Anthropogenic carbon in the ocean (Canth) is defined as the increase in dissolved inorganic carbon (DIC) in seawater attributable to anthropogenic carbon dioxide (CO₂) emissions to the atmosphere over the industrial era. As the ocean is the largest single historical sink of CO₂ and is expected to absorb most of the anthropogenic CO₂ transient on millennial scales , understanding the distribution and rates of change of Canth in the global ocean is central to informing marine climate change effects and feedbacks . On local scales, accumulation of Canth gains further relevance as a driver of ocean acidification and other ecosystem disruptions that affect important natural resources . These disruptions underlie the need for accurate and accessible methods for estimating Canth in the ocean.

Several methods for inferring Canth from observational data have been devised. These can be separated into two classes: back-calculation and inversion. Back-calculation methods such as the ΔC* and eMLR(C*) techniques seek to estimate Canth accumulation by isolating its effect on DIC from other biogeochemical processes. These techniques improved the understanding of the ocean carbon sink based on repeat hydrographic observations, but cannot extrapolate to unobserved periods, and the reliance on assumptions that complicate their interpretation including transient steady state invasion of anthropogenic signals, fixed nutrient and carbon stoichiometries, and simplified mixing models . In contrast, inversion-based methods infer the propagation of a surface response to anthropogenic atmospheric CO₂ throughout the ocean via circulation constrained by measurements of chlorofluorocarbons (CFCs), sulfur hexafluoride (SF₆), and other tracers of ocean circulation , taking advantage of similarities between the atmospheric histories of these anthropogenic gases (Fig. ). Inverted ocean tracer transport may be projected backwards and forwards in time, providing opportunities to explore changes in the ocean carbon sink and oxygen utilization . Additionally, some inventory estimates have combined elements of both back-calculation and inversion methods .

One subclass of inversion-based methods, the Transit Time Distribution (TTD), relies on a Green's function solution of the linear advection-diffusion transport equations to provide an age distribution representing the relative contributions of waters of various ages to a parcel, where age is considered to be the time since water was last at the ocean surface . This age distribution recognizes that interior ocean waters are more realistically represented as mixtures of many different water parcels of various ages carrying unique histories of atmospheric contact rather than by scalar ages . The functional form of a TTD may vary, but an inverse-gaussian (IG) distribution specified as a function of transit time t (where smaller t indicates younger waters; Eq. ) has been shown to describe tracer transport regimes of many ocean regions well in comparison with ocean general circulation models when the IG distribution is provided with optimal parameters . Its first temporal moment Γ (or mean age), and its second centered temporal moment Δ vary depending on interior location, but their ratio Δ/Γ is usually prescribed to be constant in solutions of Eq. (), as described later. 1G(t)=Γ34πΔ2t3e-Γ(t-Γ)22tΔ2 This function describes one-dimensional pipe flow along isopycnal surfaces from a single source region, neglecting diapycnal diffusion and assuming steady-state circulation. Other formulations of the distribution representing more complex regimes require additional observational constraints . Convolution of the TTD G with a surface boundary function propagates a surface signal (χs) through the ocean and allows calculation of its interior value (χ) as a function of time t at interior location r: 2χ(r,t)=∫0∞χs(t-t′)G(r,t′)dt′.

Despite the utility of TTD methods for unraveling ocean tracer transport as well as recent calls for development of Canth estimations based on transient tracers , their complex formulation and implementation has historically restricted their use. To overcome this barrier to more accessible science, an implementation of a TTD method was given by as “Tracer-based Rapid Anthropogenic Carbon Estimation version 1” (hereafter TRACEv1). Among the limitations of that implementation was its formulation using MATLAB (which while open-source is not freely available), and its dependence upon predetermined boundary conditions and TTD shape.

To address these limitations, this work describes an update of the TRACE routine and its implementation in the Python coding language. A brief overview of inherited methods is given followed by a description of new aspects of this implementation of TRACE, which encompass both practical improvements and fundamental changes to the method. This routine is validated against TRACEv1 to establish exact comparability, then used to produce an updated global gridded Canth data product using an updated integration routine. A sensitivity analysis is then carried out to explore the effect of practical improvements to the TRACE method. Finally, we consider this method's strengths, limitations, and future development.

This implementation of TRACE in Python is both an exact replication of its MATLAB-based predecessor's results as well as an improvement in function. This work inherits the IG-TTD method implemented by its predecessor in form and function, and its equivalent results and the effect of improvements are described in Sect. . Hereafter, we use “TRACE” to refer to the algorithm, “TRACEv1” to refer to its implementation in MATLAB, and “TRACE-Python” to refer to its implementation in Python, for which this study used version 1.0.0. The main steps of this routine are enumerated with inputs bolded and outputs italicized for additional clarity, then described in detail:

User-provided location (latitude, longitude, depth), salinity, temperature, and optionally Δ/Γ predict the TTD and preformed properties via pre-trained neural networks. If temperature is not provided, it is first estimated by the remaining predictors.

First, a pre-trained neural network predicts the TTD from latitude, longitude, depth, salinity, and temperature. The neural network training data consists of solutions to Eq. () optimized via the Nelder-Mead simplex optimization algorithm from paired CFC-11, CFC-12, and SF₆ observations in the GLODAPv2.2023 dataset together with age estimates from the Ocean Circulation Inverse Model . The cost function was the sum of squared normalized errors in partial pressures and age. The network architecture is composed of committees of neural networks like those used in . The shape of the IG-TTD (as specified by its first moment Γ and second moment Δ) was not originally allowed to vary from Δ/Γ=1.3; however, TRACE-Python makes Δ/Γ available as a changeable parameter, as described in Sect. . Adding this functionality required adding new neural networks for the age distributions fit to the same measurements with a set of ratios. The TRACE-Python now selects between the neural networks depending on the user provided ratio input. Similar neural networks predict preformed alkalinity, preformed phosphate, and preformed silicate “preformed” indicating the properties that interior ocean seawater mixtures had when they at last contact with the ocean surface;. Failing to input a temperature predictor for any of these networks leads to temperature being predicted from salinity and location by an additional neural network.

Next, user specification of a global mean atmospheric CO₂ trajectory guides the formulation of a surface boundary condition. Built-in atmospheric CO₂ pathways include eight shared socioeconomic pathways (SSPs): 1-1.9, 1-2.6, 2-4.5, 3-7.0, 3-7.0-lowNTCF, 4-3.4, 4-6.0, and 5-3.4 and historical data with a linear extrapolation of the present increase (denoted Historical/Linear), all spanning the years 1–2500 CE. The user may also specify a custom pathway. TRACE estimates the surface boundary condition partial pressure of carbon dioxide (pCO2oce) at a time t (in years) as a function of the time-varying atmospheric CO₂ mixing fraction XCO2atm(t): 3pCO2oce(t)=XCO2atm(t)-0.144×XCO2atm(t)-XCO2atm(t-65yr).

This was derived as an empirical relationship between atmospheric and surface ocean trends in a model-observation hybrid product , and it defines a surface boundary responsive to the rate of atmospheric increase or decrease over a 65-year lag time. Latitudinal variability in XCO2atm(t) is not considered in TRACE, as identifying a water mass source region and accompanying atmospheric boundary is beyond the application space of IG-TTD.

Finally, convoluting the surface boundary with the TTD (Eq. ) yields ocean pCO₂ for a given location and time. This is converted to DIC via inorganic carbon equilibrium calculation with provided salinity, temperature, depth, and preformed properties as previously estimated. Subtracting pre-industrial DIC (calculated from a user-provided pre-industrial atmospheric mixing ratio and the same preformed properties) leaves Canth. TRACEv1 assumed a pre-industrial XCO2atm of 280 ppm, which TRACE-Python makes more readily modifiable as an optional user input parameter, as described in Sect. .

This implementation of TRACE retains its predecessor's estimated uncertainty of Canth point estimates and inventories. The estimated 1σ uncertainty of TRACE point estimates is the root sum of squared errors derived from a Monte Carlo analysis of error propagated from training data and error associated with a model reconstruction analysis. As with TRACEv1, the resulting uncertainty in Canth likely underestimates the true reconstruction error in coastal, marginal, undersampled, and upwelling regions.

In addition to its inherited capabilities, TRACE-Python adds several features which expand its scientific applications and provide more robust results. We divide these into two categories: practical improvements (Sect. ) that improve user experience and applications, and fundamental improvements (Sect. ) that may alter the results or interpretation of the method.

Assessment of TRACE-Python sought to validate its comparability with TRACEv1, explore its sensitivity to new user parameter inputs, and finally to demonstrate its use alongside other ocean Canth data products. All estimates were produced with TRACEv1 and TRACE-Python version 1.0.0, which was developed and hosted in a Github repository containing its source code, instructions for installation, documentation, demonstration scripts, and status badges indicating that the code passes internal consistency and validation tests. Comparability with TRACEv1 was established by calculation of check values as well as global gridded Canth products using identical inputs. The two implementations were found to give identical results with precision approaching pmol kg⁻¹ levels, which when integrated into regional and global inventories led to no significant difference. Sensitivity analysis of newly-accessible parameters demonstrated increased flexibility of the TRACE-Python routine and pointed towards new directions for method development and software application.

Check values given for TRACEv1 and TRACE-Python (Table ) demonstrated results within their respective uncertainties. Precision between MATLAB and Python implementations was expected to vary depending on the exact data types and operations performed: both languages include double-precision floating point arithmetic by default, but other contributors to point estimate imprecision can be expected on the order of 10⁻⁵ µmol kg⁻¹ from inorganic carbon equilibrium calculations alone .

A global gridded Canth product was created using TRACE-Python, using seawater salinity, seawater temperature, coordinates, and depth from the GLODAPv2.2016b gridded product , which has a spatial resolution of 1°×1° and 33 depth horizons between the sea surface and 5500 m. Each of nine available atmospheric CO₂ pathways available in TRACE was employed to yield Canth estimates for the years 1750, 1800, 1850, 1900, 1950, 1980, 1994.5, 2000, 2002.5, 2007.5, 2010, 2014.5, 2020, 2030, 2050, 2100, 2200, 2300, 2400, and 2500, chosen to align with previous literature global Canth inventory estimates. These global Canth gridded estimates are archived in a Zenodo repository . Comparison of point Canth estimates to the same analysis performed by TRACEv1 demonstrated agreement within uncertainties and approaching the limits of precision imposed by inorganic carbon equilibrium calculation. Their residuals (calculated as TRACEv1 estimates subtracted from TRACE-Python), across 9 atmospheric pathways, 20 timesteps, and 1.1×106 ocean cells in the GLODAPv2.2016b gridded product, demonstrated a median error of -1.8×10-6 µmol kg⁻¹ and median error of -2.6×10-6 µmol kg⁻¹. While the total range of error was -0.02 to 0.0005 µmol kg⁻¹, 95 % of absolute error was less than 6.4×10-3 µmol kg⁻¹. TRACE-Python underestimation (relative to TRACEv1) of the global distribution of Canth was most apparent for cells with higher Canth (Fig. ) which was repeatable for all CO₂ trajectories at all calculated times (Figs. –). This apparent bias is consistent with the magnitude of expected precision of (MATLAB) CO2SYS versus PyCO2SYS as previously noted. Extrapolating the median error given above across the entire ocean yields a value on the order of 10⁻⁵ Pg, so we conclude that random or systematic biases existing between implementations of TRACE had no significant effect on inventories calculated using this gridded product, as demonstrated in the calculation of regional and global Canth inventories below.

This work described an implementation of the TRACE method for the estimation of ocean Canth in Python, incorporating several practical and fundamental improvements. The effect of these changes is to increase the accessibility and breadth of application of this tool, while providing a firmer scientific footing with clearer understanding of input parameter sensitivity. This updated version demonstrated equivalent function to the original product when given identical input, ensuring comparability across research products and users. The development of the TRACE method and its software implementations gains further currency when considered as part of a broader dialogue between scientific questions and research tools to address them. This work in particular has benefited from co-development with Empirical Seawater Property Estimation Routines (ESPER), which similarly use location, salinity, temperature, and other predictors to estimate DIC, total alkalinity, pH, nitrate, phosphate, silicate, and oxygen concentrations . This family of seawater property estimation methods is of value to scientific, marine management, and earth observing communities, who use these estimation routines to compare against observations, fill in unobserved regions, initialize models, and make informed management decisions.

The practical and fundamental improvements to TRACE described and demonstrated in Sect.  provided an opportunity to test the sensitivity of TRACE to pre-industrial XCO₂ and the shape of the TTD within the constraints of the IG framework. Global Canth inventories were sensitive to both parameters within the range of values given by previous work. The spatial distribution of mean age and Canth were similarly altered by Δ/Γ along a reconstructed meridional transect of the Atlantic Ocean. Given the variability in inferred Δ/Γ associated with different water masses cf., future work using TRACE could investigate the interaction of regionally-varying Δ/Γ on water mass age and Canth. This sensitivity analysis of ocean Canth and mean age to parameters of the TRACE method illustrates the importance of careful investigation of the assumptions of ocean state estimate routines. While TRACE-Python retains reasonable default values of these and other input parameters in common with TRACEv1, they are made accessible and tunable with the intention of aiding future investigation and expanding the applicability of this software tool.

Several other parameters and assumptions central to the TRACE method are not user-tunable, and consideration of these suggests room for continued method validation and improvement. In particular, its surface CO₂ disequilibrium does not vary in space, it prescribes transient tracer atmospheric saturation, Canth is assumed to equal the entire change in DIC since the pre-industrial era, it estimates preformed alkalinity and nutrients and assumes their invariance in time, and the IG-TTD implies steady state one dimensional pipe flow transport of transient signals into the ocean interior along isopycnals. A model-based review of uncertainties of the IG-TTD method found that transient tracer and Canth saturations were the greatest contributors to uncertainty , so continued development of TRACE and other TTD-based ocean state estimation routines can be served by targeted investigation of the transient tracer and Canth surface boundary conditions and their variability in time and space. Unfortunately, transient tracer saturations cannot yet be modified in TRACE without retraining its neural networks. These shortcomings represent a continuing opportunity for comparing TRACE output with models and ocean observations.

We emphasize that TRACE, ESPER, and their seawater property estimation peers cannot replace observation; rather, they rely on continued monitoring providing the physical and chemical basis for accurate estimation. Ocean hydrography becomes increasingly-important in the face of climate change as Earth experiences extremes moving it outside its previously-observed state captured by property estimation routines. In light of the changing and improving picture of the ocean system to be gained from future observations, TRACE will continue iteratively improving its estimation of Canth. Future GLODAP releases will better constrain TTDs with the addition of more and better tracer constraints and preformed property estimates, while the advance of global ocean circulation and biogeochemical models may indicate more accurate parameterized relationships between the atmospheric anthropogenic CO₂ increase and its ocean sink.

The development of TRACE has occurred in parallel to and in some cases dependent on related ocean chemistry software. This includes other property estimation routines , inorganic carbon equilibrium and air-sea flux calculations and seawater thermodynamic toolboxes . Further development of this suite of open-source software tools should seek to incorporate new findings and techniques, maintain dependency and interoperability, and respond to the needs of users in order to pursue high-quality and accessible ocean chemistry data practices.

It is anticipated that TRACE will continue to be developed without fundamentally altering its core approach, while continuing to reliably offer results with well-documented assumptions and consistency across implementations. Potential directions for further development include integrating future GLODAP releases in its training data, exploring the impact of other reanalysis products on estimates, including updated atmospheric CO₂ trajectories, and refining TTD shape and surface transient tracer and Canth disequilibrium assumptions. As methods for estimating Canth continue use and development, a more comprehensive understanding of their differences, assumptions, and uncertainties should be formed. This need gains currency in light of the present need to understand the effects of climate change mitigation and marine carbon dioxide removal on the ocean carbon cycle. Future work in pursuit of these needs should seek to advance the practice of Canth estimation from scientific and applied perspectives.