An empirical model for predicting low-latitude storm-time ionospheric foF2 is developed using the support vector machine technique. Considering that the ionospheric disturbances are mainly caused by interplanetary disturbances, the solar wind data are introduced as model input, as well as the ionospheric observations of Haikou (HK, with geographic coordinates of 110.3°E and 20.0°N, and geomagnetic latitudes of 8.6°N) and Chongqing (CQ, 106.5°E, 29.6°N, and geomagnetic latitudes of 18.1°N) in China. Data from 45 storms are selected as training samples to construct the model, and other 26 storms are used to validate and evaluate the model. The results indicate that the model proposed here can capture the low-latitude ionospheric disturbances most of the time. Compared with another empirical model, STORM, which has been included in International Reference Ionosphere (IRI) as storm time corrections, our model shows remarkable improvement at least for the given events.
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 The ionospheric behaviors are well known and efficiently reproduced by a number of ionospheric empirical models during quiet conditions. Currently, the most popular ionospheric model is the International Reference Ionosphere (IRI) [Bilitza, 2001], providing the monthly averages of the ionospheric parameters for magnetically quiet conditions.
 The success of high frequency communications under disturbed conditions depends largely on the ability of predicting and modeling the ionospheric responses to geomagnetic storms. During such periods, various disturbance processes occur in the ionosphere and the thermosphere, such as the dynamic process, the hydromagnetic process and the photochemical process, which lead to great difficulty in predicting the exact condition especially in the ionospheric F2 region. There have been several attempts in predicting the storm-time ionospheric foF2, such as the autocovariance method [Stanislawska and Zbyszynski, 2001] and the artificial neural network method [Wintoft and Cander, 2000;Cander and Mihajlovic, 1998]. Several years ago a notable empirical correction model, STORM [Aranjo-Pradere and Fuller-Rowell, 2002; Aranjo-Pradere et al., 2002], was developed and applied to provide a correction on the monthly median values during the disturbance time, and it was included in the international recommended standard IRI2000 [Bilitza, 2001] and its improved version by presenting a correction factor. This model relies on the theory that long-lived negative storm effects are induced by storm-time thermospheric circulations, using an integralAp index as input to denote high latitude atmospheric heating. This model has good performance in summer and equinox at midlatitude, but shows little improvement in winter or at other latitudes. Similarly, other currently existed models also can't provide precise ionospheric status at low latitudes during storm time.
 The electric field and plasma drift have long been known to play dominant roles in low latitude ionospheric dynamics, and also in the occurrence of ionospheric plasma irregularities [Abdu et al., 1991; Fejer, 2002]. Both observations [Basu et al., 2001; Kelley et al., 2003; Huang et al., 2005] and simulations [Huba et al., 2005; Maruyama et al., 2007] show that the disturbance electric field can cause great enhancement or inhibition on the equatorial ionization anomaly (EIA), depending on its magnitude and direction. Two basic sources in generating the storm time electric field disturbances have been identified as prompt penetration and disturbance dynamo. The former is induced by interplanetary electric field (IEF) penetrating into the ionosphere [Senior and Blanc, 1984], and the latter is caused by thermospheric wind driving the storm-time circulation [Blanc and Richmond, 1980]. Local time variations and time delay ones after the onset of these two fields are examined by theoretical and experimental studies, and their empirical models have been presented by Fejer and Scherliess  and Sheehan and Valladares .
 In this paper, we introduce a well developed technique, SVM (support vector machine), to build a predicting model for ionospheric foF2. This technique has been widely used in many fields, such as the solar flare [Li et al., 2007] and the magnetospheric substorm predictions [Gavrishchaka and Ganguli, 2001]. As to the ionospheric forecastings, Chen et al. [2010a, 2010b]have established several models to predict foF2 at quiet and storm time. The purpose of this paper is to introduce the SVM as a technique to forecast the low-latitude storm-time ionospheric foF2, taking both solar wind data and ionospheric observations as input. This is a useful attempt in conjugating the ionospheric disturbances with their solar wind origins.
2. Description of the SVM Technique
 SVM has recently received significant interest due to its excellent result in various applications. A brief description of the SVM is given as follow. Details can be found in the references [Vapnik, 1995; Chen et al., 2010a].
 In SVM nonlinear regression, the input xis first mapped into a high-dimensional feature space using some kind of fixed (nonlinear) mapping function, and then a linear model is constructed in this feature space. The regression modelf(x, w) is given by
where xi is the multivariate input, wi a set of weights, i = 1, …, N, N the number of input datum, b a bias, and φ the mapping function.
 SVM performs a linear regression in the high-dimension feature space usingε-insensitive loss functions and tries to reduce the complexity of a model by minimizing ∥w∥2[Yang et al., 2006]. The optimization function of SVM is formulated as minimizing the following function
where C is a pre-specified value, under the constrains
where yi is the target value of the training data set, and ξi, ξ*i are slack variables representing the upper and lower constraints on the outputs of the system.
 This constrained optimization problem can be solved mathematically [Zhang, 2006]. Then we can get the value of w and b by using
where K(xi, xj) = φ(xi)Tφ(xj) is the kernel function. This function has several forms to be chosen. Typically used are polynomial, Gaussian, and Curb kernels. The Curb kernel is selected in this study.
3. Data Sets and Model Construction
 Considering the effects of the electric field on low latitude ionosphere, the IEF calculated from the solar wind data are adopted as input to construct the model. In order to build the model, 45 intense storms (with peak Dst < − 100 nT) during 1978–1994 are selected as training samples, as listed in Table 1. Each of these storms is composed of only one main phase, which helps to avoid physical and mathematical difficulties. The hourly foF2 data are used for a 5 days-period for each event, centered at the time whenDst reaches to its minimum. In accordance with strong seasonal dependence of the ionospheric response, the storms are grouped into three seasonal bins, namely summer (from May to August), winter (from November to February) and equinoxes (March, April, September, October).
Table 1. The Geomagnetic Storms and Their Peak Dst Values Used in This Study
Aug 28, 1978
Feb 15, 1978
Oct 27, 1978
Aug 29, 1979
Nov 25, 1978
Mar 10, 1979
Jul 14, 1982
Feb 16, 1980
Sep 18, 1979
Aug 7, 1982
Dec 19, 1980
Apr 10, 1982
Jun 13, 1983
Jan 10, 1983
Sep 22, 1982
Aug 8, 1983
Nov 16, 1984
Mar 2, 1983
Aug 1, 1984
Jan 28, 1985
Sep 5, 1984
May 6, 1988
Nov 29, 1985
Apr 21, 1985
Jul 29, 1990
Feb 9, 1986
Sep 12, 1986
Jul 9, 1991
Jan 14, 1988
Mar 26, 1988
May 10, 1992
Feb 22, 1988
Mar 12, 1990
May 22, 1992
Nov 27, 1990
Oct 10, 1990
Aug 23, 1992
Nov 9, 1991
Mar 25, 1991
Feb 21, 1992
Sep 9, 1992
Nov 4, 1993
Sep 29, 1992
Feb 22, 1994
Apr 5, 1993
 Ionospheric data from two ionosondes, Haikou (HK, with geographic coordinates of 110.3°E and 20.0°N, and geomagnetic latitudes of 8.6°N) and Chongqing (CQ, 106.5°E, 29.6°N, and geomagnetic latitudes of 18.1°N) are used in this study. During quite time these two stations are situated at each side of the north crest of fully developed EIA. We use a parameter
to describe the ionospheric disturbances, where foF2obs is the observed, and foF2med is the monthly medians. The latter could be replaced by the medians of previous 30 days' observations practically.
 As to the solar wind data (in this study data from ACE are used), we estimated the magnitude of the reconnection electric field Er after [Kan and Lee, 1979]
where Vx denotes the solar wind speed in the solar to earth direction, θ the clock angle of the interplanetary magnetic field (IMF) in the Y − Z plane (GSM coordinate), By, Bz the y and z components of the magnetic field, respectively [Kelley et al., 2003]. A shielding/overshielding mechanism has been proposed for a long time to explain how IEF penetrates to the low-latitude ionosphere [e.g.,Vasyliunas, 1972; Jaggi and Wolf, 1973]. Generally the IEF has different effects on the low-latitude ionosphere at different local time. For example, the direction of the dawn to dusk penetration electric field in the ionosphere is eastward at daytime and westward at nighttime [Fejer, 2002]. In order to distinguish the direction of Er at daytime and nighttime, we convert Er during local time (LT) 20:00 to LT 05:00 to a negative one for convenience.
 The low-latitude ionospheric disturbances could last several hours longer than their disturbance electric field origins as a rule [Fejer, 2002], relying on that the recombination time is larger than the transport time [Abdu, 1997]. This indicates that some kind of integral of the electric field is more effective in ionospheric predictions than the field data itself. Based on this idea, we set up an integral of Er
to designate the accumulative effect of previous historical field disturbances. To decide these two parameters(τ0, τN), we introduce a simulation using a theoretical model, SAMI2 [Huba et al., 2000] in search of the electric field effects on the low-latitude ionosphere by exerting a disturbance field at local noon. It is found that the influence of the electric field reaches to its maximum at about 2 h later, and after about 8 h the influence decreases to a low level. It means that electric fields 8 h ahead are included with corresponding weights, and the maximum weight is the one at 2 h ahead. According to the discussion above, we chooseτ0 = 2 and τN = 8 in equation (9). This parameter of Xis used as one of the model input. It is known that the measured foF2 value at hour t has great influence on the prediction of foF2(t + 1), and foF2 values at t-23 and t-47 are of importance due to the cyclic variation of foF2 [Cander and Wintoft, 1999]. The observations at the current time and 23 h and 47 h before, denoted by Φ(t), Φ(t − 23), Φ(t − 47), are also used as input to denote the historical state of the ionosphere. Moreover, the first and the second increments of Φ(t), deduced by Φ(t) − Φ(t − 1) and Φ(t) − 2Φ(t − 1) + (t − 2), are also included in the model to reflect the recent changes of the ionosphere. Another input parameter is the local time. So there are totally seven input factors in constructing the model. The output is the value at the next hour Φ(t + 1).
 Once the input and output pairs are chosen, the SVM begin to train the data set in search of the regression relation. In the training process, support vectors (SVs) are decided automatically, which will be used in the predicting process. One example of the SVs is shown in Figure 1. The SVs are marked with stars, the trained data set with continuous line, and the predicted value with black dots. It can be found that the SVs are only a minority of the training samples. The others are abandoned by SVM and not used in the predicting process. The predicting result depends on these SVs, and the SVM algorithm complexity is largely reduced by the small amount of SVs, and the efficiency of SVM is increased by the selection of SVs.
 In order to validate and evaluate this model, 26 storms during 1997–2002 are selected as test samples, which are listed in Table 2. The root mean square errors (RMSE) of SVM and STORM are shown and compared in Table 3. Note that the calculation covers all the 5 days for each storm.
Table 2. The Storms and Their Peak Dst Values Used in the Validation
May 15, 1997
Nov 7, 1997
Oct 11, 1997
Aug 6, 1998
Nov 23, 1997
Mar 10, 1998
Aug 27, 1998
Nov 13, 1998
Sep 25, 1998
Jul 16, 2000
Feb 18, 1999
Sep 22, 1999
May 24, 2000
Jan 13, 1999
Oct 22, 1999
Aug 17, 2001
Feb 12, 2000
Apr 6, 2000
May 23, 2002
Nov 6, 2000
Sep 17, 2000
Nov 6, 2001
Oct 29, 2000
Nov 24, 2001
Apr 18, 2001
Oct 28, 2001
Table 3. The RMSE of SVM and STORM
 Here RMSE is defined as
where Φ is the measured data, Φp the predicted one, and N the number of data samples. Table 3 shows that the SVM model has less error than STORM as a whole. It has been proved by Chen et al. [2010a, 2010b] that SVM has good adaptability in ionospheric predictions. Detailed comparison shows that RMSEat CQ are smaller than those at HK in summer and equinoxes, while they are comparable in winter. This indicates that the location is important in low-latitude ionospheric disturbances. According toRishbeth , the competition between the storm-time thermospheric circulation and the background one is more complicated at low-latitude. This mechanism may lead to further complexities in the ionospheric variations, but they are excluded in this study.
 To further validate the model, six storms (two cases for each season) are selected with their results shown in Figures 2–7, namely the one occurred during February 12–14, 2000 with peak Dst value of − 133 nT and the one during November 5–7, 2001 with − 292 nT in winter, the one during August 26–28, 1998, with − 145 nT and the one during May 23–25, 2000 with − 109 nT in summer, the one during April 17–19, 2001 with − 114 nT and the one during October 21–23, 1999 with − 237 nT in equinox. In these figures, the measured data are denoted by continuous lines, the predicted values by dots, and STORM results by dotted lines, respectively.
 The disturbance event and the predicting results during February 12–14, 2000 are shown in Figure 2. It can be found that the ionosphere suffers a moderate negative disturbance over HK, but a complicated weak disturbance over CQ during the main and the recovery phases of the storm. The integral of the electric field X is mainly negative during the same period, which corresponds to a locally westward disturbance field (note that the electric field data has been converted to a negative one at local night). Though the disturbance field data X reaches to its minimum nearly 10 h later than the Dst data, the related ionospheric disturbance over HK evolutes very similar to X. This indicates that during this period the disturbance electric field does play an important role in the ionospheric evolutions over HK, and the model taking X as input does capture most of the ionospheric disturbances. However, the ionosphere over CQ suffers a negative disturbance during most time with some interruption of positive properties. This means that different disturbances other than the electric field may act on the ionosphere, leading to different evolutions over CQ. These disturbances are also predicted by our model with some discrepancy, which may be attributed to the good regression ability of SVM itself. On the other hand, STORM only gives probable estimations compared with our model, which is partly due to the fact that STORM relies on neutral composition changes and their equatorward propagations.
 As to the storm occurred during November 5–7, 2001 (Figure 3), the parameter X undergoes a long lasting positive disturbance with a weak negative disturbance at the end of the recovery phase. However, the ionospheric responses at both stations are mainly negative when X undergoes a positive disturbance, which is not consistent with that of the former case. After a careful comparison, we find that the peak value of X in this event is almost twice than that of the former. The different ionospheric variations indicated that its response to disturbance field is potentially nonlinear, as shown by Figures 2 and 3. Fortunately, the disturbances of the ionosphere are also captured by our model at most time of this event, while most details have not been represented by STORM.
Figure 4 shows the example occurred during August 26–28, 1998. The measured values in CQ are missed sometimes, as well as the predicted ones. The predicted value agrees well with the measured one on the whole in both stations.
Figure 5 shows another example for summer. No detailed discussion is presented here because of less surprising result.
Figure 6 shows an example for equinox. The disturbance field X increases gradually and then decreases after reaching to its peak value along with the storm evolving into its main and recovery phases. The data of X maintain a weak positive level during the main and recovery phases of the storm, while the ionosphere undergoes a relatively simple evolution. It can be found from Figure 6 that the ionosphere behaves quiet over HK and CQ during the initial and main phases, whereas moderate negative disturbances happens over HK and CQ during the late recovery phase. According to Huba et al.  and Maruyama et al. , the ionospheric disturbances could last several hours longer than their disturbance electric field sources as a rule. This indicates that the disturbances mentioned above are not only caused by the disturbance electric field. In other word, the ionosphere exhibits a prolonged response to its solar wind origin during this period. However, most of the disturbances are also captured by the SVM model, whereas nearly be ignored by STORM.
Figure 7 shows another example for equinox. No detailed discussion is presented here for the same reason with that in Figure 5.
 A delay of about 1h between the predicted and measured values appears sometimes in Figures 2–7. This could be attributed to the model itself. As depicted in section 3, in constructing the model the ionospheric observation one hour ahead is adopted as one of the inputs. This treatment often leads to a delay in the predictions [e.g., Francis and Cannon, 2000].
 Another phenomenon shown in Figures 2–7 is that relating to different solar wind disturbances (indicated by X), the ionosphere exhibits different variations. It is partly attributed to different local time effects of the disturbance electric fields acting on the ionosphere. During daytime the near-equatorial plasma is transported to higher latitudes, forming a giant or weak fountain [Tsurutani et al., 2008] as a result of different disturbance magnitude. The result may be totally different at night. This leads to complex effects over CQ and HK. Also, there are several other factors that lead to different variation during each storm event in both stations, such as the disturbance neutral winds [Abdu, 1997], which are beyond the object of this study.
 To better understand the performance of SVM model for the examples shown in Figures 2–7, the RMSE of SVM and STORM during these storm cases are listed in Table 4. It shows in Table 4 that the RMSE of SVM model is lower than that of STORM model during each storm event in both stations. In most cases, the RMSE of SVM in HK is lower than that in CQ, except the May 23–25, 2000 event. However, different results happen to STORM, with lower RMSE in CQ than in HK in most cases except the November 5–7, 2001 event. This result may be due to the different physical mechanism SVM and STORM model based on. If we check over Figures 2–7 again, we will find that most of large departures between observations and our model occur at local nighttime. The relevant larger RMSE are ascribed to these departures mostly. However, this could not prevent the fairly good predicting ability of SVM.
Table 4. The RMSE of SVM and STORM Model During the Storm Main Phase and All Periods of the Six Cases Selected
Feb 12–14, 2000
Nov 5–7, 2001
Aug 26–28, 1998
May 23–25, 2000
Oct 21–23, 1999
Apr 18–19, 2001
5. Conclusion and Discussion
 This paper presents a new model to forecast low-latitude storm-time ionospheric foF2 one hour ahead using SVM. The integral of disturbance electric field deduced from the solar wind data is included as input to the model, as well as the ionospheric observations. The model has been evaluated quantitatively for 26 storm events occurred during 1997–2002. The results show that the model performs well for most samples. Most of the disturbances could be captured by the model, whether the ionosphere exhibits prompt or prolonged responses to the solar wind disturbances. Compared with another empirical model, STORM, the SVM model gives smaller errors for the given samples as a whole.
 The state of low-latitude ionosphere is highly variable at different time scales ranging from years to seconds with the occurrence of ionospheric disturbances associated with geomagnetic storms. The inputs used here do not contain sufficient information for the dynamics of these events to model their rapid changes accurately. Other factors such as the neutral wind disturbances and the neutral composition changes are not included in the forecasting procedure. The knowledge of the ionospheric responses to other disturbance sources and corresponding observations is required to develop a more successful forecasting algorithm.
 The authors acknowledged the NOAA's National Geophysical Data Center (NGDC) for providing geomagnetic index Dst, the interplanetary magnetic field data, and the solar wind data. This work was supported by the Specialized Research Fund for State Key Laboratories, and the National Natural Science Foundation of China (grants 61032009, 40974092).