If you’re a marine scientist attempting to deconvolve the contributions to variations in CO2 system variables, you should be very concerned about how you compute the sensitivities of one variable to others. Such deconvolution is often done by applying a first-order Taylor series decomposition, but results don’t always add up.

For example, with , i.e., neglecting contributions for dissolved inorganic phosphorus and silicon, such a decomposition looks like

That equation sums the contributions from the simultaneous change () in each of the 4 input variables, with each being multiplied by a partial derivative (sensitivity) so that each term on the right has same units as the total change term on the left.

If results don’t add up (i.e., the right side does not equal the left), there are three potential causes:

  1. the s are not accurate enough, e.g., from imprecise measurements,
  2. the s are too large and the system is nonlinear, or
  3. the sensitivities are inaccurate.

In models, the first concern is not an issue because the s are internally consistent. The second concern might also be neglected based on results from a recent model study. Kwiatkowski and Orr (2018) deconvolved modeled seasonal amplitudes of CO2 system variables, i.e., where the s are typically quite large, but in all cases their results added up precisely. Their success appears to stem from the internal consistency and presumably the accuracy of their calculated sensitivities, an issue that has not received enough attention.

Sometimes sensitivities have been approximated by neglecting some terms to get at a straightforward analytical approximation. These approximations are useful to help understand the system. But their accuracy also matters when performing a Taylor series decomposition because there is a delicate balance between terms in some regions. Even minor imprecisions in the sensitivities of say 10% can lead to things not adding up and thus to the wrong conclusions. So if your deconvolution terms don’t add up, stop and check your sensitivities. Make sure that they are accurate.

Fortunately, it is now easy to avoid approximated sensitivities. Accurate sensitivities are provided as part of the set of new routines that were released along with the publication describing the OA-ICC-funded effort to provide uncertainty propagation add-ons in several public software packages that compute CO2 system variables (Orr et al., 2018). These packages include CO2SYS-MATLAB, seacarb, and mocsy. Clicking on those links will lead you directly to the archive where each package can be downloaded, on CRAN for seacarb and on GitHub for the other 2 packages.

Each package includes a new routine called derivnum with a suffix that depends on the computer language of each package (.m, .R, and .f90). Input arguments are just like those for the preexisting routine in each package that computes carbonate chemistry variables (CO2SYS.m, carb.R, and vars.f90, respectively) except that there is also one new argument added to the beginning of the list to specify the variable with respect to which derivatives will be taken. The derivnum.f90 routine in mocsy was used by Kwiatkowski and Orr (2018).

Besides the typical documentation for each package, the new archives for CO2SYS-MATLAB, seacarb, and mocsy also contain a notebooks directory, which itself contains jupyter notebook files with extensive examples. Jupyter has been highlighted in a recent Nature piece as the way forward for scientists dealing with such data analysis. Once in that notebooks directory, just click on the appropriately named notebook file to visualize its contents as HTML. Even better, if you download a package, those notebook files will be available locally; you may run them interactively in your browser as jupyter notebooks.

More details about these routines are available in Orr et al. (2018).

REFERENCES

Kwiatkowski, L., and Orr, J. C. (2018) Diverging seasonal extremes for ocean acidification during the twenty-first century, Nature Climate Change 8, 2, 141-145, doi:10.1038/s41558-017-0054-0.

Orr, J. C., Epitalon, J.-M., Dickson, A. G., and Gattuso, J.-P. (2018) Routine uncertainty propagation for the marine carbon dioxide system, Mar. Chem. 207, 84-107, doi:9.1016/j.marchem.2018.10.006.