## 3 Method: Real Data Analysis  
As in [1], we consider the absolute scalar data acquired by two of the Swarm satellites (Satellites Alpha and Bravo) at quasi-latitudes ranging between \(+55^{\circ}\) and \(-55^{\circ}\), and computed residuals with respect to the so-called VFM model of [7]: for the the array \(ST_{1}\) of one-day std of residuals, take a look at Figure 1 borrowed from our article. Rescale this array \(ST_{1}\) as \(r\mapsto\frac{\sigma_{1}}{r}=y\) where \(\sigma_{1}=\text{mean}(ST_{1})\) and \(r\in ST_{1}\); by virtue of Equation (2), the array \(\{y_{i}\}\) is expected to obey the lognormal distribution with parameter \(s_{1}\) (see Equation (3)) and we can directly calculate \(\sigma_{1}\) and \(s_{1}\). Now, repeat all these computations for arrays \(ST_{0.25}\), \(ST_{0.5}\), \(ST_{0.75}\) (i.e., corresponding to time intervals of 0.25 to 0.75 days). The results are as follows:  
\(ST_{1}\): Satellite A \(\sigma_{1}=2.41\)\(s_{1}=0.33\), Satellite B \(\sigma_{1}=2.40\)\(s_{1}=0.36\)  
\(ST_{0.75}\): Satellite A \(\sigma_{0.75}=2.34\)\(s_{0.75}=0.36\), Satellite B \(\sigma_{0.75}=2.35\)\(s_{0.75}=0.39\)  
\(ST_{0.5}\): Satellite A \(\sigma_{0.5}=2.22\)\(s_{0.5}=0.39\), Satellite B \(\sigma_{0.5}=2.21\)\(s_{0.5}=0.46\)  
\(ST_{0.25}\): Satellite A \(\sigma_{0.25}=1.99\)\(s_{0.25}=0.43\), Satellite B \(\sigma_{0.25}=1.96\)\(s_{0.25}=0.49\)  
Figure 1: Standard deviations (in nT) computed every day for the mid-latitude residuals of the Swarm scalar data used to compute the VFM model of [PERSON] et al. (2015). Blue large dots: data from the Swarm Alpha satellite and red dots: data from the Swarm Bravo satellite. Days are counted in Julian days, with 1 January 2000 taken as the reference. As often happens, a limited amount of lognormal data cannot provide stable statistical estimates. So, what are the \"true values\" of \(s\) and \(\sigma\)? To answer this question, let us use the following well-known method of the statistical moments of \(\theta\), namely:  
\[ \begin{split}\mathrm{E}|\theta|&=\int\limits_{- \infty}^{+\infty}|x|f_{\theta}(x)dx=\int\limits_{0}^{\infty}\left[\int\limits_{ -\infty}^{+\infty}|x|\frac{1}{t\sigma\sqrt{2\pi}}\exp\Big{(}-\frac{1}{2}\frac{ x^{2}}{t^{2}\sigma^{2}}\Big{)}dx\right]\cdot\frac{1}{ts\sqrt{2\pi}}\exp\left(- \frac{\ln^{2}t}{2s^{2}}\right)|dt\\ &=\int\limits_{0}^{\infty}\sigma\sqrt{\frac{2}{\pi}}\cdot t\frac{ 1}{ts\sqrt{2\pi}}\exp\left(-\frac{\ln^{2}t}{2s^{2}}\right)dt=\sigma\sqrt{\frac {2}{\pi}}\mathrm{E}\beta=\sigma\sqrt{\frac{2}{\pi}}\varepsilon^{s^{2}/2}\\ \mathrm{E}\theta^{2}&=\int\limits_{-\infty}^{+ \infty}x^{2}f_{\theta}(x)dx=\int\limits_{0}^{\infty}\mathrm{E}\alpha^{2}\cdot t ^{2}\frac{1}{ts\sqrt{2\pi}}\exp\left(-\frac{\ln^{2}t}{2s^{2}}\right)dt=\sigma ^{2}\mathrm{E}\beta^{2}=\sigma^{2}\sigma^{2s^{2}}\end{split} \]  
Thus, we obtain the explicit expressions of the unknown parameters  
\[ \begin{cases}s^{2}=&\ln\frac{2}{\pi}+\ln\mathrm{E}\theta^{2}-2\ln \mathrm{E}|\theta|\\ \sigma^{2}=&\frac{\pi^{2}}{4}\cdot\frac{(\mathrm{E}|\theta|)^{4}}{\mathrm{ E}\theta^{2}}\end{cases}  \tag{4}\]  
In practice, we recover from real data the estimates of the moments \(\mathrm{E}\theta^{2}\), \(\mathrm{E}|\theta|\) and then get (the estimates of) the unknown parameters. The satellite scalar data [7] cover a little less than a year (between 29 November 2013 and 25 September 2014) and were further selected following a number of criteria, among which magnetically quiet and night-time conditions, to ensure that as little as possible non-modeled external signal is included in the data. This resulted in 42,160 data points for the Alpha satellite and 42,175 for the Bravo satellite. These data can be expected to reflect the signal of the field of internal origin the model tries to model, any other source of signal being treated as a source of noise acting on top of the very low instrumental and satellite noise (less than 0.3 nT, see [8; 9; 10]).