Spectral correlation due to common temperature
On the previous page we looked at Type B methods of error correlation evaluation using the example of a rolling average. On this page, we will continue our examination of this topic by looking a common and important example of correlation between spectral channels by considering an onboard calibration target used to calibrate more than one spectral channel.
In a thermal infrared or microwave sensor, it is common to use an internal warm calibration target as a reference. The radiance of the target is given by calculating Planck’s Law for the ICWT temperature. Thus, an uncertainty associated with temperature in the ICWT affects all channels, but not equally since shorter wavelengths are most sensitive to changes in temperature than longer wavelengths. There is therefore a full correlation, because there is a common error, the error in temperature, but the sensitivity to this error is different from one channel to another. To deal with this we consider Planck’s Law, here simplifying the situation to assume that each channel is at a single central wavelength:
{L_{{\rm{ICWT}},{\rm{A}}}} = \frac{{{\varepsilon _{\rm{A}}}{c_{1,L}}}}{{\lambda _{\rm{A}}^5\left( {\exp [{c_2}/{\lambda _{\rm{A}}}T} \right] – 1)}}
{L_{{\rm{ICWT}},{\rm{B}}}} = \frac{{{\varepsilon _{\rm{B}}}{c_{1,L}}}}{{\lambda _{\rm{B}}^5\left( {\exp [{c_2}/{\lambda _{\rm{B}}}T} \right] – 1)}}
We are interested in the correlation associated with these due to the common error in temperature. Using the same formulation as we used in the rolling average example on the previous page, the covariance is given by:
u\left( {{L_{{\rm{ICWT}},{\rm{A}}}},{L_{{\rm{ICWT}},{\rm{B}}}}} \right) = \frac{{\partial {L_{{\rm{ICWT}},{\rm{A}}}}}}{{\partial T}}\frac{{\partial {L_{{\rm{ICWT}},{\rm{B}}}}}}{{\partial T}}{u^2}\left( T \right)
where:
{\left. {\frac{{\partial L}}{{\partial T}}} \right|_{\left( {{\lambda _i},T} \right)}} = \frac{{{\varepsilon _1}L\left( {{\lambda _i},T} \right)hc}}{{{\lambda _i}{k_{\rm{B}}}{T^2}(1 – {\rm{exp}}\left[ { – hc/\left( {{\lambda _i}{K_{\rm{B}}}T} \right)} \right]}}
We can convert this covariance into correlation coefficient using the expression given on the previous page.