In this paper we describe the development of two empirical models of Pc3 wave activity observed at a ground station. The models are tasked to predict pulsation intensity at Tihany, Hungary, from the OMNI solar wind data set at 5 min time resolution. One model is based on artificial neural networks and the other on multiple linear regression. Input parameters to the models are iteratively selected from a larger set of candidate inputs. The optimal set of inputs are solar wind speed, interplanetary magnetic field orientation (via cone angle), proton density and solar zenith angle (representing local time). Solar wind measurements are shifted in time with respect to Pc3 data to account for the propagation time of ULF perturbations from upstream of the bow shock. Both models achieve correlation of about 70 % between measured and predicted Pc3 wave intensity. The timescales at which the most important solar wind parameters influence pulsation intensity are calculated for the first time. We show that solar wind speed influences pulsation intensity at much longer timescales (about 2 days) than cone angle (about 1 h).

Pulsations in the Pc3 band (22–100

The relationships discovered in the 1960s and 1970s have been confirmed by
more recent studies

The empirical models developed here are based on estimating the causal relationship between solar wind plasma and magnetic field parameters and the intensity of Pc3 waves measured on the ground. Two distinct methods of modelling are employed: one model is based on NNs, and the other on MLR.

The structure of the NN-based model can be visualised as a directed graph
with input nodes connected via weighted connections to a layer of
intermediate nodes, which are in turn connected via weights to the output
node. In this case the input nodes are the SW plasma, IMF and local time
parameters and the output is Pc3 wave intensity (defined later). Network
nodes are computational structures that apply a sigmoidal function (

Multiple linear regression assumes that the relationship between

Taking the logarithm of

The models developed here use solar wind parameters and a local time
parameter as input, and a Pc3 wave activity index at Tihany, Hungary (at

The paper is laid out as follows: the data sets utilised are discussed in the next section. We also describe how measurements of solar wind parameters are shifted in time with respect to ground-based Pc3 measurements, to account for the propagation of waves through the sheath and magnetosphere. In the third section the development of the model is explained. The fourth section discusses the results, including the apparent timescales of SW influence on Pc3 waves. We conclude this investigation in Sect. 5 with a discussion of the results.

Estimating pulsation intensity from SW parameters with the empirical methods used in this study relies on analysing a large set of input (solar wind) and output (Pc3) parameter data. Measurements from the interval 2002–2007 are utilised. The data collected from the sources listed later in this section are 1 min averages of 1 s pulsation measurements and 16 or 64 s solar wind parameter measurements. The models are developed utilising 5 min running averages of the 1 min SW parameter and Pc3 pulsation data sets.

Solar wind data are collected from the high-resolution OMNI (HRO) data sets
(see

The Pc3 pulsation activity utilised in this study are recorded in the
horizontal (H) component of the geomagnetic field at the Tihany (Intermagnet
code THY) geophysical observatory at (

The perturbations that cause pulsations of the Earth's field have to
propagate from the upstream region across the magnetosheath and into the
magnetosphere for Pc3 waves to be observed on the ground. To accurately model
the relationship between upstream activity and pulsation intensity on the
ground, the upstream data set is shifted in time with respect to the
downstream data (ground measurements). The total propagation time is the sum
of the propagation from the upstream spacecraft to the bow shock,

Measurements from OMNI are shifted in time to account for the solar wind flow
from the spacecraft position to the subsolar point on the bow shock nose;
i.e. the HRO data set includes the time delay

According to

It is assumed that the fast mode wave is responsible for carrying ULF energy
through the magnetosphere at the Alfvén speed

To test the effect of applying

Correlation between Pc3

The propagation time through the sheath and the magnetosphere can also be
estimated from empirical data. It was done by a cross-correlation analysis
between the one minute resolution Pc3

Since Pc3 pulsations occur predominantly on the dayside of the Earth

During geomagnetically active periods the Pc3 band is flooded with storm time
wave activity driven by mechanisms other than those driving Pc3 waves.
Therefore, the model development is based on SW and (shifted)
Pc3

Figure

In the development of these models, we use two distinct sets of data. The training set (TRN) is used to adapt the weights during the NN training process, and to determine the constants in MLR. The test set (TST), which is distinct from the TRN set, is used to objectively gauge the performance of the models. The TRN set is compiled of selected data from 2002, 2004 and 2006 by randomly selecting 50 000 data points from each year (i.e. 150 000 in length). Similarly, 7500 data points from 2003 and 2005 are randomly selected to construct the TST set (resulting in a set of length 15 000).

Correlation between Pc3

The development of the NN and MLR models involves selecting the set of solar-wind-based parameters that best relates to Pc3 wave activity. In order to make this selection we iteratively add input parameters to the model from a set of candidate inputs. This process yields an optimal subset of input parameters from the larger set of candidates.

The set of candidate input parameters consist of six SW parameters and one
local-time-related quantity. The eight solar wind and IMF parameters are
solar wind speed

Table

Pearson correlation coefficient between Pc3

The candidate parameters listed above are used in various combinations as the
input parameters to the models with Pc3

Model fitness is quantified by the correlation coefficient between measured
and predicted output (Pc3

Distribution of Pc3

The training process is illustrated by Table

The correlation coefficient for each NN (solid line) and MLR (dashed
line) model. The winners of each round of training are indicated with
squares. The values are listed in Table

Model performance during the wrapper process. Every model (NN and
MLR) has a different set of input parameters, marked with x. The correlation

The optimal MLR model may be written as

In Fig.

This study aims to improve on the 1 h resolution model developed previously
by

In order to compare our results with a low time resolution model, like the
one developed by

The obvious improvement that the high-resolution model offers is that the
influence that

Inaccuracies in the estimates of wave propagation times from the spacecraft
to the bow shock nose (calculated by OMNI) and through the magnetosheath and
magnetosphere (described in Sect.

Measured (black) Pc3

During model development we see that the addition of

Mean rise and fall times of

In order to quantify the difference in timescale of influence we compare
moving averages of Pc3

Measured and predicted Pc3

The optimal time lag, computed by using the cross-correlation between

The time-shift process applied to OMNI data smooths the original spacecraft
measurements somewhat and hence the variance of OMNI cone angle is smaller
than what was measured at ACE, for example. That is why the correlations were
again recalculated this time using direct solar wind and interplanetary
magnetic field measurements of ACE satellite shifted in time. Convection
times were calculated based on the position of ACE, the position of the bow
shock estimated from a model and the solar wind speed. All data points were
shifted in time with the corresponding convection time, after that the time
series was resampled at a 1 min sampling rate. Magnetospheric
propagation was taking into account by another 3 min shift in time. As
expected the correlation became stronger at all timescales (except for the
shortest). Although the technique to correct for the convection time
described here keeps higher variance of the solar wind data than the OMNI
data set, it may introduce large errors by not taking into account the
orientation of the IMF, when the IMF has a significant

Correlation between Pc3

Finally, the partial correlations between Pc3

Correlation between Pc3

The correlation strength depends not only on the time resolution of the data,
but also on the length of the data set for which the correlation is
calculated (in the following part of this section all calculations were made
using the entire 2002–2007 3 min shifted Pc3

Correlation between Pc3

We created a high-resolution (5 min) model to estimate the intensity of Pc3
pulsations at a middle-latitude station (THY) from solar wind and local time
parameters. A rigorous selection procedure is applied to select the most
important input parameters to the model. Out of a set of candidate inputs,

Other extra-magnetospheric input parameters, such as the F10.7 flux, may be
included in future models.

The MLR and NN models yielded very similar results, with the same input
parameters emerging from the modelling process. Apart from the slightly
higher correlation between measured and predicted output, the MLR model
yields a relatively simple relation between input and output
(Eq.

We show for the first time explicitly the timescale at which solar wind
speed, density and IMF direction influence Pc3 wave activity. A comparison
between moving averages of the solar wind parameters, at several different
window widths, and Pc3

This research was supported by a South African–Hungarian bilateral project
funded by the Hungarian Science and Technology Foundation (OMFB-00300/2008)
in Hungary and the National Research Foundation in South Africa. The OMNI
data were obtained from the GSFC/SPDF OMNIWeb interface at