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.
- A reference set of parameters is estimated using the Matlab commandlsqnonlin (Eq. S1).