Methods and Progress in the Calculation Uncertainties

We are using a matrix method of uncertainty propagation essentially the same as that described in Appendix A of Orr et al. (ref. 1). In our case the vector of partial derivatives (their eq. A.2) consists of the differentials of the model-calculated quantity of interest (e.g., pH, one of the K* of the carbonate system) with respect to all individual Pitzer interaction parameters and thermodynamic equilibrium constants in the model. Many of these Pitzer parameters co-vary.

The second matrix contains the variances and covariances (where known) of all the interaction parameters and equilibrium constants in the model. This can be tens to hundreds of items. The values of the variances of the thermodynamic equilibrium constants are known from the literature, but this is hardly ever true of the Pitzer parameters. We have therefore developed methods of estimating variances of the Pitzer parameters by Monte-Carlo simulation. These provide valuable preliminary results, enabling us to estimate uncertainties of quantities of interest such as pH. They also identify the major contributors to those uncertainties which, in general, are relatively few. This is an important finding, because it suggests that improvements in both the speciation model and estimation of uncertainties can be made by focusing efforts on a small number of key chemical systems.

Model development

Our progress is summarised in the diagram below. There are two elements to the work, shown above and below the dashed line on the diagram:

Items 1-3: (1) writing the code for a general aqueous speciation model (an arbitrary number of species, and an arbitrary number of equilibria); (2) adding the Pitzer parameters and equilibrium constants for the two models (refs. 2, 3 below); (3) adding the ability to calculate differentials.

Items 4 and 5: the uncertainties are of three types.

• First, there are those of the logarithms of the thermodynamic equilibrium constants (ln(K)), which can be obtained from the literature.
• Second there are the variances and covariances of the Pitzer activity coefficient parameters ('Par.'). As a first step, we have estimated these by using Monte Carlo simulation and artificial data sets which consist of osmotic coefficients (a sensitive measure of water activity) over an appropriate molality range for the solute(s) being simulated. The scatter of the osmotic coefficients in the simulated datasets is that of a typical set of such measurements by the isopiestic technique (ref. 4). Many of the Pitzer interaction parameters for the electrolytes in seawater were obtained from measurements of this type.

  These estimates of covariances have been obtained for 25 °C only. As a general rule, it might be expected that uncertainties of parameter values at lower temperatures – about 5 ° and below – might be a factor of two or more greater than at 25 °C. This is because there are fewer thermodynamic data (to which the model parameters are fitted) at low T.

• Third, there are ion interaction parameters in the Pitzer model whose values are set to zero because there are no data from which to determine them. Their contributions to the uncertainty can be estimated but, for now, we have ignored these parameters.

Finally, the derivatives (item 3) and covariances (item 5) and are propagated to obtain both a total uncertainty and the individual contributions of each individual parameter and/or covarying group.

The two web based programs that we are demonstrating implement the steps described above, and output both total estimated uncertainties in the calculated quantities and the top contributors to them. The work is only partially complete at present, but demonstrates the power of the approach and likely future benefits.

References

1.  J. C. Orr, J.-M. Epitalon, A. G. Dickson, and J.-P. Gattuso (2018) Routine uncertainty propagation for the marine carbon dioxide system. Mar. Chem. 207, 84-107.

2.  J. F. Waters and F. J. Millero (2013) The free proton concentration scale for seawater pH. Mar. Chem. 149, 8-22.

3.  F. J. Millero and D. Pierrot (1998) A chemical equilibrium model for natural waters. Aquat. Geochem. 4, 153-199.

4.  J. A. Rard and R. F. Platford, Experimental methods: isopiestic. In: Activity Coefficients in Electrolyte Solutions, 2nd. edn., K. S. Pitzer (Ed.), CRC Press, Boca Raton, p.209-278, 1991.