## 6 Methodology
### Detection of Trajectory Modification  
The method of extreme event detection is based on the identification of sampling moments when a considerable modification happens to the velocity or higher-order derivatives of the velocity. Each of these time series is non-linear. Based on the first two laws of Newton, an extreme event of the trajectory is caused by force, implying velocity mod  
Figure 9: The absolute value of a (first row), j (second row) and s (third row) of Operation Phase 1 (column 1), Operation Phase 2 (column 2) and Operation Phase 3 (column 3). ification. Because the Juno object being in the solar system is always influenced by the gravitation force of the Sun, other planets (i.e., Earth, Jupiter) are in the temporal vicinity. To detect extreme modification of a signal \(u(t)\), we use the Hilbert transform (HT), which is a convolution between the signal and function \(w(t)=\frac{1}{nt}\) multiplied with the constant conforming to the following formula:  
\[\mathcal{H}\{u(t)\}=\frac{1}{\pi}u(t)*w(t)=\frac{1}{\pi}P.V.\int_{-\infty}^{ \infty}\frac{u(\tau)}{t-\tau}d\tau=\lim_{\epsilon\to 0}\biggl{(}\int_{-\infty}^{t- \epsilon}\frac{u(\tau)}{t-\tau}d\tau+\int_{t+\epsilon}^{\infty}\frac{u(\tau)} {t-\tau}d\tau\biggr{)}. \tag{9}\]  
where \"\(*\)\" is the convolution operation, _P.V._ has the meaning of the Cauchy principal value integral, because the integral diverges when \(\tau=t\). Readers should note the important properties of the HT:  
\[\mathcal{H}\{u(t+d)\}=\mathcal{H}\{u\}(t+d). \tag{10}\]  
\[\mathcal{H}\{u(\epsilon t)\}=\mathcal{H}\{u\}(\epsilon t). \tag{11}\]  
\[\mathcal{H}\{e^{i\omega t}\}=e^{i\omega(t-\frac{\omega(\omega)}{2})}. \tag{12}\]  
where \(d,\alpha,\omega\) are constant in time and \(i^{2}=-1\). The third property implies that \(\mathcal{H}\{u(t)=const\}=0\), and for \(u(t)\
eq const\) HT executes a modification with an angle \(\Delta\theta=sign(\omega)\pi/2\) of the phase of the original signal \(u(t)\). If we consider the signal as an inverse Fourier transform (FT) of the FT, \(\mathrm{F}\{u(t)\}=u(\widehat{\omega})\):  
\[u(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\widehat{u(\omega)}e^{i\omega t}d\omega. \tag{13}\]  
Then, using the relation of \(\cos\ (sign(\omega)\pi)=-1\) for any \(\omega\
eq 0\), we get another important feature of the HT. The AC (Alternate Current-dynamic part) component \(u_{AC}(t)\) of the signal \(u(t)\) is:  
\[\mathcal{H}^{2}\{\mathrm{u(t)}\}=\mathcal{H}^{2}\Bigl{\{}\mathrm{F}^{-1}\{ \mathrm{F}\{\mathrm{u}\}\}\Bigr{\}}= \mathrm{F}^{-1}\{\mathcal{H}\{\mathrm{F}\{\mathcal{H}\{\mathrm{u}\}\}\} \}=-\mathrm{u}_{\mathrm{AC}}(\mathrm{t}). \tag{14}\]  
This last property makes it possible to determine the DC (Direct Current-constant part) component \(u_{DC}(t)\) of the signal:  
\[\mathrm{u}_{\mathrm{DC}}(\mathrm{t})=\mathrm{u(t)}-\mathrm{u}_{\mathrm{AC}}( \mathrm{t})=\mathrm{u(t)}+\mathcal{H}^{2}\{\mathrm{u(t)}\}. \tag{15}\]  
We use HT to identify extreme signal \(\mathrm{u(t)}\) changes in our case. Figures 10-12 show the Hilbert transform of the acceleration signal in the first and second operation phases. The Hilbert transform may elucidate the frequency and instantaneous amplitude properties of a time domain signal. The Hilbert transform is a useful method for examining the properties of acceleration signals. The Hilbert transform generates a new signal by imposing a 90-degree phase displacement on all the positive frequencies of the preceding signal in the frequency domain. This may be characterized as a signal that has undergone a time delay. The process of combining the original signal with its Hilbert transform results in the creation of the analytical signal, which is a signal with complex values. The analytic signal preserves essential information from the original acceleration signal. To identify changes in the strength of the motion, one may use the magnitude of the analytic signal, which indicates the current level of acceleration. To ascertain the phase of the acceleration at a certain place, one may analyze the phase of the analytic signal.