Contents1. 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,

d*Dst**/d*t* = *Q*(*t*) - a*Dst**

where

*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 a*Dst** 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.