Forecasting of global data-binning parameters for high-resolution empirical geomagnetic field models

Transcription

1 SPACE WEATHER, VOL. 10,, doi: /2012sw000783, 2012 Forecasting of global data-binning parameters for high-resolution empirical geomagnetic field models M. I. Sitnov, 1 A. Y. Ukhorskiy, 1 and G. K. Stephens 1 Received 20 February 2012; revised 14 August 2012; accepted 16 August 2012; published 18 September [1] A recently developed empirical model, TS07D ( has provided for the first time a realistic description of the storm-scale geomagnetic field, free from a priori assumptions about the shape of the main magnetospheric current systems. The model uses information about the global state of the magnetosphere and its solar wind driving in the form of the Sym-H index averaged over substorm scales, its time derivative, and a similarly averaged solar wind electric field. The set of global parameters is used to bin a large number of points in a historical magnetic field database and to fit the model using only a part of the database composed of points taken at times when the global data-binning parameters were close to those at the time of interest. Transition from modeling to forecasting in such models requires a modification of their data binning procedure to use only the information about the state of the solar wind and the magnetosphere prior to the time of interest. This can be done using another (forecasting) set of binning parameters, which is optimized to provide the best prediction of the original (modeling) binning set. Several sets of such parameters are investigated. It is shown that the original (modeling) set contains spurious information, which appears because of the averaging of the solar wind electric field and Sym-H index over future moments in time. As a result, the problem of the optimization of the forecasting set becomes ill-posed, and the prediction efficiency of the original binning parameters, especially the averaged time derivative of the Sym-H index, is strongly limited. The predictions are shown to be substantially improved when the forecasting set includes an exponential decay function of the solar wind electric field proposed originally by Burton et al. (1975) with the e-folding time of the order of one hour. Citation: Sitnov, M. I., A. Y. Ukhorskiy, and G. K. Stephens (2012), Forecasting of global data-binning parameters for high-resolution empirical geomagnetic field models, Space Weather, 10,, doi: /2012sw Introduction [2] Magnetic field is the main factor that determines the structure of the magnetosphere and its dynamics during storms and substorms. Its timely, accurate, and reliable forecasting is critical for the National Space Weather Program. In recent years it has become apparent that storm-time variations in the field affect radiation belt particle motion in ways not conceived earlier. For example, the diamagnetic effect due to storm-time ring current, which can produce fast outward expansion of electron 1 Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland, USA. Corresponding author: M. I. Sitnov, Johns Hopkins University Applied Physics Laboratory, Johns Hopkins Rd., Laurel, MD , USA. (mikhail.sitnov@jhuapl.edu) longitudinal drift orbits in the adiabatic regime [Dessler and Karplus, 1960], was shown to be strongly enhanced by nonadiabatic effects leading to rapid loss of electrons through the magnetopause [Ukhorskiy et al., 2006]. This effect can explain rapid depletion of electron fluxes over a broad region of the outer electron belt during storm main phase [Turner et al., 2012, and references therein]. [3] Eventually, comprehensive modeling and forecasting of the geospace will be made using first-principles approaches, which can provide a self-consistent description of electromagnetic fields and plasma dynamics. However, at present the capabilities of the first-principles modeling are rather limited. The MHD approach [e.g., Lyon et al., 2004] does not resolve important kinetic processes associated with magnetic reconnection at the magnetopause and in the magnetotail as well as with energydependent particle drifts and wave-particle interactions in the inner magnetosphere. A special kinetic treatment of the American Geophysical Union. All Rights Reserved. 1of8

2 inner magnetosphere [Wolf et al., 2009, and references therein] remains limited to slow variations and an incomplete information on the boundary conditions. Its coupling with MHD models [e.g., Pembroke et al., 2012] has a strong potential of the comprehensive first-principles description of the magnetosphere but it also has many free parameters to be optimized. [4] Against this background, the abundance of experimental data on the geomagnetic field [Sitnov et al., 2008, and references therein] brings its empirical modeling forward as an important space weather instrument, complementary to the first-principles models. At the same time, the tremendous increase of the amount of magnetospheric data, which occurred in the last decades due to multiple missions, requires the appropriate advance in empirical modeling methods. In the case of the geomagnetic field models [e.g., Sitnov et al., 2010, and references therein] this requires a substantial increase in the model number of degrees of freedom and the use of sophisticated data mining techniques. In such techniques a small subset of the whole training database is selected, which represents only data most relevant to the event of interest. The subset must have a sufficient number of magnetic field records to match the large number of degrees of freedom in the model. At the same time, it must be selected using several global binning parameters, which determine the state of the magnetosphere (e.g., the main phase of a magnetic storm) and its solar wind driver. In the empirical model TS07D [Tsyganenko and Sitnov, 2007; Sitnov et al., 2008] (see also points of magnetometer data from many missions, such as Cluster, Geotail, Goes 8, 9, 10 and 12, IMP 8 and Polar, are binned to find a 1% subset of them, obtained when the state of the magnetosphere determined by three global parameters, the Sym-H index and its time derivative averaged over substorm scales as well as a similarly averaged solar wind electric field, was close to its state at the time of interest. Then the instantaneous spatial picture of the geomagnetic field on storm scales is obtained by fitting the model using only that subset of data. [5] The original TS07D model had been successfully tested and validated in different regimes, including the in-sample modeling, when the considered event is a part of the training database ( ) [Sitnov et al., 2008], and the out-of-sample regime, when the considered event is beyond the training database [Sitnov et al., 2010]. At the same time, the initial set of the binning parameters was neither optimized nor made suitable for forecasting purposes. In particular, the original averaging scale (of the order of the ring current decay time) was chosen to eliminate substorm effects, such as the thinning of the tail current sheet in the growth phase of a substorm and the magnetic field dipolarization in the expansion phase. This includes a time interval in the future with respect to the moment of interest, and precludes the use of the model as a practical space weather forecasting tool. [6] Elaborating a forecasting version of the model requires an investigation of a new class of the binning parameters obtained by integration in time over only previous moments in time relative to the time of interest. If the original (modeling) set can be predicted using the propagation in time of the nearest neighbors (NN) of the considered point in the new (forecasting) space of binning parameters, the optimization of that one-step prediction procedure will effectively provide a new forecasting basis for TS07D, which will be additionally optimized. The NN method was successfully used to forecast global substorm indices, such as the substorm index AL [Vassiliadis et al., 1995; Ukhorskiy et al., 2004] and the storm activity index Dst [Valdivia et al., 1996; Vassiliadis et al., 1999], as well as midlatitude magnetic fluctuations observed by ground magnetometers [Valdivia et al., 1999]. Since the data binning process to reconstruct the spatial structure of the geomagnetic field in the TS07D model [Sitnov et al., 2008] is based on a similar NN approach, the successful prediction of the original binning parameters using another NNmethod and another set of forecasting global parameters allows one to replace the original set by the forecasting one. [7] In this paper we investigate various sets of the global binning parameters, which exclude future moments in time and hence transform TS07D into a forecasting tool. Their performance is compared using a formal criterion of the average relative variance or the percentage of the data variance not represented by predictions [e.g., Vassiliadis, 2006] and the detailed analysis of predictions for individual storms. 2. Modeling and Forecasting Sets of the Global Data-Binning Parameters [8] The state of the magnetosphere is described in TS07D using three global parameters [Sitnov et al., 2010]. These are the average solar wind electric field parameter: hvb z iðþ t Z P=2 vb z ðt þ tþcosðtp=pþdt; ð1þ where B z is the z-component of the interplanetary magnetic field in the GSM coordinate system and v is the solar wind velocity (v = v x ), a similarly averaged pressure-corrected Sym-H index [Iyemori, 1990]: Z P=2 hsym HiðÞ t Sym Htþ ð tþcosðtp=pþdt ð2þ (the details of the pressure corrections can be found in Tsyganenko [1996]; they are also available at geomag_field.jhuapl.edu/model/pressure_correction.html), and another function of the Sym-H index reflecting its averagetimederivative: DSym h HiðÞ t Dt Z P=2 Sym Htþ ð tþsinð2tp=pþdt ð3þ The notation D.. /Dt is used for the third parameter to reflect the fact that in general it is not equal to the time 2of8

3 Figure 1. Filter kernels for the original set of the TS07D binning parameters vb z (t), Sym-H (t) and D Sym-H (t)/dt. derivative of the second: D.. /Dt d.. /dt. The state-space formed by parameters (1) (3) is similar to the phase space formed by the parameters vb s (B s is the southward IMF component), Dst and ddst/dt discussed in forecasting models of the Dst index and the analysis of the storm-time evolution of the magnetosphere as a dynamic system [Vassiliadis et al., 1999]. The main distinction of (1) (3) is the filtering of the original data using convolutions with kernels shown in Figure 1 to filter out the substorm-scale variations (see, in particular, Sitnov et al. [2010, Figure 13]). The specific value of the filter timescale P/2 = 6 hours was taken to be of the order of the ring current decay time [Burton et al., 1975]. [9] Note, that the normalization coefficients in equations (1) (3) are omitted because these global parameters are normalized in the binning procedure below by their standard deviations over the database. Note also, that two other important global parameters of the model, the dipole tilt angle and the solar wind dynamic pressure are built into the main spatial structure of the model. In particular, the solar wind ram pressure, P dyn, is taken into account due to pressure-dependent amplitude coefficients of the equatorial current basis functions [Sitnov et al., 2008]. Therefore the tilt angle and the dynamic pressure can be excluded from the following data mining procedure. [10] In the TS07D model, to reconstruct the spatial structure of the magnetic field by fitting its observed values with the model parameters only a small subset of L NN = 8000 points of the whole database of 10 6 points is used. Each point in the database is a point G (i) in the 3D statespace of the global binning parameters (1) (3) normalized by their standard deviations. Its position in the state-space reflects the global state of the storm-scale magnetosphere as a dynamic system at the moment when the measurement was made (e.g., the main storm phase). It is also a point in the real 3D space, and its coordinates (X (i), Y (i), Z (i) ) determine the place in the magnetosphere where the (i)-th measurement was made. The subset of L NN points (they are called the nearest neighbors or NNs) includes only the points G (i) NN, which neighbor the magnetosphere at (i) the moment of interest in the state-space R = G NN G < R NN, where i = 1,.., L NN. In the latter condition G =( vb z, Sym-H, D Sym-H /Dt) is the point of interest in the state-space, the maximum distance R NN = R NN (L NN ) is determined by the given number of NNs, and.. is the conventional Euclidean metric. The specific value L NN = 8000 chosen in the model provides one data point per cubic Earth radius R 3 E in case of a homogeneous distribution of NNs in the volume L x L y L z =20R E 20 R E 20 R E, which roughly corresponds to the modeled region of the magnetosphere [Tsyganenko and Sitnov, 2007]. [11] The forecasting of global binning parameters (1) (3), to use only past records of the magnetospheric activity and present solar wind and IMF data, can be done using various techniques. For example, predictions of the Dst index were done using linear and nonlinear filters [Burton et al., 1975; Valdivia et al., 1996; Vassiliadis et al., 1999; McPherron and O Brien, 2001; Temerin and Li, 2002; Siscoe et al., 2005], artificial neural networks [Wu and Lundstedt, 1996; Pallocchia et al., 2006] and other approaches [Klimas et al., 1998; Boynton et al., 2011]. However, the original TS07D idea of the use of data from a local vicinity of the event of interest in the state-space (1) (3) [Sitnov et al., 2008] favors methods based on the local approximation of the system evolution in the state-space, such as the locally linear autoregressive moving-average (ARMA) filters [e.g. Vassiliadis et al., 1995, 1999; Valdivia et al., 1996; Ukhorskiy et al., 2004]. [12] In the locally linear filters an additional procedure of the time delay embedding is usually employed to increase the dimension of the analysis space, at the expense of new variables obtained from the original time series shifted in time. It helps unfold possible attractors of the dynamical system [Packard et al., 1980]. The idea of the approach is that a complex nonlinear dynamical system may be described by a system of ordinary differential equations, and those of them, which are not observed directly, can be expressed in terms of the time derivatives of the known variables. Since the time derivative is proportional to the difference between the original variable and its time delayed value, new variables obtained from the original ones by the constant time delays may contain important dynamical information, which helps model the considered system and predict its future states. [13] In that approach [see, e.g., Vassiliadis, 2006] the evolution of the system is considered in an extended statespace formed by the original input (I) and output (O) parameters and their time delayed derivatives X(t) =(I(t), I(t Dt),.., I( (m I 1)DT), O(t), O(t Dt),.., O( (m O 1)DT), where Dt is the time delay while m I and m O are maximum embedding dimensions for input and output parameters. The evolution equation dx=dt ¼ FX ð Þ is linearized around the considered point X 0 = X(t 0 ) ð4þ dx=dt FX ð 0 ÞþðdF=dXÞðX X 0 Þ ð5þ Then the local model of the evolution, which allows to determine in particular the next step in time for the system 3of8

4 Figure 2. Filter kernels for forecasting parameters vb z, D vb z /Dt, D 2 vb z /Dt 2 and Sym-H (BMR). output, can be found by averaging (5) over the nearest neighbors of X 0 dx=dt hfx ð 0 Þi jnn þ hdf=dxi jnn ðx X 0 Þ ð6þ and D 2 hvb z j Dt 2 ðþ t Z 0 vb z ðt þ tþsinð3p=2 þ 4tp=PÞdt; ð9þ where.. NN is the mean over the nearest neighbors. When the NN set represents only a small fraction of the whole database, and therefore, they occupy only a small vicinity of X 0 in the phase space, the forecasting of the system, including the local approximation (5) and the explicit integration of (6), can be reduced to averaging over the nearest neighbors X X NN since their evolution is already known. [14] The locally linear ARMA filters can be further optimized by applying the principal component analysis [Broomhead and King, 1986], which helps extract the linear combinations of input and output parameters and their time delays showing the strongest variance [Ukhorskiy et al., 2004]. Their further optimization requires a proper choice of the time delay parameter Dt and the maximum embedding dimensions for input and output parameters m I and m O. This is usually done by minimizing variations between the observed and predicted values of the output parameter O(t) [e.g., Vassiliadis et al., 1995]. However, in our case such an optimization may be ill-posed because of unknown impact of substorm effects, which are taken into account neither in the model structure [Tsyganenko and Sitnov, 2007] nor in its original binning set (1) (3), and because of the spurious information in the original set of the binning parameters (1) (3) as is discussed in detail in the next section. [15] The time delay embedding method was initially proposed as a discrete analog of the original approach using higher time derivatives of the original input and output parameters [see, e.g., Vassiliadis, 2006, section 4.2] with the goal, in particular, to reduce noise in the system. However, a similar result can be achieved using filters similar to (1) (3). Let us define the new forecasting set of global parameters as follows: Z 0 hvb z jðþ t vb z ðt þ tþcosðtp=pþdt; ð7þ DvB h z j ðþ t Dt Z 0 vb z ðt þ tþsinðp=2 þ 2tp=PÞdt ð8þ which approximate past and present values of the solar wind electric field, its first and second time derivative. The corresponding kernels are shown in Figures 2a 2c. The further increase of the derivative order appears to be redundant as it brings the characteristic averaging interval down to substorm scales. Similar filters for the Sym-H index, which we denote as Sym-H, D Sym-H /Dt and D 2 Sym-H /Dt 2, differ from (7) (9) by additional 1-hour shifts back into the past to take into account that only past Sym-H data will be used for the reconstruction of the original binning parameters vb z (t), Sym-H (t) and D Sym-H (t)/dt. [16] Before we start the analysis of the forecasting method and its results we introduce one more parameter, which will be used below. This is the exponential decay function, which is a part of the formal solution of the Burton-McPherron-Russell formula (hereafter BMR) [Burton et al., 1975] hsym H j ðbmrþ ðt 0 Þ t ðþ Z t 0 vb s ðt t þ t Þexp½ðt t Þ=t 0 Šdt; ð10þ where t t 0 and t 0 is the e-folding time of the decay of the magnetospheric response to the solar wind input vb s parameter in the BMR equation. For the sake of completeness the corresponding filter kernel is shown in Figure 2d. 3. Forecasting Analysis [17] The forecasting of the binning parameters can be done using the known evolution in time of their NNs from the training database in the forecasting state-space F =( vb z, D vb z /Dt,..; SymH, D SymH /Dt,..) described above: hsym Hi ðfþ ðþ¼ t hhsym HiðÞ t i jnn ð11þ where.. NN denotes the operation of averaging over L NN nearest neighbors. To provide a quantitative assessment of the prediction quality we will use the average 4of8

5 Table 1. Average Relative Variance of the Binning Parameters for Different Forecasting State-Space Cases Case ARV 1 a ARV 2 a ARV 3 a vb z DvBz h j Dt D 2 hvbzj Dt 2 SymH DhSymHj Dt D 2 hsymhj Dt 2 (BMR) SymH (t0 =1h) a ARV 1 = ARV vbz, ARV 2 = ARV SymH, ARV 3 = ARV D SymH /Dt. relative variance ARV [e.g., Vassiliadis, 2006], which gives the percentage of the data variance not represented by the prediction model and is defined as ARV X ¼ s 2 X N 1XN i¼1 2 X i ð12þ X ðfþ i Here s X is the standard deviation of the predicted parameter X (for example, Sym-H ) and N is the number of data points used in the analysis. [18] In our study the last year 2005 of the 11-yearlong database of the TS07D model ( ) was selected as a test interval whereas the first 10 years of the database were used to search L NN nearest neighbors to forecast the binning parameters (1) (3) with a one-hour cadence and with the total number of points in the statistics (12) N = When the number of NNs becomes comparable to the number of points in the database the predictions are known to become linear and inconsistent with the expected nonlinear dynamics of the magnetosphere [e.g., Vassiliadis et al., 1999; McPherron and O Brien, 2001]. At the same time, our studies show that in the limit L NN 1 the predictions are too noisy. Since our objective is to find a suitable forecasting basis to replace the original L NN neighbors currently used in TS07D, it is sufficient to investigate the interval of NN numbers 1 L NN L NN. We have found that the forecasting results depend weakly on the value of L NN in the interval L NN = and in the following we choose L NN = 200. [19] The results of the study are summarized in Table 1. Case 1 describes the forecasting results for the set of global parameters closest to the original one (compare, in particular, Figure 1 with Figures 2a and 2b). The ARV value for the Sym-H parameter (ARV 2 = 0.07) is quite small, and Figure 3, which compares the original time series of the binning parameters and their forecasted analog, confirms that the first set provides a reasonable basis for the average Sym-H index prediction. Deviations are larger for the average solar wind parameter vb z (ARV 1 = 0.26), which is not surprising keeping in mind the inherently turbulent nature of the solar wind driver [e.g., Bruno and Carbone, 2005]. Yet, they are still relatively small, and according to Figure 3a, the characteristic amplitudes of the solar wind variations are reproduced relatively well. At the same time, the first set of the forecasting parameters substantially underestimates variations of the average time derivative of Sym-H. This is clearly seen from Figure 3c and is also reflected in the anomalously large value of the corresponding variance parameter (ARV 3 = 0.74). Case2 shows that the original set of three binning parameters used in Case 1 is only slightly better than the reduced set composed of vb z and SymH. This is consistent with the BMR formula, which suggests that a locally linear relation exists between the Sym-H index, its time derivative and the solar wind electric field. At the same time, case 3 shows that the predictions are only slightly improved in the process of the state-space extension due to higher order derivatives of Figure 3. Case 1: Original (black lines) and forecasted (red lines) time series of the main binning parameters for several sample storms in All parameters are normalized by their standard deviations with s vbz 1.85 mv/m and s SymH 22.9 nt. 5of8

6 negative pulse in the solar wind parameter vb z. It happens because the integration in (1) (3) is extended over the future time period relative to the time of interest t. The effect of this early response of the original binning parameters (1) (3) to solar wind changes is opposite to the hypothetical persistence effect in predictions, when the parameter to be predicted is propagated in time without changes X(t n+1 )=X(t n ) and is therefore delayed compared to the original time series. Note that the prediction quality of the persistence-based predictions of the binning parameters with t n+1 t n = P/2 = 6 hours would be incredibly poor with ARV 1 = 0.94, ARV 2 = 0.24, ARV 3 = 2.08, well in excess of all the corresponding variance parameters ARV 1 3 obtained using the NN-approach and shown in Table 1. [21] The early response effect is strongest in the main phase of storms when the solar wind electric field drops and the resulting changes in the Sym-H may occur on the scale as small as one hour. And it most strongly affects the third binning parameter, which reaches its extreme value in the main phase. Figure 4 suggests that variations of the binning parameters (1) (3) may also contain an Figure 4. Details of the binning parameter evolution for the 24 August 2005 storm in the format of Figure 3. The original series and their one-step predictions in Case 1 are shown by black and red lines, respectively. (a) Dotted blue line shows the original (non-filtered) solar wind electric field parameter vb z. (b and c) Vertical blue dotted lines mark the moment of the original vb z decrease in the main phase of the storm seen in Figure 4a. SymH. Although they can be further improved in case of a similar extension of the state-space in its solar wind parameter part (cases 4 and 5), the prediction efficiency remains relatively poor for the third binning parameter D Sym-H (t)/dt. (Note, that if in cases 4 and 5 the forecasting set of binning parameters does not have information about past Sym-H data provided due to the parameter SymH, the forecasted SymH parameter may have additional secular deviations similar to the offset term discussed by Temerin and Li [2006].) [20] A root of the problem can be grasped from Figure 4, which shows the details of the forecasting for the 24 August 2005 storm along with the evolution of the original (nonfiltered) solar wind electric field parameter vb z (Figure 4a). Figure 4 reveals that the original binning parameters (1) (3) contain spurious information, which makes the problem of the optimization of the forecasting basis (7) (9) ill posed. Spurious relationship between the original solar wind input and (1) (3) appears because the latter three parameters start decreasing well before the arrival of the actual Figure 5. Evolution of the original binning parameters for the 24 August 2005 storm (black lines) and their one-step predictions based on the new class of the binning parameters (7) (10) in Case 6 (red lines). Vertical blue dotted lines in all panels mark the moment of the original vb z decrease in the main phase of the storm shown in Figure 4a. 6of8

7 extension of the state-space due to the BMR-type parameter SymH (t0 ) (BMR) described by (10) with the e-folding time t 0 1 hour. As is seen from Table 1 (compare cases 6 and 7 with cases 3 and 5, respectively) and Figure 5, the latter statespace extension indeed reduces the parameter ARV 3 and substantially improves the prediction of the parameter D Sym-H (t)/dt in the non-spurious part of its variation. The improved forecasting for a group of storms presented in Figure 3 is shown for case 6 in Figure 6. Similar results have been obtained for other sample events in 2005, including January 7, May 15 and September 10 storms. For the sake of completeness we provide in Figure 7 the summary scatterplots comparing the original binning parameters and their forecasted counterparts for the whole year It should be stressed that, as is shown above, the relatively stronger dispersion of points in Figure 7c is largely caused by the significant spurious information, which is contained in the original binning parameters of the model. Two other parameters shown in Figures 7a and 7b are less contaminated by the spurious information effects and the corresponding scatterplots can be used for comparison with other models, such as the forecasting algorithms for Sym-H and Dst indices [Cai et al., 2010; Boynton et al., 2011]. Figure 6. Case 6: Original and forecasted time series of the main binning parameters for several sample storms in 2005 in the format of Figure 3. essential (non-spurious) information, such as for example, the part of the negative variation of D Sym-H (t)/dt on the right from the vertical blue dotted line in Figure 4c, that is, after the moment of the moment of the original vb z decrease in the main phase of the 24 August 2005 storm. To capture that information the forecasting set of the binning parameters must include a prompt signal of the solar wind changes. Such a signal can be provided by the further 4. Discussion and Conclusion [22] In this study we investigated the prediction algorithms for the main set of the data-binning parameters for the new generation of empirical geomagnetic field models represented by the TS07D model [Tsyganenko and Sitnov, 2007; Sitnov et al., 2008, 2010]. First tests showed that more than a half of the variation of the third binning parameter D Sym-H (t)/dt, could not be predicted using conventional time delay embedding methods. Further investigation revealed that the main part of the unpredicted variation is spurious as it represents changes of the Sym-H index before the appropriate changes of the actual (non-filtered) solar wind electric field parameter start. Spurious variations of the binning parameters arise because their filters Figure 7. Case 6: Scatterplots comparing the original binning parameters and their forecasted counterparts for the whole year of8

8 (1) (3) involve integration over the future time period relative to the time of interest. The problem can be mitigated when the set of the forecasting parameters includes the exponential decay function of the solar wind electric field similar to functions used for Dst index predictions in [Burton et al., 1975] with an extremely small e-folding time (one hour). At first sight, such a modification of the forecasting parameters contradicts the original idea to filter out from the model variations on substorm scales. However, the importance for Dst index predictions of BMR-type inputs with the decay timescales as small as 1 hour was emphasized in a number of earlier works [Vassiliadis et al., 1999; McPherron and O Brien, 2001; Temerin and Li, 2002]. Moreover, the analysis of the decay timescales for the storm-time current systems [Tsyganenko and Sitnov, 2005] shows that such timescales are not necessarily associated with the magnetotail thinning and dipolarization processes during substorms, because the smallest response timescales about 1 hour have been found for Birkeland currents (both region 1 and region 2). After all, the presence of two sets of forecasting functions with distinctly different timescales, which provide the optimum forecasting of the binning parameters (1) (3), is consistent with drastically different timescales of the magnetosphere evolution in main and recovery phases of magnetic storm. Thus, the comparative analysis of various forecasting sets of binning parameters in the TS07D model helps optimize these sets. In particular, it shows that the best forecasting sets represent combinations of the regular storm-scale filter expansions (7) (9) with a rather singular substorm-scale BMR-type filter (10), which is necessary to capture the prompt response of magnetospheric current systems in the main storm phase. [23] Acknowledgments. The authors acknowledge useful discussions with N. A. Tsyganenko. This work was supported by NSF grant ATM and NASA grant NNX10AT26G. References Boynton, R. J., M. A. Balikhin, S. A. Billings, A. S. Sharma, and O. A. Amariutei (2011), Data derived NARMAX Dst model, Ann. Geophys., 29, Broomhead, D. S., and G. P. King (1986), Extracting qualitative dynamics from experimental data, Physica D, 20, Bruno, R., and V. Carbone (2005), The solar wind as a turbulence laboratory, Living Rev. Sol. Phys., 2, 4. Burton, R. K., R. L. McPherron, and C. T. Russell (1975), An empirical relationship between interplanetary conditions and Dst, J. Geophys. Res., 80, Cai, L., S. Y. 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