## 3 Algebraic Complexity of the Schumann Resonance Amplitude Data
### The Calibration of the H-Rank Algorithm for the SRA Data  
The spectral power of the SRA is calculated according to the algorithm detailed in [21]. The algorithm involves computing the spectrogram of the SRA for consecutive one-second time intervals, determining the spectrum amplitude, and summing the spectral power values in three different frequency ranges: 0 to 1 Hz (ultralow-frequency range), 0 to 3.5 Hz (low-frequency range), and 3.5 to 36 Hz (medium-frequency range) [21; 23; 46]. The calibration of the H-rank algorithm (selection of the appropriate value \(\varepsilon\)) for the MSC logistic maps model is performed by regulating the complexity of the time series representing the difference between the logistic maps of master and slave. However, the measured SRA data cannot be regulated; these are the data streams recorded by the network of magnetometers. Therefore, a new strategy is required for proper calibration of the H-rank algorithm. Let us consider the SRA spectral power time series recorded by the Californian magnetometer in the frequency range of 3.5-36 Hz during the 28 February 2015. The length of the observation window used to construct the Hankel matrix is set to \(d=300\) data points  
Figure 3: The schematic explanation of the H-rank computations for the time series. (seconds). The raw signal normalized to the interval [0, 1] is depicted in Figure 5a; the first 300 s of the raw signal are shown in Figure 5b. Setting the value of the threshold parameter \(\varepsilon\) to 0.4, H-ranks are computed in overlapping observation windows throughout the whole day of 28 February 2015; the results are depicted in Figure 6a. Note that the H-ranks are normalized to the interval between 0% (representing the minimal H-rank equal to 0) and 100% (representing the maximal H-rank equal to 150). Figure 4: Twenty-four complexity indices computed for the MSC logistic maps (the difference time series \(x_{k}-y_{k}\)). The parameters of the H-rank algorithm are set to d = 150 and \(\varepsilon=0.15\). The coupling parameter \(\Delta\) is set to 0.1 in panel (**a)**; 0.3 in panel (**b)**; 0.4 in panel (**c)**; 0.425 in panel (**d)**; 0.45 in panel (**e**); and 0.5 in panel (**f**). The left vertical axis and the light gray line stand for the time series \(x_{k}-y_{k}\). The red vertical axis and the thin red line stand for the H-ranks computed in overlapping observation windows. Blue bars stand for the complexity indices (the magnitude of the complexity indices is measured by the same scale used to measure unsmoothed H-ranks). Next, complexity indices are computed for each hour of 28 February 2015, and we name those complexity indices Schumann Resonance Complexity Indices (SRCIs). Twenty-four SRCI are shown in Figure 6b. Note that the time scale (hours) in Figure 6 represents GMT (Greenwich Mean Time). The variation in SRCI represents the time of the day well. It is well known that the magnetic field spectral power is higher in the daytime than in the nighttime [46]. In fact, the smallest SRCI at 11:00 GMT (Figure 6b) corresponds to 03:00 PST (Pacific Standard Time). The black horizontal line in Figure 6b corresponds to the arithmetic average of 24 SRCI, calculated for the data collected by the magnetometer located in California during 28 February 2015, in the frequency range 3.5-36 Hz. Computational experiments are repeated for the same data set (California, 28 February 2015, 3.5-36 Hz) with \(\varepsilon=0.7\) (Figure 7). The variation in SRCI in Figure 7b is similar to Figure 6b. However, the arithmetic average of SRCI in Figure 7b is considerably lower compared to the average calculated at \(\varepsilon=0.4\) (Figure 6b). Now, the strategy for the calibration of the threshold parameter \(\varepsilon\) is straightforward. The aim is to choose a value of \(\varepsilon\) such that the average SRCI is exactly equal to 50%.