# Prediction of Dst during solar minimum using in situ measurements at L5

###### Abstract

Geomagnetic storms resulting from high-speed streams can have significant
negative impacts on modern infrastructure due to complex interactions between
the solar wind and geomagnetic field. One measure of the extent of this effect
is the Kyoto $Dst$ index. We present a method to predict $Dst$ from data
measured at the Lagrange 5 (L5) point, which allows for forecasts of solar
wind development 4.5 days in advance of the stream reaching the Earth. Using
the STEREO-B satellite as a proxy, we map data measured near L5 to the near-
Earth environment and make a prediction of the $Dst$ from this point using the
Temerin-Li $Dst$ model enhanced from the original using a machine learning
approach. We evaluate the method accuracy with both traditional point-to-point
error measures and an event-based validation approach. The results show that
predictions using L5 data outperform a 27-day solar wind persistence model in
all validation measures but do not achieve a level similar to an L1 monitor.
Offsets in timing and the rapidly-changing development of $B_{z}$ in
comparison to $B_{x}$ and $B_{y}$ reduce the accuracy. Predictions of $Dst$
from L5 have an RMSE of $9$ nT, which is double the error of $4$ nT using
measurements conducted near the Earth. The most useful application of L5
measurements is shown to be in predicting the minimum $Dst$ for the next four
days. This method is being implemented in a real-time forecast setting using
STEREO-A as an L5 proxy, and has implications for the usefulness of future L5
missions.

Space Weather

Space Research Institute, Austrian Academy of Sciences, Graz, Austria
Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA Institute of Physics, University of Graz, Graz, Austria Conrad
Observatory, Zentralanstalt für Meteorologie und Geodynamik, Vienna, Austria

Rachel<EMAIL_ADDRESS>
In-situ L5 data can be used for Dst forecasts at Earth and perform better than
27-day recurrence

Low consistency of Bz over multiple days limits the accuracy of Dst predicted
from L5

This method performs best when forecasting the minimum Dst during SIRs

##  1 Introduction

The solar wind has myriad effects on the Earth’s magnetic field, among them
the enhancement of the ring current around the Earth’s equator [Gonzalez .
(1994)]. Through streams of charged particles, the solar wind injects energy
into the ring current and thereby reduces the global magnetic field strength
[Daglis . (1999)]. This can have consequences on GPS and satellite
communication as well as flight operations where crew may be exposed to
greater levels of radiation [Schrijver . (2015)]. The extent of the
enhancement and resultant reduction in field strength is often given by the
disturbance storm time index, $Dst$. This is an hourly value derived from
geomagnetic field variations measured at four observatories (Honolulu,
Kakioka, San Juan and Hermanus) below $35^{\circ}$ in geomagnetic latitude
[Mayaud (1980)]. Different levels of geomagnetic activity are often described
by the minimum of $Dst$ reached, with a $Dst\leq-200$ nT denoting an extreme
geomagnetic storm, $Dst\leq-100$ nT being an intense storm, and $Dst\leq-50$
nT being only a moderate storm. A $Dst\leq-30$ nT is sometimes used to denote
weak geomagnetic storms. The largest negative values of $Dst$ are seen almost
exclusively in storms caused by interplanetary coronal mass ejections/ICMEs
[Borovsky Denton (2006)], while moderate storms occur throughout the solar
cycle [Richardson . (2001), Tsurutani . (2006)]. These more common, milder
storms are driven by high-speed streams and stream interaction regions/SIRs
[Alves . (2006), Jian . (2006)], and Richardson2000 showed that high-speed
streams lead to 70% of geomagnetic activity during solar minimum. In rare
cases SIR-driven storms can also become major geomagnetic events [Richardson .
(2006)] with an expected maximum possible storm strength of $Dst\sim-180$ nT.

Although the official Kyoto $Dst$ is derived solely from ground-based field
measurements, a very good estimate of the upcoming $Dst$ can be made based on
in situ solar wind data from satellites at the Lagrange 1 (L1) point [<]e.g.
most recently the DSCOVR satellite, see¿Burt2012. This allows for a prediction
lead time of 10–50 minutes, with the amount of time determined by solar wind
speed between L1 and Earth. The state-of-the-art approach in this respect is
the model of Temerin2006, an empirical technique that achieves a Pearson
correlation coefficient between the observed and predicted $Dst$ values of
$0.96$ for the seven years of data evaluated. This model depends solely on
solar wind input, and the variation in $Dst$ at one point depends on both the
solar wind at that point in time and past modelled $Dst$ timesteps.

For forecasts beyond a half-hour window, we look now to possible future
missions to the L5 point, which sits $60^{\circ}$ behind the Earth in its
orbit and roughly 4.5 days in advance for corotating solar wind structures. A
space weather mission at this point to perform in situ solar wind measurements
has been discussed many times before [Gopalswamy . (2011), Lavraud . (2016),
Hapgood (2017)], and presents a strong opportunity for accurate forecasts of
space weather events with a much enhanced lead time ranging from hours to
days. As shown in Thomas2018, a solar wind monitor at the L5 point can provide
very good forecasts of the ambient solar wind variations, of which high-speed
streams and SIRs are of primary interest.

In this work we show how predictions of the $Dst$ index at Earth can be made
using L5 data and discuss the applicability and accuracy of the method. This
is carried out using data from the Solar Terrestrial Relations Observatory
Behind [<]STEREO-B, NASA, see¿Kaiser2005 satellite, which crossed the L5 point
in late 2009, as a proxy for a future L5 mission. Data from this satellite is
mapped to L1 as if it had been measured there by correcting for both time
passed in solar rotation speed and solar wind expansion [Thomas . (2018)], and
a method for predicting the $Dst$ from L1 data is then applied. The accuracy
of the $Dst$ forecast is evaluated using a combination of traditional error
metrics (e.g. correlation coefficient, mean error) as well as a method
considering the prediction of events (e.g. $Dst$ minimum) without comparing
the $Dst$ development point-to-point. Past studies have looked at predicting
the general solar wind properties [Simunac . (2009), Turner Li (2011),
Kohutova . (2016), Temmer . (2018), Owens . (2019)] while this study aims
specifically to determine how well the development of $Dst$ and geomagnetic
effects can be predicted using L5 data. The results also include a brief
analysis of the sensitivity of $Dst$ prediction to offsets in measurements of
the magnetic field. This study serves as a verification for methods that are
now being implemented in real-time using STEREO-A data as it crosses the L5
point and moves towards the Earth.

##  2 Methods

###  2.1 Mapping L5 data to L1

Studies using the STEREO satellites (launched in 2006) as proxies for
satellites positioned at L5 have been undertaken in the past. Simunac2009
mapped data from L5 to L1 using a time shift determined by the synodic
rotation period of the Sun and showed good agreement between the solar wind
speed and density measured at the two locations. Turner2011 evaluated the
correlation between time-shifted measurements ahead in the Parker spiral and
L1 data. It was shown that while the correlation in solar wind speed remains
high, there is rarely much correlation in magnetic field components.
Thomas2018 continued in this thread but went a step further and carried out a
comprehensive analysis of solar wind forecasting skill using data measured
near L5.

To map data measured at L5 or thereabouts to L1, we use the same approach as
described in Thomas2018 and apply a time shift to the data measured at
STEREO-B assuming a rotation in the solar wind equivalent to the rotation
speed at the solar equator of roughly 27 days ($\Delta t_{\text{lon}}$). Here
we use a synodic rotation period of $T_{\text{syn}}=27.27$ days as given in
Owens2013. A second adjustment to the time to correct for differences in
radial distances ($\Delta t_{\text{r}}$) and solar wind expansion timing is
calculated as follows based on Eq. 1 from Simunac2009:

$\Delta t=\Delta t_{\text{lon}}+\Delta t_{\text{r}}=\frac{\Delta
lon}{\Omega_{\text{Sun}}}+\frac{(r_{\text{L1}}-r_{\text{STB}})}{v_{\text{sw}}},$
(1)

where $r_{\text{L1}}$ and $r_{\text{STB}}$ are the radial distances of L1 and
STEREO-B from the Sun, while $v_{\text{sw}}$ is the mean solar wind speed at
the time of measurement. $\Delta lon$ is the difference in longitude between
the Earth/L1 and STEREO-B. $\Omega_{Sun}$ is the variable for solar rotation
speed $360^{\circ}/T_{\text{syn}}$. The total time shift $\Delta t$, which
varies with longitudinal distance between the satellite and Earth, is then
added to the time of measurements from STEREO-B. Since this results in a new
range of times with increasing difference between the new and original values
as STEREO-B moves away, the measurements are interpolated back to periodic
hourly time values.

A second adjustment is applied to the solar wind data to account for areal
expansion of the solar wind at different radii and in the Parker spiral
[Kivelson Russell (1995)]. All variables are multiplied by a correction factor
determined by the ratio between the distance of STEREO-B and L1
$(r_{\text{STB}}/r_{\text{L1}})$. The rate of expansion for the density is
assumed to behave according to the inverse-square law [Kumar Rust (1996)],
while the magnetic fielc components scale according to factors as given in
Hanneson2020 varying between -2 and -1. In the case of STEREO-B, which was at
a distance greater than $1$ AU, this means that the solar wind was corrected
backwards and effectively compressed to L1.

###  2.2 Prediction of $Dst$ from L1

There are many different models for predicting the $Dst$ index from solar wind
measurements. Earlier models from Burton1975 and Obrien2000 achieve a
reasonable level of accuracy. When using the OMNI2 data set as input and
comparing the results to the true Kyoto $Dst$ values, they have correlation
coefficients of $0.76$ and $0.84$ respectively. One of the most exhaustive
L1-to-$Dst$ algorithms is undoubtedly the semi-empirical model developed first
in Temerin2002 for the years 1995–1999, which was later extended to the year
2002 in Temerin2006. This model has a linear correlation of around $0.95$ for
$Dst$ in the periods the model was intended for (in good agreement with the
original work), although when using it for predictions beyond this period
(2002–2019), there is a linear drift away from the real $Dst$ of about $-5.3$
nT/year due to the inclusion of a time-dependent variable in the model
[Temerin Li (2015)]. After applying a simple linear correction for the drift,
the correlation for this model in future times is still very good at 0.90 on
average. Due to the dependence of the model on local time, the input for data
mapped to L1 was the time it was expected to arrive there and not the time it
was measured.

The Temerin2006 method (henceforth called TL2006) makes very good predictions
of $Dst$ at the Earth using either data measured at L1 or data mapped to L1
using the aforementioned time-shifting method. A small improvement can be made
to the model through application of a machine learning algorithm from the
Python package SciKit-Learn to provide a correction factor for the base drift-
corrected TL2006 output. The algorithm is a gradient boosting regressor (GBR),
which develops an ensemble of basic regressors to calculate output (correction
to $Dst$) based on the provided input. The input was the same set of variables
provided to the TL2006 method (solar wind speed, density, and magnetic field
components along with time). Some feature engineering was applied to the input
variables to provide more information to the GBR such as including a solar
wind pressure term and time derivative of $B_{z}$ to evaluate which variables
improved the $Dst$ prediction, and those that did not lead to an improvement
were removed. The most important addition was the introduction a “ring-current
term” (with both current and past values from the prior 24 hours) based on the
method in Obrien2000. The ring-current term described most of the $Dst$
variation not accounted for by the TL2006 method, and the prediction of $Dst$
from solar wind measurements using the TL2006 method plus this GBR correction
value (enhanced method or ETL2006) leads to an improved average linear
correlation of 0.95 and reduced RMSE between real and predicted $Dst$ over the
time range 2000–2018. In this study we apply the model using the enhancement
throughout.

##  3 Data

The NASA OMNI2 data set was used for getting the measurements of the solar
wind in the near-Earth environment and the values for the Kyoto $Dst$ (the
real values to which all predictive models are compared). The machine learning
algorithm providing the correction to the TL2006 $Dst$ prediction method was
trained on the Kyoto $Dst$ data set for 2000-2018.

STEREO-B one-minute resolution PLASTIC [Galvin . (2008)] and IMPACT [Luhmann .
(2008)] instrument data was used as a proxy for data measured at L5. STEREO-B
differs from a true L5 mission in two ways: firstly, it was constantly in
motion and moving further away from the Earth in its orbit; and secondly, it
was also at a greater distance from the Sun than the Earth
($r_{\text{STB}}\approx 1.05$ AU). These differences were both accounted for
using the approach defined in the methods section. Beacon data, which is low-
resolution data sent soon (minutes) after measurement, was used rather than
the higher-quality science data that arrives later to simulate the forecasting
application of this model in a real-time operation scenario. This will have an
effect on the final results, although we would not expect the quality to be
greatly degraded by using beacon rather than science data. The data was also
downsampled via interpolation to one-hour resolution.

STEREO-B data is given in the reference frame STBHGRTN , a spacecraft-centric
reference frame with $x$ pointing from the Sun to the spacecraft with the Sun
as the origin, $y$ as the cross-product of the rotational axis with the
$x$-component, and $z$ as the normal to these two pointing out of the
ecliptic. While the satellite is still in the geospace environment, the
STBHGRTN coordinate system can be transformed to GSE by flipping the $x$ and
$y$ directions. For $Dst$ calculation purposes, this is converted to GSM
according to the algorithms given in Hapgood1992. The STBHGRTN frame rotates
with the spacecraft as it moves away from the Earth but we assume a rotation
of the solar wind with the mapping of data measured at STEREO-B to L1, and
therefore can perform the same coordinate transformation to quasi-GSE (as if
the measurements had been made in the geospace environment) and then to GSM.
All spacecraft positions in this work are given in heliocentric Earth
equatorial (HEEQ) longitudes and latitudes, in which the $z$-axis is parallel
to the Sun’s rotation axis and the $x$-axis is the intersection of the solar
equator and solar central meridian as seen from Earth.

Because STEREO-B is always moving and only spent a short amount of time around
the actual L5 point at $-60^{\circ}$, for the purposes of this study we
consider the location of L5 to be $-60\pm 10^{\circ}$ in longitude in addition
to evaluating overall statistics for the range $0$ to $-110^{\circ}$.
Comparisons of results in the next sections will refer to the two data sets as
the full data set and the reduced data set. The time range in the full data
set covers five years from Feb 2007 until Jan 2012. The time range for the
reduced range of angles from STEREO-B near L5 is almost six months from August
2009 until February 2010, which encapsulates almost the entire solar minimum,
meaning this is the optimal time range to evaluate L5-based ambient solar wind
prediction.

To allow comparison to a simple reference baseline model, a third data set was
created as a persistence model, as this has been shown to achieve reasonable
accuracy with a solar wind recurrence rate of 27.27 days [Owens . (2013)]. For
this, the same OMNI2 data was taken after being shifted into the future by
27.27 days. These models are referred to as OMNI, STB and PERS in short form
and plots throughout the text. In order to calculate $Dst$ reliably, any gaps
in the data were linearly interpolated over.

ICMEs that occured at one location or in one data set and not the other were
removed from the data using start and end times given in the HELCATS ICMECAT
catalogue [Möstl . (2017)], which covered the period of evaluation for both
STEREO-B and measurements near the Earth (in this case with the WIND
satellite). This was carried out for all comparison data sets after STEREO-B
had been time-shifted with ICME start and end times being corrected to the new
shifted times. In total, three sets of ICMEs were removed from all data sets:
those at STEREO-B, those at L1, and those at L1 shifted by 27.27 days for the
persistence model. This reduces the size of the total data set by 12%.

An example of solar wind speed, vertical magnetic field and $Dst$ values from
the three data sets is plotted in Fig. 1, in which the time range covered by
STEREO-B when it was near the L5 point ($-60\pm 10^{\circ}$) is shown. The
STEREO-B data has already been time-shifted according to the solar rotation
speed to Earth, and is plotted against the OMNI data for comparison. The high-
speed streams observed at STEREO-B are easy to identify and generally overlap
with those in the OMNI data, although in some cases they arrive earlier or
later than their counterparts at Earth. Gaps in the data are ICMEs that have
been removed. As can be seen in the comparison of $Dst$, the red line with
$Dst$ predicted from the OMNI data very closely matches the actual Kyoto
$Dst$, while in the mapped STEREO-B $Dst$ prediction there are still large
variations. The persistence model (grey) performs worse than the STEREO-B
approach.

Figure 1: Plot of measurements from STEREO-B (blue), the OMNI data set (red)
and the persistence model data (grey) for the time period STEREO-B was closest
to L5 ($-60\pm 10^{\circ}$) in late 2009. (a) Solar wind speed from the OMNI
data set and time-shifted data from STEREO-B. (b) The interplanetary magnetic
field northward component, and (c) the final Kyoto $Dst$ (black) plotted
against predictions made from the other data sets. Gaps in the data are ICMEs
that have been removed.

The distribution of values in $Dst$ (both observed and predicted) according to
the different models is plotted in Fig. 2. The OMNI, Kyoto and PERS data sets
have very similar distributions, but the STB $Dst$ values have a slightly
higher peak close to zero and fewer values in the positive region.

Figure 2: Comparison of density curves of $Dst$ for real (Kyoto) and predicted
(OMNI, STB and PERS) values with $binwidth=4^{\circ}$.

See Sec. 6 for a list of all data sources.

##  4 Results

The goodness of prediction of $Dst$ values predicted from data measured near
the L5 point will be evaluated in this section. First, the goodness will be
measured according to standard error metrics, and then the models will be
compared using an event-based approach looking additionally at the forecasting
of events within 24-hour windows.

###  4.1 Accuracy of prediction

We first evaluate the accuracy of a prediction using data from the L5 point
using standard metrics such as the Pearson correlation coefficient (PCC), mean
absolute error (MAE), mean error (ME) and root-mean-square error (RMSE) of the
predicted values subtracted from the observed. The results for each data set
are listed in Table 1.

Predictions from the OMNI data set consistently achieve a very good level of
accuracy compared to the real Kyoto $Dst$, which is to be expected given the
good behaviour of the ETL2006 model in general, although predictions using
STEREO-B data are considerably worse. When comparing the STEREO-B/L5
predictions to the persistence model, we see that the prediction using
STEREO-B data is better in all measures. Errors in both these models are
usually twice as large as those from the OMNI model.

Table 1: Summary of the error measures describing the accuracy of $Dst$ predicted with various methods. The mean and minimum values are taken over the whole data set, ME is mean error, MAE is mean absolute error, RMSE is root-mean-square error and PCC is Pearson’s correlation coefficient. $Dst_{\text{all}}$ lists results for the full data set, while $Dst_{\text{reduced}}$ shows results for the data while STEREO-B was near the L5 point. |  | Mean | Min. | ME | MAE | RMSE | PCC
---|---|---|---|---|---|---|---
| OMNI | -7.49 | -140.38 | -0.24 | 3.89 | 5.04 | 0.90
$Dst_{\text{all}}$ | STB | -7.28 | -172.16 | -0.21 | 8.40 | 12.25 | 0.41
| PERS | -5.38 | -164.96 | -2.11 | 9.40 | 13.42 | 0.33
| OMNI | -1.13 | -42.34 | -0.89 | 3.38 | 4.38 | 0.85
$Dst_{\text{reduced}}$ | STB | -1.45 | -40.67 | -0.56 | 6.63 | 8.89 | 0.25
| PERS | -0.83 | -62.81 | -1.19 | 8.04 | 10.90 | 0.12
Figure 3: (a) The correlation between Kyoto $Dst$ and $Dst$ predicted using
OMNI (red), time-shifted STEREO-B (blue) data as a function of $\Delta lon$
from $0^{\circ}$ to $-110^{\circ}$, plotted against results from a persistence
model (grey). (b) The RMSE for the same models over increasing $\Delta lon$.
This range of $\Delta lon$ covers a range of dates from Feb 2007 till Jan
2012. The curves denote fits to the data as a decaying function of $ae^{-bx}$.

In Fig. 3, the PCC and RMSE are both plotted as a function of longitudinal
difference $\Delta lon$ between Earth and STEREO-B. Each point was calculated
for the data measured at the point of longitude $\pm 3.5^{\circ}$, which led
to time ranges of two to seven months being evaluated because STEREO-B was not
moving away from Earth at a constant rate with regards to longitudinal
distance. As can be seen in the figure, predictions of $Dst$ using OMNI data
fairly consistently achieve a PCC of $0.90$ and an RMSE of roughly $5$ nT,
while the accuracy of the STEREO-B prediction degrades with increasing
distance from the Earth, as would be expected. Note some features of the plot
at first glance seem peculiar but can be easily explained. The sudden rise in
PCC beyond the $\Delta lon$ of -100 is a result of exceptionally quiet periods
with only small values of $Dst$. The peak in RMSE just before $\Delta lon$ of
-100 is a result of one particularly geo-effective high-speed stream
($Dst\sim-150$ nT) being observed at both STEREO-B and L1 but with an offset
of multiple days.

In Fig. 3a, the curves in the figure show fits to the PCC values as a decaying
function of $ae^{-bx}$, and we see that at a $\Delta lon$ of $-110^{\circ}$,
the predictions from STEREO-B data seem to have dropped to roughly the same
accuracy as the persistence model. The PCC here starts at $0.80$ close to the
Earth and drops to 0 at around $\Delta lon=-110^{\circ}$.

Fig. 4 explains this quick drop-off showing the correlations between solar
wind variables in OMNI and STEREO-B with increasing $\Delta lon$, both as
hourly values and the means over a window of 4 hours. The solar wind speed
remains highly correlated, which is to be expected due to the rotation of
coronal holes and the relatively slow temporal development of high-speed
streams. The correlation for total magnetic field and the $x$\- and
$y$-components also remains good throughout, but the $z$-component drops off
quickly and the correlation has already nearly reached 0 at $\Delta
lon=-10^{\circ}$. Due to the strong dependence of $Dst$ on $B_{z}$ [<]see
e.g.¿Gonzalez1987, it is not surprising that we also lose accuracy in $Dst$
prediction fairly quickly. What is however useful to note from Fig. 4 is that
the correlation in mean($B_{z}$) and the other components remains good beyond
the point at which the correlation of hourly values has dropped significantly.

Figure 4: Correlation between solar wind magnetic field measurements at L1 and
STEREO-B as a function of increasing $\Delta lon$. ’+’ markers show the
correlations for the original data, ’x’ markers correlations for 4-hour means.
The curves denote fits to the data as a decaying function of $ae^{-bx}$. Solid
curves are fits to the original 1-hour data, dashed curves are fits to 4-hour
means of the data.

In Fig. 3, the results from the persistence model are also included in grey,
and we see that there is a similar downward slope and decreasing correlation
over time even though we would expect this to stay constant. This suggests
that, for this time period at least, there may have been other effects in the
solar wind leading to a reduction in accuracy from the prediction due to more
rapid changes in the solar wind structures and high-speed streams. This is
likely explained by the increase in solar activity as the Sun entered the
rising phase of the solar cycle in 2011/2012. We would observe greater numbers
of short-duration coronal holes closer to solar maximum in addition to the
more regular coronal holes and high-speed streams that dominate the variations
otherwise. Although time does not increase linearly from point to point in
Fig. 3, on the whole it spans five years and we can look at the general
development of PCC and RMSE over time. Both show a gradual decrease in
accuracy as solar activity increases, which is most important to note for the
persistence model that serves as a benchmark for the others. When taking this
into account, a prediction using STEREO-B data at greater values of $\Delta
lon$ may also be a reasonable approach that is simply not represented well
here due to the more active period evaluated.

At the L5 point, the PCC for the STEREO-B model is around $0.30$.
Unfortunately, looking at the points alone we see that the approach performed
particularly badly at precisely this angle, although from the overall
statistics we can deduce that this was only a short-duration dissonance that
had little to do with the spacecraft position itself. This may instead be a
result of latitudinal difference between the two satellites, which we look at
in the following.

The dynamics of the outflowing solar wind in the heliospheric current sheet
have a strong latitudinal dependence, with differences of a few degrees
resulting in large variations in the shape and timing of the arriving solar
wind structures. In Simunac2009 and Thomas2018, for example, differences in
latitude between two spacecraft were shown to have a notable effect on
forecasting skill. To quantify this effect in this study, we look at the
accuracy metrics with the longitudinal dependence removed plotted against the
absolute difference in latitude between the two measuring points, STEREO-B and
Earth. See Fig. 5 for a depiction of this approach. This was achieved by
subtracting the slope of the longitudinal dependence for STEREO-B (centre
plot, light blue line) along with the slope in the OMNI values to account for
unrelated changes over time. Interestingly, few of the metrics show any
correlation with a difference in latitude, with the exception being the MAE
and RMSE, which correlate mildly with $\Delta lat$ with a PCC of $-0.35$
(increasing accuracy with increasing $\Delta lat$, a confusing result). All
other metrics have correlations $<$ $0.15$, such as the PCC showed as an
example in the figure (centre and right). If the range of $\Delta lon$ is
reduced to $0$ to $-40$ (in which the most periodic rotational behaviour was
observed), a more predictable set of behaviour emerges. The correlation values
for PCC, RMSE and MAE with $\Delta lat$ are -0.21, 0.24 and 0.15 respectively,
suggesting small dependencies on $\Delta lat$ leading to a decrease in
accuracy. Since an analysis of the whole range did not show the same pattern,
we can deduce that discrepancies caused by the increase in solar activity
later in the cycle far outweigh the effects of latitudinal difference in
regards to accuracy.

Figure 5: (a) Latitudinal difference $\Delta lat$ (green) between STEREO-B
(blue) and Earth (red) as a function of $\Delta lon$. (b) Mean error between
prediction using STEREO-B data and Kyoto $Dst$, corrected for longitudinal
effects. (c) Corrected PCC plotted as a function of absolute latitudinal
difference $\Delta lat$.

As a sanity check, the same metrics with and without ICMEs were also compared,
and the results were as expected. Predictions of $Dst$ using OMNI data were
equally good with and without ICMEs, but there was a small improvement in the
metrics for data predicted using the STEREO-B data and the persistence model
because the transient events unrelated to their measurements had been removed.
An analysis of errors with the input solar wind data split into high-speed
streams and slow solar wind was also carried out, with the results showing
that the errors in $Dst$ calculated for fast solar wind (above a threshold
1.25 times the median speed or $\sim 480$ km/s) are slightly ($\sim 20-30$ %)
larger than those for slow solar wind. In a further test, we also looked at
the same evaluation without scaling of the magnetic field components to
correct for different orbital distances to see if this step was necessary. Not
scaling the solar wind input led to a very small increase in the errors (e.g.
an RMSE of $12.25$ nT increasing to $12.83$ nT), showing that the scaling of
magnetic field to correct for a distance of 0.1 AU (at maximum) does have an
effect on the accuracy on the forecast, although it is minimal.

###  4.2 Event-based analysis

An interesting alternative to standard point-to-point measures that
quantitatively assess the magnitude of the forecast error at each time step is
to consider each time step as an event/non-event. The primary advantages of an
event-based validation analysis are as follows. Firstly, periods of weak,
moderate and enhanced geomagnetic activity are weighted equally by simple
point-to-point comparison measures. However, end-users of operational real-
time space weather models are usually more interested in accurately predicting
times of enhanced $Dst$ values, while the exact evolution in time of the $Dst$
is in most cases of secondary importance. Secondly, outliers in the predicted
times can have a significant influence on the error measures and correlation
coefficients determined. In context, an efficient approach is to label each
time step in the predicted and observed $Dst$ timeline as an event/non-event
[Owens (2018)]. An example of this approach applied to high-speed streams can
be found in Reiss2016, in which the OSEA software used for this analysis was
also used.

In this study, we define an “event” as any time step the $Dst$ exceeded (or
went below) a certain threshold. In this case we defined $-35$ nT as the
threshold for a weak geomagnetic storm. The value of $-50$ nT, usually the
threshold for a moderate storm, was also seen in the period evaluated, but did
not occur often enough to allow for a reasonable statistical analysis.
Similarly, $Dst$ values beyond $-100$ nT arose so infrequently that it was not
worth considering them here, and we restrict ourselves to looking at all
events below a level of $-35$ nT as a proxy for “mild geomagnetic activity”.
Using the corresponding threshold values, we label each time step as an event
or non-event in the measured and predicted time series, and for the case of
$Dst$ everything below the threshold was an event. By cross-checking the
events/non-events between true and predicted $Dst$, we count the number of
hits (true positives; TPs), false alarms (false positives; FPs), misses (false
negatives, FNs) and correct rejections (true negatives; TNs) and summarise
them in the so-called contingency table. From the entries of the contingency
table, we can compute different skill measures, including the True Positive
Rate TPR = TP/(TP + FN) and the False Positive Rate FPR = FP/(FP + TN). While
the TPR is the proportion of correctly predicted events among all the events,
the FPR is the proportion of non-events wrongly predicted as events.

Moreover, we compute the Threat Score TS = TP/(TP+FP+FN) as a measure of the
model performance, Bias B = (TP+FP)/(TP+FN) indicating whether the number of
observations is underforecast ($B<1$) or overforecast ($B>1$), and the True
Skill Statistics TSS = TPR - FPR as a measure of the overall model
performance. The TSS is defined in the range [-1,1] where a perfect prediction
would be equal to 1 (or -1, for a perfect inverse prediction), and a TSS equal
to 0 indicates no predictive ability of the forecast model. It is important to
note that the TSS is unbiased by the proportion of predicted and observed
events [Hanssen Kuipers (1965), Bloomfield . (2012)]. Since the number of non-
events exceeds the number of events by 10 to 1, the TSS is very well suited
for the validation analysis conducted in this study. For a more thorough
discussion of the skill measures applied here, we would like to refer the
interested reader to Jolliffe2003.

Table 2: Statistical comparison of the number of hits (true positives; TPs), false alarms (false positives; FPs), misses (false negatives; FNs), and correct rejections (true negatives; TNs), together with the True Skill Statistics (TSS), Bias and the area under the curve (AUC) for different prediction models. (A comparable AUC for the 24h minimum $Dst$ events could not be calculated due to a difference in the distribution and definition of event threshold.) |  | $N_{events}$ | TP | FP | FN | TN | TSS | Bias | AUC
---|---|---|---|---|---|---|---|---|---
| OMNI | 843 | 618 | 280 | 225 | 36622 | 0.73 | 1.07 | 0.94
$Dst_{\text{all}}$ | STB | 843 | 127 | 729 | 716 | 36173 | 0.13 | 1.02 | 0.74
| PERS | 843 | 75 | 581 | 768 | 36321 | 0.05 | 0.78 | 0.70
| OMNI | 6 | 4 | 0 | 2 | 3701 | 0.67 | 0.67 | 0.92
$Dst_{\text{reduced}}$ | STB | 6 | 0 | 1 | 6 | 3700 | 0.00 | 0.17 | 0.63
| PERS | 6 | 0 | 17 | 6 | 3684 | 0.00 | 2.83 | 0.59
| OMNI | 218 | 177 | 40 | 41 | 2886 | 0.80 | 1.00 | -
$Dst_{\text{all}}(24hMin.)$ | STB | 218 | 65 | 131 | 153 | 2795 | 0.25 | 0.90 | -
| PERS | 218 | 43 | 117 | 175 | 2809 | 0.16 | 0.73 | -

Table 2 shows the contingency table entries and the skill measures computed
over the full and reduced time ranges. We find that the prediction using
OMNI/L1 data has a very high level of accuracy in all measures, and in
comparison the prediction from STEREO-B/L5 is only somewhat better than a
persistence model. Both STB and PERS models show a tendency to underforecast
when considering the full data set.

Interestingly, we see that for the time STEREO-B was around L5 (in the reduced
data set), it managed to forecast 0 of the events below the threshold, while
the persistence model also forecast 0 but achieved an impressive number of
FPs. It is hard to carry out statistics for this period because it was an
extremely quiet time, with only 6 events below the $-35$ nT threshold over six
months. To look at a less time-sensitive approach, we consider an additional
forecasting method.

The bottom rows of Table 2 show the event-based analysis for the prediction of
the minimum $Dst$ in the next $24$ hours (with a $12$-hour resolution), and in
this way we consider the predictions while ignoring possible errors in timing
of $\pm 12$ hours. (For the OMNI model, with a maximum forecast time of 30–60
minutes, this is obviously a pointless measure, but it is left in for
comparison.) Here it becomes clear that the L5 monitor outperforms the
persistence model while also achieving a higher TSS score than when simply
applied to all data. In this case, the ratio of FPs to TPs is also greatly
reduced from around 6 to 2.

A straightforward approach to illustrate the trade-off between the proportion
of correctly predicted events (TPR) and the proportion of erroneously
predicted events (FPR) for different event thresholds is the so-called
receiver operator characteristic (ROC) curve. ROC curves are a helpful
diagnostic to compare and quantitatively assess the predictive skill of
forecast models. They illustrate how the number of correctly classified events
varies with the number of incorrectly classified non-events for each model
investigated here. In other words, they show the trade-off between the
completeness of events and the contamination with non-events. To illustrate
the predictive abilities of the different forecast models in a single summary
variable, we also compute the area under the curve (AUC) defined between 0 and
1, where the best results are equal to 1.

Figure 6: ROC curves for all three $Dst$ prediction models (OMNI: red, STB:
blue, PERS: grey) considered. The black line denotes the climatological mean
(simply using the mean $Dst$ as a prediction) as baseline comparison.

Fig. 6 shows the ROC curves for all models using the full data sets. At a
glance it is clear that the OMNI model far outperforms the models using
STEREO-B and PERS data, and the STEREO-B model is almost always somewhat
better than persistence.

###  4.3 Sensitivity of $Dst$ model to field measurements

Here we also include a brief evaluation of the sensitivity of the ETL2006
prediction model to magnetic field measurements and the effect of possible
offsets in the measurements. To achieve this, the ETL2006 model was used to
predict $Dst$ from two years of OMNI data to simply assess model sensitivity
independent of any other factors. Fig. 7 shows the dependence of predicted
$Dst$ on offsets in magnetic field measurements. An error of $\pm 1$ nT in the
$B_{z}$ component leads to an error of $+5/-6$ nT in the predicted $Dst$,
although offsets in all other components have almost no effect. This is useful
to know for future L1 and L5 missions, which rely on in-flight calibration
methods such as that described in Plaschke2019, to have an estimate of the
error in the $Dst$ prediction if the error in the magnetic field measurements
are known.

Figure 7: Plot of the offset in predicted $Dst$ as a function of offset in
magnetic field components (all). Only offsets in $B_{z}$ (purple) show a
notable effect on the value of predicted $Dst$.

##  5 Discussion and conclusions

In order to evaluate the possibility of forecasting the $Dst$ index from the
L5 point with a lead time of a few days, we have looked at in situ solar wind
data measured by STEREO-B as it crossed the L5 point. This was mapped to L1 in
time and space according to the solar rotation speed and expansion of the
solar wind and high speed streams. The mapped data is used to make a
prediction of $Dst$ and the results were then compared to the Kyoto $Dst$ as
well as $Dst$ predicted from an L1 monitor and a 27-day persistence model.
This method is useful for predicting geomagnetic effects from high-speed
streams and SIRs, but can not provide forecasts for transient events such as
ICMEs. The results can be summarised as follows:

  * •

A prediction of the $Dst$ index from data measured at L5 does not achieve a
level of accuracy similar to predictions made using L1 data, but it performs
better than a 27-day solar wind persistence model in all standard measures.
Geomagnetic effects in particular are hard to predict due to the rapidly-
changing development of $B_{z}$.

  * •

The error in the prediction can be quantified using the MAE, which at the L5
point in the STB model has an average value of 8 nT, which is double the error
from L1 at 4 nT. Offsets in $B_{z}$ measurements of $\pm 1$ nT can cause an
offset error in the predicted $Dst$ of $+5/-6$ nT.

  * •

As the values of magnetic field variations correlate much more strongly when
looking at mean values, a method of predicting the $Dst$ from means taken over
multiple hours would likely be a strong forecasting method, and this should be
considered for operational purposes.

  * •

The usefulness of L5 data in predicting $Dst$ minima in the next 24 hours is
reasonable and performs better than a persistence model.

  * •

A strong dependence of the accuracy of the predicted $Dst$ on latitudinal
difference between the L5 proxy and L1 measurements could not be determined.

Unsurprisingly, because $B_{z}$ is so badly predicted from L5 due to the rapid
changes in magnetic field [<]as discussed in detail in¿Thomas2018, it is
difficult to quantify the geo-effectiveness of a high-speed stream in advance
using data from somewhere as far-removed as L5 because of the strong
dependence on $B_{z}$ in causing geomagnetic effects [Gonzalez Tsurutani
(1987)]. This was compounded by the fact that the coherence of high-speed
streams when STEREO-B was closest to the L5 point was uncharacteristically
low, and so most of the results have been based on general trends in
forecasting capabilities across a much wider range of longitudes from the
Earth from 0 to $-110^{\circ}$, at which point the error appeared to match
that seen in the persistence model. As described in Verbanac2011, $Dst$ also
has a strong correlation with the solar wind speed, which shows that $B_{z}$
is not the only input to the model providing valuable information on how the
$Dst$ develops. A further effect that may lead to reduced accuracy is stream-
stream or stream-CME interaction. In some cases, a CME bracketed in an SIR can
give rise to enhanced geo-effectiveness in the stream [Chen . (2019)]. This
would be particularly effective if such a case were to occur at L5,
drastically changing the nature of the predicted high-speed stream that would
later arrive at Earth. Regardless, we have shown that predictions of $Dst$
from data at L5 certainly outperform a persistence model, especially when
looking at predictions of $Dst$ minimum in the next 24 hours or days.

This work was carried out to validate application of the methods in an
operational setting, and as such STEREO beacon data, which arrived in near
real-time, was used. Beacon data is, however, of considerably lower quality
than the science data, which arrives much later. Due to developments in on-
board processing since the STEREO launch in 2006, real-time data to be
expected from upcoming space weather satellites should be of better quality
than the original beacon data and may be closer in quality to the science
data. For comparison, the same parameters as discussed above were evaluated
for STEREO Level 2 scientific data, and an increase in accuracy was observed.
This was on a small scale (a correlation of $0.43$ instead of $0.41$, for
example), but shows that the data from newer satellites with more advanced
real-time data transfer should achieve a slightly higher accuracy for $Dst$
predictions than presented here.

At the moment, STEREO-A is slowly approaching the Earth in its orbit, meaning
that it can function as a proxy for real-time L5 forecasts for both the solar
wind variables and $Dst$. This study has functioned as a verification for a
real-time model currently running using STEREO-A data, which at the time of
writing is at a longitude of $-80^{\circ}$. For this forecast, we can assume
an error in $Dst$ predicted of $\pm 8$ nT from the average MAE at the L5 point
reached by STEREO-B. The knowledge gathered throughout this real-time
forecasting using STEREO-A will be invaluable in preparation for setting up
effective predictive methods relying on a real future L5 space weather
mission.

##  6 Sources of Data and Supplementary Material

Solar wind in situ data:

OMNI2: https://spdf.gsfc.nasa.gov/pub/data/omni/low_res_omni/
STEREO-B (BEACON): https://stereo-ssc.nascom.nasa.gov/data/beacon/behind/
STEREO-B (L2): https://stereo-
ssc.nascom.nasa.gov/data/ins_data/impact/level2/behind/magplasma/

Catalogues:

ICMECAT: https://doi.org/10.6084/m9.figshare.4588315.v1

Software:

Jupyter Notebook for this work: https://doi.org/10.6084/m9.figshare.11733909
Predstorm (Python, data handling): https://doi.org/10.5281/zenodo.3750749
OSEA (Matlab, statistical analysis): https://doi.org/10.5281/zenodo.3753104
HelioSat (Python, data download): https://doi.org/10.5281/zenodo.3749561

###### Acknowledgements.

The Python packages for data download and handling that this work is based on
(Predstorm v0.1, HelioSat v0.3.1) are open-source and under development and
are available on our group GitHub page (https://github.com/helioforecast). The
statistical analysis was carried out using Matlab solar wind analysis scripts
OSEA. Furthermore, the exact script that was used to create the results and
plots shown here (in Jupyter Notebook format) is available. See Section 6 for
the links to the data and scripts, and instructions for installation in the
repositories. R.L.B, C.M., M.A.R., A.J.W., U.V.A., T.A. and J.H. thank the
Austrian Science Fund (FWF): P31659-N27, J4160-N27, P31521-N27, P31265-N27. We
would like to thank the editor and reviewers for their helpful responses,
which led to improvement of this manuscript.

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