5.3.3 Time link calibration
Author(s): Dafydd W. Evans, Francesca De Angeli
As explained in Section 5.1.4, this calibration was introduced to remove the contamination signal present in the data used to define the reference system and to track the differential contamination level as a function of time.
The contamination causes a throughput loss in magnitude. Expecting this to be a function of time $t$ and source colour $C$, it is possible to express it by means of Chebyshev polynomials as basic functions:

$$\tau (t,C)=\sum {a}_{n}{T}_{n}(t)+\sum {b}_{m}{T}_{m}(C)+\sum {c}_{m}{T}_{m}(t){T}_{m}(C)$$ 

(5.1) 
where the time dependence is cubic ($n=0,\mathrm{\dots},3$), the colour dependency is linear, as well as the timecolour crossterm.
Tests were carried out to find the best dependencies.
Its variation between two observations of the same source $k$ at two times ${t}_{i}$ and ${t}_{j}$ is then

$$\mathrm{\Delta}\tau ({t}_{i},{t}_{j},{C}_{k})=\tau ({t}_{i},{C}_{k})\tau ({t}_{j},{C}_{k})$$ 

(5.2) 
The differential throughput function is solved using a selection of the 20 sources with most observations in each HEALPix pixel (of level 6),
with an uncalibrated magnitude in the range $G=[13.0,13.5]$ and an uncalibrated colour in the range
${G}_{\mathrm{BP}}{G}_{\mathrm{RP}}=[0.0,4.0]$).
For a more detailed description see also Riello et al. (2018).