Fit of VFM Scales Using Multiple Parameters

For the investigation of the correlation between the ASM-VFM scalar residuals with:

1. VFM sensor temperature, T_sensor
2. VFM electronic unit temperature, T_eu
3. Sun "inclination" angle, beta
4. Optical bench tube temperature gradient 1, T_sensor - T₀₂₉
5. Optical bench tube temperature gradient 2, T_sensor - T₀₃₂

a scalar calibration has been performed co-estimating scaling coefficients (for x and z components of the VFM) for all five of the above parameters. A moderate, homogeneous regularisation have been applied to suppress too high fluctuations; "homogeneous" implies the scaling coefficients are regularized according to the standard deviation of the corresponding parameter (i.e. there are no preferred coefficient imposed by the regularisation). The calibration has been performed for Swarm Alpha and Bravo.

The temperatures T₀₂₉ and T₀₃₂ were not readily available after June 2016, hence the parameters T_sensor - T₀₂₉ and T_sensor - T₀₃₂ were set to their mean values and the corresponding scalar residuals down-weighted in the estimation.

The following plots show the estimated scalings split into the five parameters and their sum. The top plots show the VFM x-axis scaling, the bottom plots the z-axis; please observe the small amplitude of s_z for Alpha. These plots show the very dominating role of T_eu and beta (green and red curves) for s_x, and the almost entirely dominating role of T_eu (green) for s_z.

Reducing the Number of Parameters

Given the small correlation with the temperature gradient parameters, these were next removed from the estimation leaving only parameters 1., 2., and 3. (T_sensor, T_eu, and beta).

The residual statistics obtained by these estimations were only marginally worse than the ones obtained from using all five parameters. The resulting scales are plotted next; they illustrate the noticeable effect, that the beta angle correlates significantly with s_x but only very marginally with s_z. The estimated scaling coefficients are as follows:

	Alpha	Bravo
sTsensor [10-9/°C]	6.7 27.9 49.6	-35.6 7.2 48.9
sTeu [10-9/°C]	255.7 -15.7 -465.0	622.7 -26.0 1,262.2
sbeta [10-9/deg]	-96.7 -1.6 -12.3	-66.7 4.2 8.2

Residuals

The scalar residuals of the estimations using the reduced parameter set are plotted below together with the differences in scalar residuals between the two sets of parameters (three vs. five).