This study aims to statistically estimate the errors in
local magnetic field directions that are derived from electron directional
distributions measured by Los Alamos National Laboratory geosynchronous (LANL
GEO) satellites. First, by comparing derived and measured magnetic field
directions along the GEO orbit to those calculated from three selected
empirical global magnetic field models (including a static Olson and Pfitzer
1977 quiet magnetic field model, a simple dynamic Tsyganenko 1989 model, and
a sophisticated dynamic Tsyganenko 2001 storm model), it is shown that the
errors in both derived and modeled directions are at least comparable.
Second, using a newly developed proxy method as well as comparing
results from empirical models, we are able to
provide for the first time circumstantial evidence showing that derived
magnetic field directions should statistically match the real magnetic
directions better, with averaged errors

It is well-known that energetic electrons in the Earth's outer radiation belt
– ranging from

LANL GEO satellites measure electron directional distributions.

Los Alamos National Laboratory has a long history of flying particle instruments aboard its geosynchronous satellites (LANL GEO hereinafter) to monitor the space environment since 1976. These instruments sample energetic electrons and protons from near the magnetic equator (Fig. 1a) over a wide range of energies, and the electron data used in this work are from the Synchronous Orbit Particle Analysis (SOPA) (Belian et al., 1992) as well as the Energy Spectrometer for Particles (ESP) (Meier et al., 1996) instruments. By themselves or in combination with others, LANL GEO particle data sets have been widely used in numerous studies leading to many significant discoveries, including identifying relativistic electrons as the cause of satellite deep dielectric charging (Baker et al., 1987), revealing the modulation of outer-belt electrons by solar cycle (Belian et al., 1996) and solar wind conditions (Li et al., 2005), and demonstrating the dominance of wave-particle resonance in accelerating outer-belt electrons (Chen et al., 2007a), among others. Nowadays, LANL GEO satellites provide critical complementary observations to the Van Allen Probes mission that operates inside of GEO; and in the foreseeable future, LANL GEO data sets will continue to play an irreplaceable role in scientific research as well as operational applications – such as the Dynamic Radiation Environment Assimilation Model (DREAM) (Reeves et al., 2012) – due to their long-term continuity, reliability, and high quality.

Besides resolving energy, SOPA and ESP instruments also measure particle
directional distributions (Fig. 1b). SOPA's three telescopes are mounted to
have different angles with respect to a satellite's spin axis (always pointing
toward the Earth's center). This configuration allows each telescope to sweep
out a band of the surrounding space within each spin period (

Besides turning to empirical magnetic field models, one may also derive the
local magnetic field direction using a physics-based technique that is first
proposed by Thomsen et al. (1996) and applied to Magnetospheric Plasma
Analyzer (MPA) data. This technique takes advantage of the fact that
trapped-particle directional distributions should be gyrotropic, i.e.,
rotationally symmetric around the magnetic field line, as well as symmetric
about the 90

Although the theoretical basis is solid for the above derivation technique,
determining the errors associated with this technique is still a critical
issue. This work aims to address this issue through estimating the errors in
a statistical manner. For the first time, we provide answers to the following
questions:

Does this technique outperform empirical magnetic field models?

How large can its errors be?

And do the errors depend on geomagnetic activity?

The cartoons in Fig. 1 illustrate the difficulty and our solution for this
study. Ideally, for any given instant in time, if we were able to have all
three magnetic field directions available, including the

Instrument descriptions, data, and magnetic field models are presented in Sect. 2. Section 3 explains the statistical approaches to estimate errors in derived magnetic directions. Section 4 discusses how to understand the results within context and their applications, and this report is concluded by a summary in Sect. 5.

As mentioned in Sect. 1, local magnetic field directions are derived every
4 min from spin-resolved electron measurements from each LANL GEO satellite
using the technique described in the Appendix. To get

The only real magnetic field directions used in this work are from in situ
measurements by several NOAA Geostationary Operational Environmental
Satellites (GOES). The three-axis fluxgate magnetometers, located on a boom
3 m away from the main body of each GOES satellite, provide the magnitude and
direction of the local magnetic field with a 0.512 s time resolution (Singer et
al., 1996). To get

For comparisons, we calculate local magnetic field directions from empirical
models. We always use the International Geomagnetic Reference Field (IGRF) as
the internal model, and for the external field we use three empirical models:
a static model – the quiet Olson and Pfitzer magnetic field model (OP77)
(Olson and Pfitzer, 1977); a simple dynamic and Kp-driven model – the
Tsyganenko 1989 model (T89) (Tsyganenko, 1989); and a much more sophisticated
dynamic model driven by the Disturbance Storm-Time Index (Dst) and solar wind
parameters (including the pressure, interplanetary magnetic field

In this section, we focus on data in 2004 considering the simultaneous data
coverage from a LANL GEO satellite 1991-080 and a NOAA satellite GOES-10.
During this year, 1991-080 is

Sample magnetic field directions during an 8-day period in 2004.

Figure 2 presents an 8-day period with one major storm (minimum Dst

Statistical studies comparing derived, real (measured), and model
magnetic field directions in 2004. Panels in the top row are for 1991-080.

Figure 3 presents statistical distributions of angles between magnetic field
directions. Panels in the top row present deviation angles between derived
and modeled field directions. As in panel a1, the mean deviation angle for
T01s model has a value of 4.88

When further binned to magnetic indices, deviation angle values increase with
increasing magnetic activity level, as shown by panels in the bottom row. It
is interesting to see that the

First, before applying the triangulation method, we prove that relative
positions between two magnetic field vectors have a weak azimuthal preference.
As in Fig. 4a, all GOES-10 and GOES-12 data (we include two satellites for
better statistics) in 2004 are plotted against model directions from T01s in
a coordinate system, in which the

Deviation distributions and estimating the deviation angle range
between derived and real magnetic field directions.

Then we apply the data analysis method aforementioned in Sect. 1 to both LANL
GEO and NOAA GOES data in 2004 to estimate the range of deviation angles
between

To add an extra point to the construction diagram as in Fig. 4c, we developed
a proxy method which approximates the real magnetic field direction for a
satellite using measurements from a neighboring satellite. The proxy is
derived using the equation

Validating the proxy method of using measurements from a neighboring
satellite.

Time series of deviation angles between derived and proxy magnetic
field directions (red) and deviation angles between model and proxy
directions (blue). This example covers the same 8-day period in 2004 as in
Fig. 2, which includes an intense storm with the minimum Dst

Here, we validate this proxy method using a pair of GOES satellites when they
are close enough and in situ magnetic field data are available for both. As
mentioned, GOES satellites generally have a large longitude separation of

First, locating point

Then we need to derive the value of

To answer the question, we replace the T01s model with the OP77 model and
repeat all of the steps above. As in Fig. 7c, we have different values of

Determining the position of

Determining the position of

Inspired by the proxy point

Based on this analysis, we conclude that in an average sense the derived
magnetic field directions are closer to the real magnetic field than
simulations from the three selected empirical field models used in this work.
Although the

How the

One possible major error for this study comes from the statistical approach
itself, that is, how representative the average points are in the
construction plots, such as Figs. 4, 7, and 8. For an individual case study,
each point in those figures is definite and thus the triangulation method is
valid. However, for two given distributions, the representativeness of the
calculated mean deviation points may be questionable. Indeed, considering the
variations in each distribution, the above method is only valid when the two
distributions are relatively homogeneous, which again cannot be directly
tested due to the lack of simultaneous derived and measured magnetic field
data. Nevertheless, one indirect test can give us some indications and thus
confidence for the representativeness of averages: in Fig. 8b, the distance
between

Electron PADs – based upon the derived magnetic field directions –
observed by LANL-01A SOPA during a geomagnetic storm period (7 days).

To understand the averaged deviation angle of

A direct application of the derived magnetic field direction is to sort LANL
GEO particle directional measurements into PADs, as one such example shown in
Fig. 9. During this double-dip storm period, substorm electron PADs in
panel a vary differently from those of energetic electrons in panel b. For
instance, substorm electron PADs are mainly pancake-shaped or close to
isotropic during injections (e.g. at

Results from comparing derived and model magnetic field directions
for all available LANL GEO data within 1997–2004. Panels have the same
format as in Fig. 3.

Additionally, since the deviation of

We further inspect the dependence of deviation angles on the solar cycle and
satellite positions. As in Fig. 11a, the deviation has a general growing
tendency in the rising phase of the solar cycle until reaching the maximum in

Model performance of T01s depends on solar cycle and satellite
positions.

Finally, as mentioned in Sect. 2, we only chose three representative
empirical magnetic field models without including the more recent
sophisticated TS05 model. Although previous studies have demonstrated that
T01s performs better than many other models (Chen et al., 2005; McCollough
et al., 2008), no comprehensive study has been conducted to compare between
T01s and TS05. Therefore, we cannot simply extend our conclusion to the TS05
model, although there are some clues suggesting comparable performances of
T01s and TS05 at GEO: when statistically comparing to observations dominated
by GEO data, TS05 has correlation coefficients of (0.92, 0.83, and 0.92) for
magnetic field (

This work statistically estimates the errors in the local magnetic field
directions derived from electrons' directional distributions measured by LANL
GEO satellites. First, by comparing derived and measured magnetic field
directions in GEO to outputs from empirical global magnetic field models
(including a static Olson and Pfitzer quiet magnetic field model, a simple
dynamic Tsyganenko 1989 model, and a sophisticated dynamic Tsyganenko 2001
storm model), we show that the errors in both derived and modeled directions
are at least comparable. Second, using a newly developed proxy method as well
as comparing results from multiple empirical models, we provide for the first
time evidence showing that derived magnetic field directions should
statistically outperform – with a ratio factor of

LANL GEO data used in this study are available upon request by contacting the corresponding author Y. Chen (cheny@lanl.gov).

Magnetic field directions derived from MPA aboard LANL-01A compared
to those from SOPA and ESP during a 9-day period.

The algorithm applied to the SOPA and ESP data first bins each of the three
SOPA telescopes and the lone ESP telescope into spin phase using
accumulations over a 4 min window to flesh out the distribution as a
function of spin phase. The count from each accumulation bin is either placed
into one of 32 spin-phase bins for SOPA data or into one of 180 spin-phase
bins for ESP data. Next, the
spin-phase angle,

A systematic comparison of the two methods using MPA and SOPA with ESP is outside the scope of this
work; however, it would be informative to get a glimpse of how magnetic field
directions from the two methods compare. Figure A1 presents one such example
which compares the derived (

This work was supported by the Los Alamos National Laboratory internal funding, the NASA Heliophysics Guest Investigators program (14-GIVABR14_2-0028), and the LANL Center of Space and Earth Science (CSES) program (special large project 2015-007). We want to acknowledge the PIs, instrument teams, and data support teams of LANL GEO SOPA and ESP, NOAA GOES magnetometer, as well as the data hosts CDAWeb and SSCWeb. We are grateful for the use of IRBEM-LIB codes for calculating magnetic coordinates. We also want to thank the referees for providing constructive and helpful comments that are incorporated into this paper. The topical editor, G. Balasis, thanks three anonymous referees for help in evaluating this paper.