheric mass (5.15×1018 kg), whereas we used the tropospheric mass (4.4×1018 kg). When we ran our two-box inversion with the global atmospheric mass, we also found emissions close to 600 Tg yr-1. Secondly, we could close the gap with by adjusting our a priori two-box model parameters. Specifically, when we adopted an IH exchange rate and an IH OH ratio similar to theirs (1.4 yr-1 and 1.07 respectively) in our standard inversion, we found global CH4 emissions of 595 Tg yr-1. This points to a strong sensitivity of the derived CH4 emissions to these parameters of the two-box model, which in our case are derived from full 3-D TM5 model simulations.

In our standard 3-D simulation, the IH gradient of MCF tended to be overestimated compared to observations from the NOAA network up to 2005, while global mean mixing ratios were captured much better. Translated into our two-box model, an inversion would tend to reduce MCF emissions to efficiently bring down the IH MCF gradient. To subsequently close the global MCF budget, OH will be reduced, resulting in lower global CH4 emissions in the two-box model inversion. The lower CH4 emissions we derived in our two-box model inversion are thus in line with the overestimated MCF latitudinal gradient in TM5. There are several possible explanations for this overestimate.

Firstly, MCF emissions that we used in our 3-D simulation were too high. In our two-box inversion, we found significantly lower MCF emissions (∼10%–30 %) than the prior estimate based on emission inventories, with the exception of the 2010–2014 period. also derived MCF emissions from the IH gradient and found these to be systematically lower than those based on bottom-up industrial inventories. Secondly, the NH : SH OH ratio might be higher than 0.98 (; ) and more in line with higher estimates from atmospheric chemistry simulations . Thirdly, a higher fraction of MCF emissions could be located in the SH (15 %–20 % instead of 5 %–10 %). Finally, IH exchange in TM5 could be too slow. A combination of the last two points would also arise if MCF emissions moved from NH mid-latitudes to NH low latitudes (e.g. India), since low-latitude emissions will be exchanged more rapidly with the SH. At this point it is not clear which of these explanations is most likely.

As is acknowledged in the previous two-box inversion studies of OH (; ), the problem of deriving OH from MCF and to a lesser degree from CH4 is strongly under-constrained. Therefore, many solutions fit the problem almost equally well. Moreover, a best estimate, or most likely solution, derived from a two-box model is a function of uncertain input parameters. For example, if it is assumed a priori that OH can only vary within a small band of 2 %, then a most likely solution with small OH variations will be found. In this study, we have identified a number of parameters which show variations outside of conventionally assumed bounds. As such, for these parameters, the variations we find are never fully explored in a conventional two-box model inversion, even if done as comprehensively as in . A clear example is stratospheric loss of MCF, which is generally assumed to have only small variability (10 % to 20 %). Here, we found a persistent 68 % drop in loss of MCF to the stratosphere. Potentially, this loss rate can recover if MCF emissions stop decreasing. Similarly, we find variations in transport of MCF of up to 20 % that persist for multiple years, compared to a conventional uncertainty in IH exchange of 10 %. In the 1994–1998 period, during a period of strong redistribution of MCF, the individual impact of each of the four biases was quite high, though when combined in one inversion the biases partly cancelled. During the 1998–2007 period, derived OH was less sensitive to the derived biases, likely due to a combination of a small role of uncertain emissions in the MCF budget and a period of relatively small redistribution of MCF. After this period, as MCF abundance continued to decline, we saw a growing impact from the IH exchange bias, as MCF emissions were increasingly constrained from the IH gradient, rather than from the emission inventory.

Another crucial parameter in the two-box inversion is the uncertainty in the global mean mixing ratios and in the IH gradient, as these uncertainties quantify the information content of the observational records. We provided an independent estimate of the uncertainty using 3-D model output in Sect. , summarized in Table . We can compare the uncertainties we find to observational uncertainties as derived from bootstrapping by . They find uncertainties in hemispheric means of 6–8 ppb for CH4 and of 5 %–6 % for MCF. Clearly, this is much higher than what we find, and their uncertainties seem to be an overestimate considering the limited sensitivity of our result to a different source–sink distribution. In their most likely solution, derived OH variations were such that the observed post-2007 renewed growth of CH4 coincides with a decrease in CH4 emissions. This solution does not fall within the uncertainty envelope we derived here (right panel in Fig. ). The difference in observational uncertainties is likely an important reason for this; their solution corresponds to a statistical inversion framework where less weight is given to the observations.

In the end, conclusions from our study and those drawn by and remain qualitatively similar. The post-2007 renewed growth of CH4 need not be caused by a sudden increase in emissions in 2007. Rather, emissions could have increased more gradually over the 1994–2007 period, while CH4 growth was suppressed temporarily by elevated OH levels. The lack of sensitivity of the inversion to the bias corrections and the large remaining uncertainty envelope in the final inversion both indicate that there are other parameters that result in significant uncertainties. Examples are the emission fraction in the NH, observational uncertainties, and uncertainty in the emission timing of MCF. Thus, while a first step can be made through the incorporation of 3-D model information, we confirm the conclusion drawn in and that the current state of the problem is still strongly underdetermined.

In another recent study, an effort was made to find tracer alternatives to MCF . For this, their suggested method was to use 3-D model output to improve the results of a two-box model through intelligent parametrizations. Clearly, this is similar to the work described here. For example, similar to us, they found different IH exchange timescales for different tracers. However, we explicitly resolved the two-box model in the 3-D framework, while their study focused mostly on fitting parameters empirically to find a match between two-box and 3-D model results. Additionally, for the parametrization, used hemispheric mean mixing ratios derived from the surface network, whereas we based mixing ratios on the full (hemispheric) troposphere in TM5. We identified a trend and strong, persistent variations in IH transport (CH4 and MCF) and in the surface sampling bias (MCF) which were not identified in . Additionally, they described a two-box strategy in which two tracers are used to derive the IH OH ratio, which can then also be used for other tracers. Our work suggests that there should be careful consideration of different IH OH ratios seen by different tracers and potential trends therein. A two-box inversion is sensitive to the IH OH ratio, and we have shown that the effective IH OH ratio a tracer is exposed to depends strongly on that tracer's source–sink distribution. Some of the differences between their findings and ours may be explained by the definition of a hemispheric mean mixing ratio (surface-based versus full troposphere), but further reconciliation of the two approaches in future research is necessary.

It is worth noting that the TM5 model, on which the two-box parametrization is based, has its own limitations, and so has treating TM5 as “the truth”. For example, our simulations were done on the coarse horizontal resolution of 6∘×4∘. This would have impacted how well NOAA background sites were actually situated in the background. We checked that the TM5-derived observational time series were not systematically more polluted than the real-world NOAA-GMD observations. For this, we detrended and deseasonalized the CH4 and MCF time series per surface site and quantified the spread in the residuals. At most sites, we found no offset between residual spread in the TM5-derived versus the real-world time series. At a small number of sites, TM5-derived time series showed more spread in residuals, while at others the spread was less. Therefore, we found no evidence for systematic biases in TM5-sampled observations. Additionally, any transport model is susceptible to some form of transport errors, and using a different 3-D model for the two-box parametrization will likely result in different parameters. Therefore, we are careful in suggesting quantitative interpretation of our results. Certain aspects of the biases, such as a slowdown of MCF loss to the stratosphere and the strong variations in IH transport of MCF, are likely to also be found in other 3-D transport models, as they are a direct consequence of the MCF emissions drop. Other aspects, such as the exact interannual variations of IH transport of CH4 or the 7 % offset between the physical OH ratio and the effective OH ratio, should be interpreted with more care, as these more strongly depend on the input emission and loss fields and on the exact treatment of transport in the 3-D model. Additional sensitivity tests done in multiple transport models can help in identifying sensitivities of the derived bias corrections. However, our analysis does show a potential for these biases to arise, and TM5 is a good starting point for exploring them, as TM5 has provided a strong basis for a wide variety of studies in the past (e.g. ; ; ).

In this study, we investigated variations in the global atmospheric oxidizing capacity in conjunction with variations in the global CH4 budget. We specifically revisited the use of two-box models to infer information about these quantities using global observations of MCF and CH4.

We identified four two-box model parameters that can benefit from 3-D model-derived information. Two of these are known and obvious (IH transport and surface sampling bias), while the other two are less so (stratospheric loss and IH OH ratio). Two-box model parameters for these processes that were quantified from full 3-D model output showed strong temporal trends mainly for MCF, which have not been identified in any previous research. In general, the biases resulted from a combination of variations in transport and in the spatio-temporal source–sink distributions of each tracer.

We tested the impact of each of the biases in a two-box model inversion. As expected, we found that absolute OH and thus absolute CH4 emissions show large deviations between the different inversions (∼10 %). Given that large parts of these deviations were constant through time, they do not necessarily impact conclusions of past two-box modelling studies that focused on interannual variations.

Compared to the absolute differences, we found only small differences in OH anomalies (up to 1.3 %, averaged over 1994–2015) relative to the full uncertainty envelope found here (3 %–4 %) or in (8 %). This indicates that significant uncertainties in parameters unrelated to the identified biases remain. As such, we confirm in large part the conclusions drawn by and regarding the underdetermined state of the problem. In the end, we did find that the conclusions one can draw from each individual inversion could be strongly affected by the bias corrections; in the standard inversion only 43 % of our Monte Carlo ensemble showed a positive trend in OH over the 1994–2015 period, compared to 88 % in the four-bias inversion.

The identified two-box model biases contribute to the already significant uncertainty in derived OH, and properly accounting for them can be a piece in the puzzle of improving constraints on OH. Moving forward, a likely next step is to incorporate more tracers in an effort to further tighten constraints on OH. In such a scenario, the tracer-dependent nature of the biases will likely increase the bias impact, and a proper 3-D model analysis for each tracer becomes even more important. Already, efforts have been made to do so , and in this study we provide further suggestions for such an approach. A distinct advantage in this approach is that information from multiple 3-D transport models can be used to tune the two-box inversion, making the inversion outcome less reliant on transport parametrizations of any single 3-D transport model. Additionally, computational efficiency of simple models allows for complex statistical inversion frameworks, incorporating, for example, hierarchical uncertainties .

On the other hand, the biases are often dependent on the sources and sinks used in the 3-D model simulation. As such, a feedback loop between the two-box inversion and the 3-D transport models might be necessary to correctly derive bias corrections, which makes such an analysis cumbersome. Additionally, a bias such as that in IH exchange of MCF might be difficult to resolve at all, because IH exchange of MCF is ill-defined in a two-box model (see Sects.  and S4). Therefore, we deem it important that a multi-tracer inversion in a full 3-D model should also be performed, similar to the 3-D inversion of MCF performed by but extended to more recent years. As an added advantage, a 3-D model inversion would increase the pool of potential tracers that can be implemented to constrain OH. For example, the short-lived tracer 14CO has been identified as a potential tracer to constrain OH (; ; ) but would not be implementable in a two-box model.