Assuming it has a uniform boundary condition in the surface region Ω and a constant decay rate λ, the mixing ratio ξ of a tracer may be expressed as e.g., ξ(r,t)=∫0+∞e-λτξΩ(t-τ)G(τ;r,t)dτ, where τ is the transit time from Ω to (r,t) and ξΩ is the tracer concentration at the surface. Here, G(τ;r,t) represents the distribution of transit times, i.e., the age spectrum, at time t and position r. In the following we will drop the explicit reference to r in order to simplify the notations. Equation () can be generalized to a vector equation for n different tracers (similar to our argument regarding Eq. ): ξ(t)=∫0+∞eMτξΩ(t-τ)G(τ;t)dτ, where ξ is a vector of species mixing ratios, M is the matrix of growth–decay coefficients and eMτ is the matrix exponential of Mτ. Equation () may encompass parent radioactive tracers as well as the whole associated decay chain (primary, secondary, … decay products), as explained in more detail in Sect. . It should be mentioned here that the derivation of Eqs. () and () is based on the assumption of a constant lifetime 1/λ. Although this assumption holds for radioactive tracers, the direct applicability of Eqs. () and () for the case of chemically active tracers is more questionable. This critical issue is discussed further in Sect. .

With the constant-lifetime assumption, Eqs. () and () show that the mixing ratio of any conserved or exponentially decaying (or growing) tracer may be expressed as the convolution of a generic function (involving time dependency of the source and chemistry) and the age spectrum G. The tracer content can hence be seen as a weighted average of the age spectrum, and the functions eMτξΩ(t-τ) as weighting functions (note that this perspective is reversed with respect to the general view that the age spectrum is a weighting function of the tracer boundary condition history modulated by the sink terms). However, although information on the age spectrum is contained in the tracer concentrations, it is far from being directly accessible because of this convolution with the tracer-dependent weighting functions. This limitation is evident in the case of linearly decaying tracers with a constant boundary condition at the surface and constant lifetimes τk=1/λk. For those, Eq. () simplifies as ξk(t)ξkΩ=∫0+∞G(τ;t)e-λkτdτ=G̃λk;t, where, as noted by , G̃ is the Laplace transform of the age spectrum.

Note that with the normalization of ξk(t) by its time-dependent surface value ξkΩ(t), Eq. () also applies to inert tracers exponentially increasing at the surface with growth rates λk, thus avoiding the constant-lifetime assumption.

There have been a few attempts to characterize the age spectrum from the knowledge of tracer concentrations. used time series of CO2 and N2O to diagnose the transit time distribution, assumed to be a superposition of two inverse Gaussian distributions. used water vapor time series from which they deconvolved the age spectrum by the mean of Fourier transform. However, both studies heavily relied on the assumed stationarity of the atmospheric flow. In general, stratospheric transport and the associated stratospheric age spectrum are nonstationary, as is evident from observations e.g., and model simulations e.g.,. In particular, the age spectrum exhibits seasonal and interannual variability. A few techniques have been proposed to estimate the age spectrum from tracer mixing ratios without relying on the stationarity assumption. They are briefly reviewed in the following.

A first approach, which might be referred to as the moment-estimate approach, is exposed for instance in . It is based on the relation between the moments of the age spectrum Mn, Mn(t)=∫0+∞τnG(τ;t)dτ, and the concentration χ of a passive tracer (λ=0 in Eq. ) with a boundary condition χΩ(t) evolving as a polynomial function of time t of order N, so that one may write χΩ(t-τ)=χΩ(t)+∑n=1Nαnτn, where the αn values are the coefficients of the polynomial. The relation is χ(t)-χΩ(t)=∑n=1NαnMn(t). Linearly increasing tracers constitute a particular case of Eq. () with N=1 and α1<0, such that χΩ(t-τ)=χΩ(t)+α1τ. For those, Eq. () implies that the delay time Δτ=(χ(t)-χΩ(t))/α1 is also the first moment of the age spectrum, called the mean age e.g.,: Γ(t)=M1(t)=∫0+∞τG(τ;t)dτ=Δτ. This last relation has been extensively used to derive the mean age of air from linearly increasing conserved tracers, such as SF6 and CO2 e.g.,. More generally, the moment-estimate approach builds on Eq. () to constrain specific moments of the age spectrum from tracers with different time dependency (linear, quadratic, etc.). Knowledge of given moments (e.g., the first two moments) then enables us to characterize the full age spectrum, assuming that the distribution has a given shape such as an inverse Gaussian distribution, age spectrum of the 1-D advection–diffusion process with constant advective speed and diffusivity, as performed by. This reasoning is however limited by the fact that real age spectra may exhibit a variety of shapes and are not necessarily inverse Gaussian distributions.

A second approach is the boundary impulse response (BIR) method . This method is based on a set of conserved pulse tracers, i.e., tracers which satisfy the boundary condition: χkΩ(t)=1fort∈[tk,tk+1],0otherwise. In that case, the relation between the pulse tracer mixing ratio and the age spectrum reads χk(t)=∫t-tk+1t-tkG(τ;t)dτ≃Gt-tk+tk+12;t(tk+1-tk). Thus, a set of N such tracers initialized following Eq. () with different, regular time intervals (i.e., tk=kδt) provides a resolved (though discretized) description of the age spectrum for transit times up to Nδt. The BIR method has recently been employed in atmospheric chemistry–transport models in order to gain knowledge on the model age spectrum. Though a useful diagnostic in models, the BIR method requires this specific pool of artificial pulse tracers and cannot in general be applied to standard tracers that might be available from observations.

A third approach consists in optimizing the parameters of a given function representing the age spectrum so that it best fits the observed tracer concentrations (e.g., through least-square regression). We will call that approach the parametric approach . Like the moment-estimate approach, it is based on the assumption that G has a given shape e.g., an inverse Gaussian distribution;. The technique can easily be applied to observed tracers and was employed by . Although it provides reasonable results, the parametric approach suffers from the same caveat mentioned above that the shape needs to be assumed a priori. Very recent results show that it can be substantially improved for the stratosphere by including information about the seasonality in transport .

Following , we discretize the convolution integral in Eq. () in transit time intervals [tj,tj+1] ξk(t)=∑j=0+∞∫tjtj+1eMτξΩ(t-τ)kG(τ;t)dτ≃∑j=0nLkjGj, with the k subscript indicating the kth component of the tracer species vector and the “weighting function matrix” elements Lkj and age spectrum vector Gj given by Lkj(t)=∫tjtj+1eMτξΩ(t-τ)kdτ,Gj(t)=Gtj+tj+12;t. To obtain the second equality in Eq. (), we have assumed that G is piecewise constant over the intervals [tj,tj+1]. We have also truncated the transit time axis at some tn, for practical computation reasons. In order to simplify the notation, we drop the explicit reference to t in G in the remainder of the paper, but it is implicit that the age spectrum depends on both time and location. Considering the full vector of mixing ratios, ξ, Eq. () can be written in matrix form: ξ=LG. For the special case of a suite of linearly decaying (radioactive) tracers with the unit mixing ratio at the surface, as described by Eq. (), the elements of the weighting function matrix L are simply Lkj=∫tjtj+1e-λkτdτ=τke-λktj-e-λktj+1. A piecewise constant representation of the weighting function eMτξΩ(t-τ) could also have been used if no analytical expression had been available.

In order to gain information on G from the radioactive tracers, suggested using Eq. () and constructing a square matrix L from which one could estimate G as Gest=L-1ξobs. This method is not applicable in practice, however, because the problem is ill-posed and sensitive to small perturbation of ξobs and because the matrix L is nearly singular (as demonstrated in Appendix ).

Rather than directly inverting L, it is more appropriate to consider the determination of G from the observed trace gas mixing ratios ξobs as an inverse problem, in which Eq. () is the forward model. In this formulation, the tracer content provides information on the convolution of the age spectrum with given functions. In that respect, it is similar to atmospheric soundings, for which the radiances measured at different wavelengths provide information on temperature and tracer profiles. Appropriate approaches to deal with such inverse problems are described in textbooks such as . In the following, we summarize the relevant pieces of information for the specific case considered here.

A solution to the discretized problem may be obtained through the minimization of a cost function J(G), here expressed as J(G)=(LG-ξobs)TSϵ-1(LG-ξobs)+(G-Ga)TSa-1(G-Ga). The first term Sϵ-1 is the inverse covariance matrix of the observed (or modeled) tracers. It quantifies the departure from observations and may correspond to instrumental noise or model error as well as uncertainties in the estimation of L (as, for example, uncertainties in the decay coefficients or in the boundary condition ξkΩ or even numerical errors). In our context, the second term involving the a priori Ga and its inverse error covariance matrix Sa-1 is introduced for regularization purposes (to avoid unphysical oscillations and large negative values of the retrieved G), in order to penalize solutions far from the a priori value.

Since the problem is already linear, the optimal G=Gest (which minimizes J) can be readily estimated as Gest=Ga+LTSϵ-1L+Sa-1-1LTSϵ-1ξobs-LGa. Contrary to most practical inverse problems, ours is of sufficiently small dimension (100 tracers and a few hundred points along the transit time axis at the most) so that a direct inversion of the matrix may be attempted without running into computational and memory limitations. However, similarly to most inverse problems, it is not obvious how to obtain values for the matrices Sϵ (which represents different sources of errors) and Sa (which may only be estimated from models) nor to get obtain a value for Ga. We will follow an empirical approach here for the regularization, which belongs to the class of Tikhonov regularization schemes. Specifically, we set Ga=0 and take Sϵ as σϵ2I and Sa as σa2α2I, where I is the identity matrix, σϵ2 a rough estimate of the variance of the observation (or model) error ϵ, σa2 a rough estimate of the variance of Ga and α a positive scalar. Then the cost function can be rewritten, J(G)=1σϵ2(LG-ξobs)T(LG-ξobs)+α2σa2GTG, and the optimal estimate is Gest=LTL+α2σa2σϵ2I-1LTξobs. In practice, different values of α can be tested until a reasonable retrieval is obtained. Within a certain range of values, the retrievals are only marginally sensitive to the exact value of α. The range of values yielding reasonable retrievals encompasses the ratio of variance of the observation's error to the one of the a priori.

At this point, three further remarks should be made. First, there is no guarantee that the estimated age spectra Gest are positive for all transit times. As they are a result of optimal estimation, negative values should not be discarded, but taken into account in order to obtain the most accurate average and reduce the bias. A second point is that setting Ga=0 implicitly includes a priori information regarding G, albeit limited compared to the parametric approach described above. The effect of setting Ga=0 is to favor smooth functions and implicitly penalize unphysical oscillatory solutions which would deviate significantly from the characteristics expected for a distribution (i.e., G>0 and ∫0+∞G(τ;t)dτ=1). Finally, the structures of Sϵ and Sa are merely chosen here because of their simplicity in the absence of rationale to do otherwise. One advantage is that then a unique α value needs to be tuned to perform the inversion. More complicated forms of Sϵ and Sa may be required in practical applications, especially if the error in tracer measurements exhibits covariance structures.

The first step is to construct an age spectrum and the associated tracer composition as a test bed for the retrieval method. It is straightforward to estimate the decaying tracers from the perfect knowledge of the age spectrum, either analytically or through numerical integration of Eq. () with a fine resolution along the transit time axis. For the idealized age spectrum, we use the canonical expression for 1-D advective–diffusive systems given by e.g., G(τ)=Γ34πΔ2τ3exp⁡-Γ(τ-Γ)24Δ2τ, where Γ is the mean age and Δ the age spectrum width. This functional form for the age spectrum is known as an inverse Gaussian function and has been extensively compared with model spectra e.g.,. The pseudo-observed (or modeled) mixing ratios of the tracers are derived as χobs=LhrGhr+ϵ. Here, Gjhr=Gtj+tj+12 with tj=jδt and δt=1 day. The error ϵ represents the uncertainty associated with the observation or modeling of the tracer. Here, we take ϵ proportional to the actual tracer mixing ratio, i.e., ϵk=LhrGhrk(ϵbase)k, where ϵbase is a vector of random numbers from independent uniform distributions over [-0.05;0.05]. The formulation in Eq. () is motivated by the fact that, for the tracers selected to perform the inversion, the accuracy of the measurements should be significantly smaller than their actual value; furthermore, the accuracy of trace gas mixing ratios from in situ measurements or models is in some cases proportional to their content.

Typically, two parameters need to be chosen to set up a retrieval: the resolution along the transit time axis and the strength of the regularization, i.e., the value of α. It is actually advantageous to start with a high resolution along the transit time axis (e.g., 1 month) to pinpoint the value of α before determining the effective resolution of the retrieval and adjusting the inversion to that resolution.

If the uncertainties associated with the observations or the a priori spectrum are not precisely known, there is some freedom in the choice of the optimal α. One procedure is to empirically test different values of α and choose the best fit through visual inspection of the retrieved spectrum (i.e., until complete removal of the noise oscillations). However, this leaves room for a large subjectivity; a more objective approach is the L-curve optimality criterion e.g.,. This approach consists in plotting the residual 1σϵ2(LGest-ξobs)T(LGest-ξobs) against the constraint 1σa2GestTGest for different estimates Gest obtained assuming different values of α in Eq. (). For many inverse problems and a wide range of α values, this yields an L-shaped curve, of which the corner (point of largest curvature) stands as a compromise between fidelity to the tracers and proximity to the a priori spectrum.

In order to construct the L-shaped curve and to determine an appropriate value for α, we generate a set of 100 pseudo-observations ξobs by varying ϵ in Eq. (), with the “true spectrum” Ghr taken as an inverse Gaussian distribution with Γ=2 years and Δ=1 year. For each of the 100 realizations of ξobs, a retrieval Gest is then performed using a given α in Eq. (). This procedure is carried out for different values of α, resulting in 100 L-shaped curves (for each of the 100 realizations ξobs). The average (for representativeness) of the resulting 100 L-shaped curves (i.e., <1σϵ2(LGest-ξobs)T(LGest-ξobs)> vs. <1σa2GestTGest>) is shown in Fig. . It exhibits the expected L shape and shows that α2=10-2 turns out to be a good choice for our problem.

Figure  shows the retrieved spectra obtained using α=10-1, for two typical cases, a “young-age spectrum” (Γ=2 years and Δ=1 year, and σϵ and σa set to σϵ=0.029mm-3 and σa=10-2month-1) and an “old-age spectrum” (Γ=5 years and Δ=3 years). The thin black lines are individual retrieval results for 100 retrievals from the 100 sets of pseudo-observations including noise, while the thick black lines are the averages (shaded area: ±1 standard deviation). For both idealized spectra, the averages agree reasonably well with the input (red lines). In particular, the location of the mode is found in both cases and the general shape and magnitude of the spectrum are reproduced. However, unrealistic negative values arise for small and large transit times (where the actual spectrum is close to 0), and the exact magnitude of the mode is not captured, with a ∼25% underestimation. Furthermore, there is a significant dispersion of individual retrievals around the average. This dispersion can be reduced by increasing the strength of the regularization α, but at the price of a deteriorated agreement of the multi retrieval average with the true spectrum. Conversely, a better agreement of the average spectrum with the input can be achieved by decreasing α, at the price of an increased dispersion in individual retrievals. As described above, the choice of α is a compromise between the reliability of individual retrievals and the accuracy of multi-retrieval averages.

We would like to emphasize that a different value of α may suit better when the relative strength of the noise is modified. However, as the problem is ill-posed, regularization is required even in the absence of noise (see Appendix ).

To perform the retrieval presented above, only nine tracers were used, whereas there were the 119 components of the spectrum to invert (monthly bins on a 10-year-long transit time axis). It then comes without surprise that the retrieval is strongly under-constrained and requires regularization, especially since the weighting functions all peak at τ=0. The effective resolution in transit time of the inverted spectrum can be investigated from the averaging kernel matrix A defined by Eq. () as A=LTL+α2σa2σϵ2I-1LTL. The matrix A quantifies the contribution of the value of G at different transit times to the retrieved age spectrum Gest at a specific transit time, and thus the resolution and ability to distinguish specific features. Averaging kernels peaking at one single transit time would provide the best resolution.

For our setup, the averaging kernels are displayed in Fig. . As expected from the shape of the weighting function (Fig. ), the resolution is better for short transit times, although even for those the effective resolution does not reach the 1-month-transit-time bin size chosen for the retrieval, as can be seen from the overlap of the averaging kernels. The averaging kernels also exhibit negative lobes, which are responsible for the negative values seen in the retrieval at transit times characterized by low values of G. The amplitude of the negative values may be decreased by strengthening the regularization, but this reduces the sharpness of the peak of the averaging kernels and hence degrades the resolution.

Given the redundancy visible in the averaging kernels, it is possible to use a sparser-resolution grid in transit time, which would better reflect the information available from the tracers. Although there is some freedom in the choice of the grid, we keep the linear grid spacing in the following because of its simplicity and the demonstrated feasibility of the retrievals in that setup.

As emphasized above, there is no guarantee that the retrieved age spectrum is positive for all transit times. Although negative values should not be discarded in averaging procedures, some practical applications (such as using the retrieved age spectrum to, for example, compute mean age or estimate the mixing ratio of any tracer) may impose that the retrieved spectrum fulfills the requirements of distribution functions, i.e., to have only positive values and integrate to unity. Renormalization is necessary to enforce those requirements. We propose a simple three-step procedure to obtain a normalized spectrum Gnorm from Gest.

Set all negative values to 0.