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Method for modeling of the components of ionospheric parameter time variations and detection of anomalies in the ionosphere
- Oksana V. Mandrikova^{1, 2}Email author,
- Nadejda V. Fetisova^{1, 2},
- Yuri A. Polozov^{1, 2},
- Igor S. Solovev^{1, 2} and
- Mikhail S. Kupriyanov^{3}
Received: 31 December 2014
Accepted: 4 August 2015
Published: 18 August 2015
Abstract
In this study, a new multicomponent model (MCM) to determine the time variation of ionospheric parameters is suggested. The model was based on the combination of wavelets with autoregressive-integrated moving average model classes and allowed the study of the seasonal and diurnal variations of ionospheric parameters and the determination of anomalies occurring during ionospheric disturbances. To investigate in detail anomalous changes in the ionosphere, new computational solutions to detect anomalies of different scales and estimate their parameters (e.g., time of occurrence, duration, scale, and intensity) were developed based on a continuous wavelet transform. The MCM construction for different seasons and periods of solar activity was described using ionosphere critical frequency f _{ o } F2 data from Kamchatka (Paratunka Station, 52° 58′ N, 158° 15′ E, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS). A comparison of the MCM with the empiric International Reference Ionosphere (IRI) model and the moving median method for the analyzed region showed that the suggested method was promising for future research, since it had the advantage of providing quantitative estimates for the occurrence time, duration, and intensity of the anomalies, characterizing the ionospheric state and disturbance degree with a higher accuracy. Geomagnetic storms from 17 March and 2 October 2013 were analyzed using the suggested method, and it was shown that the ionospheric disturbances were at maximum during the strongest geomagnetic disturbances. An increase in the electron concentration in comparison with the background level, under calm or weakly disturbed geomagnetic field conditions, was identified before the analyzed magnetic storms.
Keywords
- Wavelet transform
- Autoregressive-integrated moving average model
- Ionosphere critical frequency
- Ionospheric disturbances
Background
The present study aimed to develop tools for ionospheric parameter analysis and anomaly detection during ionospheric disturbances. The Earth’s ionosphere is part of the atmosphere, stretching from 80 to 1000 km and affecting radio wave propagation (Kato et al. 2009; Nakamura et al. 2009; Watthanasangmechai et al. 2012). Its structure is changeable and heterogeneous, and its investigation is based on the variation analysis of environmental registered parameters. The ionospheric parameters clearly change with the height, depend on the solar activity cycle, geomagnetic conditions, and geographic coordinates, and have characteristic diurnal and seasonal variations (Afraimovich et al. 2000, 2001; Nakamura et al. 2009; Watthanasangmechai et al. 2012; Danilov 2013). Ionospheric anomalies appear as significant deviations (increase or decrease) of the electron concentration in relation to the background level. During anomalies, local features of different shapes and durations are observed in the registered ionospheric parameters (Mandrikova et al. 2014a). In most cases, the ionospheric disturbances result from an increased solar and geomagnetic activity and, in seismically active regions, they can be observed during increased seismic activity (Afraimovich et al. 2000, 2001; Nakamura et al. 2009; Maruyama et al. 2011; Klimenko et al. 2012a, 2012b; Watthanasangmechai et al. 2012).
The most important tasks in ionospheric parameter processing and analysis are the monitoring of the ionospheric conditions and the detection of anomalies (Afraimovich et al. 2000, 2001; Liu et al. 2008a, 2008b; Nakamura et al. 2009; Watthanasangmechai et al. 2012; Danilov 2013; Ezquer et al. 2014; Zhao et al. 2014), which affect many aspects of our life and have a negative impact on satellite system operation and radio communication propagation. The problems associated with the analysis of ionospheric conditions and detection of anomalies have been addressed by many authors (Bilitza and Reinisch 2007; Liu et al. 2008a, 2008b; Nakamura et al. 2009; Maruyama et al. 2011; Klimenko et al. 2012a, 2012b; Oyekola and Fagundes 2012; Watthanasangmechai et al. 2012; Ezquer et al. 2014; Zhao et al. 2014). The main approaches include the traditional moving median method (Mikhailov et al. 1999; Afraimovich et al. 2000, 2001; Kakinami et al. 2010), ionosphere empirical models (Bilitza and Reinisch 2007; Nakamura et al. 2009; Klimenko et al. 2012b; Oyekola and Fagundes 2012; Watthanasangmechai et al. 2012), the application of adaptive algorithms based on neural networks (Martin et al. 2005; Nakamura et al. 2007, 2009; Mandrikova et al. 2012a, 2012b; Wang et al. 2013; Zhao et al. 2014), and the wavelet transform (Hamoudi et al. 2009; Kato et al. 2009; Mandrikova et al. 2012a, 2012b, 2013a, 2014a). At present, the International Reference Ionosphere (IRI) model (Jee et al. 2005; Bilitza and Reinisch 2007; Klimenko et al. 2012b; Oyekola and Fagundes 2012) is the best ionospheric empirical model. It is based on a wide range of ground and space data and, since its parameter estimation accuracy for a particular region depends significantly on the availability of local registered data, its results can largely deviate from the experimental data (Bilitza and Reinisch 2007; Ezquer et al. 2014). Therefore, the IRI-based forecasts are more accurate for mid-latitudes than for equatorial and auroral latitudes. Previous studies also showed that the accuracy of the IRI model depends on the level of solar activity, decreasing with a solar activity increase (Jee et al. 2005; Nakamura et al. 2009; Oyekola and Fagundes 2012). The recent development of empirical models using pattern recognition techniques and neural networks (Nakamura et al. 2007, 2009; Wang et al. 2013; Zhao et al. 2014) allowed for a significant improvement of the forecast quality in comparison with the IRI model, as they are easy to implement automatically and flexible enough. However, these models belong to the “black box” model class. Therefore, for feature spatial description, long training samples are required, which are prone to overfitting and can lead to unexpected results with very noisy data. The proposed multicomponent model (MCM) is based on autoregressive-integrated moving average models (ARIMA) (Box and Jenkins 1970), which allow obtaining quite accurate estimates with limited samples and, after the model identification phase, can be easily implemented automatically. Their main advantage is their mathematical basis and consequent ability to obtain results with a given confidence probability.
Previous investigation of the ionospheric parameters variation in the Kamchatka region showed a complex non-stationary structure, which significantly impedes the application of traditional classical methods for modeling and analysis of data. As the latest research (Huang et al. 1998; Odintsov et al. 2000; Rilling 2003; Huang and Wu 2008; Klionsky et al. 2008, 2009; Hamoudi et al. 2009; Kato et al. 2009; Yu et al. 2010; Akyilmaz et al. 2011; He et al. 2011; Mandrikova et al. 2012a, 2012b, 2014a; Ghamry et al. 2013; Zaourar et al. 2013) shows, the most natural and effective way of representing such data is the construction of non-linear adaptive approximating schemes. As a result, methods of empirical mode decomposition (Huang et al. 1998; Rilling 2003; Klionsky et al. 2008, 2009; Huang and Wu 2008; Yu et al. 2010) and adaptive wavelet decomposition (Hamoudi et al. 2009; Kato et al. 2009; Akyilmaz et al. 2011; He et al. 2011; Mandrikova et al. 2012a, 2012b, 2013a, 2014a; Ghamry et al. 2013; Zaourar et al. 2013) are being intensively developed at present. Given the large variety of orthogonal basis wavelets with compact support and the presence of numerically stable fast algorithms for data transformation, wavelet decomposition provides many possibilities for the analysis of data with a complex structure (Chui 1992; Daubechies 1992; Mallat 1999), including geophysical data (Hamoudi et al. 2009; Kato et al. 2009; Akyilmaz et al. 2011; He et al. 2011; Mandrikova et al. 2012a, 2012b, 2013a, 2014a; Ghamry et al. 2013; Zaourar et al. 2013). In this paper, a multiscale wavelet decomposition (MSA) of an ionospheric parameter time series was used. Based on the MSA, the time series was presented as different scale components with a simpler structure than the original series. This representation allowed the distinction of stationary components and the application of classical methods of time series modeling and analysis for their identification. As mentioned above, an ARIMA model class (Box and Jenkins 1970; Kay and Marple 1981; Basseville and Nikiforov 1993; Huang et al. 2013) was used in this study. Practical research has confirmed the power and flexibility of the ARIMA method in solving many applied problems (Box and Jenkins 1970; Basseville and Nikiforov 1993; Huang et al. 2013). At present, these methods are being developed in geophysical studies (Mabrouk et al. 2008; Huang et al. 2013; Mandrikova et al. 2013a, 2014a). However, there are some restrictions regarding their application to separate time series and determined regularities (Kay and Marple 1981; Huang et al. 2013; Mandrikova et al. 2013a, 2014a). The estimation, diagnostics, and optimization of ARIMA model parameters are based on the assumption that the data have a standard distribution, which is not always correct. Extending the application of these methods, we suggested a new MCM, based on the combination of wavelets with ARIMA models. This approach was proposed for the first time to reveal anomalies in subsoil radon data and proved to be efficient (Geppener and Mandrikova 2003). The present paper describes a method to construct and estimate an MCM. The efficiency of the suggested model was assessed using ionospheric data. The model allowed the elimination of noise, the simplification of the data structure, and the detection of stationary components liable for identification. We compared the obtained MCM with the IRI model and the moving median method, widely applied in the modeling and analysis of ionospheric parameters. The comparison showed promising results for the newly proposed method. To study ionospheric parameters in detail, we used the suggested modeling method combined with a continuous wavelet transform. The computational solutions allowed the detection of different scale anomalies in the ionosphere and the estimation of their occurrence time, duration, and intensity was based on the continuous wavelet transform.
Methods
MCM identification
where \( f\left[{2}^{-m}t\right]={\displaystyle \sum_k}{c}_{-m,k}{\varphi}_{-m,k}(t) \) is a smoothed component of a time series; coefficients c _{− m,k } = 〈 f, φ _{− m,k } 〉 and φ _{− m,k }(t) = 2^{− m/2} φ(2^{− m } t − k) are a scaling function; \( g\left[{2}^jt\right]={\displaystyle \sum_k}{d}_{j,k}{\varPsi}_{j,k}(t) \) is the detailing components of a time series; and coefficients d _{ j,k } = 〈 f, Ψ _{ j,k } 〉 and Ψ _{ j,k }(t) = 2^{ j/2} Ψ(2^{ j } t − k) are the wavelet basis.
By changing the decomposition level m, we could obtain various representations of a time series. Our task was to determine the best representation that allowed the extraction of the stationary components from the noise and the acquisition of an adequate ARIMA model. The smoothed components of the wavelet decompositions f[2^{− m } t] were less affected by the random factor than the detailing components g[2^{ j } t]. Therefore, the solution was based on the analysis of the smoothed components as follows:
Step 1. We performed multiscale wavelet decompositions of the time series to levels \( m=\overline{1,M} \) (the maximum acceptable decomposition level M was determined by the length N of the time series: M ≤ log_{2} N) and obtained a set of smoothed components: \( f\left[{2}^{-m}t\right]={\displaystyle \sum_k}{c}_{-m,k}{\varphi}_{-m,k}(t) \), \( m=\overline{1,M} \).
where \( {s}_{-m,k}={\displaystyle \sum_{l=1}^p{\gamma}_{-m,l}{\omega}_{-m,k-l}}-{\displaystyle \sum_{n=1}^h{\theta}_{-m,n}{a}_{-m,k-n}} \) is an estimated smoothed component, ω _{− m,k } = ∇^{ ν } c _{− m,k }, ∇^{ ν } is a difference operator of ν order, p and γ _{− m,l } are the order and parameters of a smoothed component autoregression, h and θ _{− m,n } are the order and parameters of a moving average of a smoothed component, and a _{− m,k } are the residual errors of the model.
where \( {e}_{k+q}^m={\left({s}_{-m,k+q}^{\mathrm{actual}}-{s}_{-m,k+q}^{\mathrm{predict}}\right)}^2 \) is the component model error at point k with time step q, \( {s}_{-m,k+q}^{\mathrm{actual}} \) are the actual values of a time series component, \( {s}_{-m,k+q}^{\mathrm{predict}} \) are the model values of a time series component, Q is the length of the data time step, and K is the length of a time series component.
Step 4. We considered that the best representation of a time series was the one corresponding to a multiscale wavelet decomposition to level m*, where \( {m}^{*}:{E}_{m^{*}}=\underset{m}{ \min }{E}_m \).
Step 5. We determined the stationary components from a set of detailing components g[2^{ j } t] and \( j=\overline{-1,-{m}^{*}} \). Applying the traditional approaches (Box and Jenkins 1970; Marple 1987), we determined the models from the ARIMA model class for the approximation of the stationary components g[2^{ j } t].
Step 6. Components g[2^{ j } t], which were not stationary, contained local features and noise and were investigated by another method.
where \( {s}_{j,k}^{\mu }={\displaystyle \sum_{l=1}^{p_j^{\mu }}}{\gamma}_{j,l}^{\mu }{\omega}_{j,k-l}^{\mu }-{\displaystyle \sum_{n=1}^{h_j^{\mu }}}{\theta}_{j,n}^{\mu }{a}_{j,k-n}^{\mu } \) is an estimated μth component, \( {p}_j^{\mu } \) and \( {\gamma}_{j,l}^{\mu } \) are the order and parameters of the μth component autoregression, \( {h}_j^{\mu } \) and \( {\theta}_{j,k}^{\mu } \) are the order and parameters of a moving average of the μth component, \( {\omega}_{j,k}^{\mu }={\nabla}^{\nu^{\mu }}{\beta}_{j,k}^{\mu } \), ν ^{ μ } is the difference order of the μth component, \( {\beta}_{j,k}^1={c}_{j,k} \), \( {\beta}_{j,k}^{\mu }={d}_{j,k},\mu =\overline{2,T} \), Τ is the number of modeled components, \( {a}_{j,k}^{\mu } \) are the residual errors of the μth component model, \( {N}_j^{\mu } \) is the length of the μth component, \( {b}_{j,k}^1={\varphi}_{j,k} \) is a scaling function, and \( {b}_{j,k}^{\mu }={\varPsi}_{j,k},\mu =\overline{2,T} \) is a wavelet basis of the μth component.
The residual errors of the μth component model were determined as the difference between the actual and predicted values for point k + q: \( {a}_{j,k+q}^{\mu }={s}_{j,k+q}^{\mu, \mathrm{actual}}-{s}_{j,k+q}^{\mu, \mathrm{predict}} \).
where Q _{ μ } is the length of the data time step based on the μth component model and T _{ μ } is the threshold value of the μth component defining the presence of an anomaly.
where \( {\psi}_{j,q}^{\mu } \) are the weighting coefficients of the μth component model, which can be determined by \( \left(1-{\gamma}_{j,1}^{\mu }B-{\gamma}_{j,2}^{\mu }{B}^2-\dots -{\gamma}_{j,{p}_j^{\mu }+{\nu}^{\mu}}^{\mu }{B}^{p_j^{\mu }+{\nu}^{\mu }}\right)\left(1+{\psi}_{j,1}^{\mu }B+{\psi}_{j,2}^{\mu }{B}^2+\dots \right)=\left(1-{\theta}_{j,1}^{\mu }B-{\theta}_{j,2}^{\mu }{B}^2-\dots -{\theta}_{j,{h}_j^{\mu}}^{\mu }{B}^{h_j^{\mu }}\right), \) where B is a backward shift operator: \( {B}^l{\omega}_{j,k}^{\mu }(t)={\omega}_{j,k-l}^{\mu }(t), \) \( {\psi}_{j,0}^{\mu }=0. \)
where u _{ ε/2} is the quantile of the 1 − ε/2 level of the standard normal distribution.
Results and discussion
Construction of the MCM for the Kamchatka region
Model identification
For the model construction, we used hourly data of the ionospheric critical frequency f _{ 0 } F2 (Paratunka station, 52° 58′ N, 158° 15′ E, Kamchatka, Russia, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS (IKIR FEB RAS)) from 1968 to 2013. To determine the degree of geomagnetic disturbance, we used the K-index based on the Paratunka station geomagnetic data. To model the ionospheric parameters for a quiet period, the time intervals for a relatively calm geomagnetic field (sum of the daily K-indices ΣK < 24), without strong seismic events (without earthquakes of Ks ≥ 12, within a 300 km radius from the station), were used as estimates.
Time intervals of the f _{ 0 } F2 data used for the construction of the multicomponent model (MCM)
Winter | Summer |
---|---|
High solar activity | |
20.12.1968–09.01.1969 | 31.07.1969–20.08.1969 |
03.01.1970–22.01.1970 | 30.07.1970–16.08.1970 |
11.02.1970–02.03.1970 | 07.06.1979–24.06.1979 |
03.01.1982–20.01.1982 | 20.07.1989–10.08.1989 |
01.01.1991–23.01.1991 | 10.07.1990–28.07.1990 |
30.11.2000–19.12.2000 | 27.06.1998–15.07.1998 |
19.12.2011–08.01.2012 | 20.06.2001–07.07.2001 |
29.12.2012–24.01.2013 | 18.06.2002–06.07.2002 |
28.05.2002–18.06.2002 | |
Low solar activity | |
09.01.1984–29.01.1984 | 24.06.1983–14.07.1983 |
11.02.1985–27.02.1985 | 26.05.1983–12.06.1983 |
22.01.1987–10.02.1987 | 20.06.1984–10.07.1984 |
29.12.2005–14.01.2006 | 28.06.2004–17.07.2004 |
15.01.2006–04.02.2006 | 13.08.2007–01.09.2007 |
09.02.2008–27.02.2008 |
The model identification was performed using the method described in “Model identification” section.
where ε[Z] is the approximation error determined as \( {\varepsilon}^{\lambda}\left[Z\right]={\displaystyle \sum_{z\notin {I}_Z^{\lambda }}}{\left|\Big\langle f,{q}_z^{\lambda },\Big\rangle \right|}^2={||f||}^2-{\displaystyle \sum_{z\in {I}_Z^{\lambda }}}{\left|\Big\langle f,{q}_z^{\lambda },\Big\rangle \right|}^2 \), where I _{ Z } is the set of indices of power Z.
where \( f\left[{2}^{-3}t\right]={\displaystyle \sum_k}{c}_{-3,k}{\varphi}_{-3,k}(t) \) is the smoothed stationary component containing periods of more than 8 h, \( g\left[{2}^{-3}t\right]={\displaystyle \sum_k}{d}_{-3,k}{\varPsi}_{-3,k}(t) \) is the detailing stationary component containing periods of 8–16 h, and \( g\left[{2}^jt\right]={\displaystyle \sum_k}{d}_{j,k}{\varPsi}_{j,k}(t) \) and \( j=\overline{-1,-2} \) are the detailing components containing the local features and noise.
The obtained approximation (Eq. 6) agrees with Shi et al. (2015), who showed that the largest variance of the ionospheric periodic oscillations ranged 2–4 days and decreased with the period increase.
Estimated parameters for the smoothed f[2^{− 3} t] component model (winter)
Time period | Constant | Parameters of the model for the smoothed component | Parameters of the model for the first difference of the smoothed component | ||||
---|---|---|---|---|---|---|---|
First parameter | Second parameter | Third parameter | First parameter | Second parameter | Third parameter | ||
03.01.1982–20.01.1982 | 16.832 | −0.195 | −0.1745 | 0.8029 | −0.5443 | −0.5505 | 0.4451 |
19.12.2011–08.01.2012 | 13.746 | −0.1677 | −0.1809 | 0.80866 | −0.5567 | −0.5744 | 0.42407 |
29.12.2012–24.01.2013 | 13.089 | −0.3028 | −0.3231 | 0.65231 | −0.5858 | −0.5936 | 0.39376 |
General model parameters (solar maximum) | 16.00 | −0.22 | −0.22 | 0.77 | −0.62 | −0.63 | 0.36 |
22.01.1987–10.02.1987 | 10.688 | −0.0931 | −0.1326 | 0.84567 | −0.4579 | −0.4669 | 0.53618 |
15.01.2006–04.02.2006 | 11.042 | −0.1716 | −0.2479 | 0.7101 | −0.7254 | −0.7371 | 0.22256 |
09.02.2008–27.02.2008 | 10.576 | −0.3156 | −0.2979 | 0.62768 | −0.7993 | −0.7656 | 0.20138 |
General model parameters (solar minimum) | 11.00 | −0.19 | −0.21 | 0.75 | −0.62 | −0.63 | 0.36 |
Estimated parameters for the smoothed f[2^{− 3} t] component model (summer)
Time period | Parameters of the model for the 1st difference of the smoothed component | |
---|---|---|
First parameter | Second parameter | |
07.06.1979–24.06.1979 | −0.4032 | −0.4099 |
20.06.2001–07.07.2001 | −0.5268 | −0.5916 |
18.06.2002–06.07.2002 | −0.6156 | −0.6099 |
General model parameters (solar maximum) | −0.50 | −0.58 |
20.06.1984–10.07.1984 | −0.8254 | −0.6382 |
28.06.2004–17.07.2004 | −0.8485 | −0.7834 |
13.08.2007–01.09.2007 | −0.9465 | −0.9156 |
General model parameters (solar minimum) | −0.83 | −0.73 |
Estimated parameters for the detailed g[2^{− 3} t] component model (winter)
Time period | Parameters of the model for the detailed component | Parameters of the model for the first difference of the detailed component | |||
---|---|---|---|---|---|
First parameter | Second parameter | Third parameter | First parameter | Second parameter | |
03.01.1982–20.01.1982 | −0.111 | −0.1471 | 0.85026 | −0.9304 | −0.9568 |
19.12.2011–08.01.2012 | −0.2325 | −0.2379 | 0.73806 | −0.997 | −0.9805 |
29.12.2012–24.01.2013 | −0.1424 | −0.1256 | 0.8261 | −0.998 | −0.9693 |
General model parameters (solar maximum) | −0.14 | −0.14 | 0.83 | −0.97 | −0.93 |
22.01.1987–10.02.1987 | −0.3439 | −0.3168 | 0.62425 | −0.9783 | −0.9476 |
15.01.2006–04.02.2006 | −0.1987 | −0.1977 | 0.79975 | −1.001 | −0.997 |
09.02.2008–27.02.2008 | −0.2114 | −0.1146 | 0.78411 | −1.007 | −0.9522 |
General model parameters (solar minimum) | −0.29 | −0.26 | 0.69 | −0.97 | −0.93 |
Estimated parameters for the detailed g[2^{− 3} t] component model (summer)
Time period | Parameters of the model for the first difference of the detailed component | |
---|---|---|
First parameter | Second parameter | |
07.06.1979–24.06.1979 | −0.8262 | −0.7529 |
20.06.2001–07.07.2001 | −0.9347 | −0.7743 |
18.06.2002–06.07.2002 | −0.841 | −0.8331 |
General model parameters (solar maximum) | −0.88 | −0.80 |
20.06.1984–10.07.1984 | −0.9272 | −0.9093 |
28.06.2004–17.07.2004 | −0.9577 | −0.9312 |
13.08.2007–01.09.2007 | −0.9342 | −0.9027 |
General model parameters (solar minimum) | −0.95 | −0.86 |
According to Tables 2 and 4, we obtained the following models for winter according to Eq. 2. Regarding the models obtained without considering the first differences and depending on the solar activity, we obtained the following equations.
For a high solar activity:
\( {s}_{3,k}^1=16-0.22\cdot {c}_{3,k-1}-0.22\cdot {c}_{3,k-2}+0.77\cdot {c}_{3,k-3}+{a}_{3,k}^1(t) \) for the estimated component f[2^{− 3} t] and \( {s}_{3,k}^2=-0.14\cdot {d}_{3,k-1}-0.14\cdot {d}_{3,k-2}+0.83\cdot {d}_{3,k-3}+{a}_{3,k}^2(t) \) for the estimated component g[2^{− 3} t].
For a low solar activity:
\( {s}_{3,k}^1=11-0.19\cdot {c}_{3,k-1}-0.21\cdot {c}_{3,k-2}+0.75\cdot {c}_{3,k-3}+{a}_{3,k}^1(t) \) for the estimated component f[2^{− 3} t] and \( {s}_{3,k}^2=-0.29\cdot {d}_{3,k-1}-0.26\cdot {d}_{3,k-2}+0.69\cdot {d}_{3,k-3}+{a}_{3,k}^2(t) \) for the estimated component g[2^{− 3} t].
For a general model for high and low solar activities, obtained by considering the first difference, we obtained:
\( {s}_{3,k}^1=-0.62\cdot {\omega}_{3,k-1}^1-0.63\cdot {\omega}_{3,k-2}^1+0.36\cdot {\omega}_{3,k-3}^1+{a}_{3,k}^1(t) \) and \( {\omega}_{3,k}^1=\nabla {c}_{3,k} \) for the estimated component f[2^{− 3} t] and \( {s}_{3,k}^2=-0.97\cdot {\omega}_{3,k-1}^2-0.93\cdot {\omega}_{3,k-2}^2+{a}_{3,k}^2(t) \) and \( {\omega}_{3,k}^2=\nabla {d}_{3,k} \) for the estimated component g[2^{− 3} t].
Our results were based on the general model for high and low solar activities. According to this model, to obtain a winter forecast, four preceding forecasts were required, taking into account the difference of order ν = 1. For the initial hourly data and a decomposition level of m = 3, this corresponded to 32 h.
According to Tables 3 and 5, we obtained the following models for summer according to Eq. 2. For a high solar activity, we obtained:
\( {s}_{3,k}^1=-0.50\cdot {\omega}_{3,k-1}^1-0.58\cdot {\omega}_{3,k-2}^1+{a}_{3,k}^1(t) \) and \( {\omega}_{3,k}^1=\nabla {c}_{3,k} \) for the estimated component f[2^{− 3} t] and \( {s}_{3,k}^2=-0.88\cdot {\omega}_{3,k-1}^2-0.80\cdot {\omega}_{3,k-2}^2+{a}_{3,k}^2(t) \) and \( {\omega}_{3,k}^2=\nabla {d}_{3,k} \) for the estimated component g[2^{− 3} t].
For a low solar activity, we obtained:
\( {s}_{3,k}^1=-0.83\cdot {\omega}_{3,k-1}^1-0.73\cdot {\omega}_{3,k-2}^1+{a}_{3,k}^1(t) \) and \( {\omega}_{3,k}^1=\nabla {c}_{3,k} \) for the estimated component f[2^{− 3} t] and \( {s}_{3,k}^2=-0.95\cdot {\omega}_{3,k-1}^2-0.86\cdot {\omega}_{3,k-2}^2+{a}_{3,k}^2(t) \) and \( {\omega}_{3,k}^2=\nabla {d}_{3,k} \) for the estimated component g[2^{− 3} t].
According to these models, to obtain a summer forecast, three preceding forecasts were required, taking into account the difference of order ν = 1. For the initial hourly data and a decomposition level of m = 3, this corresponded to 24 h.
Model diagnostics
had a distribution of approximately \( {\chi}^2\left(Z-{h}_j^{\mu }-{p}_j^{\mu}\right) \), where Z is the first autocorrelation of the μth component model residual errors, r _{ z }(a _{ μ }) is the autocorrelation of the residual error of the μth component model, and n = N − ϑ, where N is the time series length of the μth component and ϑ is the difference order of the μth component model.
where I(f _{ i }) is the periodogram of a residual error of the μth component model \( {a}_k^{\mu },k=\overline{1,n} \), n is the time series length \( {a}_k^{\mu } \): \( I\left({f}_i\right)=\frac{2}{n}\left[{\left({\displaystyle \sum_{k=1}^n}{a}_k^{\mu } \cos 2\pi {f}_ik\right)}^2+{\left({\displaystyle \sum_{k=1}^n}{a}_k^{\mu } \sin 2\pi {f}_ik\right)}^2\right],{f}_i=i/n \) is the frequency, and s ^{2} is the estimation \( {\sigma}_{a^{\mu}}^2 \) of the residual error time series of the μth component model.
Intervals of the f _{ 0 } F2 data used in the model diagnostics
Winter | Summer |
---|---|
High solar activity | |
15.12.1970–29.12.1970 | 23.06.1969–05.07.1969 |
07.02.1981–21.02.1981 | 04.06.1971–21.06.1971 |
07.02.2002–25.02.2002 | 12.06.1989–29.06.1989 |
05.12.2011–18.12.2011 | 13.08.2000–26.08.2000 |
30.01.2012–11.02.2012 | 03.08.2002–17.08.2002 |
04.02.2013–18.02.2013 | 06.07.2002–18.07.2002 |
Low solar activity | |
13.12.1983–29.12.1983 | 25.06.1987–06.07.1987 |
06.01.1993–25.01.1993 | 09.08.2010–22.08.2010 |
05.01.2007–18.01.2007 |
The tests based on the total goodness of fit (Eq. 7) showed that the resulting MCM adequately characterized the time evolution of the f _{ o } F2 data. For example, for 9–22 August 2010, the \( {Q}^1=n{\displaystyle \sum_{k=1}^{20}}{r}_k^2\left({a}_1\right)=16,15 \) (for f[2^{− 3} t]) and the \( {Q}^2=n{\displaystyle \sum_{k=1}^{20}}{r}_k^2\left({a}_2\right)=8,47 \) (for g[2^{− 3} t]) were consistent with χ _{0,05} ^{2}(20 − 2) = 28, 9 and in accordance with the total goodness-of-fit test, confirming the adequacy of the constructed models.
Error estimation for the multicomponent model (MCM) and International Reference Ionosphere (IRI) model
Analyzed period | Error sum of the squares/correlation coefficient | |
---|---|---|
MCM | IRI model | |
06.01–16.06.1991 | 132.39/1.00 | 719.7/0.89 |
03.01–13.01.2006 | 131.4/0.87 | 236.7/0.48 |
22.06–03.07.2002 | 91.4/0.69 | 183.8/0.49 |
29.06–14.07.2004 | 104.72/0.76 | 156.31/0.62 |
Ionospheric anomaly detection and estimation of their parameters based on the continuous wavelet transform and threshold functions
where the threshold T _{ a } = U * St _{ a } detects the presence of an anomaly for an a scale near point ξ included in the carrier Ψ _{ b,a } (see below), U is a threshold coefficient, and \( S{t}_a=\sqrt{\frac{1}{\varPhi -1}{\displaystyle \sum_{k=1}^{\varPhi }{\left({W}_{\varPsi }{f}_{b,a}-\overline{W_{\varPsi }{f}_{b,a}}\right)}^2}} \), \( \overline{W_{\varPsi }{f}_{b,a}} \) и \( {W}_{\varPsi }{f}_{b,a}^{\mathrm{med}} \) are the average and median for a moving time window of length Φ. Taking into account the diurnal variation of the ionospheric data, the average \( \overline{W_{\varPsi }{f}_{b,a}} \) and median \( {W}_{\varPsi }{f}_{b,a}^{\mathrm{med}} \) were calculated separately for each hour.
Given the randomness of the data, the use of any threshold T _{ a } defining the presence or absence of an anomaly is inevitably associated with the possibility of a wrong identification. To assess the quality of the decision, we used the lowest error rate, which represents the most complete data representation, i.e., the posterior risk (Levin 1963) was estimated and minimized. During the estimation of the a posteriori risk in determining the ionospheric conditions, we used ionogram data (Paratunka station, Kamchatka), which were compared with geomagnetic (K-index) and Kamchatka earthquake catalog data. A dependence of the T _{ a } threshold on the solar activity was found, with T _{ a } increasing for periods of high solar activity. Therefore, separate thresholds for years of high and low solar activity were estimated.
where the norm \( {||{W}_{\varPsi }{f}_{b,a}||}_2=\sqrt{{\displaystyle \sum_{N_a}}{\left({P}_{T_a}\left({W}_{\varPsi }{f}_{b,a}\right)\right)}^2} \), N _{ a } is the series length for scale a.
Data analysis during magnetic storms
The analysis of the magnetic storm from 2 November 2013 showed a similar nature of the processes occurring in the magnetosphere and ionosphere. The perturbations in the geomagnetic field formed during the increase in solar wind speed and were largest between 03:30 and 06:25 UT. A positive anomaly, indicated by an increase in the electron density, was observed in the ionosphere before the magnetic storm (1 November 2013) and, at the beginning of the storm, it was replaced by a medium-scale negative anomaly, which reached its maximum at night between 01:00 and 06:00 LT on 3 November. Small-scale anomalies were also observed. After the end of the magnetic storm at night on 4 November, the electron density in the ionosphere decreased significantly, as indicated by a negative anomaly.
A clear increase in the f _{ o } F2 (pre-storm enhancement) from ground measurements and total electron content (TEC) data has been observed by many authors (Danilov and Belik 1991; Danilov and Belik 1992; Danilov 2001; Burešová and Laštovička 2007; Mansilla 2007; Liu et al. 2008a, 2008b; Nogueira et al. 2011; Saranya et al. 2011; Adekoya and Chukwuma 2012). For the magnetic storms from 17 March and 2 October 2013, these effects were observed for a calm and weakly disturbed geomagnetic field, lasting from several hours to a day and a half (Figs. 9 and 10).
Conclusions
Using a newly suggested modeling method, we extracted the components that characterize the seasonal and diurnal fluctuations of the ionospheric parameter characteristics for calm conditions in the Kamchatka region. The corresponding models were also constructed. A comparison between the new model and the empirical IRI model and moving median method showed promising results from the suggested method for the studied region, which provided more reliable information about the ionospheric conditions. The computational solutions developed, based on the continuous wavelet transform, allowed the identification of different scale anomalies during ionospheric disturbances and the estimation of their duration and intensity. The ionospheric 1969–2013 data processing showed a dependence of the ionospheric anomaly intensity on the level of solar and geomagnetic activity. The largest and most intense ionospheric anomalies were observed during strong magnetic storms and were mostly characterized by a decrease in the electron density compared with the typical level.
A joint analysis of ionospheric and geomagnetic data from two strong magnetic storms that occurred on 17 March and 2 November 2013 helped to understand the processes involved and the characteristics before and during the events. A comparison between the solar wind parameters and the geomagnetic and ionospheric data processing showed a common nature for the analyzed processes. A significant increase in the solar wind speed before the main phase of magnetic storms was accompanied by disturbances in the geomagnetic field and the emergence of large-scale negative ionospheric anomalies of high intensity. During local small increases in the solar wind speed, weak perturbations were found in the geomagnetic field, accompanied by multiscale abnormal changes in the ionospheric parameters. Before magnetic storms, large-scale positive anomalies, as indicated by the increased electron density, were observed in the ionosphere, together with small-scale anomalies associated with local variations in the ionospheric electron density.
Future research includes the testing and application of the developed MCM as well as obtaining computing solutions for different data registration stations, for a more detailed analysis of the ionospheric processes during disturbances and the study of their spatial and temporal distribution.
Declarations
Acknowledgements
This work was supported by the Russian Science Foundation Grant No. 14-11-00194 and the President of the Russian Federation grant No. SP-2976.2013.5. The authors are grateful to the institutions that support the ionospheric parameters registration stations and magnetic observatories whose data were used in the research and express their gratitude to the employees of the Kamchatka branch of the Geophysical Survey of the Russian Academy of Sciences (GS RAS) that helped during the seismic data registration.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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