3844 R. Steinfeldt et al.: Anthropogenic carbon in the Atlantic with the overlying DSOW and are thus more saline and also more enriched with anthropogenic tracers. 2.3 Anthropogenic carbon inferred from the TTD method In this paper, we use a modified TTD method to infer the con- centration of Cant. This is based on the method used in Ste- infeldt et al. (2009). In addition, we explicitly allow for the admixture of old, tracer-free waters. This approach has been used before, e.g., in Steinfeldt and Rhein (2004), but there it was locally restricted to the deep western boundary current in the tropics and was not used to calculate anthropogenic carbon. Here, we introduce a new algorithm that allows us to assign the admixture of old water at any location. 2.3.1 The standard TTD method First, we explain the standard TTD method by following the procedure in Hall et al. (2002). Due to the advective– diffusive nature of oceanic transport, the water in the ocean interior consists of fluid elements with different pathways and ages (time elapsed since the water parcel left the mixed layer). The distribution of these ages, ? , is described by the TTD function G. The concentration of any conservative prop- erty C(x, t) at location x in the ocean interior and time t , which can be a particular reference year tref, is then given by (Hall et al., 2002) C(x, tref) = ?? 0 C0(tref? ?)G(x, tref,? )d?, (1) where ? denotes the age of the water. C0(t) is the concentra- tion history of the property in surface waters in the mixed layer. For upper water masses, we assume that C0(t) for CFCs and SF6 is in solubility equilibrium with the atmo- sphere. For deeper, denser waters, the saturation gradually decreases to 80 %. A detailed list of the saturation used for each density layer is given in the supporting information in Table A1. In Steinfeldt et al. (2009), the minimum saturation was chosen to be lower, i.e., 65 %. However, CFC concentra- tions measured close to the formation region of NADW tend to be larger than 65 % of the surface saturation (see Fig. B1), so the value for the minimum CFC saturation has been en- larged for all dense water data. According to Steinfeldt et al. (2009), the difference in the inferred Cant concentrations is about half the saturation difference; i.e., a saturation differ- ence of 20 % leads to a Cant difference of about 10 %. For Cant, we use a time-independent carbon disequilibrium as in Waugh et al. (2006) and Steinfeldt et al. (2009). Cant(tref) is calculated as the difference between the carbon concentra- tion at time tref and the preindustrial time (year 1800). If the carbon disequilibrium remains constant, it cancels out when calculating this difference. At the beginning of the indus- trial period, atmospheric CO2 increased by about 1 ppm per decade. This means an earlier or later start time for the begin- ning of the industrial period compared to 1800 would change the inferred Cant concentration by about 0.6 µmol kg?1 for the decades around 1800. As the Atlantic waters nowadays only contain a small fraction of water formed around the year 1800, the Cant difference arising from a slightly different def- inition of the industrial period almost vanishes. Equation (1) is used to infer concentrations of an- thropogenic carbon (Cant(x, t)) from the TTD functions G(x, t,? ). On the other hand, Eq. (1) allows us to infer the pa- rameters of the TTD such that C(x, t) is the observed tracer concentration. To do so, a certain functional form of the TTD has to be assumed. Here, we apply an inverse Gaussian func- tion as an approximation of the “real” TTD, as has been done in other studies (e.g., Hall et al., 2002; Waugh et al., 2006; Steinfeldt et al., 2009). This function only depends on two parameters: the mean age (mean value of ? ) 0 (the first mo- ment associated with the advective tracer transfer) and the width 1 (the second moment, which is related to the disper- sion or mixing on all relevant scales, including recirculation and admixtures or entrainment of older water): G(?,0,1)= ? 03 4pi12? 3 exp (?0(? ?0)2 412? ) . (2) In order to derive both parameters (1 and 0), simultaneous measurements of different anthropogenic tracers are needed (Hall et al., 2002; Steinfeldt et al., 2009; Smith et al., 2011). As these are sparse, a fixed ratio of 1/0 is often used. This ratio is a measure of the importance of mixing (higher 1/0 values imply stronger mixing). Waugh et al. (2004) inferred a ratio of 1/0 = 1 from tracer observations in the subpolar North Atlantic. In an ideal case, if G were the real TTD, the mean age 0 should be independent of the tracer from which it is inferred. Equation (1) then holds true for CFC-12, CFC-11, SF6 and any other tracer taken from the same water sample with iden- tical TTD parameters. In reality, if Eq. (1) holds true for one tracer (e.g., CFC-12) with the regional 1/0 value, applying Eq. (1) with the same parameters of G to another tracer (e.g., CFC-11) may result in deviations of the order of a few per- cent from the observed CFC-11 concentration. In this study, we preferentially use CFC-12-derived ages; we use CFC- 11 only when CFC-12 is not available. The number of the considered age data points is given in Table 1. CFC-12 and CFC-11 are the most commonly measured tracers (Table 2). The advantage of CFC-12 is that it increased in the atmo- sphere before CFC-11 and, in recent years, shows a smaller decline (Fig. C1). For young waters, this decline leads to rel- atively large errors (from measurement uncertainties and an unknown mixed-layer saturation) in the age and inferred an- thropogenic carbon (Tanhua et al., 2008). Thus, for data af- ter 2005 in the upper layers (0 < ?? < 27.6 kg m?3) with young water (central and intermediate waters) and relatively high SF6 concentrations, we use the SF6-based age estimate, Biogeosciences, 21, 3839–3867, 2024 https://doi.org/10.5194/bg-21-3839-2024