# Polar Cap Patches Scaling Properties: Insights from Swarm Data
## 2 Data and Methods
### Electron Density Scaling Exponents Estimation  
As previously mentioned, the main purpose of this work is to verify whether electron density measurements taken inside and outside PCPs have the same scaling properties. This could indirectly provide some details about the physical mechanisms involved in this particular type of ionospheric irregularity, because the scaling characteristics can offer insight into the presence of turbulence by revealing information about scale-invariance. In turbulence studies, the concepts of scale invariance and scaling properties are crucial. Scale invariance refers to the absence of dominant lengths or time scales in energy release events, resulting in the structure function conforming to a universal scaling law. Indeed, a way to investigate scale-invariance is through the estimation of the scaling exponents associated with \(q\)th-order moments of the increments of the physical quantity under investigation, in this case electron density. Details on the theory behind this method can be found in [PERSON] (1996). Mathematically this is performed by computing the \(q\)th-order structure functions, \(S_{q}(\tau)\), from \(N_{\rm e}\) by using the following formula:  
\[S_{q}(\tau)=\langle|N_{\rm e}(t+\tau)-N_{\rm e}(t)|^{q}\rangle\sim\tau^{\gamma (q)}, \tag{1}\]  
where \(\langle\rangle\) denotes the average, \(\gamma(q)\) is a scaling exponent that is a function of the moment order \(q\), and \(\tau\) is the time delay. The latter can be interpreted as the spatial scale of interest if we assume that the structure under investigation is observed for a time, in this case the satellite transit time, much shorter than the structure evolution time. This means to adopt the  
Figure 5: Space distribution of PCPs as a function of QD-latitude (\(\lambda_{\rm QD}\)) and magnetic local time (MLT), for two levels of geomagnetic activity, i.e., quiet for AE \(<\) 50 nT (**upper panels**) and disturbed for AE \(>\) 300 nT (**lower panels**). **Left** side refers to the Northern hemisphere while **right** side to the Southern one. Circles are drawn at QD-latitudes of 70\({}^{\circ}\), 75\({}^{\circ}\), 80\({}^{\circ}\), and 85\({}^{\circ}\). well-known [PERSON]'s hypothesis [40]. Here, we estimate the first-order, \(\gamma(1)\), and second-order, \(\gamma(2)\), scaling exponents. \(\gamma(1)\), also known as the Hurst exponent [41], takes values \(\gamma(1)\in[0,1]\) and provides, in our case, information on the persistent or antipersistent character of electron density fluctuations. In detail, for:  
* \(\gamma(1)=0\), \(N_{\text{e}}\) is equivalent to a white noise;
* \(\gamma(1)\in(0,0.5)\), \(N_{\text{e}}\) has an antipersistent character meaning that it is more likely that its fluctuations will tend to change their sign;
* \(\gamma(1)=0.5\), \(N_{\text{e}}\) is a completely uncorrelated time series as, for instance, a Brownian random motion;
* \(\gamma(1)\in(0.5,1)\), \(N_{\text{e}}\) has a persistent character meaning that it is more likely that its fluctuations will tend to keep a given sign;
* \(\gamma(1)=1\), \(N_{\text{e}}\) is equivalent to a linear trend and hence completely predictable. \(\gamma(2)\) gives information on the scaling features of the autocorrelation function and, most importantly, on the spectral features of the signal under investigation. Indeed, it can be demonstrated that in presence of a signal characterized by a power-law spectral density, i.e., \(\sim\)\(f^{-\beta}\) (where \(f\) is frequency), its slope, \(\beta\), is such that \(\beta=\gamma(2)+1\)[42]. As is well-known, spectral features can provide a strong indication on the occurrence of turbulent processes. To evaluate the scaling exponents we use the so-called Detrended Structure Function Analysis (DSFA) [43] since it has been proved to be the most reliable approach when dealing with measurements from satellites. The critical parameter to set for a correct application of DSFA is the width of the moving window over which the \(q\)th-order structure function defined in Equation (1) is estimated.