where m is the number of measurement times, n the number of measured compounds (n=9 in the mineral medium enrichment and n=10 in the complex medium enrichment), y is the simulated concentration value,\(\hat{y}\) is the experimental concentration value, \(\theta\)represent the calibration parameters and σ is the experimental standard deviation of the concentration values of a compound. The subscript i refers to the different measurements over time and the subscript j refers to the different compounds.
After the first parameter estimation, the reference residuals (i.e. the difference between the experimental and simulated concentration) are calculated. These residuals are used to generate new synthetic experimental data, which is then used to estimate a new set of parameters. A population of parameters is generated by iterating until convergence and can be used to determine robust estimates and uncertainty quantifications.
A Monte Carlo procedure was used to propagate the uncertainty of the estimated parameters. Samples of the parameter population are chosen using Latin Hypercube Sampling to ensure a maximal coverage of the parameter space (Helton and Davis, 2003). The Monte Carlo procedure can be briefly summarised in three steps: i) select a random sample of the estimated parameter population; ii) run the model and store the solution; iii) iterating 500 times steps i) and ii) until the distribution of model solutions converges.
  1. A reference set of parameters is estimated using the Matlab commandlsqnonlin (Eq. S1).