Virtual sounding of solar-wind effects on the AU and AL indices based on an echo state network model

. The properties of the auroral electrojets are examined on the basis of a trained machine learning model. The relationships between solar-wind parameters and the AU and AL indices are modeled with an echo state network (ESN), a kind of recurrent neural network. We can consider this trained ESN model to represent nonlinear effects of the solar-wind inputs on the auroral electrojets. To identify the properties of auroral electrojets, we obtain various synthetic AU and AL data by using various artiﬁcial inputs with the trained ESN. The analyses of various synthetic data show that the AU and AL indices 5 are mainly controlled by the solar-wind speed in addition to B z of the interplanetary magnetic ﬁeld (IMF) as suggested by the literature. The results also indicate that the solar-wind density effect is emphasized when solar-wind speed is high and when IMF B z is near zero. This suggests some nonlinear effects of the solar-wind density. This study modeled the temporal pattern of the AU and AL indices using ESN. Although the ESN model is relatively simple, it mostly accurately reproduces the variations of the AU and AL indices. We virtually sound the properties of the magnetospheric 195 system by putting artiﬁcial inputs into the trained ESN model. Our virtual sounding results show a strong impact of the solar-wind speed which was previously observed in the literature. It is also suggested that IMF B y and the solar-wind density have signiﬁcant effects, especially on the AU index. These results are consistent with other studies. In addition, an analysis of the synthetic AU and AL indices obtained from the artiﬁcial inputs suggests that the solar-wind density does not have a simple linear effect on AU and AL , but rather that some compound processes exist. According to the results, the solar-wind density 200 contributes to the auroral electrojet intensity more effectively under high solar-wind speed conditions and the solar-wind density effect becomes small under low solar-wind speed conditions. The solar-wind density effect tends to be important when IMF B z is near zero. The density effect is small on average when | B z | is large.

predicted the AL index using a model which combines the autoregressive moving average with the exogenous inputs (ARMAX) model and a neural network.
While machine learning techniques tend to be used for predictions with high accuracy, the learned relationships between 25 solar-wind inputs and auroral electrojets are of interest from the scientific perspective as well. As trained machine learning models can describe the nonlinear behaviors of the magnetospheric system, it is meaningful to analyze the input-output relationships of the trained models. Recently, Blunier et al. (2021) have identified solar-wind parameters which affect the value of geomagnetic indices by putting perturbed inputs into a trained neural network. This study takes a somewhat similar approach.
We employ an echo state network (ESN) model (Jaeger, 2001;Jaeger and Haas, 2004;Chattopadhyay et al., 2020), which is a 30 kind of recurrent neural network, to describe the relationship between various solar-wind parameters and the auroral electrojet indices AU and AL. We then virtually sound the responses of the AU and AL indices to solar-wind inputs by putting various artificial inputs into the trained ESN model and identify the properties of the auroral electrojets.

Echo state network
We model the temporal evolution of AU and AL with the ESN model because it can be easily implemented to attain a 35 satisfactory performance. The ESN is a kind of recurrent neural network with fixed random connections and weights between hidden state variables. Only the weights for the output layer are trained so that the target temporal pattern is well reproduced.
We combine the state variables at the time t k into a vector x k , where the i-th element of x k is denoted as x k,i . The number of state variables m is set at 1200 in this study. At the time step k, we update x k,i as follows: where z k is a vector consisting of the input variables. The weights w i and u i determine the connection with the other state 40 variables and input variables. The weights w i and the parameter η i are given in advance and are fixed.
It is desirable that the weights are given so as to attain the so-called 'echo state property'. The echo state property guarantees that the ESN forgets distant past inputs. Defining the weight matrix W as a sufficient condition for the echo state property is that the maximum singular value of W is less than 1. If a certain matrix W ′ is given and its maximum singular value λ ′ is computed, we can obtain the weight matrix W which satisfies this sufficient 45 condition as follows: We thus first determine W ′ randomly and obtain the weight W according to Eq. (3) with the parameter α set to 0.99. In this study, we set 90% of the elements of W ′ to be zero. Each of the remaining non-zero elements comprising 10% of W ′ is obtained randomly from a Laplace distribution for which the probability density function p(x) is written as Similarly to W ′ , 90% of the elements of u i are set to be zero and the other non-zero elements are given by the same Laplace 50 distribution. The parameter η i in Eq. (1) is obtained randomly from a normal distribution with mean 0 and standard deviation 0.3.
The output for the time t k , y k , is obtained from x k as follows: The weight β in Eq. (5) is determined so that the objective function is minimized, where d k is an observation vector consisting of the observed data. The present study aims to model the temporal 55 pattern of the AU and AL indices. Accordingly, the output vector y k consists of two elements as follows where y AU,k and y AL,k are the predicted AU and AL values at t k , respectively. In this study, 5-minute values (averages for 5 minutes) of AU and AL are used. We give the input vector z k as follows: where B z,k and B y,k denote the z and y component of the interplanetary magnetic field in the geocentric solar magnetic (GSM) coordinates at time t k , V sw,k is the −x component of the solar wind velocity in the GSM coordinates, N sw,k is the solar wind 60 density, H k is universal time (UT) in hour, and D k is the day from the end of 2000 (D k = 1 on January 1, 2001). The variables H k and D k are included for considering UT dependence and seasonal dependence (e.g., Cliver et al., 2000). The feedback of the predicted AU and AL indices which can be obtained using Eq. (5) is also included in the input vector z k . The solar wind variables B z,k , B y,k , V sw,k , and N sw,k are taken from the OMNI 5-minute data.
If z k does not contain the feedback of y AU,k−1 and y AL,k−1 , the weight β can be determined through simple linear regression because x k at each time step would not depend on β in Eq. (5). However, since the feedback of y AU,k−1 and y AL,k−1 are contained, the optimal β cannot be obtained by linear regression. We thus obtained β using the ensemble-based optimization method (Nakano, 2021).

Performance of ESN
We trained the ESN using data obtained over a period of ten years from 2005 to 2014. We used 5-minute values of the OMNI 70 solar wind data and the AU and AL indices provided by Kyoto University. Since the ESN memorizes the history of the input data, the ESN output should be compared with the observation after referring to the input data for the preceding several time steps. We then start the comparison after spin-up of the ESN for 72 steps, which corresponds to 6 hours for the 5-minute values, from the initial time of the analysis. It should also be noted that solar wind data are sometimes incomplete. If more than half of the data were missing for 1 hour, we stopped the prediction and spun up the ESN again for the subsequent 72 steps. 75 We then reproduced the AU and AL indices for the period from 1998 to 2004 and compared the outputs with the observed values. In Figure 1, the top panel shows the AU and AL reproduced by our ESN model in October 1999 with red lines and the observed AU and AL indices with gray lines for the same period. The second panel shows the three components of the IMF.
The green, blue, and red lines indicate the x, y, and z components in (GSM) coordinates, respectively. The third panel shows the solar wind speed and the fourth panel shows the solar wind density. The bottom panel shows the SYM-H index (Iyemori,80 1990; Iyemori and Rao, 1996) for the corresponding time period. High auroral activity was maintained for the period from 10 October to 17 October when high speed solar wind streams coincided with a continual southward IMF, as suggested by the literature (e.g., Tsurutani et al., 1990Tsurutani et al., , 1995. The auroral activity was also enhanced during a magnetic storm from 21 October. The model outputs mostly reproduced the observed AU and AL values well for these events. Table 1 shows the root-mean-square errors (RMSE) of the ESN prediction for each year of the period from 1998 to 2004.

85
The RMSEs were less than 100 nT for the AL index and about 50 nT for the AU index except for 2003. The RMSEs of AU and AL were larger in 2003 than in other years likely because of high auroral activity during that year. Figure    year. The fixed values of IMF B y , solar-wind speed, solar-wind density, and solar-wind temperature were 0 nT, 400 km/s, 1 /cc, and 2 × 10 5 K, respectively. We did not consider the case where the IMF B z effect was turned off because the RMSE 110 becomes very large without an accurate IMF B z input, as obviously expected from the results of many previous studies (e.g., Arnoldy, 1971;Akasofu, 1981;Murayama, 1982;Newell et al., 2007). to the efficiency of the coupling between the solar wind and the Earth's magnetosphere (e.g., Akasofu, 1981;Murayama, 1982;Newell et al., 2007). The mean deviation shown in Figure 4 indicates the bias of the ESN output, and the positive bias means that the ESN output tends to be larger than the observed AL value, which corresponds to an underestimation of |AL|.

125
The large positive bias for the case without solar-wind speed variation in Figure 4 thus suggests that a low solar-wind speed results in a small |AL|. Conversely, a high solar-wind speed activates variations of AL. We can also observe a relatively small effect of IMF B y , which would also contribute to the coupling between the solar wind and the magnetosphere. In addition, the effect of the solar-wind density can be seen for all of the years from 1998 to 2004. The large mean deviation suggests that the solar-wind density enhancement intensifies the westward electrojet as implied by some earlier studies (Newell et al., 2008;130 McPherron et al., 2015).  Figure 3. The solar-wind speed effect is again prominent. The large negative bias for the case without solar-wind speed variation in Figure 6 suggests a low solar-wind speed underestimates the AU value. In contrast with AL, AU is likely to be strongly controlled by IMF B y and the solar-wind density. In particular, 135 the mean deviation is largely negative for the case without density variation, which suggests an important effect of solar-wind density on the AU index, as discussed by Blunier et al. (2021).    Figure 1. Although the ESN output is much smoother than the observation, especially in some impulsive events which would be related to substorms, the red line reproduces the observed AU and AL indices well. In contrast, when the solar-wind speed was set to be low at 400 km/s, the ESN model clearly underpredicted the strength of AL. This suggests that a high-speed solar wind makes an important contribution to enhancing the westward electrojet. When the density effect was turned off, the ESN tended to slightly underpredict |AL| although the 145 density effect was likely to be minor in this event. Figure 8 shows the result for another event from 26 July to 30 July in 2000. In this event, since the solar-wind speed was maintained at around 400 km/s, which we set as the base level of the solar-wind speed, the green line was similar to the red line.
On the other hand, the solar-wind density effect is visible. If the density is fixed at 1/cc, the ESN tended to underpredict |AU | and |AL|. However, the relationships with the solar-wind density learned by the ESN seemed to not be linear. For example, 150 the difference between the red and blue lines tended to be larger on 29 July than on 28 July while the solar-wind density was more enhanced on 28 July than on 29 July. This might suggest some compound effects of the solar-wind density and other parameters. where AU (N = 1) and AL(N = 1) are the synthetic AU and AL indices obtained by fixing the solar-wind density at 1, /cc.
We then used ∆AU N eff and ∆AL N eff as proxies of the solar-wind density effect as a function of time. Figure 9 is a 2dimensional histogram to compare ∆AU N eff and ∆AL N eff with the solar-wind speed. As the solar-wind speed increases, ∆AU N eff increases and ∆AL N eff decreases. This suggests that the solar-wind density effect on the auroral electrojets is not independent of the solar-wind speed effect but that the solar-wind density contributes to the auroral electrojet intensity more 160 effectively under high solar-wind speed conditions. The solar-wind density effect is likely to be small when the solar-wind speed is low. Figure 10 is a 2-dimensional histogram to compare ∆AU N eff and ∆AL N eff with IMF B z . The solar-wind density effect gets large when IMF B z is near zero. The density effect is small on average when |B z | is large. The ESN model therefore suggests that the solar-wind density effect is most important when IMF B z is small.

165
It is widely accepted that auroral electrojets are mainly controlled by IMF and the solar-wind speed (e.g., Akasofu, 1981;Murayama, 1982;Newell et al., 2007). In particular, IMF B z has an essential effect on auroral activity. When IMF is directed southward, DP2 type electrojets (e.g., Kamide and Kokubun, 1996) are enhanced and contribute to both AU and AL. The substorm current wedge, which contains a westward electrojet contributing to the AL index, would also be controlled by IMF (e.g., Kepko et al., 2015). As illustrated in Figure 1, the solar-wind speed also has an important effect.

170
Although the solar-wind density effect is sometimes ignored when modeling the AU and AL indices, Gleisner and Lundstedy (1997) reported that the performance of a neural network for modeling the AE index is improved by considering the solar-wind density effect. McPherron et al. (2015) also suggested a contribution from the solar wind density to the AL index. Blunier et al. suggested that the solar-wind density has a more visible effect on AU than on AL. The stronger effect on AU suggested by 175 Blunier et al. agrees with our result shown in Figure 5. Ebihara et al. (2019) conducted simulation experiments to examine the impact of various solar-wind parameters on the SM L index (Newell and Gjerloev, 2011), which is an extension of the AL index calculated with data from a larger number of observatories. According to their result, the SM L index depends on the solar-wind density when IMF B z is weak, while it is not clearly affected by the solar-wind density when IMF B z is directed strongly southward. This simulation result is consistent with our result in Figure 10. Figure 10 may thus be regarded as 180 statistical evidence of the compound effect between IMF B z and the solar-wind density. Figure 9 shows the compound effect between the solar wind density and velocity. One plausible explanation is the effect of the solar wind dynamic pressure which is proportional to N sw V 2 sw . As some studies have suggested that field-aligned currents around the auroral latitudes are influenced by the solar-wind dynamic pressure (Iijima and Potemra, 1982;Wang et al., 2006;Nakano et al., 2009;Korth et al., 2010), it is possible that the enhancement of the field-aligned currents increases the auroral electrojets. In Figure 9, however, the density effect disappears when the solar wind velocity is around 300 km/s, which does not seem to be explained by the solar-wind dynamic pressure effect. This problem might be solved by considering the contribution of the plasma sheet condition. Sergeev et al. (2014Sergeev et al. ( , 2015 suggests that the plasma sheet temperature and density may affect the ionospheric conductivity in the region of the westward electrojet which the AL index represents. It has been suggested that the plasma sheet temperature and density depend on the solar-wind velocity and density, respectively (Terasawa et al., 1997;190 Nagata et al., 2007). The plasma sheet effect can thus partially contribute to the relationship between AL and the solar-wind density.
This study modeled the temporal pattern of the AU and AL indices using ESN. Although the ESN model is relatively simple, it mostly accurately reproduces the variations of the AU and AL indices. We virtually sound the properties of the magnetospheric 195 system by putting artificial inputs into the trained ESN model. Our virtual sounding results show a strong impact of the solarwind speed which was previously observed in the literature. It is also suggested that IMF B y and the solar-wind density have significant effects, especially on the AU index. These results are consistent with other studies. In addition, an analysis of the synthetic AU and AL indices obtained from the artificial inputs suggests that the solar-wind density does not have a simple linear effect on AU and AL, but rather that some compound processes exist. According to the results, the solar-wind density 200 contributes to the auroral electrojet intensity more effectively under high solar-wind speed conditions and the solar-wind density effect becomes small under low solar-wind speed conditions. The solar-wind density effect tends to be important when IMF B z is near zero. The density effect is small on average when |B z | is large.