Since the sensitivity coefficients tell us how much the measured value changes for a given change in a single input quantity, all three of the methods for their determination outlined on the previous page involve varying a single input quantity, whilst holding all of the others constant, and observing the resulting effect on the measured value.
Specific details of how the sensitivity coefficients are determined via each of the three methods we introduced on the previous page are given in the sections that follow.
Mathematical determination of sensitivity coefficients
The sensitivity coefficient for a given input quantity is often obtained mathematically by taking the partial derivative of the measurement model with respect to the input quantity in question, i.e.:
$$c_i = \frac{\delta f}{\delta x_i}$$
This is often the simplest method of determining sensitivity coefficients if the measurement model follows a straightforward relationship. Note that we can apply the chain rule when working with sensitivity coefficients expressed as partial derivatives. For example, with reference to the uncertainty analysis tree that we examined earlier (reproduced below), we can propagate the uncertainties in $X_1$ and $X_2$ directly to the measurand using the chain rule as follows:
$$ \begin{align} u_{X_1} (y) & = c_a c_2 u(X_a) \\ & = \frac{\delta x_2}{\delta X_a} \frac{\delta f}{\delta x_2} u(X_a) \end{align}$$
and
$$ \begin{align} u_{X_2} (y) & = c_b c_2 u(X_b) \\ & = \frac{\delta x_2}{\delta X_b} \frac{\delta f}{\delta x_2} u(X_b) \end{align}$$
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Numerical determination of sensitivity coefficients
Sometimes the most appropriate way to determine sensitivity coefficients is the through modelling. This involves establishing a model that represents the physical process and varying input parameters to understand the change in the output parameters. For example, a thermal instrument model can be used to understand the sensitivity of the instrument to a thermal gradient.
The aim is to determine the sensitivity of the calculated result to an uncertainty associated with a single parameter. Multiple parameters should only be changed at the same time if there is an error correlation between them. Changing one parameter at a time provides a sensitivity coefficient to that single parameter.
Experimental determination of sensitivity coefficients
In some cases sensitivity coefficients are determined experimentally. In this case repeat measurements are taken where the effect is changed and the change in the measured value is analysed. Generally this would be done during pre-flight calibration, where, for example the instrument nonlinearity or the instrument temperature sensitivity may be determined through experimental validation. For example, an imaging spectrometer may be calibrated in a temperature controlled chamber in which the temperature is varied over the temperature range the instrument is likely to experience in orbit. The resultant gain changes can provide a sensitivity coefficient for the uncertainty associated with temperature.