6 Sensitivity Index (SI)

Calculating a sensitivity index allows the impact of changing an independent variable (e.g. a model parameter) to be assessed on a dependent variable. Another way to think of this is how much does an output change if I change an input? Given all other variables are held constant, the SI is computed by dividing the percentage change in the simulated result (i.e. the output) by the percentage change of the input parameter:

\[\begin{equation} SI = \frac{(Output_{new}-Output_{original})/Output_{original}}{(Parameter_{new}-Parameter_{original})/Parameter_{original}} \tag{6.1} \end{equation}\]

If there is a large change in the output variable, we say that the variable is sensitive to the parameter. If there is a small change in the output variable, we say the variable is not sensitive to the parameter.

In this exercise we will assess the sensitivity of the modelled water temperature to the water clarity and the mean wind speed.

Temperature sensitivity

Calculate the sensitivity of the modelled temperature to changes in water clarity (the light extinction coefficient, \(K_w\)) and wind speed (wind_factor). These can be found in glm3.nml: &light and &meteorology, respectively.

Try increasing and decreasing the default parameter value by 0.2 (Kw) and 0.3 (wind) and see how much the output changes.

Using the SI calculation on the simulation output, assess how sensitive the water temperature is to water clarity (Kw):

GLM results Water clarity Decrease 0.37 Water clarity Original 0.57 Water clarity Increase 0.77
Average WQ_35 temperature


Assess how sensitive the water temperature is to wind speed (wind_factor):


GLM results Wind Speed Decrease 0.6 Wind Speed Original 0.9 Wind Speed Increase 1.2
Average WQ_35 temperature

Is the water temperature more sensitive to the change in wind, or the change in water clarity?

Optional: Phytoplankton (algae) sensitivity

Changing a physical parameter does not only affect the physical variables. Water clarity and wind speed directly control the temperature and mixing of the lake, but these changes then flow on indirectly to the other variables simulated by the water quality model. For example:

  • Temperature controls the growth rates of phytoplankton, so a warmer or cooler surface layer will speed up or slow down algal growth.
  • Mixing and stratification control how long algae stay in the well-lit surface layer, and how easily nutrients released from the sediments and bottom waters are brought up to the surface.
  • Water clarity also directly changes how much light is available for photosynthesis, and how deep in the water column algae can grow.

Because of these indirect pathways, a small change to a physical parameter can show up as a different algal biomass - sometimes with a larger (or smaller) sensitivity than the temperature itself.

Using the same simulations as above (with the water quality model activated, as in the previous exercise), add the phytoplankton groups (PHY_green, PHY_crypto and PHY_diatom) to your WQ_35.csv output and calculate the SI of the total phytoplankton biomass to water clarity (Kw):

GLM results Water clarity Decrease 0.37 Water clarity Original 0.57 Water clarity Increase 0.77
Average WQ_35 phytoplankton (green, crypto, diatom) biomass


And to wind speed (wind_factor):


GLM results Wind Speed Decrease 0.6 Wind Speed Original 0.9 Wind Speed Increase 1.2
Average WQ_35 phytoplankton (green, crypto, diatom) biomass

Is the phytoplankton biomass more or less sensitive than the water temperature to each parameter? Which of the pathways above do you think explains the difference?