Statistical analysis of geomagnetic field intensity differences between ASM and VFM instruments onboard Swarm constellation
 Paola De Michelis^{1}Email authorView ORCID ID profile,
 Roberta Tozzi^{1} and
 Giuseppe Consolini^{2}
DOI: 10.1186/s4062301605831
© The Author(s) 2017
Received: 5 August 2016
Accepted: 3 December 2016
Published: 1 February 2017
Abstract
Keywords
Swarm magnetic data quality Empirical mode decomposition Delayed mutual informationBackground
The Swarm mission, which consists of three identical satellites, was launched on November 2013 by the European Space Agency (ESA) with the objective to perform the bestever survey of the geomagnetic and electric fields surrounding the Earth (FriisChristensen et al. 2006). The three satellites fly on almost polar orbits (inclination being around 88\(^\circ\)) and are equipped with identical magnetometers and electric field instruments capable of providing highprecision and highresolution measurements.
One of the peculiar features of Swarm mission is the geometry of the satellite constellation. Two satellites, Alpha (A) and Charlie (C), fly in pair at an altitude that on August 2016 was approximately of 460 km. The third satellite, Bravo (B), orbits about 50 km above Swarm A and C, and it is constantly increasing its local time (LT) separation from A and C. This separation was of about 3 h on August 2016. Further details on Swarm mission can be found at http://swarmwiki.spacecenter.dk/mediawiki1.21.1/index.php/Swarm_User_Guide.
Magnetic measurements on each satellite are carried out by an absolute scalar magnetometer (ASM) for measuring Earth’s magnetic field intensity and by a vector field magnetometer (VFM) for measuring the direction and the strength of the geomagnetic field. VFM measurements are calibrated using those from the ASM similarly as already done in other satellite missions for magnetic field mapping purposes and which carried equivalent instrumentation (Olsen et al. 2003; Yin and Lühr 2011). According to the traditional inflight vector calibration of VFM fluxgate instruments, the raw data of the vector magnetometer are processed by applying an error model which takes into account of scale factors and their dependence on time and temperature, offsets and nonorthogonal angles between the sensor elements.
From the very beginning of the Swarm mission, comparisons between magnetic measurements from VFM and ASM showed discrepancies in the values of total field intensity which could not be captured by the traditional inflight calibration methods. These differences were observed in data of all satellites appearing as a disturbance in magnetic field measurements varying in strength, direction and characterized by a local time dependence. A first investigation on VFM and ASM measurements suggested that this disturbance was due to an unforeseen spurious magnetic field that contaminated the VFM measurements more than ASM ones. Successively, a comparison of ASM measurements recorded by all satellites during specific satellites maneuvers showed that the origin of this disturbance, and hence of the observed ASM–VFM total field differences, was not to be ascribed to the ASM. For this reason, it was decided to assume that only the VFM measurements had to be corrected for the presence of this undesired magnetic field whose indirect cause was attributed, after some investigations on ASM–VFM total field differences, to some thermal effect due to the varying Sun incidence angle with respect to the spacecraft.
To correct VFM measurements and consequently reduce the discrepancy between VFM and ASM total intensity of the magnetic field, a correction model was proposed (Lesur et al. 2015). This model is capable of reconstructing the vector components of the disturbing magnetic field and therefore of correcting the single vector components provided by the VFM. The model consists of a spherical harmonics expansion up to degree and order 25 in the Sun incidence angle whose coefficients are estimated iteratively by means of a least squares method. This empirical determination of the Sundriven disturbance field and its consequent removal have become part of the inflight calibration of the Swarm ensemble of magnetometers (TøffnerClausen et al. 2016). Being this disturbance field different for each satellite, coefficients are estimated separately for the three satellites. The final effect of correction on magnetic vector measurements for all Swarm satellites is quite good and substantially reduces the standard deviation of the difference between ASM and VFM geomagnetic field total intensity. The occasional spikes that are present in corrected data correspond to satellite maneuvers that the model is not able to represent.
Detailed descriptions of the correction model can be found in the ESA technical report available at https://earth.esa.int/documents/10174/1583357/Preliminary_Swarm_MagL_Data_ReleaseNotes and in TøffnerClausen et al. (2016).
To try to gain additional information on the disturbance affecting magnetic measurements and on the way correction acts on magnetic vector data, we perform an analysis based on empirical mode decomposition (EMD) method. We apply this method to the ASM–VFM total intensity differences obtained with data both uncorrected and corrected for the disturbance correlated with the Sun incidence angle, and we try to understand the nature of disturbance which, despite correction, is still partially present in VFM measurements. For this reason, we analyze also the VFM electronic unit temperature time series, which can be used as a proxy of how the temperature changes in the satellite environment along its orbit.
It is important to keep in mind that each time a correction is applied on a measurement there is a real risk to introduce spurious features that could be turned into spurious properties of the analyzed corresponding time series and, consequently, into incorrect physical interpretation of results provided by the analyses performed on manipulated data. Due to the very large community interested in Swarm data, from those studying the slow evolution of the Earth’s core to those interested in the very quickly changing magnetospheric/ionospheric environment, it is important to be aware of the effects, if any, of the performed correction. It can be important to know whether there are timescales more affected than others.
The present work is intended to provide a characterization of the nature of the residual discrepancy of the ASM–VFM total intensity measurements to understand the linear/nonlinear and/or chaotic nature of this spurious signal, leaving the discussion on any possible solution to this problem to further investigations which require a deeper knowledge of mechanical and electronic features of the two instruments. We retain that our results can give useful information to all people that work to improve the quality of Swarm magnetic data and have the right skill to try to solve the problem.
The paper is organized as follows: The next section is dedicated to the description of the analyzed datasets. It follows a section where EMD is illustrated and then applied to the selected Swarm dataset. Results from the application of EMD are displayed and discussed in the same section. Then there is a section where delayed mutual information is illustrated and applied to our dataset. In “Summary and conclusions” section, main findings are summarized and implications discussed.
Data
We consider Level 1b lowresolution (1 Hz) magnetic field data recorded onboard Swarm constellation. In detail, we use two different data groups: one consists of the difference between the total magnetic field intensity provided by the two magnetic instruments (ASM and VFM) installed onboard the three satellites from May 15, 2014, to September 12, 2014, the other one consists of both ASM–VFM total field difference and VFM electronic unit temperature recorded from January 13, 2015, to June 29, 2015, by Swarm B. Magnetic measurements used to calculate ASM–VFM total field differences are contained in files named SW_OPER_MAGx_LR_1B (x = A, B, C) and are available at ftp://swarmdiss.eo.esa.int upon registration.
In the first data group, we use both uncorrected and corrected data. Following ESA archiving nomenclature, we will refer to uncorrected data as Previous (file counter equal to 0301, 0302 and 0303) and \(\Delta F_{\mathrm{P}} (t)\) will be the corresponding series of ASM–VFM total field differences. Similarly, we will refer to corrected data as Current and \(\Delta F_{\mathrm{C}} (t)\) will be the corresponding series of ASM–VFM total field differences. We want to draw the reader’s attention that according to ESA nomenclature the most recent data version is named Current version but this has changed in the course of the investigation presented in this paper. For instance, as far as concerns magnetic data, at the time of the investigation performed on the first data group, i.e., July 2015, Current data consisted of files with file counter equal to 0405. Differently, as explained later on, at the time of the investigation on the second data group, i.e., February 2016, Current magnetic data consisted of files with file counter equal to 0408.
The standard deviation of ASM–VFM total field differences, obtained by Previous data, covering the analyzed period ranges between 0.7 and 1.3 nT (the maximum value being reached by Swarm A). When standard deviation is estimated on Current data (0405), it drops to values between 0.15 and 0.19 nT.
As in the case of magnetic data also the set of data describing the VFM electronic unit temperature is a Swarm Level 1b product available upon registration at ESA ftp. Measurements used in this work are contained in common data format (CDF) files named MAGB_CA_1B with file counter equal to 0407, the most recent version available on February 2016. Besides magnetic data (raw as well as processed VFM vector measurements and fully converted and corrected ASM measurements), these files contain VFM electronic unit and sensors temperatures. In the CDF files, electronic unit temperature corresponds to T_EU variable, while sensor temperatures correspond to T_CDC and T_CSC variables (CDC and CSC stand for Compact Detector Coil and Compact Spherical Coil, respectively).
Empirical mode decomposition analysis: a brief account and results
Data describing the dynamics of both natural and manmade systems are often characterized by a certain degree of nonstationarity and nonlinearity. This is the reason why, to describe the dynamics of these systems, it is necessary to introduce methods of analysis different from the traditional ones usually based on assumptions of linearity and stationarity of the analyzed time series. These different analytical methods are capable of representing the inherent multiscale and complex nature which characterizes the systems permitting us to gain a deep understanding of those physical processes which actually produce data. Moreover, they decompose signals using adaptive bases that are directly derived by data themselves without setting a priori assumptions.
Among these adaptive methods stand empirical mode decomposition (EMD). This technique has been introduced recently by Huang et al. (1998) as a required step to compute the instantaneous frequencies through the Hilbert transform. However, being quite intuitive and direct, it has become one of the most used adaptive methods to deal with data series originating from nonlinear and nonstationary processes. Because of its excellence, it has been applied in different physical contexts from seismology (Battista et al. 2007) to oceanography (Schlurmann 2000, 2002) without considering the applications in biomedical signal processing (Pachori 2008) and in signal denoising (Flandrin et al. 2004).
EMD has been widely used also in geomagnetism, for example to characterize the decadal periodicities of the length of the day and find a relation with torsional oscillations (Roberts et al. 2007; Jackson and Mound 2010; De Michelis et al. 2013), to study the multiscale nature of geomagnetic storms (De Michelis et al. 2012, 2015) and to analyze their impact on electric power systems (Liu et al. 2016).
The main idea behind EMD is that any timedependent data series can be written as a superposition of monocomponent signals each representing characteristics embedded in the timedependent data series. These monocomponent signals, named intrinsic mode functions (IMFs), can be directly extracted from the original time series, provided that they satisfy two important conditions. The first condition requires that number of zero crossings and of extrema are equal or differ by at most one. The second condition requires that the mean value of the two envelops fitting IMF local maxima and local minima is equal to zero. This second condition, which means that the local mean of IMF is equal to zero and guarantees that the instantaneous frequency will not have unwanted fluctuations, represents the new idea of the method. To decompose a signal via EMD, an iterative procedure must be followed. We do not describe here the details of this procedure since they are reported in many scientific papers (Huang et al. 1998, 2003; Huang and Wu 2008; Flandrin et al. 2004; De Michelis et al. 2012). This iterative procedure ends in a number of IMFs and a residue representing the longterm trend of the analyzed time series. IMFs, due to the way they are built, have each a characteristic frequency and become the basis representing the data, which is consequently obtained with no a priori assumptions on the timeseries nature. IMFs can have both frequency and amplitude modulations. A wave component with nearly constant time scale exists and dominates in each IMF, representing the carrier wave constituent at the specific time scale. In this way, we are capable of identifying the different IMFs which correspond to the different physical time scales and characterize the various dynamical oscillations in the analyzed time series.
 1.the modes with k = 1, ..., 5 (green) describe the noise associated with the signal containing 0.2% of \(\Delta F_{{\mathrm{P}}}\) total energy,$$\begin{aligned} \mathrm{Noise}_{\mathrm{P}} =\sum\limits_{k=1}^5 \mathrm{IMF}_k(t) , \end{aligned}$$(3)
 2.the modes with k = 6, ..., 11 (red) contain 92% of \(\Delta F_{{\mathrm{P}}}\) total energy and, consequently, the signal obtained by the superposition of these modes represents the main part of \(\Delta F_{{\mathrm{P}}}\) and is well representative of the ASM–VFM differences:$$\begin{aligned} \mathrm{Main}_{\mathrm{P}} = \sum\limits_{k=6}^{11} \mathrm{IMF}_k(t) , \end{aligned}$$(4)
 3.remaining modes (k = 12, ..., 19) (blue) take into account the remanent part of the signal$$\begin{aligned} \mathrm{Residue}_{\mathrm{P}} = \sum\limits_{k=12}^{19} \mathrm{IMF}_k(t) + \mathrm{res}(t). \end{aligned}$$(5)
 (a)
the modes embedded in the analyzed signals before and after correction are characterized by the same dominant periods;
 (b)
the main difference consists in the decrease in the energy associated with some modes contributing to corrected data.
It is worth recalling that since we are considering the difference between the magnetic field intensity obtained from the ASM and VFM onboard the same satellite, the common periods in the two original signals should be automatically deleted and what we obtain should describe the effect of the disturbance suffered from VFM measurements. Thus, although the disturbance is mainly characterized by modes with frequencies close to the orbital period of Swarm satellites (as it is reported in Figs. 6, 7 and 9), these frequencies are not the direct result of satellite orbiting the Earth. They are a real feature of the disturbance and hence of its source mechanism.
Due to the large number of modes necessary for the description of the disturbance, this could be interpreted in terms of a nonlinear response of the VFM to the varying environmental conditions. Anyhow, some mechanisms responsible for the observed disturbance field still exist in the Current data and are not completely removed using the empirical model (Lesur et al. 2015) built to correct VFM measurements.
The correction made on the data resulted in a significant decrease in the amplitude of the disturbance affecting VFM measurements, therefore greatly improving the magnetic vector data quality. However, our findings (see Fig. 9) suggest that the correction actually decreased the amplitude of the difference between ASM and VFM total intensity but, since the origin of the disturbance has not been identified, the spurious magnetic field affecting VFM measurements is still present, with less energy but with the same structure as before correction. To try to understand the origin of the disturbance field which still affects data also after the correction, we analyze the second data group described in “Data” section. Our hypothesis is that after correction, VFM magnetic measurements are still affected, through a mechanism we do not know, by the effects of different satellite heating due to the varying position of the Sun relative to the satellite. To verify this hypothesis, we check whether the temperature of the VFM electronic unit, which could be considered a proxy of the changes in the satellite environment along its orbit, is decomposed in a similar way as for ASM and VFM total intensity differences obtained from corrected data.
We apply EMD to the second data group, i.e., both the total field ASM–VFM differences (\(\Delta F\)) and the electronic unit temperature (\(T_{\mathrm{EU}}\)) of the VFM (see Fig. 3) onboard Swarm B. In this case, we consider a time interval longer than the previous one: We analyze about 6 months of data from January 13, 2015, to June 29, 2015. Using this different period, we can verify that our previous results remain valid independently of both the length of the analyzed time series and the specific time period considered. At the same time, we can verify the existence of a possible relation between the observed magnetic disturbance and the temperature changes the VFM has undergone and therefore indirectly the temperature changes of the environment around the satellite. Being the temperature data available with a time resolution of 15 s, we consider 1 value every 15 s also in the case of magnetic data (\(\Delta F\)).
By applying EMD to \(\Delta F,\) we decompose it into 21 IMFs and a residue. The distribution of the energy associated with the obtained modes is consistent with the previous ones, indicating that disturbance field is always characterized by the same decomposition regardless of the analyzed period, its length, the used satellite and data version (in this case 0408).
Our findings suggest that an external factor, which could be the different position of the Sun relative to the satellite, still influences both the VFM electronic unit temperature and the VFM magnetic measurements which nonlinearly and chaotically respond to this external disturbance. One possible scenario is that the VFM responds nonlinearly to electronic unit temperature variations, which are related to the position of the Sun, so that to produce biased measurements. Another possible scenario is the one according to which VFM measuring process is perturbed by external temperature variations through some mechanical deformation whose effect could manifest in terms of a spurious magnetic field. In both cases due to the nonlinear response to the external forcing, the complexity of the magnetic measurements is increased with respect to those of the inducing electronic unit temperature changes, and therefore, they are characterized by a higher number of IMFs.
To know whether these two different physical quantities (\(T_{\mathrm{EU}}\) and \(\Delta F\)) share some information, we can apply the mutual information theory which is able to measure the relationship between these two quantities. Indeed, the main advantage of the mutual information theory is that, with respect to the linear crosscorrelation function, it is able to capture both linear and nonlinear relationships between the analyzed time series.
Delayed mutual information
As already mentioned, it has been noticed (see, for example, TøffnerClausen et al. 2016) that the characteristics of ASM–VFM differences depend on local time. For this reason, we extract from magnetic data of the second data group two different time intervals: 5 days (from March 30, 2016, to April 2, 2016) when Swarm B orbits in the dawn–dusk sector and 5 days (from June 25, 2016, to June 29, 2016) when the satellite orbits in the noon–midnight sector (see Fig. 3). Figure 15 displays the two signals during two different days, one for each time interval. The shift between the two signals is not constant but characterized by a dependence on local time. Consequently, we can hypothesize that the time delay between the two signals (\(T_{\mathrm{EU}}\) and \(\Delta F\)) evaluated using the delayed mutual information may be a function of local time. For this reason, we repeat the analysis of the delayed mutual information considering the time series in the two different selected time intervals during which Swarm B orbits in different local time sectors (noon–midnight and dawn–dusk). Figure 16 shows the obtained results. The time delay between \(T_{\mathrm{EU}}\) and \(\Delta F\) changes with LT. It is about 6.5 min when the satellite orbits in the noon–midnight sector while it is about 9 min when the satellite is in the dawn–dusk sector. Also, in this case we have evaluated the level of significance at 95% through the bootstrap method. Of course, considering the average 10.5 min time delay obtained analyzing a time interval of 6 months, longer time delays are expected in the case of other LTs and perhaps due to seasonal effects. This result suggests that there could be also an anisotropy in the response of VFM instruments on the insolation side.
In conclusion, the delayed mutual information is capable of detecting the delayed shared information between the two time series (\(T_{\mathrm{EU}}\) and \(\Delta F\)) without explicitly distinguishing information that is actually exchanged from that due to a response to a common history or common input signal and permitting us to show the existence of a link between \(T_{\mathrm{EU}}\) and \(\Delta F\) without providing any information on whether the correlation comes from a linear and/or nonlinear dependence.
Summary and conclusions
The empirical model for the calibration and correction of the Swarm vector magnetic measurements, introduced by Lesur et al. (2015) and well described in a recent article by TøffnerClausen et al. (2016), has significantly reduced the scalar residuals between the ASM and VFM measurements of the geomagnetic field greatly improving the magnetic data quality. Indeed, the applied model for the calibration and correction of vector data has reduced to values below 0.5 nT the scalar differences between the Swarm magnetometers. However, the findings related to the analysis of ASM–VFM total field difference seem to suggest that the spurious magnetic field disturbing VFM measurements is still partially present in corrected magnetic vector data. ASM–VFM difference remains characterized by a structure which is different from that of a white noise even after the application of the empirical model for the calibration and correction of the Swarm vector magnetic measurements.
The empirical mode decomposition that we have used as a tool for the characterization of the ASM–VFM total field differences has provided similar decompositions when applied on uncorrected and corrected data. The energy associated with some modes of corrected data is indeed decreased, but the structure of the dependence of the energy associated with each mode on frequency (the same before and after correction) is practically identical. These modes, that explain more than 90% of the observed differences and that we identified as the main modes, exhibit mean frequencies close to the orbital period of satellite. These frequencies are not the result of the typical orbital period which is expected to be contained in the magnetic field measurements. We remind that the analyzed time series are differences of the magnetic field intensity measurements obtained from two different instruments onboard same satellite. This suggests that the discrepancy between the two instruments de facto describes the main features of the magnetic disturbance that we have characterized to gain new information about the possible disturbance sources. A first implication of our results is the possibility that some of the mechanisms responsible for the observed differences between ASM and VFM still affect the Current data regardless of the version used (either 0405 or 0408).
To understand the nature of this remaining residual after the correction of vector magnetic data, we have analyzed the VFM electronic unit temperature data which gives indirect information on the temperature changes recorded by the satellite during its orbit around the Earth. As a result of our analysis, we have found that some of the modes describe the main part of the magnetic disturbance coincide, in terms of mean frequency values, with modes obtained from the decomposition of the temperature data. This suggests that the different position of the Sun relative to the satellite, which produces temperature changes in the satellite environment, could be still responsible of a small error in the VFM magnetic measurements.
The delayed mutual information analysis confirms that there is a shared information between these physical quantities, suggesting that their response to the external disturbance source is not simultaneous and that it is characterized by a local time dependence. The response of the VFM to the external disturbance source occurs with a delay time of about 10.5 min with respect to the temperature changes of satellite environment when we consider a period of about 6 months where the different position of the satellite with respect to the Sun is not taken into account. The value of the time delay decreases to about 6 min when the satellite orbits in the noon–midnight sector or to about 9 min when it is in the dawn–dusk sector.
What is interesting to observe is that the delayed shared information between the two signals is found on data corrected by means of the empirical model adopted for the calibration and the correction of the Swarm vector magnetic measurements. These data represent the remanent part of the disturbance once its main part, which is expected to be linearly correlated and in phase with the Sun position, has been removed. Thus, we are characterizing the residual magnetic disturbance which is the result of nonlinear and chaotic processes as it has been found by analyzing the dependence of the subharmonic and superharmonic frequencies of the PSD peaks on the ratios/multiples of the fundamental frequency. Indeed, the presence of subharmonics and superharmonics in the PSD of both the magnetic (\(\Delta F\)) and temperature (\(T_{\mathrm{EU}}\)) signals suggests that the two signals may result from a nonlinear response to a common external driving (Linsay 1981) which is expected to be intense (Yen 1971). The obtained values of the time delays give us the opportunity to note that the correlation (linear and/or nonlinear) between the magnetic signal and temperature one is stronger when the satellite orbits in the noon–midnight sector than when it is in the dawn–dusk one and that the time of response of the VFM to the temperature changes of the environment around the satellite is a function of local time. For what concerns the observed time delay between the temperature and the magnetic signals, we notice that this time delay is observed on the magnetic field ASM–VFM discrepancy after having removed the Sundriven disturbance field from vector magnetic measurements, so that the long delay could be not surprising. Indeed, if the remanent magnetic discrepancy is representative of the nonlinear and/or chaotic response of the VFM instrument to the solar irradiance, it could be the result of a longterm thermal drift. This would explain the observed time delay between temperature and magnetic field ASM–VFM discrepancy. Anyway, to our opinion a reasonable possibility for the long time delay observed between temperature and magnetic field discrepancy is that this discrepancy may arise from a nonlinear response of any mechanical part. Clearly, at this stage this is only a speculation requiring more information on the mechanical mounting of the two instruments and on the response of the used material to thermal stress.
The study illustrated above is just an example of a way to try to characterize the difference of total intensity measured by VFM and ASM. Further and deeper investigations on the residual distributions and frequency structures are certainly possible and could contribute to relate the features of the observed total intensity residual to the physical characteristics of the real disturbance, thus contributing to solve this issue. In this way, it would be possible to improve the model proposed for the correction of data. The effects of correction may be extended beyond the simple reduction in the amplitude of the residual between ASM and VFM. This goal can be achieved, for instance, by analyzing whether and how the estimated mean frequencies change in the time and by analyzing in detail the response, eventually nonlinear, of the VFM to the disturbance source. Another interesting investigation could involve the indepth study of the dependence of the time delay on local time. Of course, being the remanent disturbance in the vector magnetic measurements mainly due to a nonlinear and chaotic response of the VFM to an external driving process, the task to derive a proper correction algorithm will prove to be very complicated.
Abbreviations
 ASM:

absolute scalar magnetometer
 VFM:

vector field magnetometer
 ESA:

European Space Agency
 LT:

local time
 EMD:

empirical mode decomposition
 IMF:

intrinsic mode function
Declarations
Authors’ contributions
PDM, RT, GC carried out the data analysis and participate in key discussions relative to the physical interpretation of the results. PDM and RT draft the manuscript. All the authors read and approved the final manuscript.
Acknowledgements
The results presented in this paper rely on data collected by the three satellites of the Swarm constellation. We thank the European Space Agency that supports the Swarm mission. The elaborated data for this paper are available by contacting the corresponding author (paola.demichelis@ingv.it).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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