Introduction

Natural events are interpreted by mathematicians via data. Data are converted to variables, which later provide modeling opportunities to researchers. Depending on physical conditions, variables may be separated into dependent and independent ones. When the solar wind parameters (SWp) are considered to be independent variables, the zonal geomagnetic indices (ZGi) are considered to be dependent variables. Geomagnetic activities (Akasofu 1964; Kamide et al. 1998; Rathore et al. 2014) are also such natural events. This paper tries to understand and interpret the 04 January storm (Dst =  − 71 nT), the 07 January storm (Dst =  − 99 nT), the 17 February storm (Dst =  − 64 nT), and the 24 February storm (Dst =  − 56 nT) moderate geomagnetic activities built on the SWp and the ZGi based on the cause (Bz, E, P, N, v, T)–effect (Dst, ap, AE) association. The storms are examined by an artificial neural network model (ANNm). In the ANNm, Levenberg–Marquardt (trainlm) is selected as the backpropagation (Rumelhart et al., 1986; Conway 1998) algorithm, and thirty-five neural numbers are utilized (Williams and Zipser 1989; Elman 1990; Gardner and Dorling 1998; Fausett 1994).

The sun is a plasma-dense energy and power source that produces durable magnetic waves. These magnetic waves are conveyed out to the interplanetary medium by the SWs. The SW has dense particles induced by the energy spreading from the sun (Parker 1958). Being swallowed by coronal mass ejection (CME) cloud of the earth’s magnetosphere-ionosphere and the B magnetic field’s Bz component orienting from the positive northward to the negative southward are invaluable for the geomagnetic storm. Shortly, the fast alteration in the magnetosphere of the earth governed by the SW scattering out of the sun is named a geomagnetic storm. CME is the burst of magnetically charged plasma into the interplanetary medium with high speeds. The dense magnetic field is burst in the solar corona (Lin and Ni 2018) by direct CMEs in a loop through (magnetic) reconnections (Kamide et al. 1998; Gonzalez et al. 1989; Borovsky 2012; Fu et al. 2013; 2017). Polarized magnetosonic waves and CME straightly turn out the SWp (Gonzalez et al. 1999). One of the governors’ SWp is the magnetic field Bz component. The negative Bz magnetic field replaces from the northward to the southward. This orientation and decreasing cause of the terrestrial magnetic field by causing disturbances with magnetic reconnections. Turbulences and fluctuations in the magnetic field controlled by the storm disturbance storm time (Dst) index (Dungey. 1961; Sugiura. 1964) from ZGi can be called storms after the minimum peak Bz. The course of the storm has three phases: the initial phase (sudden commencement), the main phase, and the recovery phase. In the initial phase, in which the storm begins, the Dst index decreases from positive values to negative values supporting the magnetic field. In the main phase, the Dst index indicates negative values. After the negative values show the minimum values, the recovery phase begins. Lastly, the geomagnetic storm finishes with the recovery phase when the fluctuation in the magnetic field ends and the Dst index indicates the initial values. In moderate storms, following the Bz parameter of the Dst index with a delay (Burton 1975) of 5–6 h is the response of the ring current to the SW. To better understand a geomagnetic storm (Mayaud 1980; Eroglu 2018; Eroglu 2019; Inyurt and Sekertekin 2019; Inyurt 2020; Koklu 2020; Eroglu 2020), the author considers the models between the SWp and the ZGi. This paper utilizes hourly versions of the SWp and the ZGi.

This essay tries to investigate 04 January, 07 January, 17 February, and 24 February (2015) storms based on their physical requirements via an ANN model by meticulously governing the causality principle (Eroglu 2011, 2021; Eroglu et al. 2012a; 2012b). The ANN is a precious method; it may be utilized as an operative approach for estimation in scientific disciplines (Elman 1990; Gleisner et al. 1996; Boberg et al. 2000; Gleisner and Lundstedt 2001; Karayiannis and Venetsanopoulos 2013). The use of the ANNm application has been common in recent decades, owing to its distinctive features such as learning capability, adaptation to changes, and ease of tools.

Investigations of earth–sun interaction, geomagnetic activities (Gleisner et al. 1996; Boberg et al. 2000; Gleisner and Lundstedt 2001), weather estimations, etc. with ANNs (Gardner and Dorling 1998; Lundstedt et al. 2005; Pallocchia et al. 2006) are effective in terms of space costs-times. ANN models founded by inferring geomagnetic storms’ CMEs (Uwamahoro et al., 2012; Singh and Singh 2016) give reasonable results with a high forecasting ratio of storms. Long-term time data ANN models involving SWp not only forecast the Dst index (Lundstedt, 1992; Lundstedt and Wintoft 1994; Gleisner et al. 1996; Fenrich and Luhman 1998; O’Brien and McPherron 2000; Lundstedt et al. 2002; Bala 2012; Uwamahoro 2012; Singh and Singh 2016) but also estimate geomagnetic activity phases (Gleisner et al., 1996). One may see the difficulty of estimating the Dst index without high-capacity computers with an 84% accuracy. In the 1990s, such an amazing estimation guided the new ANN models (Gleisner et al., 1996). Another ZG indicator Kp index is estimated by the SWp parameters as the proton density (N), the flux velocity (v), and the Bz magnetic field. The Kp index ANN model displays a remarkable estimation ratio (Boberg et al. 2000; Lundstedt 1992). The ap index significant model also utilizes the SWp as proton density and magnetic field (Altadill et al. 2001). The auroral electrojet index AE model uses the proton density (N), the flux density (v), and the magnetic field (B). One may realize from the ANN model estimation more than 70% of the monitored the AE index (Gleisner and Lundstedt, 2001).

This paper investigates different moderate storms separately. Governing the ANN model by the SWp, the ZGi shapes the investigation. In the examination of the first four moderate storms, the correlation matrix specifies the binary relation of the variables, and the dendrogram illustrates the hierarchical cluster of the data. The events that are visualized with graphics are exhibited to the reader. The SW scattering time from bow shock to magnetosphere-ionosphere is not considered in the study when the ZGi from ground stations is utilized.

After the literature review in the “Introduction,” yearly ZGi values appearances are seen in “Data” besides variables 5-day scattering. In “Modeling,” the ANN model and some properties of data are discussed. The paper is completed with a discussion in “Conclusion.”

Data

SPEDAS is used in this essay. Before launching the first four moderate storms of the year 2015 (day by day), one needs a glance at the annual ZGi values. One can find the observed and estimated values of the year 2015 in Fig. 1 besides their absolute errors. The estimated ZGi with the average error variance is as follows:

Fig. 1
figure 1

Annual observed-estimated Dst (nT) index, ap (nT) index, and AE (nT) index for the year 2015 and their errors

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Error

Variance

Dst (nT)

0.399 (2.77%)

0.953

ap (nT)

0.271 (2.20%)

0.272

AE (nT)

0.303 (0.14%)

0.243

Geomagnetic storms are classified according to the strength of the Dst index (Loewe and Prölss 1997). A Dst index between − 50 nT and − 30 nT indicates a weak storm, between − 100 nT and − 50 nT indicates a moderate storm, and between − 200 nT and − 100 nT indicates a strong (intense, severe) storm. Figure 2 is a 120-h view of some data related to the activities. The storm day is placed in the middle of the 5 days.

Fig. 2
figure 2figure 2

Data from top to bottom the Dst (nT) index, the Bz magnetic field (nT), the E electric field (mV/m), the dynamic pressure P (nPa), the proton density N (1/cm3), the flow speed v (km/s), and the ap (nT) index for 2015 January 02–06 (upper left side), January 05–09 (upper right side), February 15–19 (bottom left side), and February 22–26 (bottom right side)

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It would be appropriate to briefly discuss Fig. 2.

04 January 2015 storm: On 04 January at 14:00 UT, when the Bz magnetic field is at its minimum value of − 8.8 nT, the Dst orients to − 56 nT and the electric field E shows its maximum value of 3.53 mV/m. Simultaneously, the proton density N indicates to 6.9/cm3, the plasma flow speed v shows 401 km/s, and the pressure P reaches 2.63 nPa. As a response after 2 h, at 16:00 UT, the Dst index and the auroral electrojet AE index hit the minimum–maximum peak values − 71 nT and 866 nT, respectively.

On 03 January at 00:00 UT, when the first CME bursts out to the interplanetary medium, instant instability of the dynamic pressure P reaches its peak value of 8.36 nPa with the proton density N, indicating the maximum value of 33.0 1/cm3 and the flow speed v minimum value of 379 km/s.

07 January 2015 storm: After the diminishing in the flow speed v, the first CME hits with the sudden commencement (acceleration) in the dynamic pressure P and the proton density N at 16:00 UT on 06 January. On 07 January at 07:00 UT, the Dst index indicates its peak value of − 99 nT, the last CME bursts out. Meantime, the dynamic pressure P jumps from 6.05 nPa to its highest value of 12.36 nPa, and the proton density N hits its maximum value of 29.4 1/cm3. After 2 h, on 07 January at 9:00 UT, the Bz magnetic field decreases to its minimum value (− 17.04 nT), the electric field E reaches its highest value of 8.16 mV/m, the AE index shows its maximum value of 1327 nT, and the ap ZG index hits its maximum value of 94 nT.

17 February 2015 storm: In the 5 days discussion period, when the first CME bursts out on 15 February at 04:00 UT with the sudden commencement in the dynamic pressure P, the proton density N peaks immediately its maximum value of 21.9 1/cm3. On 17 February at 21:00 UT, 3 h before the Dst ZG index takes its minimum value of − 62 nT, the magnetic field Bz component indicates its peak value of − 12.0 nT and the electric field (E) hits its maximum value of 4.54 mV/m. After 3 h, at 00:00 UT, the ap index, the AE index, and the Dst index indicate their peak values of 48 nT, 579 nT, and − 64 nT, respectively.

24 February 2015 storm: In the 5 days storm period, when the first CME bursts out on 23 February at 03:00 UT with the sudden commencement in the dynamic pressure P, the proton density N immediately hits the value of 20.3 1/cm3. After 8 h at 08:00 UT, when the second CME bursts out, the dynamic pressure (P) hits 5.46 nPa and the proton density (N) indicates 20.0 1/cm3. Within 4 h, the magnetic field Bz component hits its peak value of − 7.7 nT (11:00 UT), the dynamic pressure (P) increases its maximum value of 8.81 nPa (12:00 UT), and the proton density (N) shows its peak value of 31.7 1/cm3 (at 12:00 UT).

Modeling

Table 1a and 1b demonstrates the binary relationships with the correlation matrix for the data of four moderate storms. The Pearson correlation matrix shows the relationship of variables. When the constants in the matrix are close to ± 1, mutual correlation strengthens. The correlation of the dynamic pressure P and the proton density N parameters in Table 1a and 1b seem high. The reliability of these values can be evaluated with the Cronbach’s alpha constant that should be above 60% for reliability. It is 0.645, 0.816, 0.629, and 0.640 in these tables, respectively. Dendrogram of the variables of the storms and scattering of data are specified in Fig. 3a, b and Fig. 4, respectively. The dendrogram displays the relationship of the variables by each line.

Table 1 Correlation of data
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Fig. 3
figure 3

a The cluster of the variables (left side: 04 January; right side: 07 January storm); b The cluster of the variables (left side: 17 February; right side: 24 February storm)

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Fig. 4
figure 4

Scattering of the variables (from left to right: 04 January, 07 January, 17 February, and 24 February storms)

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Figure 3a and b shows the dendrogram of the variables of the four moderate storms.

After the introductory mathematical discussion, it can be appropriate to remember the frame of the model of an ANN. The ANNs have been inspired by the working principles of the human brain. This complicated and trainable neural system, which is shaped by linking many neurons with several interface levels, imitates the brain. Studies firstly involved the mathematical modeling of the neurons in the human brain. With increasing awareness, the ANN has become a scientific discipline today with usage areas in many different fields. ANNs can observe information and data in different structures and procedures by recognizing them very quickly; they can also reveal unknown and difficult-to-notice correlations among data. They allow modeling without the necessity for any preparation or info among inputs and outputs (Elman 1990). Basically, inputs and corresponding outputs are specified to the network (Fig. 5).

Fig. 5
figure 5

The ANN framework for the estimation of ZGi

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Training or educating of the ANNs requires learning the relationship between input and output. This approach, called instructional learning, is common (Peng et al., 1992). As an architectural configuration involving some layers, the ANN uses data with a pre-determined number of artificial neural cells. The initial layer is generally the input layer. This layer is usually not numbered owing to the lack of weight factors and initiation functions of the inputs in the input layer. The second mid-layer, called the hidden layer, can be founded so many as needed. Researchers usually employ one hidden layer (Elman 1990; El-Din & Smith 2002). The layer called the output layer is the last layer. In this paper, the estimations are completed by the back propagation (Rumelhart et al., 1986) ANN algorithm. The typical back propagation algorithm applying a feedback learning structure is the gradient descent algorithm that moves the network weights in the direction of the negative gradient of the performance function. Many backpropagation algorithms appropriate for nearly all problems in ANNs are driven by standard optimization methods such as gradient descent and the Newton approach (Lipmann 1987). Feedback learning using continuous input diminishes the error caused by backward agglomeration. The author utilizes the widely performed Levenberg–Marquardt (trainlm) learning algorithm.

After the creation of the learning algorithm, the number of neurons of the hidden layer has to be specified. The number of neurons should be determined as needed. Too few neurons cause the network to be unable to learn the network pattern, while a large number of neurons cause the network to memorize. A small enough number of neurons forces the ANN to improve the generalization facility (Stern 1996). In the paper, the number of neurons in the hidden layer is determined to be thirty-five. In this neuron number, the mean square error (MSE) value begins to indicate no substantial change.

The work consists of three layers: the input layer, hidden layer, and output layer (Fig. 5). In harmony with the causality principle, the SWp (Bz, E, P, N, v, T) are the variables of the input layer and the ZGi (Dst, ap, AE) are the variables of the output layer. For the ANN model to be able to learn well without memorizing, the sigmoid transfer function is selected as the neural transition function (Fausett, 1994). A Linear transfer function is used in the output layer. Where a total of 120 (5 days) data is investigated, 84 data are utilized for training the ANN (70%), 12 data for validation (10%) and 24 data for testing (20%) (Haykin 1994).

As it may be realized from Fig. 6, the MSE values do not change after 6 updates (step, epoch) for the Dst index, after 8 updates for the ap index, and after 8 updates for the AE index in the 04 January storm (left column). In addition to this, the MSE values do not change after 7 updates for the Dst index, after 8 updates for the ap index, and after 8 updates for the AE index in the 07 January storm (right column). Therefore, learning (training) is finished. Up to these iteration totals, where the best verification performance happens, there is no monitoring of memorization owing to error constancy. Because the validation and test set errors show similar behaviors and no substantial memorization happens, the network performance is acceptable.

Fig. 6
figure 6

The chronological performance of the Dst, ap, and AE (from left to right) ANN model

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One can see from Fig. 6 that MSE values do not change after 7 updates (step, epoch) for the Dst index, after 7 updates for the ap index, and after 6 updates for the AE index in the 17 February storm (left column). In addition to this, the MSE values do not change after 8 updates for the Dst index, after 8 updates for the ap index, and after 8 updates for the AE index in the 24 February storm (right column).

Figure 7a, b and Fig. 8 visualize the results of the discussion. In Fig. 7a and b, the Dst, ap, and AE indices line up from top to bottom, respectively. While Fig. 7a and b displays the correlation, Fig. 8 exhibits the character of observed, forecasted values with their errors. Graphically, forecasting consequences are in Fig. 7a, b between the output and the target (the Dst, ap, AE indices).

Fig. 7
figure 7figure 7

a The plot of regression between the estimated and observed Dst, ap, and AE (from top to bottom) indices; b The plot of regression between the estimated and observed Dst, ap, and AE (from top to bottom) indices

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Fig. 8
figure 8

The observed and estimated Dst (nT), ap (nT), and AE (nT) indices (chronologically from top to bottom) and their errors

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Significant studies in the literature have reached remarkable results in the estimation of the Dst, the ap (or the Kp), and the AE index. In the Dst ZG index estimations, Gleisner et al. (1996) achieved an 84% accuracy, Fenrich and Luhman (1998) 79%, O’Brien and McPherron (2000) 88%, Lundstedt et al. (2002) 88%, Pallocchia et al. (2006) 90%, Bala and Reiff (2012) 86%, Uwamahoro et al. (2012) 86% (for severe storms 100%), Singh and Singh (2016) 79%, and Balan et al. (2017) 100%.

In the Kp ZG index estimations, Boberg et al. (2000) achieved a 77% accuracy, Wing et al. (2005) 94%, Bala and Reiff (2012) 96%, Young et al. (2013) 93%, Solares et al. (2016) 91%, and Wintoft et al. (2017) 92%.

In the AE index estimations, Gleisner et al. (1996) achieved an accuracy of more than 70%, Takalo and Timonen (1997) 98%, Gleisner and Lundstedt (2001) 84%, and Bala and Reiff (2012) 83%.

For the 04 January storm: The Dst, ap, and AE indices estimation models are 98.5%, 98.1%, and 98.3% (Fig. 7a, left column) reliable, respectively.

For the 07 January storm: The Dst, ap, and AE indices estimation models are 98.8%, 98.5%, and 98.4% (Fig. 7a, right column) reliable, respectively.

For the 17 February storm: The Dst, ap, and AE indices estimation models are 97.6%, 98.9%, and 98.6% (Fig. 7b, left column) reliable, respectively.

For the 24 February storm: The Dst, ap, and AE indices estimation models are 98.9%, 98.5%, and 98.6% (Fig. 7b, right column) reliable, respectively.

Forecasting model outcomes of the four moderate storms look similar. It is obvious that the ANN model displays the reliable method and the fit output for these moderate geomagnetic activities.

Estimated Dst, ap, and AE indices and their errors for the 04 January moderate storm together with actual ones from NASA are exhibited in Fig. 8, respectively. In Fig. 1 and Fig. 8, one can see the error ratio in the comparison of the estimated and monitored Dst, ap, and AE ZGi. The absolute error between the real-estimated ZGi values can be observed with the \(Error=\left|\frac{{Dst}_{est}-Dst}{Dst}\right|\), \(Error=\frac{\left|{ap}_{est}-ap\right|}{ap}\), and \(Error=\frac{\left|{AE}_{est}-AE\right|}{AE}\), where the Dstest, apest, and AEest are the estimated Dst, ap, and AE index values, respectively. The low error rate exhibits the accuracy of the estimation.

According to Fig. 8, the estimated Dst index average errors are 0.034, 0.009, 0.389, and 0.064 with 0.011, 0.002, 0.841, and 0.012 relative variance values, respectively.

Estimated ap index values and their errors for all storms together with actual ones from NASA are exhibited in Fig. 8, respectively.

According to Fig. 8, the estimated ap index average errors are 0.198, 0.265, 0.035, and 0.019 with 0.047, 0.143, 0.003, and 0.001 their relative variance values, respectively.

Estimated AE index values and their errors for all storms together with actual ones from NASA are displayed in Fig. 8, respectively.

According to Fig. 8, the estimated AE index average errors are 0.444, 0.022, 0.219, and 0.215 with values 0.340, 0.018, 0.111, and 0.056 relative variance values, respectively.

The effect of variables (for solar wind parameters) on the ANN model (Gontarski et al., 2000) can be calculated with the formula % Effect = 100.(1 − Rn/Rdif) by omitting these variables from the investigation process. The correlation coefficients govern this formula. In the formula, Rn is the correlation coefficient attained by excluding the input and Rdif is the basic correlation coefficient between estimated and observed values. Table 2a and 2b exhibits the effect of variables on the ANN model.

Table 2 Effect of each variable on the performance of the ANN estimation
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04 January storm: In the modeling of the Dst (nT) index estimation, the plasma flow speed v (km/s) indicates the main effect. The correlation coefficient diminishes by 13.70% when neglecting the plasma flow speed (Table 2a). The second-high effect belongs to the proton density N (1/cm3) and dynamic pressure P (nPa). The correlation coefficient weakens by 10.37% and 9.43% when omitting the proton density and dynamic pressure P value, respectively (Table 2a). Finally, when omitting the magnetic field Bz (nT) component, the Dst index is affected by 3.54%. The plasma flow speed v (km/s), the proton density N (1/cm3), the dynamic pressure P (nPa), and the magnetic field Bz (nT) are indispensable estimators for the Dst (nT) index (Burton et al. 1975; Gleisner et al. 1996). Physically, coronal holes created by the instability of hot particles are the origin of the flow speed v (km/s). The SW streams have high speed. The polarization of the magnetic field is indicated by the parameters of the SW speed (Tsurutani et al., 2006). Besides orienting the magnetic field Bz (nT) component from southward to northward and indicating its negative values, the flow speed v (km/s) shapes the geomagnetic storm. The flow speed v (km/s) and the Bz (nT) component, with these anomalous replies, show that the Dst (nT) index should decline to a negative minimum. By enhancing the proton density N (1/cm3), the high-density plasma dynamic pressure P (nPa) and SW suppress the magnetosphere (Tsurutani et al., 2006). The reflection of the disturbance caused by this compress governed by the flow speed v (km/s) is the Dst (nT) index. Accordingly, one of the principal motivations why the Dst (nT) index decreases to minimum value is the flow speed (Gonzalez et al. 1989; Borovsky 2012; Borovsky and Yakymenko 2017). The ANN model shapes the estimation of the Dst values with the literature (Table 2a and 2b).

The maximum impact on the ap (nT) index estimation takes into account the magnetic field Bz (nT) component, the dynamic pressure P (nPa), the flow speed v (km/s), and proton density N (1/cm3). When the magnetic field and the dynamic pressure are omitted, the correlation ratio declines by 14.66% and 12.64%, respectively (Table 2a). The flow speed v (km/s) and the proton density N (1/cm3) are also other high parameters for the ap index. If these variables are omitted from the ap estimation, the model correlation constant diminishes by 8.91% and 8.40%, respectively. Physically, magnetic field polarizations indicate parallel effects with the dynamic pressure P (nPa), the flow speed v (km/s), and the proton density N (1/cm3), while the ap index nonlinearly responses to the instabilities (Altadill et al. 2001; Eroglu 2018; 2019, 2020, 2021; Inyurt 2020; Koklu 2020). The noticeable relation between the Bz (nT) magnetic field, the dynamic pressure P (nPa), the flow speed v (km/s), the proton density N (1/cm3), and the ap (nT) index may be seen in Table 2a.

In the ANN estimation model for the AE (nT) index, the highest impact relates to the magnetic field Bz (nT) component. One may see that when neglecting the Bz (nT) magnetic field, the correlation constant decreases by 12.16% (Table 2a). The magnetic field is accompanied by the flow speed v (km/s) and proton density N (1/cm3) during the calculation of the AE (nT) index correlation ratio. It is realized that the value of R declines by 6.94% and 4.60% when the flow speed v (km/s) and proton density N (1/cm3) are subtracted from the ANN model of the AE index, respectively (Table 2a) (Gleisner and Lundstedt, 2001).

07 January storm: Parallel conclusions can also be drawn for the 07 January moderate storm. In the Dst index estimation, the highest effect belongs to the dynamic pressure P (nPa), the flow speed v (km/s), and the Bz (nT) magnetic field. The correlation coefficient diminishes by 12.35%, 10.53%, and 10.02% when neglecting the mentioned SWp value, respectively (Table 2a). According to Table 2a, the proton density N (1/cm3) affects the Dst (nT) index with a value of 8.81%.

The maximum impact on the ap (nT) index estimation takes into account the proton density N (1/cm3) and the flow speed v (km/s). When the proton density and the flow speed are omitted, the correlation constant value decreases by 17.32% and 16.04%, respectively (Table 2a). Secondly, the other main factors are the Bz magnetic field (nT) and the dynamic pressure P (nPa) for the ap (nT) index. If they are omitted from the ap (nT) prediction, the model correlation ratio decreases by 10.25% and 10.05%, respectively (Table 2a).

In the estimation of the AE (nT) index, the highest effect belongs to the proton density N (1/cm3). One can see that when neglecting the proton density, the correlation constant value decreases by 13.11% (Table 2a). Besides the proton density, it is observed that the value of R decreases by 9.76% and 9.55% when the magnetic field Bz (nT) component and the flow speed v (km/s) are excluded from the AE index model, respectively (Table 2a).

17 February storm: This moderate storm also reflects similar effects. The Dst (nT) index estimation responds to ignoring the flow speed v (km/s), the proton density N (1/cm3), the dynamic pressure P (nPa), and the Bz (nT) magnetic field. The correlation constant decreases by 8.20%, 7.48%, 7.07%, and 6.86% when neglecting mentioned SWp value, respectively (Table 2b).

The maximum effect on the ap (nT) index estimation takes into account the proton density N (1/cm3), the Bz magnetic field (nT), the dynamic pressure P (nPa), and the flow speed v (km/s). When the mentioned SWp is omitted, the correlation constant value decreases by 10.72%, 10.11%, 9.81%, and 9.00%, respectively (Table 2b).

In the estimation of the AE (nT) index, the highest effect belongs to the flow speed v (km/s), the magnetic field Bz (nT) component, and proton density N (1/cm3). One can realize that when neglecting these parameters, the correlation constant value decreases by 7.73%, 7.40%, and 6.80%, respectively (Table 2b).

24 February storm: Lastly, according to the 24 February moderate storm, the Dst (nT) index estimation is deeply related to the Bz (nT) magnetic field, the dynamic pressure P (nPa), the proton density N (1/cm3), and the flow speed v (km/s). The correlation coefficient diminishes by 23.36%, 13.04%, 8.19%, and 8.03% when neglecting these SWp, respectively (Table 2b).

The maximum impact on the ap (nT) index estimation takes into account the proton density N (1/cm3), the flow speed v (km/s), the Bz magnetic field (nT), and the dynamic pressure P (nPa). When the mentioned SWp is omitted, the correlation constant value decreases by 18.10%, 12.59%, 8.02%, and 7.41%, respectively (Table 2b). Secondly, the electric field E (mV/m) affects the ap (nT) index by 2.23% (Table 2b).

In the estimation of the AE (nT) index, the highest effect is observed by means of the proton density N (1/cm3), the Bz (nT) magnetic field, and the flow speed v (km/s). One can see that when ignoring these SWp, the correlation constant value decreases by 11.66%, 7.20%, and 7.10%, respectively (Table 2b).

Conclusion

The purpose of the study is to assess the four moderate geomagnetic storms that occurred in the year 2015. After glancing at the year 2015 as a whole, zonal geomagnetic indices (ZGi) of these first four moderate activities are estimated efficiently. When comparing the estimation of geomagnetic storms by the artificial neural network (ANN) with the literature, the conclusions are satisfactory.

For these events, it is notable that the ANN model estimates the ZGi considering the solar wind (SWp). A geomagnetic storm strength and its phases may be evaluated by discussing the ZGi. The paper is based on the method of inputting the SWp to the ANN model and yielding the ZGi as the output. The estimation performance of the ANN model with ZGi as the output is acceptable and consistent for the literature. The results demonstrate that the model is over 90% consistent in the estimation of the ZGi of these four moderate storms. The essay, in addition to the estimation, discusses the effect of the SW variables on the ANN model for these storms.

Regarding the proton density (N), the flow speed (v), the magnetic field z component (Bz), the dynamic pressure (P), and the electric field (E), according to the 04 January storm ANN model, for the Dst index, the v, N, and P have a high effect, while the Bz component has the moderate effect. Furthermore, Bz field, P, and v with N affect the ap index highly. For the auroral electrojet index AE, Bz has a high effect, v and N have a moderate effect.

According to the ANN model of the 07 January moderate storm, for the Dst index, P, v, Bz, and N have a high effect. Moreover, N, v, and Bz with P affect the ap index highly. For the AE index, N, Bz field, and v have a high effect.

According to the ANN model of the 17 February storm, for the Dst index, v, N, P, and Bz have a high effect. Additionally, N, Bz, and P with v affect the ap index highly. The electric field (E) also has a moderate effect on the ap ZG index. In calculating auroral electrojet index AE, v, Bz, and N have a high effect.

According to the ANN estimation model of the 24 February moderate storm, for the Dst ZG index, Bz has a very high effect and P, N, and v has a high effect. Furthermore, for the ap ZG index, N has a very high effect and v, Bz field, and P have a high effect besides the moderate effect of E. For the AE index, N, Bz, and v have a high effect.

The agreement between the result of this study and those of previous studies indicates the reliability of the results of this study. The author expects to contribute to geomagnetic storm estimations by making it easier to understand their dynamics. The Dst, ap, and AE indices estimated for these storms can also be estimated for other storms. With a similar approach, it will not be difficult for the ANN model to estimate the ZGi of weak, moderate, or severe storms. The indices considered together with the SWp prepare the ground for predictable storms. The author hopes to attain the same results for weak, moderate, and severe storms in further discussions.