Locally linear neurofuzzy modeling and prediction of geomagnetic disturbances based on solar wind conditions


  • Javad Sharifie,

    1. Control and Intelligent Processing Center of Excellence, Electrical and Computer Engineering Department, University of Tehran, Tehran, Iran
    2. School of Cognitive Sciences, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran
    Search for more papers by this author
  • Caro Lucas,

    1. Control and Intelligent Processing Center of Excellence, Electrical and Computer Engineering Department, University of Tehran, Tehran, Iran
    2. School of Cognitive Sciences, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran
    Search for more papers by this author
  • Babak N. Araabi

    1. Control and Intelligent Processing Center of Excellence, Electrical and Computer Engineering Department, University of Tehran, Tehran, Iran
    2. School of Cognitive Sciences, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran
    Search for more papers by this author


[1] Disturbance storm time index (Dst) is nonlinearly related to solar wind data. In this paper, Dst past values, Dst derivative, past values of southward interplanetary magnetic field, and the square root of dynamic pressure are used as inputs for modeling and prediction of the Dst index, especially during extreme events. The geoeffective solar wind parameters are selected depending on the physical background of the geomagnetic storm procedure and physical models. A locally linear neurofuzzy model with a progressive tree construction learning algorithm is applied as a powerful tool for nonlinear modeling of Dst index on the basis of its past values and solar wind parameters. The result for modeling and prediction of several intense storms shows that the geomagnetic disturbance Dst index based on geoeffective parameters is a nonlinear model that could be considered as the nonlinear extension of empirical linear physical models. The method is applied for prediction of some geomagnetic storms. Obtained results show that using the proposed method, the predicted values of several extreme storms are highly correlated with observed values. In addition, prediction of the main phase of many storms shows a good match with observed data, which constitutes an appropriate approach for solar storm alerting to vulnerable industries.

1. Introduction

[2] The electric currents in the Sun and interplanetary medium generate a complex magnetic field which extends out into interplanetary space to form the interplanetary magnetic field (IMF). The Sun's magnetic field is carried out through the solar system by the solar wind as the Sun is rotating. Geomagnetic storms occur when the magnetospheric convection is enhanced, generally by a strong continuous southward IMF. The IMF leads to a decrease of Dst from normal values (near zero), and therefore geomagnetic disturbance occurs. Dst approximately measures the strength of the ring current, and the terrestrial ring current is an electric current flowing toroidally around the Earth which is centered at the equatorial plane and at altitudes of 104 to 6 × 104 km. Changes in this current lead to global decreases in the Earth's surface magnetic field, which are known as geomagnetic storms. An enhanced ring current, indicated by a decrease of the Dst index, implies a major impact on the structure and location of the magnetosphere regions and the boundaries that separate them. On a global scale, the ring current generates a magnetic moment which augments the Earth's magnetic moment as presented to the solar wind. Intense geomagnetic storms have severe effects on technological systems, such as disturbances or even permanent damage to telecommunication and navigation satellites, telecommunication cables, and power grids [e.g., Lanzerotti, 1994; Kappenman et al., 1997]. Currently, more than 200 communication satellites circle the Earth in synchronous orbits. A large magnetic storm can greatly increase the number of fast ions and electrons which hit those satellites. One of these effects is an electric charge on the satellite, usually negative, raising its voltage to hundreds or even thousands of volts. Charging by itself has little effect on the satellite's operation, although on a scientific satellite it would seriously distort observations. However, charging different parts of the satellite with different voltages induces current between them that could cause damage. For prediction of geomagnetic storms, monitored by Dst index, an empirical linear relationship was proposed by Burton et al. [1975] from knowledge of the solar wind velocity, density, and the southward component of IMF. In particular, Burton et al. [1975] developed an equation for the rate of change of pressure-corrected Dst, showing that it was a balance between injection and decay out of the ring current. They found that decay rate for the recovery phase depends on the present strength of the ring current (Dst). An empirical physical model based on effective solar wind parameters was extended recently [Temerin and Li, 2002], where the magnetosphere is shown to be strongly controlled by the solar wind. A number of data-driven techniques is also available for Dst prediction [Kamide et al., 1998; Detman and Vassiliadis, 1997; Joselyn, 1995]. The more successful techniques include statistical time series analysis [Baker, 1986], linear prediction filters [Iyemori et al., 1979], and linear and nonlinear autoregressive and moving average filters, including local linear prediction [Vassiliadis et al., 1995]. Several of these techniques are being implemented and tested for reliable prediction of geomagnetic indices under real-time conditions, for example, a linear filter for predicting Kp. An artificial intelligence technique is used internally in the Magnetospheric Specification and Forecast Model to predict Dst [Freeman et al., 1994]. Several neural network models [Munsami, 2000; Gleisner et al., 1996; Wintoft, 1997; Wintoft and Lundstedt, 1998; Wu and Lundstedt, 1996] as well as analogue models (differential model) based upon physical knowledge [Nagatsuma, 2002; Burton et al., 1975; Fenrich and Luhmann, 1998; O'Brien and McPherron, 2000a, 2000b] are applied for geomagnetic storm prediction. Kugblenu et al. [1999] used solar wind data as external inputs to a neural network for Dst prediction. In this study a locally linear neurofuzzy with model tree learning algorithm [Nelles, 1999, 2001] as a powerful model is applied for modeling and prediction of disturbance storm time index (Dst) on the basis of its past values and solar wind conditions. The paper is organized as follows: First, a locally linear neurofuzzy model with locally linear model tree learning algorithm is introduced, and their details are discussed. Then a physical discussion about the procedure of occurrence of geomagnetic disturbance near Earth is briefly mentioned, and some analogue models are introduced on the basis of previous studies. After that, a combination of the locally linear neurofuzzy model and physical knowledge about geomagnetic disturbance is applied for modeling and prediction of geomagnetic Dst index for several storms, and finally, concluding remarks are considered.

2. Locally Linear Neurofuzzy Model With Model Tree Learning

[3] The main idea for utilizing the locally linear neurofuzzy (LLNF) model for function approximation is dividing the input space into small linear subspaces with fuzzy validity functions ϕi (u). These functions describe the validity of each linear model in its region. The validity function applied here is the normalized Gaussian function, which is defined as μ(x) = exp (−equation image), where c is the center and σ is the standard deviation of the Gaussian. The Gaussian function is the membership function (degree of membership of a specific object to the fuzzy sets) used in this study.

[4] Each local linear subspace with its validity function is called a fuzzy neuron. Thus the total model is a neurofuzzy network with one hidden layer and a linear neuron in the output layer which simply calculates the weighted sum of the outputs of locally linear models (LLMs) as

equation image

where u = [u1u2up]T is the model input, M is the number of LLM neurons, and ωij denotes the LLM parameters of the ith neuron. The validity functions are chosen as normalized Gaussians; normalization is necessary for a proper interpretation of validity functions:

equation image
equation image

Each Gaussian validity function has two sets of parameters, centers (cij) and standard deviations (σij) which are the 2Mp parameters of the nonlinear hidden layer. Optimization or learning methods are used to adjust the two sets of parameters, the rule-consequent parameters of the locally linear models (ωij) and the rule premise parameters of validity functions (cij and σij). A least squares optimization method is used to adjust the parameters of local linear models (ωij), and a learning algorithm (described below) is used to adjust the parameters of validity functions (cij and σij). Global optimization of linear parameters is simply obtained by the least squares technique. The complete parameter vector contains M(p + 1) elements:

equation image

and the associated regression matrix X for N measured data samples is

equation image
equation image


equation image

where α is the regularization parameter for avoiding any near singularity of matrix XTX and in this study is empirically set to 0.002. The remarkable properties of the locally linear neurofuzzy model, its transparency and intuitive construction, lead to the use of the least squares technique for rule-antecedent parameters. For training the model, data sets should be linearly normalized to the interval [−1, 1]. The learning algorithm is introduced as described in section 3.

3. Learning Algorithm

[5] Locally Linear Model Tree (LOLIMOT) is a progressive tree construction algorithm that partitions the input space by axis bisection in all directions of input space. It implements a heuristic search for the rule premise parameters and avoids a time-consuming nonlinear optimization.

[6] The LOLIMOT algorithm is described in five steps according to Nelles [1999, 2001].

[7] 1. Start with an initial model: Start with a single LLM, which is a global linear model over the whole input space with Φ1(u) = 1, and set M = 1. If there is a priori input space partitioning, it can be used as the initial structure.

[8] 2. Find the worst LLM: Calculate a local loss function, for example, mean square error (MSE), for each of the i = 1, …, M LLMs and find the worst performing LLM.

[9] 3. Check all divisions: The worst LLM is considered for further refinement. The hyperrectangle (more than a three-dimensional rectangle or cube) of this LLM is split into two halves with an axis orthogonal split. Divisions in all dimensions are tried, and for each of the p divisions, the following steps are carried out. First, construct the multidimensional membership functions for both generated hyperrectangles and construct all validity functions: In part a, only the membership function of LLM that is split would change and the membership function of other neurons do not change, but all of the validity functions change that must be updated for all LLMs by equation (2). Second, estimate the rule-consequent parameters for newly generated LLMs. Third, calculate the loss function for the current overall model.

[10] 4. Find the best division: The best of the p alternatives checked in step 3 is selected, and the related validity functions and LLMs are constructed. The number of LLM neurons is incremented M = M + 1.

[11] 5. Test the termination condition: If the termination condition is met, then stop; otherwise, go to step 2.

[12] The termination condition is reaching to a predefined error between output (y) and LLNF output with M neuron (equation image), that is, when the condition: ∥yequation image∥ ≤ ɛ is satisfied. In practice we used a predefined number of neurons to LOLIMOT, plotted the error as a function of this number, and kept increasing the number of neurons until satisfactory performance was obtained. A suitable number of LLMs would be fit to training data on the basis of a validation set. The best number of LLMs is that in which the root mean square error (RMSE) for the validation set starts to increase. Details can be found in work by Nelles [2001].

[13] In each iteration, the worst performing locally linear neuron is determined to be divided. All the possible divisions in the p-dimensional input space are checked, and the best is selected. The splitting ratio can be simply set to 0.5, which means that the locally linear neuron is divided into two halves. The fuzzy validity functions for the new construction are updated; their centers are the centers of the new hypercubes (more than a three-dimensional cube), and the standard deviations are usually set to 0.7 times the width of the hypercube in that dimension. Figure 1 illustrates the operation of the LOLIMOT algorithm in the first four iterations for a two-dimensional input space. In iteration 1, a global linear model is fit to data. Then for refinement, input space is split into halves, and a local linear model is fit in each hyperrectangle. In iteration 2, first, the best possible splitting method is selected (e.g., in Figure 2, iteration 2 splitting along the u2 axis is assumed to be better), then in the selected model, the worst LLM should be used for further refinement (blue rectangle or 2-1, for instance), and the algorithm continues with a default number of LLMs.

Figure 1.

Illustration of LOLIMOT algorithm for two-dimensional input space.

Figure 2.

Reconnection between solar wind magnetic field and the Earth's magnetic field. This phenomenon was originally proposed by Dungey [1961]. Reconnection opens the magnetosphere and allows the entry of plasma, momentum, and energy. Magnetospheric convection is indicated by open arrows.

4. Geomagnetic Disturbance Storm Modeling and Prediction

[14] The solar wind is an extended ionized gas with very high electrical conductivity. It drags some magnetic fluxes out of the Sun. A geomagnetic storm occurs when the solar wind causes deep and intense circulation of the magnetospheric plasma. This usually excites by reconnection (the merging of oppositely directed magnetic fields) between the IMF and the Earth's magnetic field as illustrated in Figure 2. The IMF must be strong and steadily southward for several hours [Russell et al., 1974]. The dynamic pressure compresses the magnetosphere and increases the field strength on the Earth's surface. A sudden compression of magnetosphere size by an increase of solar wind dynamic pressure leads to a sudden variation in both surface field and Dst index. Furthermore, it has been shown that a northward IMF component could carry the solar wind plasma to the Earth's magnetosphere [Hasegawa et al., 2004].

4.1. Analogue Models for Dst Index

[15] Burton et al. [1975] proposed an empirical linear model for predicting Dst index from solar wind parameters. Their model indicates that ring current and magnetopause current are proportional to the square root of the solar wind dynamic pressure. In addition, their equation contains the energy-coupling functions for southward and northward interplanetary magnetic fields, which are the product of the outward solar wind velocity and the northward component of the interplanetary magnetic field. Many modeling methods [Vassiliadis et al., 1999; McPherron, 1997; Klimas et al., 1997] have related the Dst index to the negative part of Bz (Bs) and have used them as important input parameters to the system. Temerin and Li [2002] recently developed an empirical physical model based on effective solar wind parameters and showed that the magnetosphere is strongly controlled by the solar wind. Thus magnetic storms are highly predictable from solar wind measurements. Thus it is reasonable to predict Dst on the basis of its past values and other geoeffective solar wind parameters. An intense geomagnetic disturbance storm has four region of activity as follows: impulse or storm commencement (Dst > 0, d(Dst)/dt > 0), return from impulse (Dst > 0, d(Dst)/dt < 0), main phase of the storm commencement (Dst < 0, d(Dst)/dt < 0), and the recovery phase (Dst < 0, d(Dst)/dt > 0).

[16] The study of Wang et al. [2003] implies that the ring current injection increases when the magnetosphere compresses by high solar wind dynamic pressure. Their results demonstrate that the prediction of Dst by using O'Brien and McPherron's [2000b] model is improved, especially for intense geomagnetic storms when the influence of the solar wind dynamic pressure on the decay and injection of ring current is taken into account.

4.2. Nonlinear Modeling and Prediction of Dst Index

[17] Geomagnetic storms dominantly originate from solar wind flows. On the basis of physical research and our studies about geomagnetic storm and solar terrestrial physics, it is obvious that Dst index modeling based on solar wind conditions is nonlinear (may be a local dynamical model). It is reasonable to utilize a locally linear model for Dst index modeling such that several local lines are fit for each region of data set. In addition, soft switching between each local linear model with a membership function causes avoiding from discontinuity. The hourly omni-directional data are downloaded from the National Space Science Data Center and are used here. History of the Dst index, history of Bs, the Dst derivative, and the square root of dynamic pressure past values are used as geoeffective inputs to LOLIMOT for modeling and prediction of the next Dst values. The solar wind dynamic pressure is computed as follows:

equation image

where n and V are the density and velocity of solar wind, respectively. For the modeling and prediction of geomagnetic storms that is studied here, inputs to the LOLIMOT model depend on physical backgrounds mentioned before, and they are selected as follows:

equation image

For selecting the appropriate number of LLMs for prediction of storms in order to obtain the maximum generalization, the error criteria with respect to number of LLMs must be computed and plotted; the appropriate number of LLMs is that where the error criteria in the validation set start to increase. Some other criteria that are computed for predictions studied here are the average relative variance (ARV), normalized mean square error (NMSE), mean square error (MSE), and root mean square error (RMSE), which are computed as follows:

equation image
equation image
equation image
equation image

where y, equation image, and equation image are observed, predicted, and average observed values, respectively. The average accuracy of the forecasts in the sample relative to the accuracy of forecasts produced by a reference method is called skill, which can be measured by any number of so-called skill scores (SS). On the basis of MSE and RMSE, the skill scores are defined as

equation image
equation image

where MSE and RMSE are the mean square error of the sample forecasts and MSEr and RMSEr are the mean square error of the reference forecasts.

[18] The data before storms are predicted are used for model selection. About 80% of this data is used for training the model, and about 20% is used for validating (selecting the number of neurons) in order to predict the test data set. The test set differs from both the training and the validating sets. The data set from 1998 to 6 April 2000 is applied for model selection to predict the storm of 6–8 April 2000. Three LLMs are applied for prediction of this storm, and a correlation coefficient of 0.982 is achieved in this case. The data set from 1998 to 20 June 2002 is used for model selection for predicting the storms from 21 June 2002 to the end of 2002. In order to achieve maximum generalization, in Figure 3, NMSE is plotted with respect to the number of LLMs for both training and validation data; the LLNF model for best structure must have 11 LLMs. A correlation coefficient of 0.988 between observed Dst and predicted Dst for storms from 1 September 2002 to the end of 2002 is achieved that implies almost perfect prediction of Dst index with the LOLIMOT model on the basis of geoeffective solar wind conditions and Dst past values. In Figure 4, observed versus predicted values are depicted for prediction of these storms that shows a compact error bar and a little deviation between predicted values and observed values. In Figures 5 and 6, two storms are shown with higher timescale resolution for two models: in model 1, only the Dst past values are used in LOLIMOT and in model 2, both Dst past values and solar wind data are applied. As depicted in Figures 5 and 6, main and recovery phases except in commencement of the storm are predicted almost perfectly and have no time delay between predictions and observations, owing to dynamic modeling of the geomagnetic disturbance Dst index on the basis of geoeffective data and Dst dynamic data by using LOLIMOT. In model 1 (using only Dst past values), predictions in most phases are approximately the same as in model 2, but the prediction in the main phase has delay time. Correlation coefficients between predicted and observed Dst index for the storms in Figures 5 and 6 by model 2 are 0.99 and 0.987, and the ARVs of these predictions are 0.008 and 0.025, respectively, implying that 99% and 97.5%, respectively, of these storm variances are predictable. The correlation coefficients of model 1 in Figures 5 and 6 are 0.985 and 0.960, and the ARVs are 0.083 and 0.028, respectively. As found from this result, by using physical background for prediction of these storms in the LOLIMOT model, not only does the prediction accuracy increase but also the time delay for the main phase is eliminated, but both models have delay for prediction of storm commencement. Kugblenu et al. [1999] employed solar wind data as external inputs to a multilayer perceptron (MLP) neural network for Dst prediction. Their network has three layers. They have used intense storms from years 1972, 1973, 1974, 1975, 1977, 1978, 1979, 1981, and 1982 for training the MLP network to predict the geomagnetic storm on 19 December 1980, and their method achieved a correlation coefficient of 0.96, ARV of 0.04, and RMSE equal to 11 nT. Although we have not used their data set, we have trained a similar MLP model with our data set with the same input parameters to both MLP and LOLIMOT models for comparison. The data sets from 1978 and 1979 and also part of 1980 prior to this event are used for model selection in order to predict this storm. By using the MLP neural network, the best number of hidden neurons is seven, and for the LOLIMOT model it is two. The LOLIMOT method achieved a correlation coefficient of 0.987 for prediction of this storm, and two LLMs was the best number of neurons to achieve maximum generalization. The prediction result is depicted in Figure 7. With the LOLIMOT method, the ARV is 0.024, which means that 97.54% of observed Dst variance is predictable from both Dst and the other geoeffective parameters. RMSE with the LOLIMOT model is 8.61, which is equivalent to SSRMSE = 0.217 with respect to the MLP model [Kugblenu et al., 1999]. The correlation of 0.985, ARV = 0.030, and RMSE = 9.61 nT are obtained with the MLP network, and SSRMSE of LOLIMOT with respect to the MLP model is 0.104.

Figure 3.

Error criteria for selecting the appropriate number of LLMs for prediction of storms in year 2002: data from first of year 1998 to 20 June 2002 are used for model selection and from September 2002 to end of 2002 for validation; appropriate number of LLMs to achieve maximum generalization is 11 with RMSE = 4.7 nT.

Figure 4.

One-step predicted versus observed Dst index with LOLIMOT method with 11 LLMs, based on solar wind conditions for the first of September to the end of December 2002. Data from year 1998 prior to September 2002 are used for model selection. Correlation coefficient = 0.988.

Figure 5.

One-step prediction of geomagnetic storm on 2–3 September 2002 by LOLIMOT with two sets of input parameters: model 1, only Dst past values (correlation = 0.985, ARV = 0.03, NMSE = 0.022, and RMSE = 8.43 nT) and model 2, both Dst past values and solar wind parameters (correlation = 0.995, ARV = 0.008, NMSE = 0.006, and RMSE = 4.55 nT).

Figure 6.

One-step prediction of geomagnetic storm on 8–9 September 2002 by LOLIMOT with two sets of input parameters: model 1, only Dst past values (correlation = 0.960, ARV = 0.083, NMSE = 0.017, and RMSE = 12.04 nT) and model 2, both Dst past values and solar wind parameters (correlation = 0.986, ARV = 0.029, NMSE = 0.006, and RMSE = 7.10 nT).

Figure 7.

One-step prediction of storm on 19 December 1980 with LOLIMOT based on Dst past values and geoeffective parameters. Correlation coefficient = 0.987 and ARV = 0.024.

[19] A question may arise whether the complex modeling and extrapolation method (locally linear with LOLIMOT) could perform better than linear extrapolation method. As mentioned in section 2, first, the model fits a global linear model, and if the performance is good, then the algorithm stops, and if not, the algorithm continues for constructing a locally linear model. In Figure 3, an error criterion is plotted as a function of LLM number. LLM = 1 indicates a global linear extrapolation model that is less efficient with respect to a locally linear model with LLM = 11 for this data set (see caption for Figure 3). In some other data sets, one linear model might be the best model, but such nonlinear processing is essential to find how many locally linear models have the best structure for applied data for modeling.

[20] Gleisner et al. [1996] have shown that the solar wind history for 18–24 hours is needed as input to a trained network in order to reproduce all phases of a geomagnetic storm. They developed a time delay feed forward neural network based on a temporal sequence of solar wind data. Their networks showed better performance with larger temporal size of the input data sequence and were able to reproduce 85% of the Dst variance, which is quite an improvement on the earlier work of Lundstedt and Wintoft [1994]. Wu and Lundstedt [1996], attempting to reproduce the recovery phase that was difficult to model with the networks of Lundstedt and Wintoft [1994], used Elman recurrent networks. The Elman network is an extension of the multilayer back propagation networks with an addition of a feedback connection from hidden to input layers which allows the network to detect and generate time-varying patterns. Their Elman network model obtained a relatively high correlation coefficient, that is, 0.91, for a very long period of Dst (900 hours). For each storm period, however, their model does not always give a good fit to the observations, but the LOLIMOT method with geoeffective data shows very good performance for prediction of several intense geomagnetic storms. In Table 1, correlation coefficient, RMSE, NMSE, and ARV for prediction of several intense geomagnetic storms with the LOLIMOT method on the basis of solar wind data and Dst past values are reported. According to Table 1, prediction of several storms has a high correlation coefficient with observed values, and ARV for all of the case studies is less than 0.082, which shows that even in the worst case, 92% of storm variance is predictable, and in the best case, 98% of storm variance is predictable. In addition, the best number of LLMs for achieving maximum generalization is at least two for prediction of storms on 15–18 July 2000, and the other geomagnetic storms have different numbers of LLMs as reported in Table 1; this implies that the geomagnetic disturbance Dst index based on solar wind data is a nonlinear dynamic system, which could be considered as an extension of Burton et al.'s [1975] empirical linear model.

Table 1. Real-Time Prediction of Some Geomagnetic Storms With LOLIMOT Models Based on Solar Wind Data
Geomagnetic Storm DateCorrelationRMSE, nTNMSEARVBest Number of LLMs
6–8 Apr 20000.98213.4580.0110.0353
15–18 Jul 20000.96027.4460.0280.0812
11–13 Aug 20000.9848.6910.0070.0324
31 Mar to 2 Apr 20010.98915.8520.0090.0213
1–2 Oct 20000.9819.4510.0060.04111

[21] Figure 8 shows the one-step prediction of a very intense geomagnetic storm from 1200 UT on 31 March to 1200 UT on 2 April 2001. The data set from 1998 prior to this event is utilized for model selection to predict the event. In training and validating the data set, the value of events does not exceed −301 nT, but in the test set, a geomagnetic storm with value −387 nT, is predicted perfectly. The minimum value of the event is −387 nT, and the prediction of this value is −372.51 nT. A correlation coefficient of 0.989 and an ARV of 0.021 are achieved for prediction of this extreme storm, which indicates that almost 98% of Dst variance of this storm is predictable; these results imply near-perfect prediction of this event without delay, which is depicted in Figure 8. As found from Figure 8, by using the solar wind parameters, the main phase of the storm is not only predicted without time delay owing to utilization of the southward IMF, but also the prediction of the recovery phase of the storm has an almost perfect match with observed values owing to the use of dynamic pressure history as an external input to the novel nonlinear LOLIMOT model. By using only Dst past values, the prediction in the main phase not only has a 1 hour delay but also in the recovery phase has less accuracy with respect to a situation using the solar wind parameters as external physical parameters. The data set from 1995 to 1999 is fitted with three LLMs, and a linear correlation coefficient of 0.984 and a RMSE of 4.31 nT are gained. In addition, the model is tested for the first half of the year 2000, and a correlation coefficient of 0.983 and a RMSE of 4.38 nT are obtained for this interval. It is worth mentioning that we do not compare these results with Temerin and Li's [2002] model through (15), since we have used a different set of inputs, which makes the comparison unfair.

Figure 8.

One-step prediction of geomagnetic storm from 1200 UT on 31 March to 1200 UT on 2 April 2001 with LOLIMOT with two sets of input parameters: model 1, only Dst past values (correlation = 0.981, RMSE = 22.68 nT, and ARV = 0.041) and model 2, both Dst past values and solar wind parameters (correlation = 0.989, RMSE = 15.852 nT, and ARV = 0.021).

[22] To highlight the contribution of physical parameters for Dst prediction with the LOLIMOT model, first, the LOLIMOT model is trained and validated with both solar wind and past values of measured Dst, then the model is applied for prediction of further values of Dst by using only the observed solar wind data and predicted values of Dst from previous steps of the LOLIMOT model instead of using the measured Dst. In other words, the input to the LOLIMOT model in prediction mode for predicting the further time steps of Dst index is as follows:

equation image

where equation imagest(t − 1), equation imagest(t − 2), and equation imagest(t − 3) are the Dst predictions at times t − 1, t − 2, and t − 3, respectively. Figure 9 shows the prediction result of the storm on 19 December 1980. It can be seen from Figure 8 that the peak of the storm and its vicinity are predicted well. More significantly, we see that before the commencement of the storm and decreasing of the measured Dst index, the prediction starts to decrease, and this indicates that the magnetosphere and Dst index are controlled by solar wind. This result is compatible with the result of recent physical models [Temerin and Li, 2002]. The result of Figure 9 also implies that the LOLIMOT model could correlate the dynamic of solar wind data to Dst index and have a correlation coefficient of 0.88 when prediction is done only with observation of solar wind condition. However, prediction with LOLIMOT for a long time by using the predicted Dst instead of measured Dst as input to LOLIMOT is not possible, and after a time, prediction diverges to infinity. An important point is that when solar wind is southward for several hours, then the entry of plasma to magnetosphere starts, and the change in the ring current leads to the occurrence of a geomagnetic storm. We can predict the Dst index that in this situation assumes an abnormal negative value by using solar wind parameters. The large-amplitude oscillation around the mean Dst behavior, however, indicates that even with the inclusion of the solar wind data, the model loses forecasting value significantly when predicted past values of Dst are used instead of the observed past values. The model predicts Dst of −100 nT, indicating a major geomagnetic storm, about 8 hours before the actual commencement of the storm. But this essentially renders the LOLIMOT prediction without using the observed data worthless since 8-hour error is well beyond any tolerance level of usable Dst forecast. It can therefore be concluded that although physical data is very important for achieving good prediction and alerting capability, the identification of the internal dynamics of the Dst time series via advanced nonlinear identification and prediction techniques is a crucial factor, and without the use of observed data, one has absolutely no faith in the changes of the modeled Dst that could signal, for example, storm commencement or the beginning of the recovery phase.

Figure 9.

Prediction of storm on 19 December 1980 by using predicted Dst (not the measured Dst) for prediction of consequent Dst values. Correlation coefficient = 0.88.

[23] Then using the solar wind parameters in the LOLIMOT model causes that model to detect the decrease in the Dst index more promptly, and as a result, the contribution of physics or solar wind parameters is in eliminating the time delay in the main phase of the storm and also increasing the prediction accuracy. If only the past values of the Dst index are utilized for the LOLIMOT model, then the prediction, especially in the main phase of the storm, has longer time delay, and this kind of prediction has less worth, and the model may in some crucial cases perform slightly more than attributing the present observed Dst value as the prediction of the next hour.

5. Multistep Prediction of Dst Index

[24] In order to investigate the model accuracy with the aid of external physical parameters (e.g., solar wind parameters), multistep prediction of the Dst index is tested here. The data set in 1995–1999 is applied for model selection in order to predict the Dst index during the first half of the year 2000 with different prediction horizons. For k step prediction, the input to the LOLIMOT model is as follows:

equation image

Can we use as much of the available data as possible to build a single LOLIMOT model that should perform well for years to come instead of creating a bunch of specialized models that describe little more than the idiosyncrasies of a particular storm or time period? In Table 2, the real time prediction for various prediction horizons is reported. Here 10 years of the data set were used for modeling, and half a year was used for prediction. The model with three locally linear models is found to be appropriate for different time steps. It can be concluded that a single LOLIMOT model comprising three locally linear models will model Dst index in the long run. This could be considered as a refinement of a global linear model discussed by Burton et al. [1975]. Another conclusion is that physical parameters like solar wind data can be used in addition to the lagged Dst values for improving prediction accuracies, especially in time intervals where there are time lags in predictions, though they do not always and completely eliminate the time lag problem. Better understanding of the relationship between Dst and physical processes is surely necessary for constructing better predictors.

Table 2. Multistep Prediction of Dst Index in the First Half of the Year 2000 With Modeling From Data Set in Years 1990–1999
Step Ahead PredictionCorrelationNMSERMSE, nTBest Number of LLMs
1 hour ahead0.9830.0234.383
2 hours ahead0.9510.0677.433
3 hours ahead0.9090.121103
4 hours ahead0.8700.17011.833

[25] The correlation of test data with their lag values for 1–4 hour lagged values is {0.973, 0.930, 0.886, and 0.845}. The correlation between observed and predicted Dst for 1–4 hour–ahead prediction as reported in Table 2 is more than the correlation of test data with their lagged values. For instance, 1 hour prediction might have a correlation not much more than the correlation of Dst index with its past values, and someone who assumes the present value of Dst index as the prediction of the next hour might obtain a high correlation coefficient, but such insight has a harmful effect. For intense storms, despite a high correlation between predicted and observed Dst, detecting the sudden decrease in index is very important for alerting technological systems such as satellites and power systems.

6. Conclusions

[26] A locally linear neurofuzzy model was used for modeling and prediction of geomagnetic disturbance storms on the basis of solar wind conditions. A geomagnetic disturbance storm was depicted as a nonlinear dynamical system. We fit in each region of geomagnetic activity a number of locally linear models with Gaussian soft switching between local models. This led to a global nonlinear dynamical modeling of the Dst index based on some of its past values and solar wind conditions, which might be considered as an extension of empirical linear models [e.g., Burton et al., 1975]. The proposed method was applied to modeling of the Dst index and prediction of several extreme storms. Prediction results showed no delay in predicting the main phase of the storm, which was more critical for alerting the vulnerable technological systems. The proposed method presented an important step forward with respect to past works on prediction of geomagnetic and solar indices on the basis of time series autoregression coupled with advanced statistical and neurofuzzy techniques [Gholipour et al., 2004, 2005; Sharifie et al., 2006]. The physics of the magnetosphere and its interactions with solar wind plasma had an important effect on most of the intense geomagnetic storms and their models. The LOLIMOT model could be considered as a nonlinear (locally linear) extension of some linear empirical physical models, including those of Burton et al. [1975], McPherron [1997], and O'Brien and McPherron [2000a, 2000b]. This interpretation also leads to the appreciation that it has been shown that the nonlinear extension can improve the Dst predictions beyond, for example, simple linear extrapolation. It is also shown that although solar wind data by themselves cannot achieve more, their addition can further improve the predictive and alerting power. In our model, the relation between solar wind parameters and Dst index was nonlinear. The model furnished a good prospect for prediction of many other extreme geomagnetic storms. As a result, it could serve as a tool for alerting human technological systems to geomagnetic anomalies. Future progress in this field could be achieved through a better understanding of the physical processes as well as more creative use of intelligent and statistical techniques for prediction and alerting.


[27] The authors wish to express their appreciation for the very helpful comments made by two anonymous reviewers. The revisions made in accordance to their suggestions have significantly improved the presentation as well as the scientific content of the paper.