### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Results of ERR Based Data Analysis
- 3. Interpretation of the ERR Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[2] The magnetosphere is a very complex dynamical system because its evolution under the influence of the solar wind involves an enormous number of coupled physical processes operating on spatial scales from meters to tens of thousands of kilometers. Recently the terrestrial magnetosphere has been subjected to comprehensive experimental studies that combined multi-spacecraft missions such as THEMIS and Cluster together with ground based observations. However, the resulting vast amount of data has not led to the development of a comprehensive model of the magnetosphere deduced from first principles. Any major geomagnetic disturbance, which can affect modern technological systems, such as a magnetic storm, is a result of a complex chain of events. At present, even many basic processes that form some of the links of this chain are not well understood. This can be illustrated on the part of the magnetospheric dynamics that manifests itself in the *D*_{st} index, by asking questions such as: What processes cause the decay in the ring current? How it is possible to explain observed energies of oxygen in the ring current? These questions form only a small part of all the unanswered questions related to the ring current dynamics.

[3] Because it is impossible to deduce a model of the magnetosphere without finding answers to these and many other questions, an alternative approach can be used that treats the magnetosphere as an input-output black box system, with unknown yet physical state space, that evolves under the influence of the solar wind. In such a treatment inputs to the magnetosphere should be some function of the solar wind parameters, while measured characteristics of the state of the magnetosphere (e.g., geomagnetic indices) should be regarded as the system outputs. This approach allows the implementation of the methodology that was developed for nonlinear system forecasts: local linear filtering, neural networks, evolutionary algorithms etc. to predict the magnetospheric dynamics. However, application of all these methods to the dynamical systems requires a-priori knowledge about the input(s) into this system. The search for the inputs can be based either on data analysis or on physical considerations from first principles. Previously the correlation function was the main tool in the data based search for “the most appropriate inputs” into the magnetospheric system. However, it is a commonly accepted view that the magnetosphere is a nonlinear system [e.g., *Klimas et al.*, 1996]. Application of the correlation function designed to identify linear relations for nonlinear systems can be misleading. To illustrate this let us consider a simple example of a quadratic system *x* *x*^{2}. If this system is driven by a zero-mean stochastic signal, the correlation between the input *x* and the output *x*^{2} will be zero. A high correlation between two data sets indicates some causal relation. However a simple comparison of the values of the correlations (as it was done for example by *Newell et al.* [2007]) can be misleading in the quest for the “input” with the best predicting capability. Strictly speaking such a quantitative assessment requires not only the linear correlation between the input *X* and the output *Y* but also an infinite number of correlations between *X*^{2} and *Y*, *X*^{3} and *Y* etc. It is only the cumulative of all these correlations that should be compared to find the input with the best predicting capability.

[5] In addition to the Akasofu proposed ε parameter a number of other CFs have been either derived from basic principles, from dimensional considerations, or have been deduced using multilinear regression for example by *Scurry and Russell* [1991]. *Newell et al.* [2007] studied correlations of various geomagnetic indices and a number of previously proposed CFs in order to find the most optimal selection: , [*Vasyliunas et al.*, 1982], [*Scurry and Russell*, 1991], [*Temerin and Li*, 2006], [*Newell et al.*, 2007], and [*Wygant et al.*, 1983] (where *n* and *p* are the solar wind density and dynamical pressure.) The existence of so many expressions for the solar wind magnetosphere CF indicates the failure to deduce such a function analytically. Linear correlations employed by *Newell et al.* [2007] to differentiate between the efficiency of these CFs suggested as the optimal coupling function. However as explained above the application of correlation functions to nonlinear systems can be misleading.

[6] The main goal of the present letter is, to continue the quest for solar wind-magnetosphere coupling functions, to show how proper data treatment techniques that have been developed for the analysis of nonlinear dynamical systems can complement analytical studies of complex nonlinear phenomena. The most powerful methodology in data based approaches to complex nonlinear systems is based on the Nonlinear Autoregressive Moving Average models (NARMAX) [*Billings et al.*, 1989; *Boaghe et al.*, 2001; *Balikhin et al.*, 2001]. NARMAX is able to provide physically interpretable results that can be directly compared with and which can be used to validate and enhance analytical models. The main area of application of NARMAX methodology is to identify, from input-output data sets, mathematical models that govern the evolution of dynamical systems so complex that first principles based models are incomplete or are not known at all. Since its development the application of NARMAX methodology has been enormously successful in many areas of medicine [*Zhao et al.*, 2007], science [*Boaghe et al.*, 2002; *Wei and Billings*, 2006] and engineering [*Nehmzov et al.*, 2010]. The concept of the Error Reduction Ratio (ERR) [*Billings et al.*, 1989] has been developed to identify the structure of the model at the initial stage of the NARMAX. ERR ratio can be used to measure the contribution of model terms (inputs) on the evolution of the nonlinear dynamical system and to identify the significance of casual relations between particular functions and the system output. In the present paper ERR is exploited in order to identify the forecasting potential of various CF.

[7] Comprehensive descriptions and the rigorous mathematical foundations of NARMAX in general and ERR in particular are beyond the scope of the present letter and can be found elsewhere [see, e.g., *Billings et al.*, 1989]. We provide here only a simplified, brief explanation of the method. The NARMAX approach for discrete data sets of inputs *I*(*t*_{i}) and the output *O*(*t*_{i}) is based on a very general assumption that the output of the system can be expressed as some function of all previous values of inputs, outputs and some error *O*(*t*_{i}) = *F*[*I*(*t*_{i}), *I*(*t*_{i−1}), *I*(*t*_{i−2}),…, *O*(*t*_{i−1}), *O*(*t*_{i−2}),.*e*(*t*_{i}), *e*(*t*_{i−1})…]. Here the error function *e*(*t*_{i}) accounts not only for measurements errors, but for effects of unknown inputs as well. In the framework of the NARMAX approach the function *F*[…] is not searched for in an explicit form, but is decomposed in some complete basis (e.g., polynomial). The mathematical derivation of NARMAX takes into account both measurement noise and error resulting from incomplete knowledge of all inputs [*Billings et al.*, 1989]. A simplified description of NARMAX is provided here for illustrative purposes and does not account for these factors. In the case of the polynomial basis, a primary NARMAX approach can be formulated as a technique to find coefficients *s*_{k} in the decomposition of an unknown function *F* in polynomials basis *q*_{k}: *O*(*t*_{i}) = ∑_{k}*s*_{k}*q*_{k}, where the sum ∑_{k}*s*_{k}*q*_{k} is a polynomial representation of unknown *F* function, and *q*_{k} are monomials with respect to values of inputs *I* and previous values of the output *O*. As the first step of NARMAX orthogonalisation of basis *q*_{k} is performed resulting in the new basis functions *w*_{k} such that 〈*w*_{l}*w*_{j}〉 = = 0 if *l* ≠ *j*. In the new orthogonal basis

and the coefficients *g*_{k} can be estimated separately one by one, as

The above relation is derived by multiplication of (1) by *w*_{k} and exploiting the orthogonality of the *w*_{k} basis.

[8] The relative contribution of the basis function *w*_{k} into the evolution of *F* can be estimated as *Billings et al.* [1989]:

and is called the Error Reduction Ratio (ERR). It is obvious that the larger the value of ERR the larger is the contribution of *w*_{k}. Different ways of NARMAX implementation are given by *Billings and Zhu* [1995] and *Aguirre and Billings* [1994].

[9] A comprehensive analysis of ERR calculations for a combinations of solar wind parameters and previously proposed relationship between CF functions as inputs to the magnetosphere system with the *D*_{st} index as the output has been performed and is described by R. J. Boynton et al. (Using the NARMAX OLS-ERR algorithm to obtain the most influential coupling functions that affect the evolution of the magnetosphere, submitted to *Journal of Geophysical Research*, 2010). In the present letter only this part of the ERR based treatment of the magnetosphere system will be discussed to illustrate how the approach by Boynton et al. (submitted manuscript, 2010) can assist in the analytical derivation of the CF from first principles.

### 2. Results of ERR Based Data Analysis

- Top of page
- Abstract
- 1. Introduction
- 2. Results of ERR Based Data Analysis
- 3. Interpretation of the ERR Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[10] The solar wind data obtained far upstream at the L1 point and available from the OMNI WWW site together with the *D*_{st} data series have been used to apply the NARMAX methodology to the quest of determining the “best input” corresponding to the *D*_{st} dynamics. Data over the time interval from the start of 1998 to the end of 2008 in GSM coordinate frame have been used. As NARMAX application requires about 1000 pairs of continuous input-output measurements, the initial time interval has been subdivided into 64 continuous data intervals of simultaneous solar wind and *D*_{st} of 1000 data points each. The time intervals where data gaps occur in the solar wind measurements have been excluded from the initial data set. For each interval, the ERR that is due to a particular combination of solar wind parameters and previous *D*_{st} values, and the sum of ERR for all terms identified by the deduced NARMAX procedure, *ζ*_{NM}, have been calculated. It is well known that the value of the *D*_{st} over the previous hour provides very good predictions for the current *D*_{st} and so should provide a significant ERR *ζ*_{Dst}. That is why the contributions of all other possible inputs have been normalized on the remainder *ζ*_{NM} − *ζ*_{Dst}. This Normalized ERR will be referred to as NERR. For every form of input considered in this study NERR have been averaged over 64 time intervals and the results compared to identify the form of the input that has the highest potential contribution to the variation of the output *D*_{st}.

[11] The NERR has been investigated for all coupling functions that are given in the Table 1. The NERR has been compared for 40 monomials: 7 model expressions for CF (Table 1) and *D*_{st}, with 5 possible delays each: *t* − 1, *t* − 2,…, *t* − 5. To compare just these coupling functions the order of monomials have been limited to one. The resulting NERR for the five CFs with highest NERR values are summarized in the Table 2. Thus in creating Table 2, the NERR has been used in a similar way to the correlation function. Note that sum of all 40 NERR should be equal to 100%.

Table 2. Top Five Coupling Functions Rated by Their NERR ValueCoupling Function | NERR (%) |
---|

*p*^{1/2}*VB*_{T} sin^{6}(θ/2)(*t* − 1) | 31.32 |

*VB*_{s}(*t* − 1) | 12.76 |

*n*^{1/6}*V*^{4/3}*B*_{T} sin^{4}(θ/2)(*t* − 1) | 10.30 |

*p*^{1/2}*VB*_{T} sin^{4}(θ/2)(*t* − 1) | 8.37 |

*D*_{st}(*t* − 2) | 7.23 |

[12] These results suggest the CF proposed by *Temerin and Li* [2006], as having the most promising forecasting abilities and shows when averaged over 64 intervals that the normalized ERR value exceeds 31%. This selected term is followed by the *VB*_{s} with very drastic change in the NERR value, which for the second term is almost 3 times lower 12.76. The remaining 3 terms listed in the table show relatively slow decrease in the NERR value from 10.30 for one of the expressions for a CF proposed by *Vasyliunas et al.* [1982] to 7.23 for the two hours lagged *D*_{st} index. The sharp decline in NERR value from the first to the following terms indicates a significant superiority of the *Temerin and Li*'s [2006] CF in accounting for the evolution of the *D*_{st} index. It can be seen that the expressions for the coupling functions that are in Table 2, in addition to the simple solar wind parameters, have more complex factors such as , , etc. The superiority of the NARMAX methodology is that it is able to automatically combine preselected factors to form a monomial term that corresponds to the largest ERR value. To exploit this advantage of the NARMAX algorithm, the following factors of the top coupling functions from Table 2 have been used to identify a new set of CF with better forecasting abilities: *p*^{1/2}, *n*^{1/6}, *V*, *V*^{4/3}, *B*_{s}, and . The 4th order limit has been imposed on the monomials build of these factors, as higher orders are computationally are much more expensive. This is also the reason why terms and , have been included rather than *B*_{T} and . The standard NARMAX procedure of replacing the original set of monomials composed of these factors by an auxiliary set of orthogonal polynomials had been implemented. For each time period the ERR ratio have been calculated for these auxiliary polynomials and afterwards recalculated for the original monomials, similar to the procedure described by *Billings et al.* [1989]. The monomials identified by the NARMAX algorithm that correspond to the CF functions with highest NERR are given in Table 3.

Table 3. Top Five Coupling Functions Rated by Their NERR Value, Identified Coupling Functions Selected by the NARMAX Algorithm Using the Decomposed Parameters From the Best Coupling Functions and *Dst*Coupling Function | NERR (%) |
---|

*p*^{1/2}*V*^{4/3}*B*_{T} sin^{6}(θ/2)(*t* − 1) | 5.46 |

*p*^{1/2}*V*^{2}*B*_{T} sin^{6}(θ/2)(*t* − 1) | 3.18 |

*n*^{1/6}*V*^{2}*B*_{T} sin^{4}(θ/2)(*t* − 1) | 3.15 |

*D*_{st}(*t* − 2) | 2.96 |

*p*^{1/2}*VB*_{T} sin^{6}(θ/2)(*t* − 1) | 2.77 |

[13] Surprisingly both top CFs in the Table 3 have a factor . NERR values in the Table 3 are considerably lower in comparison to those in the Table 2. Such a significant difference is due the number of possible expressions for CF taken into account. while it was only 40 for the first table (7 model expressions for CF and *D*_{st}, times 5 possible delays: *t* − 1, *t* − 2,…, *t* − 5), a few thousands terms corresponding to all possible monomials and time delays have been rated in the second case.

### 3. Interpretation of the ERR Results

- Top of page
- Abstract
- 1. Introduction
- 2. Results of ERR Based Data Analysis
- 3. Interpretation of the ERR Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[14] As noted above the aim of this paper is not to find a single expression for a possible unique set of CF. CF's should be related to the physical processes of solar wind magnetosphere interaction. It is clear that the physics of the interaction should depend upon the interplanetary conditions. The mechanisms of this interaction during the prolonged period of northward IMF differs drastically from the time period of strong southern IMF component. It is quite possible (until it is proven otherwise) that, for example, different levels/durations of southern IMF direction will also lead to different interaction mechanisms. For example, polar cap potential saturation [*Borovsky et al.*, 2009] should be related to the initiation of some underlying physical processes and may potentially result in a different overall set of CF during saturation times. The aim of the study was to identify the potentially most promising relations for CF and to use these data identified expressions to look back into the initial analytical models for coupling functions. The top two expressions for coupling functions in Table 3 have a similar structure. Both have *B*_{T} as a factor, and pressure or density (it depends upon how to account for a factor of velocity)in a fractional power that is less then one. Both top CFs have a velocity term in a power greater than 4/3 but smaller then than 2.5, and both have a factor .

[15] In contrast to many other techniques such as Neural Networks, NARMAX leads to physically interpretable results. It should therefore be regarded as a toolkit that can aid mathematical modelling and the physical interpretability of such models. The fact that both top relations for CF possess the same factor , not as it was originally proposed in Akasofu parameter posses the question where in the previous analytical models the 4th power comes from? In *Perreault and Akasofu*'s [1978] original study this factor appears as the simplest continuous dependence of coupling function upon clock angle. *Vasyliunas et al.* [1982] speculated about coupling functions from the point of view of physical dimensions. Such dimensionless factors as could not be identified in the frame of *Vasyliunas et al.* [1982] approach. *Scurry and Russell* [1991] applied data fitting and used only as a factor. Again the task to distinguish between the fourth or other power for factor was beyond the scope of *Scurry and Russell* [1991]. It is KL's paper that based on first principles provides the reasoning for the Akasofu parameter and therefore for the term. To understand how the extra factor can be justified in the framework of KL's model, the arguments of KL should be revisited. Figure 1 (right) shows the model adopted by KL. The reconnection electric field at dayside magnetopause is parallel to the reconnection line *PO* and is equal to [*Sonnerup*, 1974]

where *B*_{ms} is the magnetic field on the magnetosheath side and *V*_{ms} is the component of plasma flow velocity perpendicular to the magnetic field there. This reconnection field *E*_{r} is the only component of the magnetosheath electric field to penetrate into the open magnetosphere. While *E*_{∥} = 0 on the magnetosheath side *E*_{∥} ≠ 0 in the magnetosphere as magnetospheric magnetic field is not perpendicular to the X-line [*Kan and Lee*, 1979]. Let the length of the reconnection line *PO* = *l*_{0}. The length of the reconnection line *l*_{0} is constant in the frame of *Kan and Lee*'s [1979] model and therefore does not depend on θ. The potential difference Φ across the polar cup is measured between the points *P* and *Q*, where the line *PQ* is perpendicular to the magnetic field. This potential difference is

One of the factors sin(θ/2) was lost by KL, apparently in the substitution of the integration path *PQ*, as they multiplied the electric field magnitude by a factor sin(θ/2), but did not scale the integration path with the same factor. For a potential difference that corresponds to the perpendicular electric field component only the perpendicular component of the reconnection line should be taken into account, so *l*_{0} also should be multiplied by sin(θ/2). This last factor is missing in KL's calculations. In the framework of KL's approach the expression for the potential Φ can be used to deduce the power produced by the solar wind dynamo as *P* = Φ^{2}/*R*, where *R* is the total equivalent resistance connected to the solar wind-magnetosphere dynamo [*Kan and Lee*, 1979]. It is also assumed that *V*_{ms}*B*_{ms} = *VB* due to the magnetic flux conservation. Eventually, taking into account a correct scaling for *l*_{0}, an accurate relation for CF in the framework of KL's model is given by: