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.

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?

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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