Operational Dst forecasts
Hans Gleisner, hans@irfl.lu.se

1. Introduction
2. Why make Dst forecasts?
3. Neural networks
4. Linear filters
5. Differential equations
6. References

1. Introduction

The radiation belts are regions of the near-Earth space where charged particles become trapped on closed geomagnetic field lines. The trapped particles drift around the Earth - positively charged particles westward and negatively charged particles eastward - and create the ring current which is a permanent feature of the magnetospere. During certain conditions in the interplanetary environment, the ring current occasionally becomes strongly enhanced. This is detected at the Earth's surface as a depression of the geomagnetic field at low- and mid-latitudes. If the depression is sufficiently strong we refer to the event as a magnetic storm.
    By definition, the Dst index is the longitudinally averaged field depression at low latitudes. It provides a simple measure of the strength of the ring current. Over the years, there have been many attempts to develop models that take solar-wind data as input and give Dst as output. The most widely used models are those based on first-order differential equations that followed on the study by Burton et al. [1975], but also linear filters and, more lately, neural networks have been used. The common theme of these models is that they are data-based, i.e. the model coefficients have been derived from observed data.
    The reason to undertake these modelling efforts has mostly been to investigate the dynamics of the ring current, or to study the role of the solar wind as a driving force for geomagnetic storms. However, now when solar-wind data is available in real time from a location upstream in the solar wind, these models can equally well be used to make forecasts or to specify the state of the ring current.

2. Why make Dst forecasts?

It is a well known fact that rapid variations of the geomagnetic field can be harmful to technical systems. But the ring current variations are normally not very rapid compared to the substorm effects at higher latitudes. Further, the magnitude of Dst variations are smaller than the magnitude of high-latitude magnetic disturbances. Why then should we make Dst forecasts?
    Dst measures the strength of the ring current, and the ring current is one of the major current systems of the magnetosphere. Without an accurate knowledge of the state of the ring current, our view of the near-Earth space environment would be incomplete. An enhanced ring current, indicated by a decrease of the Dst index, have a major impact on the structure and location of the magnetospheric 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. For these reasons, many specification models - i.e. models that specify the state of the near-Earth space environment - are parameterized in Dst. Operational Dst forecasts provide us with inputs to such models, and tell us much about current and upcoming space weather conditions.
    Another reason to make Dst forecasts is that some of the most adverse effects of space weather take place during magnetic storms, and the ring-current strength is the basic defining property of a magnetic storm. The Dst index can thus be used as a proxy for many type of disturbances that occur during a storm, even though the space-weather effects are not directly caused by the ring-current magnetic field variations.

3. Neural networks

The geomagnetic disturbances detected at the Earth's surface can be interpreted as an output signal from a physical system driven by an external input, namely the solar wind. It is a very complex output signal, and to further complicate matters the underlying physical system might even be non-stationary. Nevertheless, if we assume that Dst is a well-defined output, and that we can measure all relevant inputs, then we can hope to find a valid representation of the system dynamics. We have done this using neural networks.
    The Lund Dst model is based on a neural network which take solar-wind data at time t as input and produce Dst at time t+1 as output. Alternatively, depending on the network setup, it produces Dst at the same time t as the input. The solar-wind input data are normally combinations of density N, velocity V, and magnetic field components By and Bz. Since the network contains feed-back connections from a set of internal state variables back onto the input, it is able to describe a nonlinear, dynamic mapping. In the case of the Lund Dst model, this mapping represents the magnetosphere dynamics. When driven by an observed solar-wind time series, the model responds dynamically and produces a time series of predicted Dst.
    One of the major advantages with neural networks is that we do not have to assume very much about the underlying mapping, such as the functional form of the dynamic relation between input and output. It is an essentially non-parametric method. The model properties is governed by a set of model coefficients, or weights, that have been determined through a training procedure.

4. Linear filters

If the response of the Dst index to some solar-wind parameter was truly linear, an auto-regressive moving-average (ARMA) filter would be perfectly sufficient as a Dst model. This is, however, not the case. The introduction of certain nonlinearities clearly improve the predictions. Nevertheless, in many cases the solar wind-Dst relation is sufficiently simple for a linear ARMA filter to provide reasonably accurate predictions. In 1986, McPherron et al. described the use of linear ARMA filters in real-time predictions of Dst, and ARMA filters have frequently been used in studies that involve the Dst index. Currently no operational forecasts of Dst are made with linear filters.

5. Differential equations

In 1975, Burton et al. described the evolution of the ring current by a simple first-order differential equation,

    dDst*/dt = Q(t) - aDst*


    Dst* = Dst - b sqrt(Pdyn) + c

is a Dst index that has been corrected for variations in the solar-wind dynamic pressure Pdyn. In the differential equation above, Q represents the injection of particles to the ring current, and aDst* represents the loss of particles with an e-folding time 1/a. Based on this simple first-order differential equation, several Dst models have been suggested. The models differ (a) in the way they correct for variations of the solar-wind dynamic pressure, (b) the dependence of the source term on solar-wind conditions, and (c) the dependence of the decay rate on solar-wind conditions and Dst itself. In the original paper by Burton et al. the decay time is assumed to be constant, 7.72 hours, while the source term is an almost linear function of VBs. In 1998, Fenrich and Luhmann presented a slightly modified version of the Burton et al. model in which the source term depends on both VBs and Pdyn, and the decay time is set to 3.0 hours when the interplanetary electric field Ey exceeeds a certain threshold. Compared to the original model, these modifications give some improvements, particularly for high geomagnetic activity levels.
    Other modifications to the original model were proposed by O'Brien and McPherron in 2000. In their model, the decay time vary continuously from 2.4 hours for high activity levels to 19 hours at low geomagnetic activities. The pressure correction is slightly different from the original model, being somewhat less responsive to variations of the solar-wind dynamic pressure. This model clearly outperforms the older Burton-type models.

6. References

Neural networks

H. Lundstedt and P. Wintoft:
Prediction of geomagnetic storms from solar wind data with the use of a neural network,
Annales Geophysicae 12, 19-24, 1994.

H. Gleisner, H. Lundstedt and P. Wintoft:
Predicting geomagnetic storms from solar-wind data using time-delay neural networks,
Annales Geophysicae 14, 679-686, 1996.

J.-G. Wu and H. Lundstedt:
Prediction of geomagnetic storms from solar wind data using Elman recurrent neural networks,
Geophysical Research Letters 23, 319-322, 1996.

H. Lundstedt, H. Gleisner, and P. Wintoft:
Operational forecasts of the geomagnetic Dst index,
Geophysical Research Letters 29, submitted, 2002.

Linear filters

R.L. McPherron, D.N. Baker, and L.F. Bargatze:
Linear filters as a method of real time prediction of geomagnetic activity,
in Solar Wind Magnetosphere Coupling, edited by Y. Kamide and J.A. Slavin, p. 85-92, 1986.

K.J. Trattner and H.O. Rucker:
Linear prediction theory in studies of solar wind-magnetosphere coupling,
Annales Geophysicae 8, 733-738, 1990.

Analogue models   -   differential equations

R.K. Burton, R.L. McPherron, and C.T. Russell:
An empirical relationship between interplanetary conditions and Dst,
Journal of Geophysical Research 80, 4204-4214, 1975.

F.R. Fenrich and J.G. Luhmann:
Geomagnetic response to magnetic clouds of different polarity,
Geophysical Research Letters 25, 2999-3002, 1998.

T.P. O'Brien and R.L. McPherron:
An empirical phase space analysis of ring current dynamics: solar wind control of injection and decay,
Journal of Geophysical Research 105, 7707-7719, 2000.

T.P. O'Brien and R.L. McPherron:
Forecasting the ring current index Dst in real time,
Journal of Atmospheric and Solar-Terrestrial Physics 62, 1295-1299, 2000.

For information or comments, please contact:
Hans Gleisner, hans@lund.irf.se