#### 2.1. Inverse Method

[11] The inverse problem that is considered here can be written as:

Here, χ_{s,t} and σ_{s,t} (pmol/mol) denote time series of the measured monthly mean CH_{3}CCl_{3} concentrations and corresponding standard deviations at the five ALE/GAGE/AGAGE stations; *F*(**P**, **b**) denotes the model simulated CH_{3}CCl_{3} at the stations. These values depend on the state vector **P** and on other parameters that are collected in vector **b**. To optimize the OH temporal or spatial variations, parameters that are linked to these variations are included in the state vector. The other parameters, for example, those linked to transport, emissions, and CH_{3}CCl_{3} destruction in the stratosphere and oceans, are included in **b**.

[12] The optimization of the parameters in the state vector is based on minimization of the cost function, which is defined as:

where ϵ is the absolute measurement calibration; vector **P** refers to *n*_{p} free model parameters; Λ scales the *prior* information given by ϵ_{ap} and *P*_{ap} with standard deviations σ_{ϵ} and σ_{P}. Note that the absolute measurement calibration is always optimized along with the parameters in the state vector **P**.

[13] The first term of equation (2) sums the squared differences between model and measurements, weighted with the measured standard deviation during a month. Owing to declining emissions in the 1990s, these standard deviations have substantially reduced since the 1980s. The current approach does not use polynomial fitting like in the work of *Krol et al.* [1998]. These polynomials filter out seasonal and interannual variability. One reason to use polynomial fitting was the inability of our climatological model to simulate interannual variability. However, the extension of the ALE/GAGE measurements with the AGAGE data hamper the use of low order polynomials. Moreover, when polynomials are used, the weighting at the beginning and the end of the time series might be slightly different compared to the original data [*Prinn and Huang*, 2001]. Although we showed that the derived OH trend and concentration are not influenced by the use of polynomials [*Krol et al.*, 2001], we decided to use the monthly averages and standard deviations directly in the optimization. Since the model uses monthly averaged winds, pollution events are excluded from the measurements [*Prinn et al.*, 2000].

[14] Differences between model and measurements are caused by inaccuracies in the model transport, the applied CH_{3}CCl_{3} emissions, the OH distribution and its temporal variation, other CH_{3}CCl_{3} sinks (stratosphere, oceanic sink), the CH_{3}CCl_{3} measurement calibration and chemical rate constants. Moreover, representation errors occur because point measurements are compared to large grid boxes in the model. It is assumed that the variations in the mixing ratios during a month (σ_{s,t}, pollution events excluded) provide an estimate for these representation errors [*Prinn et al.*, 2001]. It is not likely, however, that σ_{s,t} provides a reliable estimate of modeling errors caused by the parameters (e.g., transport, emissions, collected in **b**) that are not optimized. One option would be to increase the observational uncertainty σ_{s,t} to account for these modeling errors. Another, which is employed here, is to introduce a weight in front of the *prior* information in the cost function (see below).

[15] In the period 1978–2000 1248 (*N*) monthly averaged CH_{3}CCl_{3} concentrations at the ALE/GAGE/AGAGE stations are publicly available (http://cdiac.ornl.gov/ftp/ale_gage_Agage/). The resulting correspondence between measurements and model will be reported as the value of the cost function divided by the number of measurements (*J*/*N*). *J*/*N* ≈ 1 thus corresponds to an average deviation of about 1.4σ between model and measurements.

[16] The second term in equation (2) represents the costs related to deviations from the *prior* information. The use of *prior* information has two well-known advantages [see, e.g., *Tarantola*, 1987]: (1) The optimized parameters are constrained to values that are physically realistic, and (2) the solution is stabilized. An unstable solution may occur if a parameter in the state vector is strongly correlated to other parameters. In that case, a range of parameter values corresponds to the same minimum of the cost function. The use of *prior* information resolves this ambiguity, since the optimized parameters should remain consistent with their *prior* information. Correlations between the determined parameters can be inferred from the *posterior* covariance matrix.

[17] The model relation *F*(**P**, **b**) is not exact and the resulting model errors will be highly correlated. This would require the evaluation of the full *prior* covariance matrix σ_{s,t} in the first term of equation (2). Moreover, the number of observations is much larger than the number of unknowns in the current inversion, which would put a relatively large weight on this first term. To resolve these problems, a factor Λ is introduced to scale the *prior* information in the cost function. Without this factor, estimated parameters can deviate a few standard deviations from the *prior* information without contributing significantly to the cost function, which would make the solution unrealistic. Introduction of Λ = 100 guarantees that, in this inverse problem, the free parameters are optimized well within their σ_{ap} values. Parameters that are not constrained by *prior* information are given large σ_{ap} values. On the other hand, small σ_{ap} values can be used to constrain a parameter to a predefined value. No *prior* constraints are placed on parameters that are related to OH. The absolute calibration uncertainty σ_{ϵ} is set to 5% [*Prinn et al.*, 2001] and whenever emissions are estimated, the associated 1σ uncertainty is used as a *prior* constraint.

[18] The effect of the use of Λ instead of a more realistic *prior* covariance in equation (2) is that the *posterior* errors in the optimized parameters will only reflect the errors that are associated with the inverse procedure, that is, assuming a perfect model. Moreover, the average deviation between the measurements and the optimized model will generally be larger than 1σ since the σ_{s,t} values in equation (2) only partly reflect the true modeling errors.

[19] The cost function is minimized by varying the state vector **P** until a minimum is obtained. Practically, the model is linearized around a solution **P**_{0}:

After minimization of the cost function (equation (2)) **P**_{0} is replaced by the new solution and the procedure is repeated until convergence is reached. Owing to the linear nature of the perturbations, convergence is usually obtained in one iteration. The error covariance matrix of the estimated state vector is obtained from the linearized model, the measurement errors, and the *prior* errors in the state vector [*Tarantola*, 1987]. It is assumed that the measurement errors are uncorrelated and Gaussian. The same assumptions apply to the *prior* errors.

[20] The procedure outlined here leads to results that are identical to our earlier approach which employed an ensemble smoother [*Krol et al.*, 1998], but is computationally more efficient. Additionally, the absolute calibration ϵ of the measurements is now optimized along with the other parameters in the state vector.

#### 2.2. Model Description

[21] Two models have been used to perform the CH_{3}CCl_{3} simulations. The TM3 model [*Dentener et al.*, 1999; *Lelieveld and Dentener*, 2000; *Peters et al.*, 2001] is employed to perform simulations using 6-hourly analyzed wind fields from the European Centre for Medium range Weather Forecasts (ECMWF). The TM3 model is operated with a relatively coarse horizontal resolution of 7.5° latitude and 10° longitude and with 19 layers in the vertical. The upper five layers represent the stratosphere in which, apart from OH oxidation, photolysis of CH_{3}CCl_{3} is important. Photolysis rates in the stratosphere are calculated using a look-up table approach. The DISORT model [*Stamnes et al.*, 1988] has been used to construct the look-up table. Cross-sections and quantum yield are taken from *DeMore et al.* [1994]. Monthly averaged OH fields as calculated for the year 1986 are used to oxidize CH_{3}CCl_{3} [*Lelieveld and Dentener*, 2000]. The 1986 meteorological fields are used in all years of the 1951–2000 simulation period.

[22] The calculated lifetime for stratospheric loss (total atmospheric burden over stratospheric loss) calculated by TM3 amounts to about 45 years, in excellent agreement with other work [*WMO*, 1999]. The corresponding stratospheric loss rate (0–100 hPa burden over 0–100 hPa loss) amounts to about 2.5 year. Two reasons hamper the use of the TM3 model for CH_{3}CCl_{3} inversions: (1) The TM3 model is computationally too expensive, and (2) owing to the more detailed meteorology in TM3, pollution events will be partly resolved. A full simulation of all the pollution events, for example, at the Ireland station [*Ryall et al.*, 2001] would require the use of all years of meteorological data and a higher resolution of the model. Therefore, the TM3 simulations are used to tune the stratospheric loss in the MOGUNTIA model and to check the CH_{3}CCl_{3} budget (see Appendix A).

[23] Similar to *Krol et al.* [1998] the MOGUNTIA model is used in the inversion of the CH_{3}CCl_{3} observations. The replacement of the Oregon station (45°N) by the California station (41°N) does not change the sampled grid box in the MOGUNTIA model. In MOGUNTIA, monthly averaged wind fields of the year 1986 are used to transport CH_{3}CCl_{3}. Emissions have been adapted from *McCulloch and Midgley* [2001] and the spatial distribution is similar to *Krol et al.* [1998]. For the recent years the emissions are distributed similar to that given by *Prinn et al.* [2001].

[24] In previous work [*Krol et al.*, 1998] we accounted for the stratospheric sink by applying hemispheric mean and monthly varying loss rates at 100 hPa (the top of the model). Before 1990, CH_{3}CCl_{3} emissions increased continuously and caused a large concentration difference between the troposphere and the stratosphere. However, the emissions declined strongly in the 1990s and it is therefore expected that during this period the stratosphere becomes a less important sink for troposphere CH_{3}CCl_{3}. For this reason, we modified the stratospheric loss parameterization. Two reservoirs which represent the stratosphere (0–100 hPa) in the NH and the SH are added on top of the MOGUNTIA model. Exchange between the stratospheric reservoir and the upper model layer is calculated from the concentration difference multiplied by an exchange rate. The SH exchange rate is taken 33% slower than the NH value. The exchange rate and the stratospheric CH_{3}CCl_{3} lifetime are obtained from a simulation with the TM3 model as described previously. A more detailed discussion of the stratospheric parameterization is given in Appendix A.

[25] Uptake and subsequent hydrolysis of CH_{3}CCl_{3} in ocean water is modeled by a first order loss process [*Krol et al.*, 1998; *Kanakidou et al.*, 1995; *Kindler et al.*, 1995]. The loss rate is based on measurements reported by *Butler et al.* [1991], who observed negative saturation anomalies in the tropical pacific ocean in 1990. Hydrolysis of CH_{3}CCl_{3} is most efficient in warm ocean waters but CH_{3}CCl_{3} is much more soluble in the colder oceans at higher latitudes. Hydrolysis in these waters proceeds much slower owing to the lower temperatures. Hydrolysis in cold oceans [*Gerkens and Franklin*, 1989] is expected to be even slower than the observed atmospheric decay time since 1992. Cold oceans may therefore have acted as a buffer for the atmosphere since dissolved CH_{3}CCl_{3} may have returned to the atmosphere. This idea is highly speculative and not tested while biological degradation in the productive waters at high latitudes cannot be ruled out [*Butler et al.*, 1991]. Possible consequences for the CH_{3}CCl_{3} inversion, which appear to be small, are discussed in Appendix B.