All publicly available software packages (Table ) take two ocean carbonate system variables as input and compute the others from basic thermodynamics. All 10 packages were first downloaded in November 2012. Our earliest findings led developers to update two packages (CO2calc and seacarb). Another three packages (csys, mocsy, and CO2SYS-Excel-Pelletier) were updated following publication of our discussion paper . Different results from some older versions of these packages are briefly shown in one figure to illustrate the discrepancies associated with running software that is out of date. The remaining comparison refers only to the latest version of each package.

To compare packages, it was necessary to define a common reference. Although check values exist for most of the equilibrium constants , none are available for computed variables. Hence we chose CO2SYS as a relative reference for three reasons: (1) it was the first publicly available package; (2) its core routines already serve as the base code for two other packages (CO2calc and ODV); and (3) its documentation and code reveal the intense effort that its developers have put into ferreting out the right coefficients from the literature and the most appropriate version of formulations for the constants.

Our choice of the reference had to be refined, however, because, as mentioned, CO2SYS comes in four variants: the original in QBasic running on DOS , two others that run with Excel , and finally MATLAB code . The original variant is still used by some, but it does not provide options to use formulations for K1 and K2 from , as recommended for best practices . Thus, we reduced our choices for the reference to the other three variants of CO2SYS (MATLAB and both Excel variants), all of which provide options to use all constants recommended for best practices (with one minor exception). All three versions give nearly identical results (Fig. ). But they differ significantly from the original version run with the closest substitutes for 's K1 and K2, namely the previous refits by of the same measured constants from (DM87). With that older set of K1 and K2, however, the original version produces results that match those from the Excel-Pierrot version when also run with DM87. Another requirement was efficiency, since one aspect of our comparison required use of a global-scale input data set with nearly 1 million records (Table ). Thus we further narrowed our choice of the reference to the most efficient variants, CO2SYS-MATLAB and CO2SYS-Excel-Pelletier. Finally, we chose CO2SYS-MATLAB as the reference because it was the most efficient and because its source code was easiest for us to inspect, modify, and rerun for sensitivity tests. This arbitrary choice of the relative reference was necessary for our comparison, but it does not imply that the chosen reference is necessarily error free.

The 10 packages compute results from the same thermodynamic equilibria, but software features differ, including available input pairs, pH scales, and constants. Package diversity covers all commonly used operating systems: all packages run on Windows, eight packages on Mac OSX, and five packages on Linux and Unix (Table ). Source code is available in seven packages in standard programming languages (QBasic, Visual Basic, MATLAB, R, Fortran 95), thereby allowing code validation and improvements by users on all three operating systems mentioned above. The number of possible input pairs of carbonate system variables varies widely between packages, from 1 to 20 (Table ). The mocsy package treats only one input pair: AT–CT, the two carbonate system variables carried by models. The four CO2SYS variants and its two derivatives (CO2calc and ODV) allow the user to select from six commonly measured pairs (AT–CT, AT–pH, AT–pCO2, CT–pCO2, CT–pH, and pH–pCO2); in addition, they allow equivalent pairs where fCO2 replaces pCO2. The csys package provides 10 more input pairs by allowing pair members to include one or more of the three inorganic carbon species CO2*, HCO3-, and CO32-. Although the two former species can only be calculated, promising new techniques are being developed to measure the latter . Yet despite csys's enhanced number of input pairs, it limits pCO2 to be used as input only when combined with pH. The two remaining packages, seacarb and swco2, include the same 16 pairs as csys but also add four others, all including pCO2.

Computed variables are affected by the choice of the pH scale and the constants. All packages allow users to work on the total pH scale as recommended for best practices (Table ) and as used for this comparison. The mocsy package provides only the total scale, while the others allow for conversion to the free scale. The others also allow users to work on the seawater scale except for csys. The 4 CO2SYS variants as well as CO2calc and swco2 also offer the NBS scale. The choice of the pH scale affects the values of the constants for which H+ is part of the equilibrium equation. For K1 and K2, the CO2SYS variants and derivatives offer a large range of choices (Table ). Yet most of those may now be considered out of date, having been replaced by more recent assessments, sometimes with some of the same data. All packages except CO2SYS-QBasic offer the K1 and K2 formulations from , as recommended for best practices. Six packages also offer the most recent formulations for K1 and K2 that have been proposed as more appropriate for low-salinity waters . The formulations for K1 and K2 from the two latter studies are used individually in this comparison to assess associated differences between packages. For the other constants, all packages provide the formulations recommended for best practices, except for KF, a difference shown later to have no consequence.

Some packages also offer additional features. For example, CO2SYS variants and CO2calc allow users to compute variables at a temperature that differs from the in situ value. Some also distinguish different components of total alkalinity, including those from total B, P, and Si. The seacarb package provides explicit functions to the user to allow conversion of pH and constants between the free, total, and seawater scales; other packages make such conversions internally but do not provide user-callable functions. The seacarb package also offers functions to help design perturbation experiments to investigate effects of ocean acidification . Two packages, mocsy and seacarb, allow users to account for pressure effects on subsurface fCO2 and pCO2 following ; other packages neglect these pressure effects.

To compare packages, we used two different kinds of input data. A first analysis compared variables computed in each package as a function of latitude and depth using as input the three-dimensional gridded data products for AT and CT from the Global Ocean Data Analysis Project GLODAP; combined with comparable products from the 2009 World Ocean Atlas (WOA2009) for T , S , and concentrations of total dissolved inorganic phosphorus PT and total dissolved inorganic silicon SiT . We will refer to this combined gridded input data as GLODAP-WOA2009. A second analysis focused on comparing packages while separating the effects of physical input variables (T, S, and P) on computed variables. For that, we started with five commonly used input pairs: AT–CT, AT–pH, AT–pCO2, CT–pCO2, and CT–pH. Then for each pair, we computed the other carbonate system variables over ranges of T, S, and P, assuming zero nutrient concentrations. More precisely, all other carbonate system variables were first calculated with one package (seacarb) from AT=2300 µmolkg-1 and pCO2=400µatm at global average surface conditions (T=18 ∘C, S=35, and P=0db). Then two surface data sets were produced for each pair (and each package) by varying T and S, individually, and recalculating all other carbonate system variables from the fixed input pair. For the first, T was varied from -2 to 50 ∘C, while for the second S was varied from 0 to 50. In both cases, pressure was held at 0 db. To assess how packages differ below the surface, we used the same approach, varying pressure between 0 and 10 000 db and maintaining S=35. But pressure corrections are highly sensitive to temperature (Sect. ), so for each package we made two data sets: (1) holding T=2 ∘C (typical of the deep open ocean) and (2) holding T=13 ∘C (typical of deep waters of the Mediterranean Sea).

Comparisons were made using the total pH scale and constants recommended for best practices by . The equilibrium constant for the solubility of CO2 in seawater K0 is from . The equilibrium constants K1 and K2 are from , who refit the constants determined by to the total pH scale. The formulation for KB is from and is also on the total pH scale. Formulations for KW, K1P, K2P, K3P, and KSi are from , who provides equations for the seawater scale, and those are converted to the total scale. The formulation for KS is from on the free scale (see above). The solubility products for aragonite KA and for calcite KC are from . All these are equilibrium constants given in terms of concentrations, not activities. The only constant for which the formulation was not that recommended by is KF, because that best-practices formulation is not offered by most CO2SYS variants, CO2calc, nor ODV. Instead, we used the KF formulation by on the total scale, which is offered by all packages and recommended by . state that results from the two formulations are in reasonable agreement.

Additionally, all packages used consistent formulations for total concentrations of boron , sulfur , fluoride , and Ca2+ , each proportional to salinity. Nine packages compute saturation states for aragonite ΩA and calcite ΩC from the product of the concentrations of Ca2+ and CO32- divided by the corresponding solubility product, either for aragonite KA or calcite KC , respectively. Only the csys package does not provide output for ΩA and ΩC. To simplify comparison, most figures plot results as absolute differences: computed values are shown after subtracting off corresponding results from the reference.

The most extensive comparison was made with AT–CT as input, the only pair that is available in all packages. With that pair, packages were also compared in terms of how their computed variables were affected by nutrient concentrations, i.e., by varying PT and SiT across their observed ranges in the ocean. Additionally, the same pair was used to quantify effects of two important developments since the best practices were published in 2007. For the first, we quantified effects on computed variables of Lee's () assessment that the total boron concentration in the ocean may be 4 % larger than considered previously . For the second, we assessed impacts of using Millero's () new K1 and K2 formulations, which are designed to cover a wider range of input S and T relative to the intended range for recommended constants .

To better assess the most likely causes of differences in computed carbonate system variables, we also compared associated constants. For the four packages where source code was available and easily modified (CO2SYS-MATLAB, csys, seacarb, mocsy), we used existing routines or slightly modified versions to output all the constants for the same physical input data (T, S, and P) that we used for computing variables. For CO2SYS-Excel-Pelletier, the constants are also available. For swco2, we retrieved its constants using its documented parameter numbers and its routine to extract anything with a parameter number. For packages where source code was not available, we computed constants from output variables when possible. With output from CO2calc and CO2SYS-Excel-Pierrot, we computed its K0, K1, K2, KB, KW, KA, and KC; from ODV output, we computed its K0, K1, K2, KA, and KC.

Until recently, no public package accounted for pressure effects on K0 (needed to convert CO2* to fCO2) and the corresponding fugacity coefficient Cf (needed to convert fCO2 to pCO2) as originally proposed Eqs. 5 and 9. Instead, the total pressure term in those equations was simply assigned to be that of the atmosphere only (1 atm). Hence, their computed subsurface fCO2 and pCO2 may be considered as being referenced to the surface. These pressure effects are accounted for, however, in the latest versions of two packages, mocsy 2.0 and seacarb 3.0.6. Both allow users to compute fCO2 and pCO2 in three ways: (1) the same “common” approach that computes K0 and Cf with total pressure of 1 atm and in situ T, (2) the “potential” approach that likewise uses atmospheric pressure only but also uses potential temperature θ instead of in situ T, and (3) the “in situ” approach that uses the true total pressure (atmospheric + hydrostatic) and in situ T. Other packages offer only the first approach.

For the other equilibrium constants, all packages make pressure corrections following the approach of . That is, the effect of P on each equilibrium constant Ki is given by the equation lnKiP/Ki0=-ΔVi/RTkP+0.5Δκi/RTkP2, where the left-hand side contains the ratio between Ki at depth (P in bars) and at the surface (P at 0 bars), R is the gas constant, Tk is temperature in K, ΔVi is the partial molal volume, and Δκi is the change in compressibility. The latter two variables differ for each constant and were fitted empirically by to be quadratic in temperature: ΔVi=a0+a1Tc+a2Tc2,Δκi=b0+b1Tc+b2Tc2, where Tc is temperature in ∘C. Some of these original coefficients Table 9 contained typographical errors as identified in the code and documentation of CO2SYS-QBasic Appendix. Nonetheless, these errors have persisted in some of the packages as well as in the literature e.g.,. To help amend this situation, Table  lists these coefficients for each constant where known errors have been corrected. To determine the fidelity of packages to this array of coefficients, we studied available source code and evaluated patterns of discrepancies in results by carrying out sensitivity tests to decipher fingerprints characteristic of previous errors.

Although the same approach is used by all packages to make pressure adjustments (Eqs.  and ), it is based on extremely limited data. Thus it may not be particularly accurate. For example for K1 and K2, there are differences of 3 and 8% between adjusted values from and data from for deep water at 2∘C at 10000 dbar. Although improving the accuracy of the pressure adjustments to equilibrium constants should be a high priority for future research, our aim here is to assess package precision.

If software packages with identical input cannot agree to within much less than the measurement precision of a computed variable (e.g., pCO2), then their varied use would add substantially to the total uncertainty. To avoid this situation, it is necessary for these tools to have a numerical precision that is far superior to the measurement precision. By numerical precision, we mean their agreement, including all coding differences and errors as well as the usually much smaller numerical round-off error. Therefore, we arbitrarily define the cutoff level for numerical precision to be 10 times smaller than the best measurement uncertainty Table 1.5. A package that agrees with a given variable from the reference package within the numerical cutoff specified in Table  will be referred to here as having a negligible discrepancy relative to the reference; conversely, a package with a greater difference for a given variable will be considered to have a significant discrepancy.

In this section, all variables are computed from the GLODAP-WOA2009 gridded input data. With those data, we first compare two new approaches to the common approach of computing subsurface fCO2 and pCO2, in two packages. Then we expand comparison to all packages and other variables.

The mocsy and seacarb packages offer the three approaches to compute fCO2 and pCO2 (Sect. ). Both packages agree within 0.008% (0.03 µatm at the surface) for each approach for each of the two variables (Tables  and ). While the two packages always compare well throughout the water column, the three approaches diverge as depth increases, as detailed in our companion paper for one package Figs. 1 and 3. Differences between common and potential pCO2 reach 7 µatm at 5000 m, while differences between potential and in situ pCO2 are much larger. The latter is 5% greater than the former at 100 m but 18 times larger at 5000 m. Differences between potential and in situ fCO2 are smaller because they involve pressure corrections only to K0 and not Cf. Yet they still differ by more than a factor of 2 at 5000 m. Subsequent comparison of pCO2 shows just the common approach, the only one offered by all packages.

More generally, surface zonal means from all packages agree within 0.2 µatm for pCO2, 0.006 µmol kg-1 for CO2*, 0.0002 units for pH, 0.1 µmol kg-1 for CO32-, 0.004 for ΩA, and 0.1 for the Revelle factor (Fig. ). Packages diverge as pressure increases, but agreement generally remain within a factor of 2 of that seen at the surface (Fig. ). There are two exceptions: the disagreement in pH is 5 times larger at 5000 m, where csys is 0.001 larger than other packages, which agree within 0.0003; for the Revelle factor Rf, packages agree within 0.02 throughout the water column except for seacarb, whose discrepancy grows to 0.2 at 5000 m. Although seacarb computes Rf with an efficient analytical formula , that approach neglects effects of PT and SiT on total alkalinity, unlike the less efficient numerical approach used in other packages . Overall, discrepancies among packages are larger at depth, but they remain negligible (Table ) except for pH and pCO2.

Yet agreement was not always so close. For some perspective, the same CO2SYS-MATLAB reference was also compared to older versions of four packages: CO2calc (version 1.0.4 revised on 18 June 2013), csys (version revised on 3 February 2010), seacarb (version 2.3.3 revised on 2 April 2010), and an early predecessor of mocsy developed by but not released publicly. Discrepancies relative to the same reference were once larger, e.g., more than 10 times as much for pCO2, pH, and CO32- (Fig. ). With the mocsy precursor, there are significant discrepancies in pCO2 reaching up to 1.5 µatm at the surface. Those grow with depth, e.g., reaching 4 µatm at 5000 m. At the same depth, there are discrepancies in CO32- reaching 0.5 µmolkg-1 and in pH up to 0.007. Subsurface discrepancies are mainly due to two common modeling approximations that were corrected in the first public release of mocsy . With CO2calc v1.0.4, surface discrepancies reach up to 2 µatm in pCO2, up to 1.3 µmolkg-1 in CO32-, and up to 0.007 in pH. Those discrepancies are associated with coding errors in the K1 and K2 formulations from , errors that were corrected in CO2calc version 1.2.0. With the previous version of csys, surface pCO2 is about 1 µatm lower than the reference because that variable was mislabeled; it was actually fCO2. As for seacarb v2.3.3, there are no significant discrepancies. However, with an even earlier version of seacarb (v2.0.3 released in 2008, not shown), the only package that maintains public access to all previous versions, discrepancies at depth are much larger (e.g., -7 µmolkg-1 in CO32- and -0.165 in pH at 4000 m). Because earlier versions of packages often have much larger discrepancies, users would be wise to keep their carbonate system software up to date.

Previous analysis has illustrated how discrepancies vary spatially across the global ocean, but the realistic gridded input data sets that were exploited did not allow us to isolate how discrepancies vary with individual physical variables and chemical input options. We will now focus on those factors, individually, by exploiting simple artificial input data.

Packages were compared with five common input pairs with the same simple data sets where T, S, and P were varied individually. All packages were compared with the AT–CT pair. Comparison with the four other pairs excluded the mocsy package, which is designed to use only AT–CT. Comparison with two of the pairs, AT–pCO2 and CT–pCO2, excluded the csys package, which does offer pCO2 as an input variable but only when paired with pH.

In Sect. , we compared differences among packages while varying physical input for different input pairs. Here we assess differences due to chemical factors, namely accounting for alkalinity from silicic and phosphoric acids (nutrient alkalinity) and opting for potentially important developments since publication of the best-practices guide .

Besides the accuracy and precision of the different packages that compute carbonate system variables, some users with large data sets may be concerned with computational efficiency. To assess differences in computation time between packages, we chose to use the global gridded data set described in Sect. . With nearly 1 million ocean grid points, the computational time needed to compute all carbonate system variables varies by more than a factor of 1800 between packages (Table ). The slowest package (swco2-Excel) required more than half a day, while the fastest (mocsy) needed 30 s. Packages based on spreadsheets are generally slower than those run by directly calling routines with programming languages (Fortran 95, MATLAB, Visual Basic). The latter are usually coded so that the equilibrium calculations are made one time for each set of input data, whereas spreadsheets often repeat the same set of calculations for each computed variable (each cell). An exception is CO2SYS-Excel-Pelletier, whose execution time rivals that of CO2SYS-MATLAB. Fortunately, even with the slowest of the packages shown in Table , the computational time is trivial for most observational analysis efforts, because the number of samples is much smaller. Hence developers of most packages have not concerned themselves with computational efficiency. Nonetheless, for very large data sets and for models, which may have millions of grid cells to treat every time step, computational efficiency is critical.

A computed variable y is affected by errors in all input variables (including constants as well as members of the input pair), each denoted here as xi (for i=1,2,…n). Thus, we calculated the sensitivity ratio as the relative change of y to the relative change in each xi, namely ∂y/y:∂xi/xi. These sensitivity ratios were determined numerically in three successive steps. First, we calculated variables with seacarb under our standard conditions (S=35, T=18∘C, P=0 db, and zero nutrients, along with CT=2058.185 and AT=2300 µmolkg-1). Then, we increased each input variable by 1 % (∂xi/xi=0.01), individually, and recalculated output variables with seacarb for each perturbation. Finally, we took the difference between the first and second computations to obtain the proportional change in the computed variable ∂y/y.