The model was calibrated using the “controlled random search with local mutation” (CRS2-LM) global optimisation algorithm (Kaelo and Ali, 2006; Johnson, 2013). CRS2-LM is randomly initialised and introduces random “mutations” in further iterations, so that repeated calibrations may lead to different local optima. The pa- rameters’ distributions from repeated calibrations can be examined to give insight into the reliability of any individual calibration. Since this algorithm does not allow for the consideration of “inequality constraints” (e.g., the constraint of Qmin< Qmax,
or kc < Qmin), care must be taken when choosing lower and upper bounds for the pa-
rameter ranges, so that meaningless combinations of parameter values are excluded. The years 1991 to 1999 were used as the calibration period, and the years 2000 to 2009 as the validation period. The objective function was defined to be the mean absolute deviation between modelled and observed values, which here is deemed the preferred measure of model performance, compared to, for example, the root mean square deviation (Willmott, 1982; Willmott and Matsuura, 2005). The algorithm was terminated when the relative change in the cost function, one or more of the parameters, or both, was less than 0.1%, or when the number of model evaluations was ≥ 20 · k.
The tissue nitrogen quota equilibrium solution Q∗was evaluated independently
for each point of the timeseries of NH4+, NO−
x, T emp and Irr data for the sensitivity
analysis and calibration as well as the final model evaluations. Absolute values of model Q∗are shown as time series, however, to ease interpretation and comparison with the observation data time series.
Model parameter ranges (Table 2.2) and default values (Table 2.1) were sur- veyed in the literature, from published models utilising the same mathematical description of tissue nitrogen quota dynamics as described here. Ranges for the environmental forcing nutrient concentrations (NH4+ and NO−x) were based on the
local observation data from Tauranga Harbour. The lower bounds were set to zero, and the upper bounds to the maximum of the 95th percentiles calculated separately
for six different sites around the harbour, from nutrient monitoring data based on samples taken in the period from April 1991 until August 2009. Arithmetic mean values for for Tauranga Harbour were used as defaults for other forcing variables (NH4+= 0.025 mg l−1, NO−x = 0.05 mg l−1, T emp= 16.7
◦
C and Irr= 60000 lux). To assess the relative importance of uncertainty in physiological parameters of the model, as well as uncertainty and natural variability in the environmental forcing data (NH4+, NO−
x, T emp and Irr), a sensitivity analysis (SA) was carried out (Saltelli
et al., 2000; Loucks et al., 2005). Due to the relatively low number of parameters
Table 2.2: Model parameter ranges derived from the literature. Rates per day were con- verted to rates per hour for calculations. In cases where only a single value was found, the range was set to that value ±10%.
symbol value unit reference
VmNH+4 2 mg N (g dw)−1h−1 Coffaro and Bocci (1997), Bendoricchio et al.
(1994), Guimaraens et al. (2005) 5.2 mg N (g dw)−1h−1 Solidoro et al. (1997)
8.5 mg N (g dw)−1h−1 Solidoro et al. (1995) kNH+4 0.1 mg N l−1 Solidoro et al. (1995)
0.5 mg N l−1 Coffaro and Bocci (1997), Bendoricchio et al. (1994), Guimaraens et al. (2005)
0.7 mg N l−1 Solidoro et al. (1997) VmNO−
x 0.45 mg N (g dw)
−1h−1 Solidoro et al. (1995)
0.7 mg N (g dw)−1h−1 Coffaro and Bocci (1997), Guimaraens et al. (2005) 0.9 mg N (g dw)−1h−1 Solidoro et al. (1997)
kNO−
x 0.05 mg N l
−1 Solidoro et al. (1995)
0.07 mg N l−1 Solidoro et al. (1997)
0.25 mg N l−1 Coffaro and Bocci (1997),Guimaraens et al. (2005) Qmax 42 mg N (g dw)−1 Solidoro et al. (1995)
45 mg N (g dw)−1 Solidoro et al. (1997)
40 mg N (g dw)−1 Coffaro and Bocci (1997), Bendoricchio et al. (1994)
Qmin 10 mg N (g dw)−1 Solidoro et al. (1995), Solidoro et al. (1997), Coffaro
and Bocci (1997), Bendoricchio et al. (1994)
µmax 0.3 day−1 Öberg (2005)
0.36 day−1 Lapointe and Tenore (1981), Guimaraens et al. (2005)
0.4 day−1 Coffaro and Bocci (1997)
0.45 day−1 Solidoro et al. (1995), Solidoro et al. (1997), Ben- doricchio et al. (1994)
0.5 day−1 Henley and Ramus (1989) kc 8 mg N (g dw)−1 Solidoro et al. (1997)
2. Algebraic equilibrium solution of tissue nitrogen quota
in the model, an initial screening to reduce the number of parameters considered in further analyses was not necessary. Following Wainwright et al. (2013), we carry out a local, one-(factor-)at-a-time (OAT) SA as a first step, followed by the global, variance-based method of Sobol’ (1990, 1993). This method has been successfully applied to similar models examining Ulva in coastal lagoon systems, for example by Pastres et al. (1999) to a dynamic simulation model with higher complexity (15 state variables) on a shorter time scale (one year). Although computationally more efficient algorithms are available, for example the Fourier amplitude sensitivity test (FAST, Cukier et al., 1973), since evaluating the algebraic steady-state solution in this study requires very little computation time, preference was given to the straight- forward sampling and analysis procedure of Sobol’.
Sobol’ first-order sensitivity indices Si for each of the k model parameters may
be interpreted as quantifying the fraction of total model output variance which would disappear if that parameter were fixed. In this method, all Si values are
normalised by the total variance of model output, so that their sum is equal to one. “Total-order” or “total-effect” sensitivity indices ST i indicate the total contribution
of a parameter to model output variance including all interaction effects with other parameters. A sample size of N = 10000 was deemed adequate after examining both the convergence of the sensitivity indices for increasing N (in intervals up to N= 20000) as well as the absolute values of the indices’ bootstrapped (1000 resam- ples) 95% confidence intervals. Using the modified quasi-Monte Carlo sampling method of Saltelli (2002), estimates for first-order as well as total sensitivity indices were calculated for the k = 10 model parameters using N · (2 · k + 2) = 220000 model evaluations.
The Sobol’ SA was performed for four different scenarios. In scenario 1, the physiological parameters were varied by ±10% around their default values, while the DIN concentrations were held fixed at default values for Tauranga Harbour. In scenario 2, both the physiological parameters as well as the DIN concentrations were varied by ±10% around their default values. In scenario 3, the physiological model parameters were varied over their literature range, while the DIN concentra- tions were again fixed at default values as in scenario 1. In scenario 4, the physio- logical parameters were varied over their literature range as in scenario 3, and the DIN concentrations were varied over the range for Tauranga Harbour. Scenario 4 thus represents most accurately the degree of uncertainty in our present knowledge of the physiological model parameters as well the range of variability in environ- mental conditions at the study sites.
To examine the uncertainty in the calibration results, the calibration procedure
with the initial setup was repeated 100 times with random initialisation and local mutation. Additionally, 100 calibrations were run with “scaling factors” sNH+4 for
NH4+and sNH+4 for NO−x added to the set of existing parameters. The NH+4 and NO − x
values were multiplied by the corresponding scaling factor. The calibration range for both scaling factors was set to the interval between 0 and 2.
2. Algebraic equilibrium solution of tissue nitrogen quota