ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-34-759-2016On determining fluxgate magnetometer spin axis offsets from mirror mode observationsPlaschkeFerdinandferdinand.plaschke@oeaw.ac.athttps://orcid.org/0000-0002-5104-6282NaritaYasuhitoSpace Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, AustriaFerdinand Plaschke (ferdinand.plaschke@oeaw.ac.at)16September201634975976628May20169August201630August2016This 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/34/759/2016/angeo-34-759-2016.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/34/759/2016/angeo-34-759-2016.pdf
In-flight calibration of fluxgate magnetometers that are mounted on
spacecraft involves finding their outputs in vanishing ambient fields, the
so-called magnetometer offsets. If the spacecraft is spin-stabilized, then
the spin plane components of these offsets can be relatively easily
determined, as they modify the spin tone content in the de-spun magnetic
field data. The spin axis offset, however, is more difficult to determine.
Therefore, usually Alfvénic fluctuations in the solar wind are used. We
propose a novel method to determine the spin axis offset: the mirror mode
method. The method is based on the assumption that mirror mode fluctuations
are nearly compressible such that the maximum variance direction is aligned
to the mean magnetic field. Mirror mode fluctuations are typically found in
the Earth's magnetosheath region. We introduce the method and provide a first
estimate of its accuracy based on magnetosheath observations by the THEMIS-C
spacecraft. We find that 20 h of magnetosheath measurements may already be
sufficient to obtain high-accuracy spin axis offsets with uncertainties on
the order of a few tenths of a nanotesla, if offset stability can be assumed.
Magnetospheric physics (magnetosheath; instruments and techniques)Introduction
In situ magnetic field measurements in space are a key observational element
in space plasma and planetary physics: magnetic fields of planetary,
interplanetary, or solar origin impose structure on space plasmas. The
behavior of waves in these plasmas, characterized by propagation speed and
direction, is governed by the ambient magnetic field, e.g., due to the
Alfvén velocity being proportional to its strength.
Fluxgate magnetometers have so far been most widely used for direct
measurements of magnetic fields in space. They can be miniaturized and only
require small amounts of electric power. If well calibrated, they are able to
accurately determine three components of the DC and low-frequency magnetic
fields e.g.,. Calibration
activities need to be performed on ground (pre-launch) and routinely in space
(post-launch); they imply finding the components of a calibration matrix
M and an offset vector O that convert raw instrument
output Braw into magnetic field vectors B in
meaningful units e.g.,:
B=M⋅Braw-O.M is composed by three gain values and six angles that define the
magnetometer sensor directions. The 3-D offset vector O is the
magnetometer output in vanishing ambient fields. Changes in offsets reflect
not only drifts in the magnetometer sensors or electronics but also changes
in spacecraft generated fields over time. Long-term offset drifts with rates
of up to several tenths of a nanotesla per year and medium-term (seasonal)
variations with peak-to-peak amplitudes of up to a few tenths of a nanotesla
have been reported by , for instance. These offset
changes occur on timescales of years and months, respectively. Furthermore,
there are short-term offset variations with timescales on the order of an
hour, triggered by temperature changes at terrestrial/planetary periapsis
passes or in eclipse phases. These variations vanish on timescales of hours
as temperatures revert to equilibrium values. Clearly, there is a need to
monitor the offsets continuously and update them more frequently than the
other parameters.
In total, M and O are composed by 12 independent
parameters. In space, 8 of those parameters can be relatively easily
determined if the respective magnetometer is mounted on a spinning
spacecraft, as their choice will influence the content of spin tone and
harmonics in the de-spun magnetic field data. The two spin plane offsets
belong to this set.
The spin axis offset, instead, has to be determined by other means.
Classically, nearly incompressible (Alfvénic) fluctuations of the
interplanetary magnetic field are used to estimate spin axis offsets
. Therefore, the spacecraft are required
to take measurements in the pristine solar wind. Furthermore, independent
measurements in magnetic field magnitude and/or direction by other
instruments may be used for spin axis offset determination: the calibration
process of the fluxgate magnetometers of the recently launched
Magnetospheric Multiscale (MMS) mission involves, for the
first time, routine cross-calibration with observations by the electron drift
instruments . These instruments yield
electron gyro-times and, hence, absolute measurements of the ambient magnetic
field magnitude that can be used to estimate spin axis offsets, as detailed
in and . Unfortunately,
magnetospheric spacecraft rarely feature electron drift or other instruments
that yield measurements suitable for cross-calibration with fluxgate
magnetometers.
Here we introduce a new method to determine magnetometer spin axis offsets.
It is based on the idea that the mirror mode exhibits mostly compressible
fluctuations: the fluctuating fields appear to be nearly parallel to the
local mean magnetic field e.g.,. Hence,
it is possible to approximately determine the mean magnetic field direction
only from the fluctuation sense of the mirror mode.
Mirror modes and ion-cyclotron waves appear to be the two most important wave
modes in the (Earth's) magnetosheath e.g.,, which is
confined between the bow shock and the magnetopause. Both wave modes are fed
from the temperature anisotropy that is characteristic to the magnetosheath:
higher temperature in the plane perpendicular to the magnetic field.
Lower and higher plasma-β conditions favor the growth of the
ion-cyclotron and mirror mode, respectively. Typical high plasma-β
conditions in the (middle) magnetosheath make the mirror mode dominant there,
although it may often be superposed and, therefore, masked by other
coexisting waves.
Using magnetosheath observations by a Time History of Events and Macroscale
Interactions during Substorms (THEMIS) spacecraft ,
we lastly address the question of how accurate spin axis offsets can be
determined routinely by the mirror mode method.
Mirror mode method
A time series of magnetosheath magnetic field observations (including mirror
modes) shall be available in a spin axis aligned and de-spun (inertial)
coordinate system. The spin axis should point in the z direction. The
magnetic field measurements should be calibrated (B in Eq. ) but for the spin axis offset Oz.
Sketch illustrating the magnetic field and maximum variance
directions B and D, their projections onto the spin plane
Bxy and Dxy, and the angles θB, θD, and
ϕ.
Within mirror modes, the mean spin axis offset-corrected magnetic field
vector Bc, given by
Bc=BxByBz-Oz,
should ideally point in the same direction as the maximum variance direction
D determined by principal component analysis. By assuming that this
is the case, Oz can be obtained by equating the elevation angles
θBc=arctan((Bz-Oz)/Bxy) and θD=arctan(Dz/Dxy), yielding
Oz=-DzDxyBxy+Bz=Bxy(tanθB-tanθD),
where Bxy=Bx2+By2, Dxy=Dx2+Dy2, and
θB=arctan(Bz/Bxy). Figure illustrates the
directions of the magnetic field B and of the maximum variance
D, their projections onto the spin plane Bxy and Dxy, and
the angles between them θB and θD.
An uncertainty ΔOz of Oz may be estimated by
ΔOz=(tanθB-tanθD)ΔB2+BxyΔθBcos2θB2+BxyΔθDcos2θD2
with ΔB=|B|Δg+ΔBn being the
uncertainty of individual B components see.
Here ΔBn is a constant uncertainty related to noise and
also to errors in the offsets other than Oz; Δg denotes a relative
uncertainty associated with the magnetometer gain values and with the angles that
define the magnetometer sensor directions. Deviations in these quantities
produce errors in the components of the calibrated magnetic field
measurements that are proportional to the magnetic field strength. The
uncertainty ΔθB is computed via
ΔθB=ΔB1+(Bz/Bxy)21Bxy2+BzBxy22,
and ΔθD=arctan(λ2/λ1) results from the
ratio of intermediate to largest eigenvalues (λ2/λ1)
pertaining to the maximum variance analysis.
As can be seen in Eqs. () and (), it is
favorable for the Oz determination if B and D are located
near the spin plane so that the effect of a non-vanishing Oz is largest
on θB and, hence, on the difference tanθB-tanθD.
Consequently, we are interested in finding mirror mode intervals with small
Dz within a given time series of magnetosheath magnetic field
observations. Therefore, the entire time series is divided into small
(overlapping) subintervals of length tint. The aforementioned
quantities are computed for each of the subintervals, from which we select a
subset based on the threshold values Cxy, Cϕ, CB, and CD, as
defined in the following paragraphs.
Mirror modes are characterized by large amplitude fluctuations in magnetic
field magnitude δBc/Bc. We can only ascertain the spin plane part of that
expression and, hence, require
δBxyBxy:=Bxymax-BxyminBxymean>Cxy.
Here Bxymax, Bxymin, and Bxymean
denote the maximum, minimum, and mean magnetic field magnitudes in the spin
plane within a subinterval.
Furthermore, mirror modes are identified by the angle between D and
B that should stay below a certain value .
That angle, however, is directly dependent on
Oz and, hence, should not be directly constrained. Instead, we require for
the angle ϕ between D and B in the spin plane (see
Fig. ) that is independent of Oz:
|ϕ|<Cϕ.
We then simply restrict θB and θD individually by
|θB|<CB|θD|<CD
so that B and D are both closer to the spin plane than to the
spin axis but relatively unrestricted in pointing direction along z with
respect to each other. By setting Cxy, Cϕ, CB and CD, we
select a subset of the Oz and ΔOz estimates.
A final spin axis offset Ozf value may then be computed from
the selected Oz estimates by finding the maximum of the probability
density distribution P of those estimates. Therefore, the kernel density
estimator (KDE) method may be applied. Using a Gaussian kernel, P is
computed by the KDE method via
P(Õz)=12πNh∑i=1Nexp-12Õz-Ozih2.
Here N is the number of selected Oz estimates and h is the bandwidth
or smoothing width. The final offset Ozf shall then be given by
Õz for which P maximizes.
It should be noted that it is beneficial if the spin axis offset stays
constant over the entire magnetosheath observation time, from which Oz
estimates are computed. Otherwise, P will be broadened and the final offset
Ozf will represent an average spin axis offset.
Application and accuracy
We apply the mirror mode method to magnetosheath observations of one month
(July 2008) of THEMIS-C (THC) fluxgate magnetometer (FGM) data
in the despun sun-sensor L-vector (DSL) coordinate system
that is inertial (de-spun) and aligned with the spin axis (z direction).
The FGM data are continuously available in spin resolution (spin period:
∼3 s). In July 2008, THC's orbit around Earth was nearly equatorial
and highly elliptical. The orbit period was almost 2 days long and the apogee
was located in the dayside afternoon sector, upstream of the bow shock.
Hence, THC crossed the dayside magnetosheath twice per orbit and once per
day. The THC calibration tables reveal that the solar wind intervals were
used to routinely update the spin axis offset, on a monthly basis, to correct
for the long- and medium-term offset variations .
Hence, we expect the remaining Oz to nearly vanish. It should be noted
that THC's spin axis offset drift was only about 0.2 nT over the month of
July 2008, which is comparable to or even less than the uncertainty thereof.
Thus, we do not expect the correction of that spin axis offset drift to
affect/facilitate the application and/or enhance the accuracy of the mirror
mode method in any significant manner. Furthermore, we would like to point
out that the short-term offset variations occurring after perigee passes and
eclipse intervals are unimportant to this study, as sufficient time (roughly
6 h) passed after these events before THC entered the magnetosheath on the
outbound orbit legs.
The entire month is divided into overlapping tint=3 min long subintervals, shifted by 10 s. We determine
whether THC was in the sheath by checking whether the mean ion density of each
subinterval as measured by THC's electrostatic analyzer
is larger than twice the mean solar wind density as provided by NASA's OMNI
high-resolution data set for the same subinterval. This
criterion proved to work remarkably well for the identification of subsolar
sheath intervals in . Note that the particle data are
just used to automate the magnetosheath interval selection/identification
but are not at all required for the mirror mode method as the turbulent
sheath intervals may also be easily manually/visually identified from
magnetic field measurements only. In total, THC was approximately 154 h
in the magnetosheath in July 2008.
For each subinterval, we compute the quantities introduced in the previous
section, in particular Oz and ΔOz. As in ,
we choose Δg=10-4 and ΔBn=10 pT for the
computation of ΔB. Magnetic field measurements involved in computing
ϕ, θB, θD and Bxy are averaged component-wise over
each subinterval. Following and , we
select Oz estimates by setting the following threshold values: Cxy=0.3, CB=CD=30∘, and Cϕ=20∘. Therewith, we obtain
7831 Oz estimates and associated uncertainties ΔOz from as many
subintervals.
Figure shows a common example interval from 2 July 2008, which
contributes a larger number of 535 Oz estimates. At the beginning of the
interval, THC was in the magnetosphere; at the end it senses the solar wind
upstream of the bow shock. The magnetosheath observations are clearly
identified using the density criterion outlined above, as apparent from panel
b. For large parts of these observations, the criteria on ϕ, θB,
θD and Bxy are fulfilled (see panels e–h). Where this is the
case, Oz and ΔOz estimates are shown (panels c and d). It is also
apparent that the fluctuation level in B (black trace in panel a) is
significant throughout the entire magnetosheath observation time. This is a
typical characteristic of the magnetosheath that we are making use of here.
When zooming in, e.g., to the time around 14:30 UT, it can be seen that the
THC-measured magnetic field strength and the ion density are anti-correlated,
which is expected only for clear mirror mode signatures that are not masked
by other superposed fluctuations.
Example interval of THC data of 2 July 2008. Panels from top to
bottom: (a) THC FGM magnetic field components in DSL coordinates,
(b) THC ESA ion density and twice the OMNI ion density (red line),
(c)Oz, (d)ΔOz, (e)ϕ and
Cϕ in red, (f)θD and CD in red,
(g)θB and CB in red, and (h)δBxy/Bxy and Cxy in red.
All selected 7831 Oz over ΔOz estimates are depicted in the top
panel of Fig. . It is apparent that the amount of spread of Oz
around 0 nT of the individual Oz estimates is considerable: the
standard deviation of Oz is 4.09 nT. Hence, individual Oz values are
generally rather poor estimates of the spin axis offset. The large spread in
Oz values comes from the expected statistical deviation of vectors
B and D. In the z direction, that deviation is expressed by
the difference θB-θD. The average value of |θB-θD| pertaining to all selected Oz values is 11.7∘, which
is rather typical. , for instance, reported changes in
magnetic field direction across magnetosheath mirror mode structures to be <10∘. More recently, presented cases of
mirror mode storms in the solar wind and found angular deviations as large as
18∘. investigated properties of mirror mode
structures observed in the Jovian magnetosheath. They found deviations <10∘ “during the intervals of large-amplitude field fluctuations”.
use 20∘ as threshold for the angle between
B and D for the selection of mirror mode intervals.
As |Oz|≤ΔOz mostly holds (crosses mostly confined between red
lines in Fig. ), ΔOz may be regarded as a suitable
estimate for the accuracy of the individual Oz estimates. The average
values of the three terms that contribute to ΔOz in Eq. () are
|tanθB-tanθD|ΔB=0.002nTBxyΔθBcos2θB=0.012nTBxyΔθDcos2θD=7.287nT.
It is apparent that the error in Oz corresponds to the uncertainty in
θD that stems from the accuracy of the D determination by the
principal component analysis.
Although the spread in Oz is large, Ozf=0.21 nT as of
Eq. () using h=1 nT is found to be remarkably close to
the expected value of 0 nT and certainly within the limits of the spin
axis offset accuracy of the THC FGM calibration. The probability density
distribution P is shown in the bottom panel of Fig. . It is
highly symmetric around ∼0 nT. The mean and median values of the Oz
distribution are 0.08 and 0.20 nT, respectively. Hence, they come even
closer to 0 nT than Ozf.
Top panel: Oz over ΔOz estimates obtained from
magnetosheath magnetic field observations by THC in July 2008. The red lines
depict Oz=±ΔOz. Bottom panel: corresponding probability
density distribution P determined by the KDE method using a Gaussian kernel
and a bandwidth of h=1 nT.
We check whether a change to Bz by adding a fixed magnetic field value of
5 nT is correctly recovered by the mirror mode method. This is indeed the
case, as we obtain Ozf=5.17 nT. The resulting distributions
of Oz versus ΔOz and P are shown in Fig. . As can
be seen in the top panel, the distribution of Oz shifts to larger values
while the spread approximately remains. In the lower panel of Fig. ,
P is shown, which exhibits a large skewness. This may stem from
the fact that a positive non-vanishing Oz leads to θB being
systematically higher than θD. As we need to restrict θB and
θD by CB and CD to ensure selecting compressional signals with
variations close to the spin plane, the distribution of θB-θD
and, hence, Oz∼tanθB-tanθD cannot be symmetric any
more. Furthermore, a secondary, minority population of Oz estimates
centered on 0 nT seems to be apparent in the top panel of Fig. .
We speculate that this population originates from other (than
mirror mode) waves that feature a combination of compressional and transverse
magnetic field fluctuations. If we assume their maximum variance directions
to be approximately uniformly distributed, then the corresponding
distribution of the θB-θD and Oz values should be
symmetric around 0∘ and 0 nT, respectively. That secondary Oz
population contributes to the asymmetry in the total distribution of Oz
estimates, resulting in mean and median values of 3.71 and 4.47 nT,
respectively. Both values deviate more substantially from the added offset of
5 nT than Ozf, justifying the evaluation of P for its
determination. Note that the offsets are independent of the other
calibration parameters (matrix M in Eq. ). Hence,
adding an artificial, constant spin axis offset value to fully calibrated
data yields a data set that is equivalent to one which has been calibrated
except for the spin axis offset, if that offset can be assumed to be
constant.
Same as Fig. but determined from THC FGM measurements to
which 5 nT were added in the Bz component. Red lines in the top panel
depict Oz=±ΔOz+5 nT.
So far, the final spin axis offset results pertain to one month of THC data.
We can check how the uncertainty/spread in Ozf increases with
the cadence, i.e., by decreasing the sample size available for the
determination of P. Therefore, we compute P and, thereof,
Ozf for each day in July 2008, individually. The results are
shown in Fig. .
Daily spin axis offsets Ozf(a), standard
deviation σ of contributing Oz estimates (b), number N of
contributing Oz estimates (c), and σ/N(d).
Red crosses pertain to THC data with Bz component shifted by 5 nT.
The figure shows Ozf, the standard deviations σ of
contributing Oz, their numbers N, and the quotients σ/N,
determined from daily THC FGM data in black and from Bz shifted data (by
5 nT) in red. Apparently, daily Ozf values scatter very
significantly around 0 and 5 nT, respectively; the standard deviations
of the Ozf values shown in Fig. a are 2.2
and 2.8 nT, respectively. However, it can also be seen that deviations
from the expected offset values tend to be larger for larger σ and/or
lower N, as expected. Indeed, σ/N seems to be a good proxy
for the uncertainty associated with Ozf. This can be seen in
Fig. , which depicts daily Ozf values as a
function of σ/N: the spread in Ozf increases with
σ/N.
Daily spin axis offsets Ozf as a function of σ/N. Red crosses pertain to THC data with Bz component shifted by
5 nT. The black and red dotted lines depict Ozf=0 and
5 nT, respectively
Lowest σ/N with N>1000 are obtained for 7 July. For this
day, we find highly accurate Ozf of -0.01 and 4.97 nT
respectively. As stated above, THC spent approximately 154 h in the
magnetosheath in July 2008, from which we obtained 7831 Oz estimates.
That is ∼51 estimates per hour. If we regard N>1000 as a sufficient
number to accurately determine P and, consequently, Ozf, then
about 20 h of magnetosheath measurements should be enough to accomplish
this task.
It should be noted, however, that this latter statement is based on the
analysis of a particular set of THC magnetic field observations in the
magnetosheath, which feature only a small offset drift. Larger offset drift
rates may require the computation of less accurate Ozf values
on a higher cadence. If, instead, the direction of the magnetic field in the
magnetosheath in not suitable for spin axis offset determination over longer
time spans (|θB|>CB), then much more than 20 h of
magnetosheath measurements may be needed to obtain accurate Ozf
values. Hence, the stated minimum time length of 20 h can only be
regarded as a rough estimate.
Summary and conclusions
Making use of the compressible nature of the mirror mode opens the door to
determine and correct for the spin axis offset Oz in fluxgate
magnetometer data. We have introduced the mirror mode method, by which spin
axis offset estimates can be routinely obtained from magnetic field
measurements in the magnetosheath region, where mirror modes are dominant.
Furthermore, we have provided a first test of the accuracy of this method
using one month (July 2008) of THC FGM data.
From the entire month of data, we obtain an overall spin axis offset value of
Ozf=0.21 nT. This value is quite close to 0 nT, which is
the expected value, as the THEMIS data have already been spin axis
offset-corrected by making use of Alfvénic fluctuations in the solar wind
. The uncertainty of that
correction is on the order of a few tenths of a nanotesla. Hence, our
monthly value of Ozf is in excellent agreement therewith and we
can expect the uncertainty in Ozf to be on the order of a few
tenths of a nanotesla, as well.
Daily Ozf values, however, exhibit a significantly larger
spread; deviations on the order of several nT are found. We find,
though, that these deviations/uncertainties are related to the quotient
σ/N of the standard deviation σ of contributing Oz
estimates and their number N. Hence, it is possible to either select
accurate daily Ozf or to combine data from several days in
order to push σ/N to acceptable levels. In essence, rather
accurate spin axis offset determinations with the mirror mode method are
possible on cadences of a few days or above for spacecraft with similar daily
magnetosheath dwell times as THC in July 2008, if offset stability can be
assumed. That corresponds to a minimum of approximately 20 h of
magnetosheath observations.
Finally, we would like to point out that it should be possible to extend the
mirror mode method so that it becomes able to determine all three offset
components of magnetometers that are mounted on three-axis stabilized spacecraft.
As stated in Sect. , the offset determination is based on the
analysis of systematic differences in direction between B and
D during mirror mode intervals. If B and D are
approximately pointing in the x direction, for instance, then the offset
components pertaining to the directions perpendicular to x, Oy and
Oz, may be obtained with high accuracy, while Ox has to be assumed
constant. Different mirror mode intervals with B and D
pointing in different directions will yield estimates for all offset
components Ox, Oy, and Oz. Furthermore, mirror modes are not
restricted to the Earth's magnetosheath but have also been observed in other
solar system environments e.g.,.
Hence, magnetic field measurements in distant
regions of the solar system may benefit from offset determinations by the
mirror mode method, as well.
Data availability
Data from the THEMIS mission, including THC level 2 FGM and ESA data, are
publicly available from the University of California Berkeley and can
obtained from http://themis.ssl.berkeley.edu/data/themis. The solar
wind data from NASA's OMNI high resolution data set (1 min cadence) are also
publicly available and can be obtained from
ftp://spdf.gsfc.nasa.gov/pub/data/omni.
Acknowledgements
We acknowledge valuable discussions with Werner Magnes and Rumi Nakamura on
fluxgate magnetometer calibration procedures. In addition, we
acknowledge constructive criticism and valuable comments by Adam Szabo,
who reviewed an earlier version of this article.
We acknowledge NASA contract NAS5-02099 and V. Angelopoulos for use of data
from the THEMIS mission. Specifically, we acknowledge C. W. Carlson and
J. P. McFadden for use of ESA data and K. H. Glassmeier, U. Auster, and
W. Baumjohann for the use of FGM data provided under the leadership of the
Technical University of Braunschweig and with financial support through the
German Ministry for Economy and Technology and the German Center for Aviation
and Space (DLR) under contract 50 OC 0302. The topical editor, G. Balasis, thanks C. Carr and two
anonymous referees for help in evaluating this paper.
ReferencesAcuña, M. H.: Space-based magnetometers, Rev. Sci. Instrum., 73,
3717–3736, 10.1063/1.1510570, 2002.Alconcel, L. N. S., Fox, P., Brown, P., Oddy, T. M., Lucek, E. L., and Carr,
C. M.: An initial investigation of the long-term trends in the fluxgate
magnetometer (FGM) calibration parameters on the four Cluster spacecraft,
Geosci. Instrum. Method. Data Syst., 3, 95–109, 10.5194/gi-3-95-2014,
2014.
Angelopoulos, V.: The THEMIS Mission, Space Sci. Rev., 141, 5–34, 2008.
Auster, H. U., Glassmeier, K. H., Magnes, W., Aydogar, O.,
Baumjohann, W., Constantinescu, D., Fischer, D., Fornaçon, K. H.,
Georgescu, E., Harvey, P., Hillenmaier, O., Kroth, R., Ludlam, M.,
Narita, Y., Nakamura, R., Okrafka, K., Plaschke, F., Richter, I.,
Schwarzl, H., Stoll, B., Valavanoglou, A., and Wiedemann, M.: The
THEMIS Fluxgate Magnetometer, Space Sci. Rev., 141, 235–264, 2008.Balogh, A., Carr, C. M., Acuña, M. H., Dunlop, M. W., Beek, T. J., Brown,
P., Fornacon, K.-H., Georgescu, E., Glassmeier, K.-H., Harris, J., Musmann,
G., Oddy, T., and Schwingenschuh, K.: The Cluster Magnetic Field
Investigation: overview of in-flight performance and initial results, Ann.
Geophys., 19, 1207–1217, 10.5194/angeo-19-1207-2001, 2001.Belcher, J. W.: A variation of the Davis-Smith method for in-flight
determination of spacecraft magnetic fields, J. Geophys. Res., 78,
6480–6490, 10.1029/JA078i028p06480, 1973.Burch, J. L., Moore, T. E., Torbert, R. B., and Giles, B. L.:
Magnetospheric Multiscale Overview and Science Objectives, Space Sci. Rev.,
199, 5–21, 10.1007/s11214-015-0164-9, 2016.Enríquez-Rivera, O., Blanco-Cano, X., Russell, C. T., Jian,
L. K., Luhmann, J. G., Simunac, K. D. C., and Galvin, A. B.:
Mirror-mode storms inside stream interaction regions and in the ambient
solar wind: A kinetic study, J. Geophys. Res., 118, 17–28,
10.1029/2012JA018233, 2013.Erdős, G. and Balogh, A.: Statistical properties of mirror mode
structures observed by Ulysses in the magnetosheath of Jupiter, J. Geophys.
Res., 101, 1–12, 10.1029/95JA02207, 1996.Glassmeier, K.-H., Motschmann, U., Mazelle, C., Neubauer, F. M.,
Sauer, K., Fuselier, S. A., and Acuna, M. H.: Mirror modes and fast
magnetoacoustic waves near the magnetic pileup boundary of comet P/Halley,
J. Geophys. Res., 98, 20955–20964, 10.1029/93JA02582, 1993.
Hedgecock, P. C.: A correlation technique for magnetometer zero level
determination, Space Sci. Instrum., 1, 83–90, 1975.Kepko, E. L., Khurana, K. K., Kivelson, M. G., Elphic, R. C., and
Russell, C. T.: Accurate determination of magnetic field gradients from
four point vector measurements. I. Use of natural constraints on vector data
obtained from a single spinning spacecraft, IEEE T. Mag., 32, 377–385,
10.1109/20.486522, 1996.King, J. H. and Papitashvili, N. E.: Solar wind spatial scales in and
comparisons of hourly Wind and ACE plasma and magnetic field data, J.
Geophys. Res., 110, A02104, 10.1029/2004JA010649, 2005.
Leinweber, H. K., Russell, C. T., Torkar, K., Zhang, T. L., and
Angelopoulos, V.: An advanced approach to finding magnetometer zero levels
in the interplanetary magnetic field, Meas. Sci. Technol., 19, 055 104,
2008.Lucek, E. A., Dunlop, M. W., Balogh, A., Cargill, P., Baumjohann,
W., Georgescu, E., Haerendel, G., and Fornaçon, K.-H.: Mirror mode
structures observed in the dawn-side magnetosheath by Equator-S, Geophys.
Res. Lett., 26, 2159–2162, 10.1029/1999GL900490, 1999a.Lucek, E. A., Dunlop, M. W., Balogh, A., Cargill, P., Baumjohann,
W., Georgescu, E., Haerendel, G., and Fornaçon, K.-H.:
Identification of magnetosheath mirror modes in Equator-S magnetic field
data, Ann. Geophys., 17, 1560–1573, 10.1007/s00585-999-1560-9,
1999b.McFadden, J. P., Carlson, C. W., Larson, D., Ludlam, M., Abiad, R.,
Elliott, B., Turin, P., Marckwordt, M., and Angelopoulos, V.: The
THEMIS ESA Plasma Instrument and In-flight Calibration, Space Sci. Rev.,
141, 277–302, 10.1007/s11214-008-9440-2, 2008.Nakamura, R., Plaschke, F., Teubenbacher, R., Giner, L., Baumjohann, W.,
Magnes, W., Steller, M., Torbert, R. B., Vaith, H., Chutter, M.,
Fornaçon, K.-H., Glassmeier, K.-H., and Carr, C.: Interinstrument
calibration using magnetic field data from the flux-gate magnetometer (FGM)
and electron drift instrument (EDI) onboard Cluster, Geosci. Instrum. Method.
Data Syst., 3, 1–11, 10.5194/gi-3-1-2014, 2014.Plaschke, F., Hietala, H., and Angelopoulos, V.: Anti-sunward high-speed jets
in the subsolar magnetosheath, Ann. Geophys., 31, 1877–1889,
10.5194/angeo-31-1877-2013, 2013.Plaschke, F., Nakamura, R., Leinweber, H. K., Chutter, M., Vaith,
H., Baumjohann, W., Steller, M., and Magnes, W.: Flux-gate
magnetometer spin axis offset calibration using the electron drift
instrument, Meas. Sci. Technol., 25, 105008,
10.1088/0957-0233/25/10/105008, 2014.Price, C. P., Swift, D. W., and Lee, L.-C.: Numerical simulation of
nonoscillatory mirror waves at the earth's magnetosheath, J. Geophys. Res.,
91, 101–112, 10.1029/JA091iA01p00101, 1986.
Russell, C. T., Song, P., and Lepping, R. P.: The Uranian magnetopause
– Lessons from earth, Geophys. Res. Lett., 16, 1485–1488,
10.1029/GL016i012p01485, 1989.Russell, C. T., Anderson, B. J., Baumjohann, W., Bromund, K. R.,
Dearborn, D., Fischer, D., Le, G., Leinweber, H. K., Leneman, D.,
Magnes, W., Means, J. D., Moldwin, M. B., Nakamura, R., Pierce, D.,
Plaschke, F., Rowe, K. M., Slavin, J. A., Strangeway, R. J.,
Torbert, R., Hagen, C., Jernej, I., Valavanoglou, A., and Richter,
I.: The Magnetospheric Multiscale Magnetometers, Space Sci. Rev., 199,
189–256, 10.1007/s11214-014-0057-3, 2016.Schmid, D., Volwerk, M., Plaschke, F., Vörös, Z., Zhang, T. L.,
Baumjohann, W., and Narita, Y.: Mirror mode structures near Venus and Comet
P/Halley, Ann. Geophys., 32, 651–657, 10.5194/angeo-32-651-2014, 2014.Schwartz, S. J., Burgess, D., and Moses, J. J.: Low-frequency waves in
the Earthś magnetosheath: present status, Ann. Geophys., 14, 1134–1150,
10.1007/s00585-996-1134-z, 1996.Torbert, R. B., Vaith, H., Granoff, M., Widholm, M., Gaidos, J. A.,
Briggs, B. H., Dors, I. G., Chutter, M. W., Macri, J., Argall, M.,
Bodet, D., Needell, J., Steller, M. B., Baumjohann, W., Nakamura,
R., Plaschke, F., Ottacher, H., Hasiba, J., Hofmann, K., Kletzing,
C. A., Bounds, S. R., Dvorsky, R. T., Sigsbee, K., and Kooi, V.: The
Electron Drift Instrument for MMS, Space Sci. Rev., 199, 283–305,
10.1007/s11214-015-0182-7, 2016.Tsurutani, B. T., Lakhina, G. S., Verkhoglyadova, O. P., Echer, E.,
Guarnieri, F. L., Narita, Y., and Constantinescu, D. O.: Magnetosheath
and heliosheath mirror mode structures, interplanetary magnetic decreases,
and linear magnetic decreases: Differences and distinguishing features, J.
Geophys. Res., 116, A02103, 10.1029/2010JA015913, 2011.Violante, L., Cattaneo, M. B. B., Moreno, G., and Richardson, J. D.:
Observations of mirror waves and plasma depletion layer upstream of Saturn's
magnetopause, J. Geophys. Res., 100, 12047–12055, 10.1029/94JA02703,
1995.