ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-34-831-2016Determination of errors in derived magnetic field directions in
geosynchronous orbit: results from a statistical approachChenYuecheny@lanl.govCunninghamGregoryHendersonMichaelLos Alamos National Laboratory, Los Alamos, New Mexico, USAYue Chen (cheny@lanl.gov)21September201634983184324December201522August201631August2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/34/831/2016/angeo-34-831-2016.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/34/831/2016/angeo-34-831-2016.pdf
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 <∼ 2∘, than those from
the three empirical models with averaged errors >∼ 5∘. In
addition, our results suggest that the errors in derived magnetic field
directions do not depend much on magnetospheric activity, in contrast to the
empirical field models. Finally, as applications of the above conclusions, we
show examples of electron pitch angle distributions observed by LANL GEO and
also take the derived magnetic field directions as the real ones so as to
test the performance of empirical field models along the GEO orbits, with
results suggesting dependence on solar cycles as well as satellite locations.
This study demonstrates the validity and value of the method that infers
local magnetic field directions from particle spin-resolved distributions.
Magnetospheric physics (energetic particlestrapped; storms and
substorms; instruments and techniques)Introduction
It is well-known that energetic electrons in the Earth's outer radiation belt
– ranging from ∼ 3 to 8 Earth radii (RE) – are highly
dynamic and present storm-specific behaviors (e.g. Reeves et al., 2003;
Chen et al., 2007b; Tu et al., 2014). Thus, monitoring, understanding, and
forecasting the variations of outer-belt electrons are central topics for the
space weather community. To address these topics, one basic imperative is to
have long-term continuous observations with high quality and good coverage
over key areas, particularly regions close to the low-altitude boundary
(i.e., the lower thermosphere and mesosphere where originally trapped
electrons precipitate), the internal plasma boundary (i.e., the plasmapause
where wave-electron resonance prevails), as well as the high-altitude
boundary (i.e., the magnetopause separating the enclosed drift shells from
open ones). Among those regions, satellites in the geosynchronous orbit (GEO,
a geo-equatorial circular orbit with geocentric distance of
∼ 6.6 RE) play a unique role by monitoring the corridor through which substorm particles are injected into the inner
magnetosphere, while radiation belt electrons can also be diffused outward towards the
magnetopause.
LANL GEO satellites measure electron directional distributions.
(a) Side view of the GEO orbit. A LANL GEO satellite is usually
close to but not exactly in the geomagnetic equator due to the tilted
geomagnetic dipole field. (b) Rotation of the satellite platform
allows the three SOPA telescopes (T1, 30∘ to the spin axis z;
T2, 90∘, and T3, 120∘) to sample directional
distributions of electrons; meanwhile, the unit local magnetic field from the
empirical model (Bm in blue), the one derived from electron
distribution (Bd in red), and the real direction
(Br in black, if measured) can be different. The goal of
this work is to determine the angle between Bd and
Br (indicated by the question mark). (c) One
example electron distribution measured by SOPA. Count rates are sorted by the
roll angle (defined as the azimuthal angle in the satellite spin plane:
0∘ along the x (due east) direction and 90∘ along y (due
south)), and a derived magnetic field direction from symmetry of the
distribution is marked by the white cross. The very low counts for T2 are
measured close to the loss cone. (d) In a unit sphere, the three
magnetic vectors form a polar triangle ΔBdBrBm‾, whose side lengths
(a, b, and c) are proportional to the angles between each pair of unit
vectors. (e) Polar triangle ΔBdBrBm‾ can be approximated by
the planar triangle ΔDRM‾ in this study.
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 (∼ 10 s),
and different pointing directions make each telescope sample different pitch angle ranges. Since the average magnetic field direction is more or less
perpendicular to the spin axis, telescopes T1 and T3 will usually not
be able to measure electrons near the loss cone (aligned with the magnetic
field direction) as the example distributions in Fig. 1c show. Thus, measurements
from all telescopes form a spin-resolved distribution for each energy
channel. For higher energies, ESP has a single telescope that points
perpendicular to the spin axis and provides additional directional
measurements. However, without a magnetometer on board, extra measures are
needed to convert the directional distribution from SOPA and ESP into a more
useful pitch angle distribution that is often used to characterize radiation
belt dynamics (e.g., see the introduction and references in Chen et al.,
2014).
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∘ pitch angle. Thus, applying a principal-axis analysis
to the MPA plasma directional distributions, one may generate three
eigenvalues and eigenvectors and choose the most unique eigenvector as the
one that is parallel or antiparallel to the local magnetic field direction
(see Thomsen et al., 1996, or Chen et al., 2005, for detailed descriptions of
the algorithm). Considering the fact that the low-energy plasma is often
nearly isotropic on the nightside during substorm injections (e.g., Meredith
et al., 1999), a related technique is developed to apply to the
spin-resolved energetic electron distributions measured by SOPA and ESP. In
the same vein, this technique searches for the symmetric direction in
particle distributions, and details can be found in the Appendix. One example
magnetic field direction derived using this technique is marked in Fig. 1c.
This study focuses on the latest method of using SOPA and ESP measurements,
and testing the MPA method is left to the future (further discussions on this
can be found in the Appendix).
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 Bd
derived from particle distribution, the Bm calculated from
an empirical model, and the “real” magnetic direction Br
from an in situ measurement, they would usually point in different directions
(panel b). If plotting those directions inside a unit sphere as in panel d,
the three points form a polar triangle with each of the side lengths proportional
to the angles between each pair of unit vectors. This way, we may simply
compare the length a of the side BdBr to
the length b of the side BmBr to draw a
conclusion. Unfortunately, in our case, the main barrier is the unknown
position of Br due to the lack of in situ magnetic field
measurements, and thus both values of a and b in panel d are undetermined.
To overcome the barrier, we replace individual directions with statistical
averages, assuming similar statistical distributions and average values for
neighboring satellites, and use a triangulation method to determine the
location of Br. That is, starting from two points with
positions known, we first calculate their distances to Br
using statistical averages from other resources; then, we draw a circle
around each of the two points with a radius of the calculated average, and the intersection of circles will reveal the position of
Br. In addition, since the angles between magnetic
directions are mostly smaller than 10∘, we use planar triangle
ΔDRM‾ to approximate the spherical triangle ΔBdBrBm‾ (panel e), which brings an
ignorable error <∼ 0.5 %. Essentially, our primary goal in this
work is to determine the position of R and then the length of
DR‾. More details will be discussed in Sect. 3. Hereinafter,
Bd, Br, and Bm always
refer to the statistically averaged directions of derived, real, and modeled
magnetic field (i.e., unit vectors), respectively, unless being specified
otherwise, and they are often shortened to D, R, and M
in triangulation plots.
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.
Resources: instruments, data, and empirical global magnetic field
models
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 Bd in this
work, long-term LANL GEO data sets are used, ranging over 1996–2004 from
seven satellites (1989-046, 1990-095, 1991-080, 1994-084, LANL-97A, LANL-01A, and
LANL-02A) distributed globally with different geographic longitudes.
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 Br in this work, GOES data are downloaded
from the Coordinated Data Analysis Web (CDAWeb), including from GOES-08, 09 (in 1995 and 1997) and GOES-10,
-12 (2004). After removing the offsets in GOES data (Tsyganenko et al.,
2003; Chen et al., 2005), the downloaded 1 min resolved GOES data are
rebinned to 4 min to match LANL GEO data. Generally there are two
GOES satellites in operation simultaneously: one at ∼ 285∘ and
the other at ∼ 225∘ longitude. Occasionally data are available
with longitudinal separations smaller than ∼ 60∘ when a third
GOES satellite is being activated or changing station.
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 y and z
components, and interplanetary indices G2 and G3) – the Tsyganenko
2001 storm model (T01s, also called TSK03) (Tsyganenko et al., 2003). Our
selection of these models is based upon previous studies (e.g., Chen et al.,
2007b; Huang et al., 2008; McCollough et al., 2008), with the expectation
of differing performance and the best performance of T01s from the model list
in Chen et al. (2007b). It should be mentioned that we do recognize the
existence of other magnetic field models (e.g., the more recent TS05 by
Tsyganenko and Sitnov, 2005), and we will have more discussions on this in
Sect. 4.
Error estimation in derived magnetic field directions using
statistical approaches
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 ∼ 30∘ west of GOES-10. Here
we show both individual data examples and their statistical distributions.
Sample magnetic field directions during an 8-day period in 2004.
(a) Polar angles of derived magnetic field directions (red) from
1991-080 particle data are compared to those calculated from T01s model
(blue), both plotted as a function of time. Polar angle is defined as the
angle between a magnetic field direction and the z axis of GSM coordinate
system. (b) Polar angles of observed magnetic field directions
(black) by GOES-10 compared to those from T01s model (blue).
(c) Angles between (b/t) derived and model magnetic vectors for
1991-080. Gray (black) symbols are for data in dayside (nightside).
(d) Angles between measured and model vectors for GOES-10.
(e) The Dst (black) and Kp (gray) indices. A major storm occurs on
22 January during the period.
Figure 2 presents an 8-day period with one major storm (minimum Dst
∼-150 nT as in the last panel) for a glimpse of how the data,
derivation, and model results compare. Panel a plots the time series of polar
angles in the geocentric solar magnetospheric (GSM) system for magnetic field
directions derived from 1991-080 particle distributions in comparison to
polar angles of T01s model outputs. In the same format, panel b plots polar
angles for measured magnetic field directions by GOES-10 in comparison to
those from T01s model. Panel c depicts the angles between derived and model
directions for 1991-080, while panel d presents angles between real and model
directions for GOES-10. Comparing panels a to b and c to d, one can see the
similarities between LANL GEO and GOES data sets, such as the diurnal
variations and large deviations in storm main phase. Clearly, angles in
panel c and d are smaller in dayside than nightside for each satellite (a
spatial feature), while angles increase significantly and simultaneously for
both satellites during active times (a temporal feature).
Statistical studies comparing derived, real (measured), and model
magnetic field directions in 2004. Panels in the top row are for 1991-080.
(a1) Accumulative percentage vs. deviation angles between derived
and modeled directions for the three empirical models T01s (black), T89
(green), and OP77 (gray). Mean angle values as well as satellite coordinates
are also presented. (b1) Normalized percentages vs. deviation angles
for T01s. (c1) Deviation angles are binned to MLT for the three
models, and the vertical gray bars are the errors for T01s model. Panels in
middle row are for GOES-10 in the same format except for comparing real and
modeled directions. (a3) Deviation angles are binned to Kp for
1991-080 (black) and GOES-10 (red) using T01s model. Again the vertical bars
are errors for each. The gray dotted line plots data sample number in each
bin (read by the vertical axis on right). (b3) Deviation angles are
binned to Dst. (c3) Deviation angles are binned to the Auroral
Electrojet Index (AE).
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∘ that is the line segment length
between D and M in Fig. 1e. Besides the mean values,
distributions show that more than 90 % of the angles are below
10∘ for dynamic magnetic field models (panel a1), while a small
portion has large angle values as the long tail of the distribution in
panel b1. In general, the mean angle values get smaller (with a minimum
∼ 2∘) in dayside and larger in nightside (with a maximum
∼ 7∘ for T01s), and the sizes of error bars determined from
root mean squares are comparable to the mean values (panel c1). From panels
in the middle row comparing measured and modeled field directions, we see
similar distributions, while the mean deviation angles have slightly smaller
values (panel a2) and higher percentages for low angle values (panel b2).
Here the mean deviation angle for T01s has a value of 3.81∘ that is
the line segment length between R and M in Fig. 1e. We should
note that a larger value of DM‾ than RM‾ does not
necessarily indicate a large value of DR‾.
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 DM‾ (black) and RM‾
(red) curves trace each other very closely, and their separations are almost
independent of the activity index, except for the highly active categories
for which data sample numbers are too small (< 100) to make statistically significant. Results from all three magnetic field models show a similar closeness
between DM‾ and RM‾ (not shown here), leading us to
the hypothesis that the dependence of deviation angles on magnetic activities
is merely caused by the degrading performance of each empirical field model, and the barely changing separations between DM‾ and
RM‾ suggests small values for DR‾ all the time. This
hypothesis will be addressed next.
Determining the range of DR‾
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 z axis (pointing out of paper) is always the
local field direction calculated from the model (M), the x axis is in
the z–xGSM plane and points to the Sun, and the y axis completes
the right-handed orthogonal set. Thus, the position of each data bin is
determined by its distance to the origin M, i.e., the deviation angle
between real field direction (R) and modeled field direction
(M), as well as the azimuthal angle of R with respect to the
x axis. The color in each bin indicates the count of data points (distributions
with deviation angles > 20∘ are not plotted here), the red circle
plots the mean of all deviation angles, and the white curve shows the
directional mean of deviation angles in each radial direction. Although data
samples are highly unevenly distributed azimuthally, the directional mean
values are still very close to the mean of all with an average absolute
fluctuation level of ∼ 11 %. Therefore, we conclude that, given a
statistically averaged distance of RM‾, we may draw a circle
around the point M for all possible positions for the point
R, whose exact location is, however, undetermined unless additional
information is provided. Similarly, the distribution comparing the M
and D from two LANL satellites (1991-080 and LANL-02a) in Fig. 4b
also shows no significant azimuthal preference with an average absolute
fluctuation level of 6 %. Thus, we assume that there is weak azimuthal
preference for any pair of two directions in this study.
Deviation distributions and estimating the deviation angle range
between derived and real magnetic field directions.
(a) Distributions of real directions (R) relative to model
directions (marked by the white “M” in the origin pointing out of paper).
The radial distance from any point to M is the deviation angle between a pair
of model and real directions, and the azimuthal angle is determined in a
modified local B-GSM coordinate system (and thus is not local time).
Color in each bin indicates the count of data points. The overplotted white
curve indicates the directional mean of deviation angles in each radial
direction, compared to the red circle showing the mean of all deviation
angles. (b) Distributions of derived directions (D) relative
to model directions (M), directional means, and the mean circle in the
same format. (c) Given averaged deviation angle values for
DM‾ (4.88∘) and RM‾ (3.81∘) in
Fig. 3, we may estimate that the range of DR‾ is between
[1.07∘, 8.69∘], that is [0.28, 2.28]×DM‾.
The imaginary point P and circle in gray, if available, will help
pinpoint the position of R.
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 D and R. First, based on the comparison of 1991-090
data in 2004 to T01s model, we draw a line connecting D and M
in Fig. 4c with the segment length of 4.88 in between, which is the average
value for deviation angles between the two directions as discussed in the
beginning of this section. (Hereinafter all length values between two points
have the unit of degree.) Then, based on analysis of GOES-10 data in 2004, a
half circle is drawn around M (the lower half can be omitted due to
symmetry) with a radius of 3.81 – the average value between M and
R calculated from above. Point R can be anywhere on this
circle, from which we estimate the median (minimum, maximum) angle between
the derived direction D, and the real direction R is 6.19
(DR‾min=1.07, DR‾max=8.69). That is, the
averaged deviation angles between D and R range within [0.28,
2.28] times of DM‾ with a median value of 1.62×DM‾. Therefore, at least we can first conclude that the errors
between derived and real magnetic directions are comparable to that between
model and real directions. However, to further locate the exact position of
R, an extra point (e.g., the imaginary point P in Fig. 4c) as
well as its distance to R is needed for triangulation.
Locating point R using proxy magnetic field
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 Bpt-Bmt=Brs-Bms, where Bpt is proxy
magnetic vector for the target satellite, Brs is the real
magnetic field from a neighboring (source) satellite, and
Bmt(Bms) is the magnetic vector calculated
from an empirical magnetic field model (T01s is used here) for the target
(source) satellite, and all vectors vary with time. Since deviations in the modeled magnetic field are from both temporal and spatial features, the above
equation assumes that the deviations in two neighboring satellites are
homogeneous due to their proximity. Obviously, the validity of this assumption
degrades with increasing longitude separation between two satellites.
Validating the proxy method of using measurements from a neighboring
satellite. (a) In this 1-month period, deviation angles between the
proxy magnetic field direction and in situ measurements (green) along the
GOES-09 orbit are plotted as a function of time, compared to angles between
T01s model and measurements (blue). (b) During the period, the
relocation of GOES-09 makes its longitude separation with GOES-8 varying from
∼ 15∘ to up to 40∘. (c) Dst (black) and Kp
(gray) indices. Minor and moderate magnetic activity is observed during the
period.
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
∼-150 nT on 22 January. The proxy magnetic field for LANL-GEO
1991-080 is derived from in situ measurements of NOAA GOES-10 with a
∼ 30∘ longitude separation.
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
∼ 60∘, but this separation can be smaller when a GOES satellite
is relocated, although observation data during those periods are rarely
available. We were fortunate enough to identify a short period with available
data in 1995 when GOES-09 was moved from longitude 270 to 244∘. This
movement makes the longitude separation between GOES-09 and GOES-08 increase
from initially ∼ 15 to ∼ 40∘. Therefore, after applying
the above equation to approximating GOES-09 magnetic field using GOES-08
measurements, proxy magnetic field directions are validated by GOES-09
measurements, as the green curve in Fig. 5a. For comparison, deviation angles
between GOES-09 measurements and T01s model are also plotted. It is clear
that the proxy outperforms the T01s model significantly when the longitude
separation between satellites is <∼ 30∘, and both perform
similarly even when the separation goes beyond ∼ 40∘ by the end
of the period. Therefore, since the longitude separation between GOES-10 and
1991-080 is ∼ 30∘ in 2004, this proxy method can add the point
P to the plot by using GOES-10 to derive proxy for 1991-080.
First, locating point P requires knowing the lengths of
DP‾ and MP‾. Therefore, the proxy field directions
are compared to derived and model field directions for 1991-080, and Fig. 6
presents a short interval as an example. A statistical study gives out an
averaged DP‾ value of 5.34 and an MP‾ value of 4.11.
Thus, we are able to plot the point P for proxy directions in Fig. 7b.
Then we need to derive the value of PR‾ for the circle radius.
Besides the above validation using the pair of GOES-08, -09 in December 1995,
we also use data from two other periods: GOES-08, -09 between 20 November and
1 December 1997, and GOES-10, -12 between 1 March and 1 April 2004, both with
a longitude separation of 60∘. Deviation angles between derived and
proxy field directions for all three periods are plotted against longitude
separation in Fig. 7a, overplotted by averaged angle values ranging from
∼ 2 to 5. Here we use the average value of 3.50 for the segment length
PR‾. As in Fig. 7b, the circle ⊙P with a
PR‾ radius of 3.50 intersects with the circle ⊙MT01s (with a radius of 3.81): the intersection point
R1 has a distance of 1.83 from D, and the intersection
point R2 has a distance of 7.71. So now the question is which
point – R1 or R2 or both – is real.
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
DM‾, MP‾, and MR‾ due to the different
model but the same values of DP‾ and PR‾, and again
there are two intersection R points. However, DR‾1 in
both panels b and c has the same values but the DR‾2 values
are different, which serves as the first piece of evidence that R1
should be the real R point since we do expect the DR‾
values to be independent of empirical magnetic field models.
Determining the position of R point(s) using proxy magnetic
field. (a) Deviation angles between proxy and measured field
directions, in three selected periods, are plotted against the longitude
separations between each pair of GOES satellites. Overplotted data symbols
are averaged angles for binned longitude separations. (b) The
introduction of the point P and the PR‾ circle generate
R1 and R2 intersection points when T01s model is used.
(c) The introduction of the point P and its circle generate
another pair of R1 and R2 intersection points when OP77
model is used.
Determining the position of R point(s) using multiple
empirical models. (a) A circle is drawn around each of the points
MT01s and MOP77 from the two models, and
the intersections give two candidate points R1 and R2.
(b) The introduction of another point MT89 (from
the T89 model) and its circle generate R1a and
R1b points very close to the R1, as well as
R2a and R2b points spreading away from
R2. (c)R1 and R2 points are
determined for different magnetic activity categories. Since the
DR‾m1 and DR‾m2 basically stay
constant with magnetic activity, the grouping of R1 points
should be much tighter than that of R2 points.
Locating point R from grouping of points
Inspired by the proxy point P added in Fig. 7, we speculate that an
alternative way of using two empirical models should also be able to add an
extra point. As in Fig. 8a, after using the T01s model to place the baseline
DM‾T01s and drawing the circle ⊙MT01s, an extra point Mop77 from the model
OP77 can be located from the segment lengths of DM‾op77
and Mop77MT01s‾ using 1991-080
data. Then the second circle ⊙MOP77 is drawn with the
radius of MOP77R determined from GOES-10 data. Again
the two circles have two intersection points: the R1 point with a
distance of 1.10 from point D, and R2 with a distance of
8.73. To differentiate R1 and R2, we introduced the
second extra point MT89 using the model T89, whose position
is located from the segment lengths of DM‾T89 and
MT89MT01s‾ using 1991-080 data
(panel b). And the third circle ⊙MT89, with a radius
of 3.81 from GOES-10 data, intersects with the circles ⊙MT01s (⊙MOP77) at points
R1a and R2a (R1b and
R2b). In the ideal case, R1,
R1a, and R1b should overlap (the same is
true for R2, R2a, and R2b),
though it is natural to see that they do not do so exactly since
statistically averaged values are used here. However, points R1,
R1a, and R1b in panel b are indeed tightly
clustered but not points R2, R2a, and
R2b, which serves as the second piece of evidence that
R1 points should be very close the real position of the R
point, instead of the widely spreading points R2,
R2a, and R2b.
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 DR‾1 values are not the same in Fig. 7 (1.83) and
Fig. 8 (1.10), this can be explained by the uncertainties in the numbers used
here. For example, the 3.50 for PR‾ used in Fig. 7 may have
larger errors than other numbers due to the limited available data. However,
from both DR‾1 values, it is reasonable to state that the
average deviation angle between derived and real magnetic field directions is
smaller than the value between model and real directions by a factor of >∼ 2.
How the DR‾ value varies with magnetic activities can be learned in a
similar way, by taking advantage of the fact that length difference between
the DM‾ and RM‾ length stays almost unchanged with
magnetic activity levels as discussed in the beginning of this section. A
qualitative instead of quantitative method is employed in this step, which
should guarantee that our conclusion is reliable. As in Fig. 8c, we draw a diagram using two
magnetic field models (1 and 2) for two different activity categories (a and
b). As just mentioned, since the distances from D to the circles
around M1a and M1b along the
DM‾1a line have the constant value of
DR‾m1, and the circles around M2a and
M2b (not drawn here) will both be at a tangent with the small
circle D with a radius of DR‾m2, we can see
that the intersection points R1a (between ⊙M1a and ⊙M2a) and R1b
(between ⊙M1b and ⊙M2b) stay
very close to each other while R2a and
R2b are well separated. Therefore, because we already know
that the R1 group is close to the real R point, we
conclude that DR‾ values are not sensitive to the magnetic
activity levels. This supports our hypothesis that the observed increasing
DM‾ values with elevated activity levels in Figure 3 should be
mainly due to the degrading performance of empirical models, as discussed in the beginning of this section.
Discussion and applications
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 MT89 and MOP77 can be measured
from the plot to be 5.49. Compared to the calculated value of 4.66, this
indicates a ∼ 18 % error that should be acceptable.
Electron PADs – based upon the derived magnetic field directions –
observed by LANL-01A SOPA during a geomagnetic storm period (7 days).
(a) Pitch-angle-resolved fluxes for low-energy (131 keV) electrons
evolve with time. (b) Pitch-angle-resolved fluxes for high-energy
(1.2 MeV) electrons evolve with time. (c) Dst (black) and Auroral Electrojet (AE) (gray)
indices for the period. The time bin size for each PAD is 4 min. LANL-01A
reaches the noon local time position at ∼ 23:00 UT each day during this period.
To understand the averaged deviation angle of <∼ 2∘ in derived
magnetic field directions, we need to discuss what are the possible error
sources for this method. The most likely error source is the large size of
angular bins used to sort measured particle counts. In our case, the largest
angular bin size can be ∼ 11∘ so that the assigned pitch angles
can have errors as large as ∼ 5.5∘. The second error source may
be at times when particle distributions are close to isotropic. This can be
significant for low-energy plasma particularly during substorm injections but
should be alleviated for energetic radiation belt electrons (typically with
several hundred keV to > 1 MeV energies like the SOPA E5 and ESP E1
channels selected for this work). For example, according to a recent pitch
angle distribution (PAD) statistical study, PADs for ∼ 150 keV
electrons at L∼ 6 are statistically very close to isotropic during
substorms as shown in Fig. S2b, panels A2 and B2, in the Supplement of Chen
et al. (2014), while PADs for ∼ 1.5 MeV electrons at the same L are
statistically highly anisotropic as in Fig. S2b, panels A2 and B2. Another
possible error source is the intrinsic asymmetry in PAD due to either the
statistical fluctuations in counts registered by instruments or some process
that breaks down the particles' bounce movement. The former occurs when MeV
electron fluxes drop significantly during storm main phases; the latter may
also be possible for electrons close to the loss cone but can be ignored for
stably trapped populations that make up the LANL GEO observations. All these
could contribute to the small but existing errors we found here.
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 ∼ 40 and 125 h), while MeV
electrons show intriguing sustained butterfly PADs in the early phase of
radiation belt enhancements (e.g., throughout the day 19 March). This
difference suggests that the two populations should have experienced
different physical processes. Therefore, as discussed in the “Introduction”
section, LANL GEO measurements have high energy and pitch angle resolutions
and are distributed over multiple longitudes at GEO; thus, they are highly
valuable for studying radiation belt dynamics, particularly together with
simultaneous observations from Van Allen Probes inside GEO.
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. (a) Accumulative percentage vs. deviation
angles between derived and simulated directions for three empirical models
T01s (black), T89 (green), and OP77 (gray). Mean angle values are also
presented. (b) Normalized percentage vs. deviation angles for T01s
model. (c) Mean deviation angles are binned to MLT for the three models,
and the vertical gray bars are the error ranges for T01s model.
(d) Deviation angles are binned to Kp ranges using different colors
for three models. Again, the vertical bars are error bars for T01s model. The gray
dotted line plots data sample numbers (read by the vertical axis in right).
(e) Deviation angles are binned to Dst. (f) Deviation
angles are binned to the Auroral Electrojet Index (AE).
Additionally, since the deviation of Bd is small, we may
use the derived directions as real ones to test the performance of empirical
models over the long term (1997–2004). Figure 10 presents the distributions in
the same format as in Fig. 3. Percentage distributions in panels a and b are
similar to those in Fig. 3 except getting slightly flatter, which is
consistent with the slightly increased mean values in the magnetic local time (MLT) distributions
in panel c. The small spikes at noon are mainly from data before 2000, and
how realistic they are will be left to future investigation by examining
individual events. This larger data set allows better coverage with
statistical significance extending to higher magnetic activity categories in
panels d, e, and f. From low- to moderate-activity categories, dynamic models
persistently perform better than the static model; however, an interesting
reverse can be seen in distributions for which T01s model has the largest
deviation for the very high activity range.
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
∼ 2002 and then declines afterwards. Also, with different geographic
longitudes, LANL GEO satellites are located at different magnetic latitudes or
equivalently at different L shells (Chen et al., 2005). By plotting the
mean deviation values vs. the Lm (McIlwain L shell) calculated from T01s
model, we do see a general trend of increasing deviation values with
increasing Lm by linearly fitting those data points. Indeed, the calculated
Pearson's correlation coefficient has a nontrivial value of 0.41. All these
suggest that the model T01s performance degrades with increasing L shells
(or, latitudes), which is consistent with our general impression of empirical
models.
Model performance of T01s depends on solar cycle and satellite
positions. (a) Annual average angles are plotted as a function of
years for each LANL GEO satellite. (b) Average angles are plotted as
a function of Lm (McIlwain L shell) for each LANL GEO satellite.
Data points are fitted by the gray straight line, whose equation is given on
the top of the panel, with a Pearson's correlation coefficient value of
0.41.
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 (x, y, z) components, while T01s has values of (0.91,
0.82, and 0.90) (Tsyganenko and Sitnov, 2005). We decide to leave the
inclusion of the TS05 model to the future.
Summary
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 >∼ 2 between
magnetic field deviation angles – the three selected empirical models
(including T01s) in matching the real magnetic directions. Additionally, our
results suggest that errors in derived magnetic directions are not so much
dependent on magnetospheric activities as the empirical field models. At
last, after showing electron PADs observed by LANL GEO satellites, we further
use the derived magnetic field directions for testing the performance of
empirical field models, with results showing dependence on solar cycles as
well as GEO satellite positions. This study for the first time demonstrates
the validity and the value of using the symmetric nature of particle
spin-resolved distributions for deriving local magnetic field directions.
Data availability
LANL GEO data used in this study are available upon request by contacting the
corresponding author Y. Chen (cheny@lanl.gov).
Inferring magnetic field directions from LANL GEO SOPA and
ESP measurements
Magnetic field directions derived from MPA aboard LANL-01A compared
to those from SOPA and ESP during a 9-day period. (a) The time
series of magnetic field directions' polar angle theta (θ) derived
from MPA (red) are compared to those from SOPA and ESP (blue). LANL-01A
crosses midnight at ∼ 23:00 UT each day in 2005. (b) Time series of field directions' azimuthal
angle Phi (ϕ). (c) Dst (black) and Auroral Electrojet (AE) (gray) indices for the
period. A double-dip storm occurred during the period with the minimum Dst
∼-135 nT reached on 30 May (DOY 149).
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, ϕ, is found, about which the particle distribution
measured by the ESP E1 (0.7–1.8 MeV) channel is most symmetric. This angle
points parallel or antiparallel to the projection of the background magnetic
field into the plane perpendicular to the spin axis, or, for certain particle
distributions, points 90∘ perpendicular to the magnetic field. These
ambiguities are cleared up in the second stage of the analysis, wherein every
angle, θ, measured from the spin axis, is tested in 2∘
increments as a potential field line direction when combined with ϕ.
The pair (ϕ, θ) specifies a tested magnetic field direction, and
the spin-resolved SOPA E5 (225–315 keV) electron channel counts are binned
into pitch angles under the assumption that this pair is the correct one. A
smooth polynomial function is fitted to the pitch angle binned counts, and
the root mean squared error (RMSE) of the fit is calculated. The pair
(ϕ, θ) that produces the lowest RMSE is chosen as the field
direction. Because the three telescopes for SOPA may not be perfectly
calibrated to one another, multiplicative constants for T1 and T3 are
found that map the pitch angle binned counts for T1 and T3 so that they
best match those from T2. This “calibration” of T1 and T3 is done
separately for each 4 min time bin, each energy channel, and each
hypothesized magnetic field direction (ϕ, θ). SOPA Channel E5
was chosen to estimate the magnetic field direction because it had the best
combination of anisotropy and count rate over the broadest range of
conditions, but a better algorithm could be devised that analyzes all energy
channels simultaneously, as in Thomsen et al. (1996), or selects the best
energy at any given time.
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 (ϕ, θ) values from two methods along
the orbit of LANL-01A during a 9-day period. It can be seen that in panels a
and b, although directions from both methods agree well with each other
mostly during quiet times, values from MPA experience large fluctuations when
the satellite travels through the midnight sector (at ∼ 23:00 UT each
day) with substorm injections indicated by high Auroral Electrojet Index (AE) values (e.g., on
DOY 147–150), which is consistent with the discussion in Sect. 4. In
comparison, directions from SOPA and ESP do not have as many large
fluctuations (e.g., during the first small dip of Dst on DOY 147–148).
Similar results have been seen for other LANL GEO satellites for different
periods (not shown here). Indeed, if necessary, directions derived from MPA
measurements can also be statistically studied using the same approach
presented in the current work.
Acknowledgements
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.
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