Scholarly article on topic 'Interinstrument calibration using magnetic field data from the flux-gate magnetometer (FGM) and electron drift instrument (EDI) onboard Cluster'

Interinstrument calibration using magnetic field data from the flux-gate magnetometer (FGM) and electron drift instrument (EDI) onboard Cluster Academic research paper on "Physical sciences"

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Academic research paper on topic "Interinstrument calibration using magnetic field data from the flux-gate magnetometer (FGM) and electron drift instrument (EDI) onboard Cluster"

Geosci. Instrum. Method. Data Syst. Discuss.,3,459-487, 2013 www.geosci-instrum-method-data-syst-discuss.net/3/459/2013/ doi:10.5194/gid-3-459-2013 © Author(s) 2013. CC Attribution 3.0 License.

Geoscientific 0 Instrumentation As Methods and 8

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This discussion paper is/has been under review for the journal Geoscientific In-trumentation, Methods and Data Systems (GI). Please refer to the corresponding final paper in GI if available.

Inter-instrument calibration using

magnetic field data from Flux Gate

Magnetometer (FGM) and Electron Drift

Instrument (EDI) onboard Cluster

1 1 12 2 1 R. Nakamura , F. Plaschke , R. Teubenbacher , L. Giner , W. Baumjohann ,

W. Magnes1, M. Steller1, R. B. Torbert3, H. Vaith3, M. Chutter3, K.-H. Fornagon4,

K.-H. Glassmeier4, and C. Carr5

1 Space Research Institute, Austrian Academy of Sciences, 8042 Graz, Austria 2Graz University of Technology, 8010 Graz, Austria 3University of New Hampshire, Durham, NH 03824, USA

4Institut für Geophysik und extraterrestrische Physik, Technische Universität Braunschweig, 38106 Braunschweig, Germany

5Blackett Laboratory, Imperial College London, London, UK

Received: 31 May 2013 - Accepted: 4 July 2013 - Published: 30 July 2013

Correspondence to: R. Nakamura (rumi.nakamura@oeaw.ac.at)

Published by Copernicus Publications on behalf of the European Geosciences Union.

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FGM-EDI-calibration onboard Cluster

R. Nakamura et al.

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Abstract

We compare the magnetic field data obtained from the Flux-Gate Magnetometer (FGM) and the magnetic field data deduced from the gyration time of electrons measured by the Electron Drift Instrument (EDI) onboard Cluster to determine the spin axis offset of the FGM measurements. Data are used from orbits with their apogees in the magne-totail, when the magnetic field magnitude was between about 20 nT and 500 nT. Offset determination with the EDI-FGM comparison method is of particular interest for these orbits, because no data from solar wind are available in such orbits to apply the usual calibration methods using the Alfvén waves. In this paper, we examine the effects of the different measurement conditions, such as direction of the magnetic field relative to the spin plane and field magnitude in determining the FGM spin-axis offset, and also take into account the time-of-flight offset of the EDI measurements. It is shown that the method works best when the magnetic field magnitude is less than about 128 nT and when the magnetic field is aligned near the spin-axis direction. A remaining spin-axis offset of about 0.4 ~ 0.6 nT was observed between July and October 2003. Using multi-point multi-instrument measurements by Cluster we further demonstrate the importance of the accurate determination of the spin-axis offset when estimating the magnetic field gradient.

1 Introduction

Magnetic field and plasma environments of the Earth and other bodies in the solar system have been studied in-situ since decades (Balogh, 2010). Therefore, magnetic field experiments onboard of spacecraft are of primary importance. Most commonly, flux-gate magnetometers are used, due to their high accuracy, measurement range, resolution, and stability, paired with reasonable mass, power consumption, level of complexity, and overall costs (Acuña, 2002).

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FGM-EDI-calibration onboard Cluster

R. Nakamura et al.

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A flux-gate magnetometer (FGM) that is able to measure the strength and direction of the ambient magnetic field (B) with high precision, require extensive pre-flight (ground-based) and in-flight calibration (e.g., Glassmeier et al., 2007; Auster et al., 2008). The aim of the calibration is to determine 12 parameters needed to convert raw measurements (Braw) into components of a magnetic field vector (Bcal) in a usable coordinate system (e.g., Kepko et al., 1996). The calibration parameters are: six angles describing the orientation of the sensor axes in, e. g., a spacecraft-fixed frame of reference (constituting matrix M), three gain values (elements of a diagonal matrix G), and three zero level offset values (elements of vector O). Therewith, the conversion of Braw into Bcal is given by (e. g., Kepko et al., 1996; Acuña, 2002; Auster et al., 2008):

G ■ M ■ Braw - O

Despite pre-flight calibration under a variety of conditions (magnetic fields, temperatures), in-flight calibration remains necessary to account for slight changes of the calibration parameters during launch, instrument drifts overtime while the mission proceeds, and, most importantly, spacecraft-caused disturbances which are beyond the scope of ground-based tests.

Variations in ambient magnetic field strengths and temperatures may have a minor influence on gain levels (G) and orientations (M) of the sensor axes relative to the spacecraft body. Spacecraft generated fields (e.g., due to electrical currents or magnetic materials) strongly contribute to the zero level offsets (O), as these offsets represent the field values measured under the absence of an external magnetic field. Influence of the spacecraft on the magnetic field measurements can be reduced either by placing the FGM sensor on a long boom (e. g., Dougherty et al., 2004), hence, furthest possible away from the spacecraft main structure, or by implementation of a magnetic cleanliness program (e.g., Ludlam et al., 2008). Unfortunately, both measures tend to be extremely expensive.

Spin stabilization of the spacecraft greatly supports the in-flight calibration process, as the presence and content of spin tone and/or higher harmonics in the magnitude

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and/or spin axis component of Bcal is influenced by 8 of the 12 calibration parameters (see, Auster et al., 2002), namely the spin plane components of O (which shall be O1 and O2),the ratio of the spin plane components of G (i.e., G11/G22), and five elements of M (all but the angle defining the absolute orientation of the two spin plane axes within that plane).

The in-flight determination of the spin axis component of O (which we denote with O3) is often dependent on the availability of prolonged solar wind observations, where Alfvenic fluctuations are prevalent. These fluctuations are characterized by rotations in magnetic field while the field strength (|B|) remains constant. Hence, O3 can be determined by minimization of variance of |Bcal| while observing Alfvenic fluctuations, as proposed in Hedgecock (1975). Improvements to his method are discussed in Leinweberet al. (2008) and, more recently, in Pudney et al. (2012).

If solar wind measurements are not available, O3 may be determined with the help of complementary magnetic field observations, for instance from an electron drift instrument (EDI), which is the main subject of this paper. The EDI (Paschmann et al., 1997, 2001) onboard Cluster consists of two electron gun/detector units placed on opposite sides of the spacecraft, similar to that flown on the Equator-S spacecraft (Paschmann et al., 1999). Amplitude-modulated electron beams are fired by the two guns in specific directions. They perform one (or more) gyrations due to the ambient magnetic field and are eventually collected by the detectors after times T1 and T2. The primary objective of the EDI is to measure the drift of the electrons caused by electric fields or magnetic field gradients.

The drift step d = vdTg during the gyration time Tg (drift velocity: vd) is a direct result from EDI measurements: small d can be determined by triangulation, based on the two beam-firing directions (for a detailed description see, Paschmann et al., 1997; Quinn et al., 1999). Large d are more accurately determined by time-of-flight observations of the two beams (Paschmann et al., 1997; Vaith et al., 1998). These times are different for electron release in parallel or anti-parallel directions to vd: T12 = Tg(1 ±|vd|/|ve|), where ve is the electron velocity dependent on their (known) kinetic energy: the sum

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of T1 and T2 yields twice the gyration time Tg, their difference is proportional to d S (Paschmann et al., 1999). Use of different electron energies further allows to distinguish drifts caused by electric fields or magnetic field gradients (see, Paschmann et al., 1997).

Since the gyration time Tg is inversely proportional to the magnetic field strength |B|, EDI measurements allow for a determination of ambient |B|:

2nme eTn

where me is the electron mass and e the elementary charge. These values are practically not influenced by spacecraft fields, as electrons perform most of their gyration at sufficient distances from the spacecraft. Hence, they are ideally suited as a reference for FGM measurements. Comparison of EDI and FGM magnetic field data yields FGM zero level offset vectors O and, in particular, their spin axis components O3, as shown by Georgescu et al. (2006).

Their methods were developed further by Leinweber et al. (2012) in order to obtain absolute spin plane and spin axis FGM gains (i.e., G11 and G22 with constant ratio G11/G22, and G33), in addition to O3, with the help of EDI time-of-flight |B| values. Note that the spacecraft spin does not support calibration of any of these three parameters, as they do not influence the content of spin tone or higher harmonics in Bcal.

Both studies (Georgescu et al., 2006; Leinweber et al., 2012), however, do not take into account that the time-of-flight measurements themselves are known to be subject to offsets (Georgescu et al., 2012). T1 and T2 values differ systematically from the respective true values; deviations depend on instrument mode as we will show later.

Accurate calibration of FGM gains and zero level offsets with EDI |B| measurements is only possible if electron time-of-flight offsets are previously determined and corrected for. In this paper, we show how this can be achieved by using Cluster data from EDI and FGM (Balogh et al., 2001) and present the possible schemes of inter-instrument calibration. We further examine the characteristics of the FGM spin-axis offsets in the low

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field region and demonstrate the importance of accurate calibration when determining magnetic field gradient using multi-point Cluster measurements.

2 Method of analysis: inter-instrument calibration

Since our main interest is to determine the spin-axis offset component, we use the flux-gate spin reference (FSR) coordinates, where Z points along the spin-axis and X and Y are the spin-plane components. Here we assume that except for some residual spin-axis offset, ABZf all the calibration parameters have been accurately determined. Since the time of flight data provide magnitude of the magnetic field, Bedi, from (2), we use the spin-plane components of the FGM data to deduce the spin-axis component,

BZ edi:

BZedi2 = BL - BX2gm - BYfgm2-

The spin-axis offset, ABZf can then be obtained from

\BZ edi1 = \BZ fgm + ABZ fgm1,

if the spin-axis component of the magnetic field deduced from the EDI time of flight measurements and the spin-plane component of the FGM magnetic field are obtained with sufficient quality. For determining Bedi, we have simply used all the time of flight data from the two gun-detector units, GDU1 and GDU2, without identifying the pairs of long and short time of flight to obtain the gyration time from their average such as described before, based on an assumption that the usage of large numbers of data of both time of flight is equivalent to effectively averaging the measurement pairs. We use the high resolution FGM data (22.4 Hz for normal mode) and match them with the nearest neighbor to the EDI time of flight data. The EDI time of flight data are irregularly spaced data with a smallest interval of 16 ms, but are sparse compared to the FGM data, since detection of the returning electron beam is required.

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In this study we use Cluster data from July-October 2003 and from July-October 2006, when apogee of Cluster orbit is at night side. The inter-spacecraft distance was on the order of 200 km in 2003 and 10 000 km in 2006. During these summer seasons, when Cluster stayed in the magnetosphere and no solar-wind data were available, it is of particular interests to determine the FGM offset using the EDI measurements since the Hedgecock method (Hedgecock, 1975) cannot be applied. Furthermore, one of the scientific interests in the tail region is the magnetic reconnection process, for which the magnetic field component normal to the current sheet, corresponding to the spin-axis component, is key in detecting the process. Hence an accurate determination of the spin-axis component is crucial in this region.

Since both the FGM and EDI instruments are designed to obtain optimized field measurements in different regions of space, the digital resolution of the measurements change. In this study we analysed magnetic field data with magnitudes less than 600 nT. For FGM, within our region of interest, this corresponds to 3 different ranges, i.e., digital resolutions, changing from 7.813 pT to 0.125 nT depending on the field magnitude as will be discussed later. The EDI time-of-flight measurement, on the other hand, is operated by tracking electron beams that are amplitude-modulated with a pseudo-noise (PN) code, with a certain code period, TPN, or alternatively represented as the code repetition frequency (CRF), which is 1 /TPN. The PN code consists of either 15-chip or27-chip with different code chip length, Tchip. The accuracy of EDI measurements depend on the Tchip, and therefore TPN or CRF, which is usually given in unit of kHz. TPN varies between 30 ^ and 2 ms for the data set used in this study. The time resolution of EDI is defined by the shift-clock period, which is the shift in the PN code to track small time-of-flight variations, that varies from 1.907 ^ to 0.119 ^ depending on the magnetic field (see more detail in Georgescu et al., 2006). Further details about these parameter and the EDI operation schemes are given by Vaith et al. (1998) and Paschmann et al. (2001). Here we call the different measurement settings of the EDI as "CRF-mode" for convenience. As will be discussed later in more detail, these different resolutions/modes need to be taken into account when data are calibrated.

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Figure 1 shows FGM and EDI magnetic field magnitude data during a quiet interval of about 3 min from Cluster 3 for different FGM calibration schemes. The FGM data shown in the left three panels a-c use the orbit calibration file provided for the Cluster Active Archive (CAA) data set (Gloag et al., 2006), the mid three panels d-e use the daily calibration file (Fornagon et al., 2011) used for the Cluster Prime Parameter (PP) and Summary Parameter (SP) data set in the Cluster Science Data System (CSDS), and the right three panels g-i use the fine-tuned calibration file using the daily calibration file as an input. Figure 1a shows the magnetic field magnitude data estimated from EDI and FGM, in which the latter data are time-matched data to EDI using the nearest neighbor data selected from the high-time resolution (22.4 Hz) data shown in Fig. 1b. Although the example shown here is from a period when the number of the returning beam are quite evenly distributed all the times, EDI data depend on the availability of the returning beam and can be also sparse in time. Hence it is essential to compare EDI data with the time-matched FGM data. Figure 1c shows 1 Hz averaged data for both FGM and EDI. It can be seen that both data sets have small standard deviation (about 0.1 nT) during this interval and there exists a clear difference between FGM and EDI magnitude of about 0.5 nT. The same comparison has been done for data calibrated using the daily calibration file (Fig. 1 d-f). The 22.4 Hz data have slightly larger standard deviation compared to the CAA data, but the difference between EDI and FGM are smaller, about 0.14 nT. The relatively large scatter of the 1 Hz data (Fig. 1f) comes from the spin-tone, which can be more clearly seen in the 22 Hz data (Fig. 1e). Data shown in Fig. 1 g—i are using the same daily calibration file, as was used for data in Fig. 1 d-f, as input and then further refined the calibration file to reduce the spin tone. This additional procedure, however, has little effect on the average FGM-EDI difference as can be seen in the numbers obtained for the high-resolution data (Fig. 1d and g) and for the 1-Hz data (Fig. 1f and i). Note that for following discussions on offset calibration procedure we use the daily calibration file, prepared since the Cluster launch by the Technical University of Braunschweig Cluster Co-I team. That is, we use the same data set as shown in Fig. 1 d-f. It should be therefore noted that when we write "spin-

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axis offset" in this paper, we are not speaking about an offset from the raw data as given in the Eq. (1), but about a remaining offset correction from an already in-flight calibrated data set.

Figure 2a shows the number of EDI time of flight data points from Cluster 1 in August 2003, when corresponding FGM data were available, binned by the magnitude of the field, Bfgm. The size of the bins is 16 nT. The number of points are grouped by different CRF modes. Note that these different CRF modes generally correspond to data from different field magnitude regions, which is marked with R1-R6 next to the legend. More details of the meaning of these different magntitude regions, R1-R6, and the EDI measurement resolution are explained later (Fig. 3). It can be seen in the histogram that for smaller field regions, in particular, the EDI observations have been made with several different CRF modes. Figure 2b shows the differences between the \BZedi\ and \BZfgm\. The bin averages (dotted line) and medians (solid line) are also depicted in the figure. When both BZ values are positive, it corresponds to the spin axis offset. It can be seen that the values are widely scattered, particularly with increasing magnitude of the field. Also, instead of seeing a constant offset value of FGM, the difference is increasing with magnetic field magnitude but not monotonically. As will be discussed below, these variable differences can be due to: (i) the effects of different magnetic field angles relative to the spin-axis, (ii) the different CRF modes of the instruments and different offsets, and (iii) the effects from variable calibration parameters other than the offsets considered here. In the following we mainly examine the first two effects when obtaining the spin-axis offset of FGM and further discuss the possible effect due to (iii) based on the obtained offsets.

Since we are interested in the spin-axis offset, it is important to use measurements with sufficient magnitude of the spin-axis direction. As mentioned before, a meaningful comparison of the two spin-axis components using Eq. (4) can be only performed when both have the same (positive, for majority of the data used in this study) sign even when the possible offset values are subtracted, because Eq. (3) does not provide the sign of the magnetic field along the spin axis. The unknown sign of the BZedi will lead to

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miscalculation when the spin-axis offset effect will change the sign of the spin-axis component. This corresponds to cases when the expected spin-axis offset becomes significant compared to the spin-axis component of the magnetic field. Considering that we use an already calibrated data set as an input, a typical offset value is expected to be small, i.e., less than a couple of nT. For the Cluster data we are examining in this paper, such offset can be more than 10% of the field magnitude. Hence we need to take into account only data when |cosb\ = \BZfgm/Bfgm\ is sufficiently large so that the offset subtraction will not make any difference in sign change. As we will show later, \cos b\ > 0.4, would typically work for the analysis.

In this study we consider a time of flight offset of EDI, ATedi, which is expected to have different values for different CRF modes. For simplicity we assume the same offset value for the time of flight measurements from GDU1 and GDU2. That is, when calculating the magnetic field from EDI measurement we use

e(Tedi+ATedi)'

to determine both ATedi and ABZfgm from the data, instead of Eq. (2).

Significance of the EDI and fGm offsets varies for different field magnitude as is shown in Fig. 3. The four solid curves in Fig. 3a show the effective spin-axis offset value caused by an EDI time of flight offset, ATedi = 0.5 that will appear when the EDI and FGM measurements are compared such as in Fig. 2. They are plotted for different angle of the magnetic field, cosb. Here, the effective EDI magnetic field measurement resolution based on the digital resolution of the EDI measurements discussed by Georgescu et al. (2006) is also given as a dashed curve for the different magnetic field regions, R1-R6, as indicated at the bottom of Fig. 3b. The borders of R0-R6 are shown with the vertical dotted line, which corresponds to 16, 32, 64, 128, 164, 326 nT. The horizontal brown line indicates the 0.5 nT level, as a typical number for the spin-axis offset of FGM. In a similar way, we plotted the effective time of flight offsets caused by a FGM spin axis offset of ABZfgm = 0.5 nT. The dashed lines indicate the same EDI

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FGM-EDI-calibration onboard Cluster

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digital resolution of the time of flight measurement as given in Fig. 3a. The horizontal brown line shows 0.5 as a typical number for the time of flight offset of EDI. It can be immediately seen that the time of flight offset will have no effect in the small field region regardless of the angle to the magnetic field (brown line located above the curves in Fig. 3). These curves show therefore that the different angle of the fields as well as the time of flight offset can easily cause the large scatter of points in Fig. 2b. One can also conclude that for determining the offset in BZ in a given field magnitude, it would be most effective to use data from large cosb, since the relative importance of the EDI time of flight offset would be smallest. The large cosb Furthermore, in the low-field region, a time of flight offset of about 0.5 ^ will have only negligible effect in the spin-axis component of the magnetic field, which is a value below the instrument resolution. In the high-field region, on the other hand, a 0.5 nT spin axis offset is a negligible value in the time-of-flight data and comparable to the resolution of the EDI measurement. It is also important to note that when we determine A7"edi, it is most efficient to use data with low cosb, i.e., when the field direction is mainly along the spin plane direction. Vice versa, ABZf should be determined for large cosb as mentioned before. Due to these variable effects over the field magnitude, we need to consider different approaches for different magnetic field magnitudes depending on the importance of the offset. In Sect. 3 we demonstrate an example of a calibration in which all the different offsets are obtained using a large number of points and for different magnetic field magnitude regions. We also specifically use data from the low-field region to examine the possibility for estimating an offset with a small number of samples.

3 Example of inter-instrument calibration

Figure 4a shows the number of EDI measurements from Cluster 1 in August 2003 in the same format as Fig. 2a, but only for cosb > 0.7. As discussed before, this condition angle allows to select data when the relative importance of the BZ offset is higher than the possible time-of flight offset as discussed before in addition to fulfill the condition of

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FGM-EDI-calibration onboard Cluster

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the same positive sign of FGM and EDI in spin-axis component. As discussed before, EDI is operated with different CRF modes in different magnetic field regions. For this field angle, data were available only between regions of R2 and R6 (See Fig. 3 for definition of the regions). The FGM range changes at 256 nT, which is a value within R5. Depending on the importance of the offset we determined ATedi or ABZf in the following way:

- Low-field region (R1-R3), when the effect of ABZf is important: ABZf is first determined for cos b > 0.7. ATedi is then determined using data when cosb < 0.1 and is obtained for R1-R3 separately.

- Mid-field region (R4), when both effects from EDI time of offset and FGM offset in spin-axis component are comparable: ATedi is determined for cosb < 0.1 using ABZf determined for R2-3. Since there are two different CRF-modes used for EDI measurements in this region, we calculated the time of flight offsets for each-CRF mode separately.

- Mid-field region (R5), when both effects are comparable and FGM range changes within the same EDI CRF-mode: same method as R4 is used for data with Bfgm < 256 nT. Determine ABZf for cos b > 0.7 using ATedi determined for R5 data with Bfgm < 256 nT.

- High-field region (R6), when the effect of ATedi is important: determine ATedi taking into account the FGM offset determined for R5. Since the effect of spin-axis offset is not important regardless of cosb all data are used.

Figure 4b shows the FGM and EDI differences of original calibrated data as shown in Fig. 2 except for cosb > 0.7. The bin averages and median are shown as solid lines, although the difference between the two are hardly recognizable in this plot. The average profile in Fig. 4b shows some jumps coinciding with CRF-mode change and more monotonic increase in the high-field region within the same CRF-mode as expected in the curve shown in Fig. 3a. Figure 4c shows the results of the calibration procedure

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FGM-EDI-calibration onboard Cluster

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for August 2003. The points are the differences between the offset-corrected FGM and EDI data. The lines again show the bin average and the median of the differences of the offset-corrected FGM and EDI data. Here again the differences between the two lines are hardly seen. It can be seen that the bin average (or median) are running at almost zero level except for some fluctuations of < 0.1 nT in the higher field region. The nearly zero level of the bin average (or median) profile suggests that the spin-axis component difference between EDI and FGM was well explained due to the spin axis offset of FGM and time of flight offset of EDI.

Table 1 provides the monthly average results of the different offsets between July and October 2003: ABZfgm for low field range (< 256 nT) and high field range (> 256 nT) and A7edi for different CRF-modes, corresponding to R2-R6 (as given in the legend of 4a). Although we used all the available data without selecting, for example, quiet time data, it can be seen that ABZfgm determined from the low field region (R2-R3), which corresponds to B ~ 32-128 nT, stays at a about 0.4-0.6 nT with a relatively small standard deviation. The standard deviation is quite large for the FGM offset at high-field region (R5), while the values stays at a similar value to the low field region within 0.1 nT during all the four months. A7edi, on the other hand, are stably obtained only in the field region larger than about 128 nT (R4-R6), while the time of flight offsets could be poorly determined with large standard deviation only in the low field region. This behaviour can be understood with the characteristics of resolution of the EDI measurements (Fig. 3), i.e. finer B resolution of EDI for the smaller field region, smaller (larger) effect of A7edi in smaller (larger) field region relative to the effect of ABZfgm. Except for the poorly determined A7edi (R1-R3), the values shown in Table 1 were used to calculate the gray points in Fig. 4b.

We have performed the same procedure for every orbit in August 2003 for Cluster 1 and the results are shown in Fig. 5. ABZfgm for low field (< 256 nT) and high field (> 256 nT) and their corresponding numbers of points are shown in Fig. 5a and b, respectively. As described before, low field data points are from EDI CRF-modes R2 and R3 (see Fig. 4b), while high field data points are from EDI CRF modes R5. A7edi

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for each orbit for R2, R5, and R6 and corresponding numbers of data points are shown in Fig. 5c and d. Note that measurements in low field regions took place not every orbit for this month and therefore values using those data points can be seen only every second or fourth orbit. It can be seen that ABZf obtained from the low field region are relatively stable compared to that obtained from high-field region. As for ATedi, the values of R6 is most stable among the three offsets. ATedi are larger for R2 compared to R5 and R6. Yet the effect from ABZf still can be expected to dominates in R2 for these values (see Fig. 3).

The spin-axis direction, which is approximately the Z direction in geocentric solar ecliptic (GSE) coordinates, is closely aligned to the normal component of the current sheet in the magnetotail, where the apogee is located for Cluster between July and October. This normal component drops to zero when magnetic reconnection occurs, which is an important science target in magnetospheric missions such as Cluster as well as for upcoming Magnetospheric Multi Scale (MMS) mission. To detect the process accurately, therefore, it is required that the spin-axis offset has been corrected. It is therefore desirable that the calibration will take place close to such target intervals, that is, in a relatively small field region when the disturbance of the field is small. Below we use Cluster data for a short interval, i.e. several minutes, in a small field region such as the example shown in Fig. 1 to examine the effect of the spin-axis component offset in the difference between FGM and EDI magnetic field. We searched for quiet and constant field intervals using data between July and October 2003 in small field region (R2), corresponding to magnetic field between about 30 and 60 nT. A quiet field short time interval is defined as an interval with standard deviation less than 0.1 nT. We chose time period of 7min. We obtained 579 such intervals for C1 during the four months. Figure 6a and b shows the magnitude difference, AB = Bedi - Bfgm, and difference in the spin axis components, ABZ = \BZedi\ - \BZf \, plotted vs. the field angle, cosb. On average, the magnitude difference is small when the magnetic field is nearly aligned to the spin plane (small \ cos b\) within an error of about 0.1 nT and justifies our assumption that the main discrepancy between the two datasets are attributed to the

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spin axis offset. When the \cosb\ is small, \cosb\ < 0.1, it is not possible to obtain the correct sign of ABZ. In such case comparison between the spin-axis component will contain large errors. That is, we may obtain the sums of the two measurement instead of differences, meaning that the ABZ will rather become twice an average of the spin-axis component value (2Bcosb). If we assume, for example, that such errors happen about half of the cases we can expect an average to be estimated as B cos b. For the field magnitude in this data set, i.e., B = 30-60 nT, a "wrongly" estimated ABZ of <36 nT can be expected for cos b < 0.1, which was in fact the case as shown in Fig. 6b. On the other hand, the spin axis offsets are more stable for larger cosb, i.e., cosb < 0.4, indicating the importance of preselection of the angle of the field when determining the spin-axis offset.

The essential advantage of a multi-point measurement such as Cluster is the ability to determine spatial gradients. We finally examine the possible effect of the offset calibration by comparing the magnetic field gradient (differences between two spacecraft) for Bedi, Bfgm, and an empirical magnetic field, i.e., combined IGRF and Tsyganenko 89, Kp = 2, as shown in Fig. 7. Here we select again quiet time intervals, when standard deviation of Bedi < 0.07 nT for 5min interval and when data from both C1 and C3 are available. Cluster data are used from an interval between July and October in 2003, when the interspacecraft distance was about 200 km, and between July and October in 2006, when the interspacecraft distance was about 10 000 km. Figure 7a shows the spacecraft differences, ABedi,C1-C3 (black), ABfgm,C1-C3 (red), and model (green) plotted again over cos b (of Cluster 1) observed at locations shown in Fig. 7c for the events in 2003. The model provides a reference value of the magnetic field profile and is constructed based on fitting a number of previous satellite data. Therefore we can expect that the model represents some averages of randomly distributed different "offsets" among the different previous measurements providing an empirical value of the field. ABedi,C1-C3 and model generally agrees well. This suggests that ABedi,C1-C3 provides closer values to an empirical value of the magnetic field. ABfgm,C1-C3 shows smaller difference in the small cosb region, which corresponds to magnetic field direction where

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spin-axis component does not play the role, suggesting that the spin-plane components are well calibrated. The differences, however, become larger for larger cos b indicating that the effect of the spin-axis offset is apparent and causing these larger differences. Figure 7b and d show the results of the same analysis performed for the data in 2006 for comparison. In contrast to 2003, the gradients obtained from the two measurements show similar values, while the model values are deviating from these two. The inter-spacecraft distance of 200 km is small enough that the effect of the offset calibration exceeds the magnetic field gradient, while such offset determination plays no difference for the interspacecraft distance 10 000 km. Hence, depending on the interest of the gradient scales it will become essential to perform special offset calibrations when determining the gradient of the magnetic field.

4 Discussion

Not taken effects are: based on a simple comparison between the magnetic field of FGM and the magnetic field deduced from the time of flight of the EDI measurements, we have shown that the remaining spin-axis offset of FGM data can be well determined from the calibrated data set by selecting the appropriate interval, by taking into account the measurement conditions such as the angle of the magnetic field relative to the spin-plane, magnetic field magnitude, and by also considering the effect of the time-of-flight offset of the EDI measurement. While the effect of the time-of-flight offset was unimportant in determining the spin-axis offset in the low field region, it was the major source of the discrepancy between the two data sets in the large field region. Once the effects of these two offsets are taken into account, the difference between the two measurements are reduced to be well below 0.1 nT level. Note that there is a tendency of somewhat larger fluctuations superposed with negative trend for larger field region (R6) in Fig. 4. This might suggest that some additional FGM gain correction needs to be considered. The current offset-correction does not take into account any gain correction. If there is a gain error, it should appear as a linear trend if all the other

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calibration parameters are perfectly determined. Such gain error curve, however, is difficult to differentiate from the EDI time-of-flight profile particularly for a low-resolution measurement. Therefore each EDI range may show different resultant curve and may not appear a continuous line in Fig. 4c even if there is a gain error. In the low field region, we cannot see any systematic trend, for example. If we take the ~ -0.1 nT deviation in the R6 region (covering about 200 nT wide region), as a observed number, it will correspond to a linear gain correction of 0.0005. Such change in the gain may likely happen due to the change in the temperature. Indeed if we use the ground-calibration result from one of the Cluster ground sensors, i.e., 0.00004 K-1 (Othmer et al., 2000), this corresponds to a gain drift for a temperature change of about 12°, which would not be an unrealistic variation within an orbit. For an accurate determination of the gain from these comparisons, however, only a statistical approach is possible because in this high field region, EDI can measure the field only with about 1 nT resolution, while the effects expected from gain errors would be less than 0.1 nT scale, which is also below the FGM resolution in this range and therefore fluctuations are unavoidable.

While we demonstrate that the simple comparison is overall working, particularly for the spin-axis determination in low field regions, once we are interested to determine also other parameters, such as time-of-flight offsets throughout the EDI CRF modes or FGM offsets and gain factors for high field ranges, further investigations would be necessary. For example, our simplified approach of pre-selecting the data set based on specific conditions in angle and magnitude of the field, as discussed in Sect. 3, limits the number of useful data. Instead one may consider to use all the data from different field magnitude (and therefore with different EDI CRF modes) and try to determine the EDI and FGM offsets at once by applying appropriate weighting factors, that depend on the contribution of the EDI and FGM offsets in the measurement, and by minimizing the differences between the two measurements. Furthermore determining the EDI time of flight offset for the two GDUs, separately, may be also important particularly for mid and high field region.

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In this study we only used the time of flight data of the EDI measurements to compare with the FGM measurements. Another useful approach is to use the direction of the EDI electron beam, uedi, which should be perpendicular to the ambient magnetic field, and use the condition of uedi ■ (Bfgm + Ofgm) = 0, to determine the offset of the FGM measurement, Ofgm. A combination of these two methods will further improve the accuracy of the offset determination.

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5 Conclusions

We have shown that the concept of determining the spin axis offset of a flux gate magnetometer (FGM) using absolute field magnitude data determined from the electron gyration time data of the electron drift instrument (EDI) works best when the magnetic field magnitude is small, i.e., less than about 128 nT corresponding to the EDI modes for low field, so that the EDI time of flight offset is negligible, and when the spin-axis component becomes the major component (cosb > 0.7). A remaining spin-axis offset of about 0.4 ~ 0.6 nT was observed between July and October 2003, which is important for studies using the magnetic field component normal to the current sheet in the central plasma sheet such as magnetotail reconnection or thin-current sheet dynamics or particle trajectories near the center of the current sheet.

When the effect of time-of-flight offset from EDI is taken into account, it is shown that data from higher field can be also used for calibration. It is shown that additional determination of the gain factor of the FGM instrument would most likely be also possible.

The EDI-FGM comparison method is of particular interest for the observations, when no solar-wind data are available for calibration. It will play an essential role for accurate determination of the small normal component (and its reversals) in the current sheet required for studying magnetic reconnection, which is the main objective of NASA's Magnetotspheric Multi-Scale (MMS) mission.

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Acknowledgements. This papers is dedicated to the memory of Edita Georgescu, Max Planck Institute for Solar System Research, Germany. Georgescu started initially the EDI-FGM comparison work. Without her inspiring comments and suggestions this study would not have been done. We thank P. Daly, H. Eichelberger, G. Laky, and the CAA team for the valuable suggestions/comments and supporting the data analysis. This research was partly supported by the Austrian Science Fund FWF I429-N16 and I23862-N16. KHF and KHG were financially supported through grants 500C1102 and 500C1001 by the German Bundesministerium für Wirtschaft und Technologie and the Deutsches Zentrum für Luft- und Raumfahrt.

References

Acuña, M. H.: Space-based magnetometers, Rev. Sci. Instrum., 73, 3717-3736,

doi:10.1063/1.1510570, 2002. 460, 461 Auster, H. U., Fornacon, K. H., Georgescu, E., Glassmeier, K. H., and Motschmann, U.: Calibration of flux-gate magnetometers using relative motion, Meas. Sci. Technol., 13, 1124-1131, doi:10.1088/0957-0233/13/7/321, 2002. 462 Auster, H. U., Glassmeier, K. H., Magnes, W., Aydogar, O., Baumjohann, W., Constantinescu, D., Fischer, D., Fornacon, 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, doi:10.1007/s11214-008-9365-9, 2008. 461 Balogh, A.: Planetary magnetic field measurements: missions and instrumentation, Space Sci.

Rev., 152, 23-97, doi:10.1007/s11214-010-9643-1, 2010. 460 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, doi:10.5194/angeo-19-1207-2001, 2001. 463 Dougherty, M. K., Kellock, S., Southwood, D. J., Balogh, A., Smith, E. J., Tsurutani, B. T., Gerlach, B., Glassmeier, K.-H., Gliem, F., Russell, C. T., Erdos, G., Neubauer, F. M., and Cowley, S. W. H.: The Cassini magnetic field investigation, Space Sci. Rev., 114, 331-383, doi:10.1007/s11214-004-1432-2, 2004. 461

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Fornagon, K.-H., Georgescu, E., Kempen, R., and Constantinescu, D.: Fluxgate magnetometer data processing for Cluster, Tech. Rep., Institut für Geophysik und extraterrestrische Physik, Technischen Universität Braunschweig, 38106 Braunschweig, Germany, CL-IGEP-SN-0001, 2011. 466

Georgescu, E., Vaith, H., Fornacon, K.-H., Auster, U., Balogh, A., Carr, C., Chutter, M., Dunlop, M., Foerster, M., Glassmeier, K.-H., Gloag, J., Paschmann, G., Quinn, J., and Torbert, R.: Use of EDI time-of-f light data for FGM calibration check on CLUSTER, in: Cluster and Double Star Symposium, Vol. 598 of ESA Special Publication, ESTEC, Noordwijk, the Netherlands, 2006. 463,465, 468

Georgescu, E., Puhl-Quinn, P., Vaith, H., and Matsui, H.: Cross Calibration Report of the EDI Measurements in the Cluster Active Archive (CAA), http://caa.estec.esa.int/documents/CR/ CAA_EST_CR_EDI_v14.pdf (last access: 29 July 2013), 2012. 463 Glassmeier, K.-H., Richter, I., Diedrich, A., Musmann, G., Auster, U., Motschmann, U., Balogh, A., Carr, C., Cupido, E., Coates, A., Rother, M., Schwingenschuh, K., Szegö, K., and Tsu-rutani, B.: RPC-MAG The Fluxgate Magnetometer in the ROSETTA Plasma Consortium, Space Sci. Rev., 128, 649-670, doi:10.1007/s11214-006-9114-x, 2007. 461 Gloag, J. M., Carr, C., Forte, B., and Lucek, E. A.: The status of Cluster FGM data submissions to the CAA, in: Cluster and Double Star Symposium, Vol. 598 of ESA Special Publication, ESTEC, Noordwijk, the Netherlands, 2006. 466 Hedgecock, P. C.: A correlation technique for magnetometer zero level determination, Space

Sci. Instrum., 1, 83-90, 1975. 462, 465 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, doi:10.1109/20.486522, 1996. 461 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, 055104, doi:10.1088/0957-0233/19/5/055104, 2008. 462 Leinweber, H. K., Russell, C. T., and Torkar, K.: In-flight calibration of the spin axis offset of a fluxgate magnetometer with an electron drift instrument, Meas. Sci. Technol., 23, 105003, doi:10.1088/0957-0233/23/10/105003, 2012. 463

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Ludlam, M., Angelopoulos, V., Taylor, E., Snare, R. C., Means, J. D., Ge, Y. S., Narvaez, P., Auster, H. U., Le Contel, O., Larson, D., and Moreau, T.: The THEMIS magnetic cleanliness program, Space Sci. Rev., 141, 171-184, doi:10.1007/s11214-008-9423-3, 2008. 461 Othmer, C., Richter, I., and Fornagon, K.-H.: Fluxgate magnetometer calibration for Cluster II, Tech. Rep., Institut für Geophysik und Meteologie, Technischen Universität Braunschweig, 38106 Braunschweig, Germany, CL2-IGM-TR-0008, 2000. 475 Paschmann, G., Melzner, F., Frenzel, R., Vaith, H., Parigger, P., Pagel, U., Bauer, O. H., Haeren-del, G., Baumjohann, W., Scopke, N., Torbert, R. B., Briggs, B., Chan, J., Lynch, K., Morey, K., Quinn, J. M., Simpson, D., Young, C., McIlwain, C. E., Fillius, W., Kerr, S. S., Mahieu, R., and Whipple, E. C.: The electron drift instrument for Cluster, Space Sci. Rev., 79, 233-269, doi:10.1023/A:1004917512774, 1997. 462,463 Paschmann, G., Sckopke, N., Vaith, H., Quinn, J. M., Bauer, O. H., Baumjohann, W., Fillius, W., Haerendel, G., Kerr, S. S., Kletzing, C. A., Lynch, K., McIlwain, C. E., Torbert, R. B., and Whipple, E. C.: EDI electron time-of-flight measurements on Equator-S, Ann. Geophys., 17, 1513-1520, doi:10.1007/s00585-999-1513-3, 1999. 462, 463 Paschmann, G., Quinn, J. M., Torbert, R. B., Vaith, H., McIlwain, C. E., Haerendel, G., Bauer, O. H., Bauer, T., Baumjohann, W., Fillius, W., Förster, M., Frey, S., Georgescu, E., Kerr, S. S., Kletzing, C. A., Matsui, H., Puhl-Quinn, P., and Whipple, E. C.: The electron drift instrument on Cluster: overview of first results, Ann. Geophys., 19, 1273-1288, doi:10.5194/angeo-19-1273-2001, 2001. 462, 465 Pudney, M. A., Carr, C. M., Schwartz, S. J., and Howarth, S. I.: Automatic parameterization for magnetometer zero offset determination, Geosci. Instrum. Method. Data Syst., 1, 103-109, doi:10.5194/gi-1-103-2012, 2012. 462 Quinn, J. M., Paschmann, G., Sckopke, N., Jordanova, V. K., Vaith, H., Bauer, O. H., Baumjohann, W., Fillius, W., Haerendel, G., Kerr, S. S., Kletzing, C. A., Lynch, K., McIlwain, C. E., Torbert, R. B., and Whipple, E. C.: EDI convection measurements at 5-6 RE in the post-midnight region, Ann. Geophys., 17, 1503-1512, doi:10.1007/s00585-999-1503-5, 1999. 462 Vaith, H., Frenzel, R., Paschmann, G., and Melzner, E.: Electron gyro time measurement technique for determining electric and magnetic fields, American Geophysical Union Geophysical Monograph Series, 103, 47, 1998. 462, 465

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Table 1. Average offsets determined for different modes/ranges.

Parameters July 2003 August 2003 September 2003 October 2003

Aß. AßZ

Z fgm,/

JZfgm,ft [nT] ATedi,R1 [Ms] ATedi,R2 [Ms] ATedi,R3 [Ms] ATedi,R4a [Ms]

AT( ATe

edi,R4b

edi,R5 edi,R6

[Ms] [Ms] [Ms]

0.51 ± 0.15 0.40 ± 0.99 2.92 ± 5.77 1.81 ± 2.42 0.38 ± 1.15 0.21 ± 0.25 0.65 ± 0.97 0.55 ± 0.42 0.28 ± 0.19

0.46 ± 0.41 ± 1.90 ± 1.60 ± 1.03 ± 0.15 ± 0.63 ± 0.55 ± 0.26 ±

0.16 0.99 4.77 1.96 1.27 0.20 0.96 0.43 0.19

0.64 ± 0.17 0.57 ± 1.04 2.87 ± 6.42 1.81 ± 1.89 0.70 ± 0.89 0.19 ± 0.16 0.50 ± 0.97 0.59 ± 0.46 0.27 ± 0.19

0.57 ± 0.17 1.00 ± 0.19 1.92 ± 4.98 1.85 ± 2.40 1.20 ± 1.04 0.05 ± 0.23 0.48 ± 1.00 0.57 ± 0.50 0.26 ± 0.20

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2003 0810 C3 CAAcal

<Bedi> =82.0 ± 0.13 nT <Bfgm>=81.5 ± 0.11 nT

2003 0810 C3 daily-cal

<Bfgm>=81.9 ± 0.12 nT

2003 0810 C3 daily+sp-cal

<Bfgm>=81.9 ± 0.12 nT

<ESedi> =82.0 ± 0.12 nT <Bfgm>=81.5 ± 0.12 nT <Bedi-Bfgm> = 0.496 nT :

<Bfgm>=81.9 ± 0.14 nT

<Bfgm>=81.9 ± 0.11 nT

: 1Hz EDI <Bedi-Bfgm> = 0.147 nT

<Bedi-Bfgm> = 0.145 nT :

■oSv&nte*

Fig. 1. FGM and EDI magnetic field magnitude data during a quiet interval from Cluster 3 using FGM data with different calibration schemes: the orbit calibration, method used for CAA data set (a-c); daily calibration method used for CSDS dataset (d-f); and refined calibration applied to daily calibration input (g-i). The upper three panels (a), (d), and (g) show high-resolution EDI and time-matched fGm 22.4 Hz data, the middle three panels (b), (e), and (h) show the 22.4 Hz FGM data, and the lower three panels (c), (f), and (i) show 1 Hz averaged data for both FGM and EDI.

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1Hz EDI

1Hz FGM

1Hz FGM

1Hz FGM

Fig. 2. (a) Number of points for all available Cluster 1 EDI data in August 2003, binned by the magnitude of the field Bfgm. The size of the bins is 16 nT. The number of points are grouped for different CRF modes (see details in text). (b) Differences between \BZedi\ and \BZfgm\ for the same data set. The solid line shows the median and the dotted line shows the average of the data within each bin. Here every 20th points from the entire dataset shown in (a) are plotted.

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10.00^

. 0.01 a) 0

10.00P

B (nT)

ATfgm for ABzfgm,off = 0.5 nT (cosb = 0.1, 0.4, 0.7, 1)"]

b) ' 0

B (nT)

Fig. 3. (a) The effective spin-axis offset value caused by an EDI time of flight offset, ATedi = 0.5 ^s, that will appear when the EDI and FGM measurements are compared, plotted for selected angles of the magnetic field, cosb. The dashed lines show the resolution of the EDI magnetic field measurement. The horizontal brown line shows 0.5 nT level, which represents a typical number for the spin-axis offset of FGM. (b) The effective time of flight offsets caused by a FGM spin axis offset, ABZf = 0.5 nT, plotted for selected values of cos b. The dashed lines indicate the EDI digital resolution of the time of flight measurement. The horizontal brown line shows 0.5 ^s level, which represents a typical number for the time of flight offset of EDI. The vertical dotted lines indicate the border of different EDI measurement settings, R0-R6. See text for further details.

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Fig. 4. (a) Number of points for Cluster 1 EDI data in August 2003 binned by the magnitude of the field ßfgm as in Fig. 2 except for cos b> 0.7. Difference between spin axis component EDI and FGM fields for cos b > 0.7 (b) for the original calibrated data and (c) for the offset-corrected data. Bin average and median for the original and offset-corrected data are shown in each panel. Note that both curves are nearly identical and their differences can only be therefore hardly seen. As in Fig. 2, every 20th points from the corresponding datasets given in (a) are plotted. Dashed lines indicate -0.5, 0.0, 0.5 nT levels.

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E „ 0.6 h-

N m 0.4

<D 1.5-105

c £ 1.0-105

a 5.0-104

0.5 Q.Q

Cluster 1

August, 2003

- o— (c)

F(d) ^

— low field (R2,R3)

—— high field (R5 with B>256 nT)

CRF 8.7 (R2) 1.0 (R5) . 2.1 (R6) kHz

Orbit number

Fig. 5. (a) ABZfgm determined for every orbit for low field (< 256 nT) and high field (> 256 nT) and (b) the corresponding numbers of data points from Cluster 1 in August 2003. (c) A7"edi for each orbit for R2, R5, and R6 and (d) corresponding numbers of data points.

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2003 07 - 2003 10

2003 07 2003 08 2003 09 2003 10

+ +*Î + ++

-+/++ ++

+ + + +1

-0.2 0.0

cosb = <Bxfgm /Bfgm >

: 2003 07

: 2003 08

j 2003 09

3^ 2003 10

-0.2 0.0 0.2 0.4

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Fig. 6. (a) Average magnitude difference, AB = Bedi - Bfgm, and (b) average difference in the spin axis components, ABZ = \BZedi| - \BZfgm|, plotted vs. the field angle, cosb, obtained using quiet, low field (30-60 nT), short time interval (7min) data sets in July-October 2003. The vertical bars in (b) show the standard deviation.

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2006 07-10

(X2+Y2)1/2 [Re]

(X2+Y2)1/2 [Re]

Fig. 7. Average magnetic field differences between C1 and C3 for Bedi (black cross) and Bfgm (red cross), and a model magnetic field (green cross) during quiet time intervals (standard deviation of Bedi < 0.07 nT for 5 min interval) plotted vs. cos b for data from (a) July-October in 2003, when the interspacecraft distance was about 200 km, and from (c) July-October in 2006, when the interspacecraft distance was about 10000 km. The location of the spacecraft in GSM coordinate during these two sets of intervals are shown in (b) and (d), respectively.

03 T3 CD

03 T3 CD

03 T3 CD

03 T3 CD

FGM-EDI-calibration onboard Cluster

R. Nakamura et al.

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2003 07-10