Geomagnetic Storm Prediction Based on the Neural Network Digital Processing of Joint Observations of the URAGAN Muon Hodoscope and Neutron Monitor Stations

Abstract—

A method for predicting geomagnetic storms based on the neural network digital processing of joint observations of the URAGAN muon hodoscope and the international system of neutron monitor stations has been proposed. A time series of Dst indices are used. Formulas for extrapolating model estimates of Dst indices have been developed. A fully-connected feed-forward neural network has been used. Prediction decision rule has been implemented. The probability characteristics of geomagnetic storm prediction have been estimated. An experimental study of the prediction method confirmed its effectiveness. It has been shown that the observations of the hodoscope–monitor system increased the probability of correctly predicting geomagnetic storms compared to using each of the observations separately.

1 INTRODUCTION

Geomagnetic disturbances usually occur during extreme events in the heliosphere, which are the consequence of plasma formations in the form of solar coronal mass ejections (CMEs). Geomagnetic storms (GMS’s) are geomagnetic disturbances with an amplitude greater than a given one. GMS prediction is an urgent scientific problem, which to date has not been exhaustively solved.

Geomagnetic activity is characterized by various geomagnetic indices, among which the k-, kp-, and ap-indices are quite common (Menvielle et al., 2011). This article is based on Dst indices, which are frequently used in geomagnetism practice (Sugiura and Kamei, 1991). These indices are measured in nT and are determined by hourly averaging of the values of the meridional geomagnetic field components for magnetic observatories located at the Earth’s equator. For quiet states of the magnetosphere, Dst indices are mainly in the range –50…+20 nT; for GMS, Dst indices are in the range –150…–50 nT, and, in exceptional cases, they are outside this range.

This article uses information from the following sources:

1. Time series of matrix observations from the database (Real-time URAGAN, 2015) of the URAGAN Muon Hodoscope (MH) constructed at the MEPhI (Yashin et al., 2015). Here, we consider one possible simplification of the GMS prediction problem: the time series of matrix observations of the MH is converted into a scalar time series obtained by averaging the MH observation matrices. This time series is formed by the MH-observations proportional to the intensity of the muon fluxes detected by the URAGAN hodoscope. Matrix time series of MH-observations for GMS prediction will be the subject of research in the following publications.

2. Time series of scalar observations of the worldwide network of neutron monitors (NMs) (NMDB, 2021). The functions of the isotropic components of the NM observations derived from the global survey method are used here (Abunina et al., 2018). This time series is formed by the values of NM observations proportional to the intensity of neutron fluxes detected by neutron monitors.

3. Time series of scalar Dst indices at the website of the World Data Center of Geomagnetism (WDCG), Kyoto (World Data Center, 2021).

GMS predictions are determined by the type of used information sources, hardware, mathematical methods, and variants of the implemented applied problems. There are a number of services of different departmental and state affiliation that publish information on GMS predictions.

For the Russian Federation, the IZMIRAN’s Space Weather Prediction Center (Gaidash et al., 2016; Prediction Center, 2016; Gaidash et al., 2017) should be noted, which offers more than 20 types of products related to space weather and GMS prediction to consumers, including a 3-day forecast of 3-hour k- and kp-indices, an 8-day geomagnetic forecast based on average daily ap-indices and probabilities of the highest kp-indices per day, geomagnetic activity forecasts based on ap-indices for 55 days, etc.

The activity of the Laboratory of X-ray Astronomy of the Physical Institute in solving the problem of GMS prediction using solar radiography technologies should be noted. The website (Laboratory of X-ray, 2017) provides calculations of kp-indices for 3- and 27-day prediction intervals, which are carried out taking into account estimates of solar wind parameters.

It is necessary to mention the Space Weather Prediction Center of the National Oceanic and Atmospheric Administration (NOAA, United States) (NOAA/NWS, 2021), which deals with variants of GMS predictions. The NOAA website publishes data on kp- and ap-indices for 1–4-week prediction intervals. Images from the Extreme ultraviolet Imaging Telescope (EIT) and the Large Angle and Spectrometric Coronagraph (LASCO) (NOAA/NWS, 2021) are used for the GMS predictions in order to estimate solar corona heating characteristics and CME processes and to determine corona effect on the solar wind. LASCO images are used for the WSA-Enlil prediction model (NOAA/NWS, 2021), which has been in operation since 2011. This large-scale physical model of the heliosphere is designed to provide 1–4-day advance warning of changes in solar wind structures and CME motion directions. The Advanced Composition Explorer (ASE, United States) spacecraft (NOAA/NWS, 2021) launched in 1997 is used, which is permanently located at the Lagrange point at a distance of 1.44 million km from Earth on a straight line connecting the Earth and the Sun. This spacecraft, which can be taken as a patrol satellite, continuously records parameters of the solar wind and the interplanetary magnetic field and automatically transmits them to the Earth; the recorded parameters are published on the NOAA websites for further processing.

GMS prediction can be implemented on the basis of a number of methods, e.g., using probabilistic models or recognizing anomalies in time series. Thus, in (Dobrovol’skii et al., 2019; Chinkin et al., 2019), the use of special 2D variation functions of muon fluxes and indicator matrices was proposed for matrix MH observations.

Neural networks (NN) are widely used in solar-terrestrial physics in the problems of predicting (recognizing) extreme events in the heliosphere and magnetosphere (Barkhatov and Revunov, 2010). A number of publications related to NNs, Dst indices, and GMS’s differ in the variants of the used methods, software products, and information sources (databases). These circumstances introduce significant variation in the formulation of the problems.

In (Wu and Lundstedt, 1997; Stepanova and Pérez, 2000; Barkhatov et al., 2001; Dolenko et al., 2005; Pallocchia et al., 2006; Shirokii, 2015; Efitorov et al., 2018; Gruet et al., 2018) are materials concerning research into the GMS prediction using NN variants.

The aim of this article is to develop a GMS prediction method based on Dst index models, the combined use of MH and NM observations, extrapolation technology, and NNs. The proposed formulation in some details can be considered additional to the listed publications. The results obtained in the article on GMS prediction based on NN model estimates of Dst indices with extrapolation are intended for a number of scientific and technical applications, e.g., in case of possible sudden absence (omission) of Dst indices, GMS prediction can be implemented based on prebuilt models of Dst indices working only on MH and NM observations. The proposed GMS prediction can prove to be an alternative in case of possible patrol satellite failures.

2 ANALYSIS OF Dst INDICES AND MH, NM OBSERVATIONS

All variables used for NNs in this article were synchronized and sampled in hourly steps on a single UTC time scale. The time index k defined the sample points Tk, T = 1 h. The Dst indices Y_D(k) and NM observations Y_N(k) were implemented on the time interval January 1, 2002–December 31, 2018, and the MH observations Y_M(k), in the interval January 1, 2008–December 31, 2018. Initial and final indices for Y_D took values k₀ = 1, k_f = 149 016, for Y_M, indices k₀₁ = 52 285, k_f = 149 016, and for Y_N – k₀₂ = 1, k_f = 149 016.

Figure 1 contains examples of the plots of fragments of the initial variables Y_D = Y_D(k), Y_M = Y_M(k), and Y_N = Y_N(k) for the 6-month time interval (April 1, 2012–September 30, 2012) as a function of time Tk. GMS events were determined by the inequality Y_D(k) ≤ Y_D0. For the threshold Y_D0 = –50 nT, nine GMS events occurred in this interval. The analysis of Y_D(k) in Fig. 1 led us to conclude that the average duration of GMS was on the order of 2–2.5 days. The consideration of the initial variables made it possible to conclude that for them the average period of additive uninformative low-frequency trends to be filtered was about 60–75 days. In Fig. 1, the uninformative trends Y_DΦ0(k), Y_MΦ0(k), and Y_NΦ0(k) are shown as smooth lines.

Fig. 1.

Graphs of fragments of the initial MH and NM observations Y_M(k), Y_N(k), and Dst indices Y_D(k).

We can see from the graphs that variables Y_D(k) and Y_N(k) can be represented as a sum of informative low-frequency trends and high-frequency noise. The variable Y_M(k) can be represented as a sum of the informative low-frequency trend, interference components from daily fluctuations and high-frequency noise. The analysis of changes in informative low-frequency trends of the variables Y_M(k), Y_N(k) allowed us to conclude that these, in some cases, behave almost identically in time.

3 FORMULATION OF THE GEOMAGNETIC PREDICTION PROBLEM

In the practice of analyzing geomagnetic observations, it is sometimes customary to make a conclusion about the GMS prediction based on criteria that are formed on the basis of different variants of geomagnetic indices. A criterion based on comparing Dst indices with a given threshold is quite common and, to a certain extent, reliable in GMS prediction. However, sometimes the direct use of Dst indices for prediction can be problematic due to the fact that there can be none of them at the current and subsequent points in time, for various reasons.

There is a well-defined functional relationship between the Dst indices and the values of MH, MN observations that can be represented by a model scheme based on (Borog, 2008). The Dst index functions act as GMS indicators and their values are determined by the intensity of the interaction of the CME formations with the Earth’s magnetosphere. The travel time of CMEs from the Sun to the Earth and the time of the start of the GMS, if it is counted from the time of emission of CMEs, is usually 1.5–2 days. Simultaneously, CMEs modulate intensities of relativistic proton fluxes approaching the Earth; their travel time from the Sun to the Earth is about 8 min. Proton fluxes enter into nuclear reactions with atoms of substances of the upper atmosphere, as a result of these reactions, muon and neutron fluxes are formed, the intensity of which is detected by the URAGAN hodoscope and neutron monitors. Modulations of proton flux intensities under certain conditions cause modulations of intensities of generated muon and neutron fluxes, i.e., values of MH, MN observations. The modulations of muon and neutron fluxes occur significantly earlier than the GMS; this fact is the basis for the proposed method for predicting GMS.

We will assume that:

— The current considered time is given, to which corresponds the time index k which satisfies the inequalities k_f0 + 1 ≤ k ≤ k_f. The preceding times are those to which the time indices k_f0 + 1, …, k – 1 correspond, where k_f0 is the given time index; the initial index k₀, the final index k_f, and the given index k_f0 are related by the inequalities k₀ < k_f0 <k_f.

— In the interval k₀₁ ≤ k ≤ k_f0, a time series of MH observations and a time series of Dst indices are implemented; in the interval k₀₂ ≤ k ≤ k_f0, a time series of NM observations and a time series of Dst index values.

— In the interval k_f0 + 1, k – 1, k only the time series of MH and NM observations are implemented.

We will consider GMS prediction with respect to the current times with indices k on k_e indices forward.

The aim of the work is to create a system of model estimates of Dst indices in the interval k_f0 + 1 ≤ k ≤ k_f based on the implemented times series of Dst indices and MH, NM-observations in the intervals k₀₁ ≤ k ≤ k_f0, k₀₂ ≤ k ≤ k_f0 and use them for GMS prediction. For this problem, we apply a full-connected feed-forward NN and extrapolation technique in forming model estimates of Dst indices.

4 THE GENERAL PLAN FOR SOLVING THE GMS PREDICTION PROBLEM AND EXTRAPOLATION FORMULA

4.1. The general plan for solving the GMS prediction problem taking the formulation in item 3 of the formulation into account, is divided into five items, including

(1) algorithms for preliminary digital processing of initial Dst indices and MH, NM-observations in order to isolate significant informative components in them;

(2) algorithms for training and validation of NNs;

(3) algorithm for extrapolation of MH, NM-observations;

(4) a calculation-test algorithm for model estimates of Dst indices with NN-based extrapolation using extrapolated MH, NM-observations;

(5) a decision rule algorithm for GMS prediction.

Figure 2 presents a diagram of computational operations that explains the solution to the GMS prediction problem. The computational operations are subdivided into digital preprocessing block 1, blocks 2.1, 3.1 of MH, NM NN training, extrapolation blocks 4, 5, blocks 2.2, 3.2 for computing model estimates of Dst indices for MH, NM, and block 6 for decision making for GMS prediction.

Fig. 2.

A diagram of computational operations for solving the GMS prediction problem.

Digital preprocessing algorithms for the initial Dst indices Y_D = Y_D(k) and MH, NM observations Y_M = Y_M(k), Y_N = Y_N(k) implement their filtering (Filter Design, 2021) in order to remove high frequency noise and daily variations as well as to remove low-frequency uninformative trends and scale in order to ensure commensurability of the initial variables, which is necessary for efficient NN operation.

The results of the digital preprocessing are denoted as Y_DC = Y_DC(k), Y_MC = Y_MC(k), and Y_NC = Y_NC(k). The scaling operation brings the preprocessing results to a single range: Y_min ≤ Y_MC(k) ≤ Y_max, Y_min ≤ Y_NC(k) ≤ Y_max, Y_min ≤ Y_DC(k) ≤ Y_max, Y_min = 0.7, Y_max = 1.3. Figure 3 shows example graphs of fragments of the variables Y_MC = Y_MC(k), Y_NC = Y_NC(k), and Y_DC = Y_DC(k) obtained from the numerical preprocessing on the monthly time interval April 1, 2012–April 30, 2012.

Fig. 3.

Graphs of the variables Y_MP(k), Y_NP(k), and Y_DP(k) derived from the numerical preprocessing on the monthly interval.

The NN training stage for MH variables Y_MC1(k) is implemented in the interval with indices k₀₁ + dk₀₁ + 1 ≤ k ≤ k_f0, for NM variables Y_NC1(k), on indices k₀₂ +dk₀₂ + 1 ≤ k ≤ k_f0, variables Y_DC(k) are implemented in these intervals. The values of dk₀₁, dk₀₂ are set. The sequences of ∆k-dimension vectors moving in unit steps and formed by these variables are fed to the NN input. The validation stage of NN models after training is performed for the indices k₀₁ ≤ k ≤ k₀₁ + dk₀₁ for MH variables and for the indices k₀₂ ≤ k ≤ k₀₂ + dk₀₂ for NM variables. As a result of the training and validation stages, MH, NM NN models for Dst indices are created. The testing stage is dedicated to computing model Dst indices on the basis of NN models in the interval with the indices k_f0 + 1 ≤ k ≤ k_f. For testing, an extrapolation algorithm over the variables Y_MC2(k) and Y_NC2(k) is implemented in order to further use the extrapolations in the generated NN models.

Figure 4 shows examples of the plots of variables Y_MC = Y_MC2(k), Y_NC = Y_NC2(k) obtained by filtering and scaling on the 4-day interval of April 1, 2012—April 4, 2012, and prepared for extrapolation. It can be concluded that the variables Y_MC2(k), Y_NC2(k) are well suited for extrapolation: it can be seen that their successful extrapolation based on parabolic functions is quite acceptable at least for k_e = 5–10 steps ahead.

Fig. 4.

Graphs of variables Y_MP2(k), Y_NP2(k) obtained from filtering and scaling operations on the 4-day interval (Apr. 1–Apr. 4, 2012).

4.2. Extrapolation formulas are implemented based on approximation parabolic models with respect to the current index k; the extrapolation parameter k_e, the number of extrapolation steps k + 1, k + 2, …, k + k_e, is set. The number of indices in the interval k, k – 1, …, k – k_a + 1, in which the approximation model is supposed to be built, is set. Parabolic approximation functions with parameters $c_{1}^{T} = ({{c}_{{01}}},{{c}_{{11}}},{{c}_{{21}}})$, $c_{2}^{T} = ({{c}_{{02}}},{{c}_{{12}}},{{c}_{{22}}})$ and approximation functionals are formed

$$\begin{gathered} F({{c}_{1}},{{Y}_{{MC2}}},{{k}_{a}},k) \\ = \sum\limits_{s{\kern 1pt} = {\kern 1pt} k{\kern 1pt} - {\kern 1pt} {{k}_{a}}{\kern 1pt} + {\kern 1pt} 1}^k {({{Y}_{{MC2}}}(s) - {{c}_{{01}}} - {{c}_{{11}}}s - {{c}_{{21}}}{{s}^{2}}} {{)}^{2}}, \\ F({{c}_{2}},{{Y}_{{NC2}}},{{k}_{a}},k) \\ = \sum\limits_{s{\kern 1pt} = {\kern 1pt} k{\kern 1pt} - {\kern 1pt} {{k}_{a}}{\kern 1pt} + {\kern 1pt} 1}^k {({{Y}_{{NC2}}}(s) - {{c}_{{02}}} - {{c}_{{12}}}s - {{c}_{{22}}}{{s}^{2}}} {{)}^{2}}. \\ \end{gathered} $$

Optimal parameters of the approximation models are determined by minimizing the introduced functionals

$$\begin{gathered} c_{1}^{^\circ } = \arg \{ \mathop {\min }\limits_{{{c}_{1}}} F({{c}_{1}},{{Y}_{{MP2}}},{{k}_{a}},k)\} , \\ c_{2}^{^\circ } = \arg \{ \mathop {\min }\limits_{{{c}_{2}}} F({{c}_{2}},{{Y}_{{NP2}}},{{k}_{a}},k)\} . \\ \end{gathered} $$

The extrapolated models Ȳ_MC2 = Ȳ_MC2(k, k_e), Ȳ_NC2 = Ȳ_NC2(k, k_e) are formed on ∆k-dimension intervals moving in unit steps. For ${{k}_{e}}$ indices $k + 1,\,\,k + 2,...,\,\,k + {{k}_{e}}$, the extrapolated models are represented by the parabolic approximation functions

$$\begin{gathered} {{{\bar {Y}}}_{{MC2}}}(k,{{k}_{e}}) = c_{{01}}^{^\circ } + c_{{11}}^{^\circ }k + c_{{21}}^{^\circ }{{k}^{2}}, \\ {{{\bar {Y}}}_{{NC2}}}(k,{{k}_{e}}) = c_{{02}}^{^\circ } + c_{{12}}^{^\circ }k + c_{{22}}^{^\circ }{{k}^{2}}. \\ \end{gathered} $$

(1)

For $\Delta k - {{k}_{e}}$ indices $k + {{k}_{e}} - \Delta k + 1,$ $k + {{k}_{e}} - \Delta k + 2,...,k$, the extrapolated models are represented by formulas

$$\begin{gathered} \bar {Y}_{{MC2}}^{{}}(k,{{k}_{e}}) = {{{\bar {Y}}}_{{MC2}}}(k), \\ \bar {Y}_{{NC2}}^{{}}(k,{{k}_{e}}) = {{{\bar {Y}}}_{{NC2}}}(k). \\ \end{gathered} $$

(2)

Moving extrapolated model estimates of Dst indices are computed using NN models for the extrapolated MH, NM observations Ȳ_DM = Ȳ_MC2(k, k_e), Ȳ_DN = Ȳ_NC2(k, k_e) in the interval k_1T ≤ k ≤ k_2T. In order to calculate k_1T , k_2T, the following inequalities derived from (1)–(2) are written

$${{k}_{{f0}}} + 1 \leqslant k + {{k}_{e}} - \Delta k + 1,\,\,\,\,k + {{k}_{e}} \leqslant {{k}_{f}}.$$

The solution of these inequalities makes it possible to determine the boundary indices

$${{k}_{{1T}}} = {{k}_{{f0}}} - {{k}_{e}} + \Delta k,\,\,\,\,{{k}_{{2T}}} = {{k}_{f}} - {{k}_{e}}.$$

(3)

The decision rule algorithm for predicting GMS’s is based on the computed moving extrapolated model estimates of Dst indices Ȳ_DM, Ȳ_DN and comparing them to a given threshold Y_D0. GMS prediction decision making is implemented in the interval k_1T ≤ k ≤ k_2T (3).

5 NN ARCHITECTURE

The results obtained above are based on a fully-connected feed-forward NN generated using a generic software product (Deep Learning, 2021). The use of NN of this type is due to the fact that in this case the raw data and observations are scalar time series.

The architecture of the used NN is presented in Fig. 5. Four connected layers (CLs) with input variables Y_DC, Y_MC1, Y_NC1 were implemented, on the basis of which the ∆k-dimension input vector sequences were formed. Activation functions f(x) = 1, x > 0, f(x) = 0, x ≤ 0 were used. The outputs from the four layers CL1–CL4 were fed to the summing fully-connected layer (FCL). The NN output at the training stage is the generated NN models.

Fig. 5.

NN architecture.

The indices of the intervals of initial variables took numerical values in accordance with item 2 and were assigned dk₀₁ = 12 000, dk₀₂ = 15 000. The index k_f0 = 131 810 corresponding to the date of January 1, 2017, was assigned, which defined the 8-year training interval (January 1, 2008–December 31, 2016) for the MH and the 14-year interval (January 1, 2002–December 31, 2016) for the NM. The 2-year interval (January 1, 2017–December 31, 2018) was allotted to compute model estimates of Dst indices based on MH, NM observations using NN model structures.

6 THE GMS PREDICTION DECISION RULE

We reduce the GMS prediction method to a classification procedure (Bishop, 2006; Fomin, 2010) based on comparing extrapolated model estimates of Dst indices to a given threshold for the current index k given the extrapolation steps k_e.

We create a decision rule for GMS prediction decisions based on the joint use of MH, NM model estimates of Dst indices, which is that if at least one or both conditions for the current index k are met

$$\begin{gathered} \bar {Y}_{{DM}}^{{}}(k,{{k}_{e}}) \leqslant {{Y}_{{D0}}}\,\,\,\,{{{\text{and}}} \mathord{\left/ {\vphantom {{{\text{and}}} {{\text{or}}}}} \right. \kern-0em} {{\text{or}}}}\,\,\,\,\bar {Y}_{{DN}}^{{}}(k,{{k}_{e}}) \leqslant {{Y}_{{D0}}}, \\ {{k}_{{1T}}} \leqslant k \leqslant {{k}_{{2T}}}, \\ \end{gathered} $$

(4)

then we will make a GMS prediction decision for the index k + k_e; in other cases, we will make the opposite decision.

Let us provide clarifications on the implementation of the prediction. We assume that the possible GMS was implemented on a time interval with indices that satisfied the inequalities k_GS1 ≤ k ≤ k_GS2. We will still assume that the prediction is performed for a point in time with the index k for k_e indices forward. We consider two possible cases: 1. When the time k of making a prediction is outside the interval with the GMS; 2. When the time k of making a prediction is inside the interval with the GMS. In that case, even if both prediction variants are implemented, which is equivalent to the inequality k_GS1 ≤ k_e ≤ k_GS2, the variants are significantly different. For the first variant, the implemented variant is quite complete, because the prediction was implemented in advance with respect to the GMS. In the second variant, the prediction took place when the GMS had already occurred. Figures 6a and 6b show the predictions for the first and second variants. The following inequalities correspond to the first and second variants of the GMS prediction

$$k \leqslant {{k}_{{GS1}}} \leqslant {{k}_{e}} \leqslant {{k}_{{GS2}}},\,\,\,\,{{k}_{{GS1}}} \leqslant k < {{k}_{e}} \leqslant {{k}_{{GS2}}},$$

Fig. 6.

First and second variants of GMS predictions.

7 CALCULATING ESTIMATES OF THE PROBABILITIES OF CORRECT AND FALSE GMS PREDICTIONS

GMS prediction is usually accompanied by errors, such as incorrect forecasting and generation of false predictions. These errors depend on the probability characteristics of the variables Ȳ_DM, Ȳ_DN. Let us use the results of (Merkov, 2014; Mikhailov and Voitishek, 2006) for their approximate calculation. Let us form estimates of prediction error probabilities, in which we use the initial Dst indices Y_D(k), model estimates of predicted Dst indices Ȳ_DM(k, k_e), Ȳ_DN(k, k_e), which are random. We implement calculations for a number of variants of extrapolation steps ${{k}_{e}}$ and use decision rule (4).

We fix the threshold Y_D0 and consider the time with the index k + k_e in which GMS takes place if the inequality Y_D(k + k_e) ≤ Y_D0 is satisfied. We calculate the number N_GS of the states with the GMS which are determined by the fulfillment of this inequality in the interval k_1T ≤ k ≤ k_2T using the following sum

$${{N}_{{GS}}} = \sum\limits_{k{\kern 1pt} = {\kern 1pt} {{k}_{{1T}}}}^{{{k}_{{2T}}}} {{\text{sgn}}({{Y}_{{D0}}} - {{Y}_{D}}(k + {{k}_{e}}))} ,$$

(5)

where sgn x = 1, x ≥ 0, sgnx = 0, x < 0. Let us determine N_M,GS, which is the number of correct GMS predictions using Ȳ_DM(k, k_e) based on (4), and find β_M, which is an estimate of the probability of correct forecasting

$$\begin{gathered} {{N}_{{M,GS}}} = \sum\limits_{k{\kern 1pt} = {\kern 1pt} {{k}_{{1T}}}}^{{{k}_{{2T`}}}} {{\text{sgn}}({{Y}_{{D0}}} - {{Y}_{D}}(k + {{k}_{e}}))} \\ \times \,\,{\text{sgn}}({{Y}_{{D0}}} - \bar {Y}_{{DM}}^{{}}(k,{{k}_{e}})),\,\,\,\,\beta _{M}^{^\circ } = \frac{{{{N}_{{M,GS}}}}}{{{{N}_{{GS}}}}}. \\ \end{gathered} $$

(6)

We count the number N_N,GS of correct GMS predictions using Ȳ_DN(k, k_e) using (4) and determine an estimate of the probability of correct forecasting β_N

$$\begin{gathered} {{N}_{{N,GS}}} = \sum\limits_{k{\kern 1pt} = {\kern 1pt} {{k}_{{1T}}}}^{{{k}_{{2T}}}} {{\text{sgn}}({{Y}_{{D0}}} - {{Y}_{D}}(k + {{k}_{e}}))} \hfill \\ \times \,\,{\text{sgn}}({{Y}_{{D0}}} - \bar {Y}_{{DN}}^{{}}(k,{{k}_{e}})),\,\,\,\,\beta _{N}^{^\circ } = \frac{{{{N}_{{N,GS}}}}}{{{{N}_{{GS}}}}}. \hfill \\ \end{gathered} $$

(7)

The estimate of the probability of correct GMS predictions β_MN when Ȳ_DM(k, k_e), Ȳ_DN(k, k_e) and (4) are used together is found as follows

$$\begin{gathered} {{N}_{{MN,GS}}} = \sum\limits_{k{\kern 1pt} = {\kern 1pt} {{k}_{{1T}}}}^{{{k}_{{2T}}}} {{\text{sgn}}[{{Y}_{{D0}}} - {{Y}_{D}}(k + {{k}_{e}}))} \\ \times \,\,{\text{sgn}}({\text{sgn}}({{Y}_{{D0}}} - \bar {Y}_{{DN}}^{{}}(k,{{k}_{e}})) \\ + \,\,{\text{sgn}}({{Y}_{{D0}}} - \bar {Y}_{{DM}}^{{}}(k,{{k}_{e}}))],\,\,\,\,\beta _{{MN}}^{^\circ } = \frac{{{{N}_{{MN,GS}}}}}{{{{N}_{{GS}}}}}. \\ \end{gathered} $$

(8)

The numbers N_0GS, N_M,0GS, N_N,0GS, N_MN,0GS and the probabilities of false predictions of the GMS $\alpha _{M}^{^\circ }$, $\alpha _{N}^{^\circ }$, $\alpha _{{MN}}^{^\circ }$ will be calculated by formulas similar to (5)–(8). Calculations by formulas (5)–(8) are implemented for the set of extrapolation steps k_e. We determine the probabilities of omitting predictions for MH, NM, and MH|NM by the differences $1 - \beta _{M}^{^\circ }$, $1 - \beta _{N}^{^\circ }$, and $1 - \beta _{{MN}}^{^\circ }$.

8 EXPERIMENTAL STUDY OF THE GMS PREDICTION METHOD

8.1 Estimating the Probability of Correct and False GMS Predictions Depending on the Extrapolation Step k_e

In an interval with indices k_1T ≤ k ≤ k_2T, the time series Y_D(k) of Dst indices from the database (World Data Center, 2021) was examined. For the given integer parameters k_e, extrapolated model estimates Ȳ_DM(k, k_e), Ȳ_DN(k, k_e) were calculated and compared with the Y_D0 threshold, and the probabilities of correct and false GMS predictions were determined using formulas (5)–(8) as a function of the parameter k_e for the range ${{k}_{{e,mn}}} \leqslant {{k}_{e}} \leqslant {{k}_{{e,mx}}}$, ${{k}_{{e,mn}}} = 0,$ ${{k}_{{e,mx}}} = 9$ for Y_D0 = –50 nT.

Figures 7a and 7b present graphs of the results of calculations of the estimates of probabilities of correct forecasting $\beta _{M}^{^\circ }$(MH), $\beta _{N}^{^\circ }$(NM), $\beta _{{MN}}^{^\circ }$ (MH|NM) and probabilities of false prediction $\alpha _{M}^{^\circ }$(MH), $\alpha _{N}^{^\circ }$(NM), $\alpha _{{MN}}^{^\circ }$ (MH|NM) depending on k_e. From the presented graphs with the results of the calculations, it can be seen that the combined use of MH and NM observations increased the efficiency of GMS prediction. From Fig. 7a, we can conclude that for the prediction parameter k_e = 5, the probability of correct joint GMS prediction took the maximum value and was $\beta _{{MN}}^{^\circ } \approx 0.65$, which is greater than the corresponding value $\beta _{N}^{^\circ } \approx 0.57$ by 13–15%. The probability of false joint prediction, according to Fig. 7b, did not exceed $\alpha _{{MN}}^{^\circ } \approx 0.06$.

Fig. 7.

(a) Estimates $\beta _{M}^{^\circ }$, $\beta _{N}^{^\circ }$, $\beta _{{MN}}^{^\circ }$ of probabilities of correct GMS forecasting. (b) Estimates $\alpha _{M}^{^\circ }$, $\alpha _{N}^{^\circ }$, $\alpha _{{MN}}^{^\circ }$ of probabilities of false GMS forecasting.

8.2 Calculating GMS Predictions for a Given Time Interval

An initial time interval with indices k_1T ≤ k ≤ k_2T was considered, on which model estimates of Dst indices were computed and the GMS prediction problem was solved. In this interval, the time series Y_D(k) was formed and extrapolated model estimates Ȳ_DM(k, k_e), Ȳ_DN(k, k_e) were calculated for the given values of the extrapolation parameter k_e.

The control interval with indices k_1C, k_2C that satisfied the inequalities k_1T < k_1C ≤ k ≤ k_2C < k_2T was considered. Sequences of intervals k_GS1 ≤ k ≤ k_GS2 in which GMS occurred were determined by checking inequalities for moving k and the given value of the extrapolation parameter k_e

$$Y_{D}^{{}}(k + {{k}_{e}}) \leqslant {{Y}_{{D0}}},\,\,\,\,{{k}_{{1C}}} \leqslant k \leqslant {{k}_{{2C}}}.$$

Sequences of intervals k_M1 ≤ k ≤ k_M2 and k_N1 ≤ k ≤ k_N2, in which GMS predictions were made according to decision rule (4), were determined by checking the following inequalities

$$\bar {Y}_{{DM}}^{{}}(k,{{k}_{e}}) \leqslant {{Y}_{{D0}}},\,\,\,\,\bar {Y}_{{DN}}^{{}}(k,{{k}_{e}}) \leqslant {{Y}_{{D0}}}.$$

The GMS predictions took into account the mutual arrangement of intervals (k_GS1, k_GS2) and (k_M1, k_M2), (k_N1, k_N2). Time differences Tdk_M, Tdk_N for separate observations and differences Tdk_MN, where k_MN = max(k_M1, k_N1) for joint MH, NM observations were introduced to assess the effectiveness of GMS predictions

$$\begin{gathered} Td{{k}_{M}} = T{{k}_{{GS}}}_{1} - T{{k}_{M}}_{1},\,\,\,\,~Td{{k}_{N}} = T{{k}_{{GS}}}_{1} - T{{k}_{N}}_{1}, \\ Td{{k}_{{MN}}} = T{{k}_{{GS}}}_{1} - T{{k}_{{MN}}}. \\ \end{gathered} $$

Obviously, GMS predictions make sense when the inequalities Tdk_M > 0, Tdk_N > 0, and Tdk_MN >0 were satisfied in separate and joint observations.

Depending on the positions (k_GS1, k_GS2) and (k_M1, k_M2), (k_N1, k_N2), there are obviously possible implementations of correct forecasting (cf), miss of correct forecasting (mcf), and false forecasting (ff) of GMS’s. The larger the values of the introduced differences, the more advance the GMS prediction is. Negative values for dk_M, dk_N, dk_MN can be assumed to correspond to missing forecasts, mfc.

For the first example, a reference time interval of 5 months (May 1, 2017–September 30, 2017) was considered; Y_D = Y_D(k) and Ȳ_DM = Ȳ_DM(k, k_e), Ȳ_DN = Ȳ_DN(k, k_e) with k_e = 5 were used. Figure 8a shows graphs for these variables. The dotted line indicates the assigned threshold Y_D0 = –52.5 nT. The results of the prediction calculations are presented in Table 1; for space considerations, the year 2017 is omitted from the dates; the columns contain the calendar variables, such as month, day, and hour, derived from the conversion of the k indices. Columns 2, 3 contain times of the intervals (Tk_GS1, Tk_GS2) of start and end of the events due to Y_D(k) reductions, which can turn out to be GMS events; the first column defined numbers N₀ of intervals with GMS events. Columns 4, 5 and 7, 8 contain interval times (Tk_M1, Tk_M2), (Tk_N1, Tk_N2) with 1-hour accuracy. A total of 5 GMS events took place. From the contents of column 10 with prediction time Tdk_e = max(Tdk_M, Tdk_N), we can conclude that correct GMS predictions were made in four events, N_cf = 4: for no. 1, prediction was made in 3 h, no. 2, in 4 h, no. 3, in 2 h, and 4, in 5 h. For event no. 5, there was a miss of correct forecasting N_mf = 1. There were 0 false forecasts, N_ff = 0.

Fig. 8.

(a) Graphs of the variables Y_D(k), Ȳ_DM(k, k_e), and Ȳ_DN(k, k_e), k_e = 5. (b) Graphs of the variables Y_D(k), Ȳ_DM(k, k_e), and Ȳ_DN(k, k_e), k_e = 7.

Table 1.   Extrapolation k_e = 5, 2017

For the second example, the 3-month reference interval (August 1, 2015–October 31, 2015) was considered and Y_D = Y_D(k) and Ȳ_DM = Ȳ_DM(k, k_e), Ȳ_DN = Ȳ_DN(k, k_e) with k_e = 7 were used; the threshold Y_D0 = –52.5 nT was assigned. Figure 8b shows the corresponding graphs of the variables with the dotted line indicating the threshold. The results of the calculations are presented in Table 2. Three GMS-events took place during the 3-month interval under consideration. We can conclude from the contents of column 10 that correct GMS predictions were made in two cases, N_cf = 2: for no. 1, prediction was made in 1 h and for no. 2, in 7 h. There was one false forecast, N_ff = 1, and one miss of correct forecasting, N_mf = 1.

Table 2.   Extrapolation k_e = 7, 2015

In order to approximate the effectiveness of the GMS prediction method, we introduced coefficients of correct and false forecasting ε_cf and ε_ff, whose quite obvious physical meaning is derived from the following formulas

$${{\varepsilon }_{{{\text{cf}}}}} = {{{{N}_{{{\text{cf}}}}}} \mathord{\left/ {\vphantom {{{{N}_{{{\text{cf}}}}}} {({{N}_{{{\text{cf}}}}} + {{N}_{{{\text{mf}}}}})}}} \right. \kern-0em} {({{N}_{{{\text{cf}}}}} + {{N}_{{{\text{mf}}}}})}},\,\,\,\,{{\varepsilon }_{{{\text{ff}}}}} = {{{{N}_{{{\text{ff}}}}}} \mathord{\left/ {\vphantom {{{{N}_{{{\text{ff}}}}}} {{{N}_{{{\text{cf}}}}}}}} \right. \kern-0em} {{{N}_{{{\text{cf}}}}}}}.$$

The calculation of the coefficients on the basis of Tables 1 and 2 allowed us to establish their average values ${{\bar {\varepsilon }}_{{{\text{cf}}}}} \approx 73\%$, ${{\bar {\varepsilon }}_{{{\text{mf}}}}} \approx 25\%$, which can be taken as first approximations of estimates of the effectiveness of the proposed prediction method.

9 CONCLUSIONS

1. The proposed method for predicting geomagnetic storms (GMS) on the basis of joint observations of the URAGAN muon hodoscope and neutron monitors using a fully-connected feed-forward NN and extrapolated model estimates of Dst indices is operational.

2. The study of the proposed method of GMS prediction on observations of the muon hodoscope, neutron monitors, and Dst indices obtained for 2008–2018 and 2002–2018 showed its effectiveness.

3. The conducted calculations of the combined use of MN and NM observations showed that for the optimal prediction parameter k_e = 5, the estimated probability of correct GMS prediction is $\beta _{{MN}}^{^\circ } \approx 0.65$, which is greater than the corresponding value $\beta _{N}^{^\circ } \approx 0.57$ by 13–15%; the probability of false forecasting does not exceed $\alpha _{{MN}}^{^\circ } \approx 0.06$.

4. The consideration of examples of experimental data on 5-month intervals showed that it is possible to implement the coefficients of correct and false forecasting ≈73%, ≈25% using the proposed method, which indicates acceptable capabilities of the proposed method.

5. The proposed GMS prediction method has a lot of room for improvement, in particular, in terms of further optimization of the NN parameter settings in order to increase correct prediction probabilities and decrease false prediction probabilities.

6. The proposed GMS prediction method has a favorable outlook for its use in applied geophysical problems.