Use of the Comprehensive Inversion method for Swarm satellite data analysis
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 27 March 2013
Accepted: 9 September 2013
Published: 22 November 2013
An advanced algorithm, known as the “Comprehensive Inversion” (CI), is presented for the analysis of Swarm measurements to generate a consistent set of Level-2 data products to be delivered by the Swarm “Satellite Constellation Application and Research Facility” (SCARF) to the European Space Agency (ESA). This new algorithm improves on a previously developed version in several ways, including the ability to process ground-based observatory data, estimation of rotations describing the alignment of vector magnetometer measurements with a known reference system, and the inclusion of ionospheric induction effects due to an a priori 3-dimensional conductivity model. However, the most substantial improvements entail the application of a mechanism termed “Selective Infinite Variance Weighting” (SIVW), which mitigates the effects of non-zero mean systematic noise and allows for the exploitation of gradient information from the low-altitude Swarm satellite pair to determine small-scale lithospheric fields, and an improvement in the treatment of attitude error due to noise in star-tracking systems over previously established methods. The advanced CI algorithm is validated by applying it to synthetic data from a full simulation of the Swarm mission, where it is found to significantly exceed all mandatory and most target accuracy requirements.
The European Space Agency (ESA) is scheduled to launch the Swarm mission (Friis-Christensen et al., 2006) in 2013, a constellation of three satellites to map the Earth’s magnetic field to unprecedented accuracy. During its multi-year lifetime, two low orbiting spacecraft will act as a magnetic gradiometer while a third at higher altitude monitors the main and external fields at other local times. ESA has established the Swarm “Satellite Constellation Application and Research Facility” (SCARF) for the purposes of generating derived Level-2 products from the single-satellite Level-1b data. The “Comprehensive Inversion” (CI) method of Sabaka and Olsen (2006) is a major processing chain of SCARF, producing one version of each of five defined items: core, lithospheric, magnetospheric, and ionospheric spherical harmonic expansions, time-varying when appropriate, and Euler angles describing the alignment between the vector fluxgate magnetometer frame (VFM) system and that of the Common Reference Frame (CRF) system of the star imager.
The basic CI algorithm is presented in Sabaka and Olsen (2006) where the magnetic fields from all major near-Earth current sources are parameterized and then co-estimated to obtain optimal field separation. This co-estimation approach is the key to the superior results obtained because it eliminates ambiguities between parameters spaces. Technically, the basic CI algorithm is an iterative Gauss-Newton (GN) least-squares estimator (Seber and Wild, 2003), which derives the model parameters from Swarm vector magnetometer measurements. While its application to the Swarm E2E simulator (Sabaka and Olsen, 2006) showed promising performance in field recoverability, it still lacked several features that would render it a truly competent algorithm for the generation of actual Level-2 products. For instance, the basic algorithm did not allow for surface measurements such as observatory hourly-means (OHM) data, which are known to greatly enhance field separation. Estimation of the rotation between the VFM and CRF system for vector measurements mentioned above was not included in the basic algorithm. The a priori conductivity model assumed for ionospheric induction was 1-dimensional (1D) rather than 3-dimensional (3D) in its variation. Finally, the formal treatment of measurement and theory errors was very primitive and not considered versatile enough for actual mission application.
In this paper an attempt will be made to remedy the aforementioned inadequacies of the basic algorithm by developing an advanced CI algorithm. While this new algorithm now admits the OHM data, estimates the Euler angles describing the VFM to CRF rotations, and includes ionospheric induction due to 3D conductivity structure, its greatest advancement is in the area of formal error treatment. Here a methodology, termed “Selective Infinite Variance Weighting” (SIVW), is developed to handle non-zero mean error due to, for instance, theory inadequacies through the use of bias estimation which exploits Signal-to-Noise ratio (SNR) levels in different data subsets in order to extract the best models. In addition, the methodology of Holme and Bloxham (1995, 1996) and Holme (2000) to account for attitude error present in vector magnetometer measurements due to star tracker instabilities is revised in order to improve parameter estimates in the CI algorithm. The entire error treatment mechanism is then placed in the context of robust estimation by applying a Huber weighting scheme to mitigate the effects of outliers (Constable, 1988; Walker and Jackson, 2000; Olsen, 2002).
The structure of the paper is as follows: a brief overview of the basic CI algorithm will be presented in Section 2, followed by the development of SIVW in Section 3. In Section 4 the improved attitude error framework will be derived, followed by the development of the advanced CI algorithm in Section 5. The results of the application of the advanced algorithm to synthetic data from a mission simulation, known as “Version-2” (V2) (Olsen et al., 2013), and a discussion are in Section 7, followed by conclusions in Section 8. Finally, Appendices A and B are provided that contain some technical information and derivations of formulae presented in Sections 4 and 5, respectively.
2. Overview of the Basic CI Algorithm
What is not included in the basic CI is a mechanism to exploit the enhanced lithospheric SNR in the differences of the vector measurements made by the Swarm low satellite pair. It is more complicated than simply using only the low pair vector differences to make models since the complementary data set (the low pair vector sums) is needed in order to resolve other field constituents. In the next section such a mechanism is indeed developed. The application of this mechanism to Swarm will be discussed in Section 5.
3. Selective Infinite Variance Weighting
The near-Earth magnetic field is a highly dynamic system containing signals that vary over a large range of spatial and temporal scales. Even a constellation like Swarm cannot completely decouple all of the space and time modes. A prominent example is the mis-modeling of time-varying external fields which can manifest itself as spurious static structure, for instance, in the lithospheric field. While this structure may be spatially broad-scale, it can vary rapidly in time and represents a systematic noise bias that can contaminate the nominal estimate of the lithospheric field. Different philosophies exist on how to enhance the recovery of signals of interest, like the lithosphere, while mitigating the effects of unwanted or contaminating signals in the estimation procedure. Several recent models have employed direct data selection techniques to derive good descriptions of core and crustal fields (Maus et al., 2007, 2008; Thomson and Lesur, 2007; Lesur et al, 2008; Olsen et al, 2011). The “Comprehensive Modelling” (CM) approach (Sabaka et al., 2002, 2004) has not generally relied upon this practice, with the exception of gross outliers, etc., but rather has focused on using as much data as possible to ensure a stable co-estimation of parameters from all sources. Comparison of the CMs with these models, however, has revealed effects of contamination, particularly of ionospheric “leakage” into lithospheric fields.
The Swarm constellation offers the ability of taking differences between the vector magnetometer measurements of the satellite low pair, which effectively eliminates most of the broad-scale external contamination, thus leading to a high SNR of the crustal field signal. However, the complementary data set, i.e., the summation of the vector magnetometer measurements of the satellite low pair, is necessary for a full co-estimation of the other field constituents, but it has a much lower SNR for its lithospheric field because it suffers from the aforementioned systematic noise bias. With this in mind, a more sophisticated error treatment will be required of the CI models in order to exploit the high SNR in one data set while limiting damage done by retaining low SNR data. The SIVW mechanism is now developed by first showing its ability to eliminate bias from systematic noise of a particularly common form that would otherwise contaminate estimates if treated in traditional ways, and secondly, how to combine this property with selection of data subsets that exhibit different SNRs for different parameter sets in order to obtain optimal solutions.
3.1 Mitigating biases
This challenge can be stated mathematically by how best to handle additive systematic error terms so as to not bias the estimation of signals of interest. The name “systematic” is used here to describe a vector error term that has the form z = By, where B is a matrix having more rows than columns and y is a vector of Gaussian random errors having a mean vector μ and covariance matrix Q, indicated by . Thus, z represents noise that cannot be reduced to arbitrary levels by repeated experiments because it has a non-zero mean, but can be eliminated in certain subspaces due to the fact that q = dim (z) > p = dim (y).
There are two interesting properties about the estimate given in Eq. (14) that should be mentioned. First, there is no need to actually specify Q, that is, the covariance of the systematic noise term. The weight matrix W∞ gives zero weights to the directions defined by the column space of B rendering a specification of Q completely unnecessary. It is also interesting to note that W∞ not only eliminates Bμ in the mean, but also individual realizations of the systematic error z.
3.2 Subset selection
Clearly a hierarchy of nominal/nuisance parameter combinations could be distributed over the observation equations in order to mitigate systematic errors. Obviously, in extreme cases where a data subset is contaminated by such errors in all parameters subsets, it would be wise to simply eliminate that data.
4. Attitude Error
The observation equations relating spherical harmonic coefficients in a local spherical system (North, East, Center or NEC (Olsen et al., 2013)) to the VFM will rely upon coordinate transformations provided by star-imagers (STRs). As such, these transformations will be degraded by random errors due to physical limitations of the STRs and should therefore be accounted for in the error analysis of the estimators. In a series of papers (Holme and Bloxham, 1995, 1996; Holme, 2000) a mechanism was developed in order to account for these errors that will henceforth be referred to as “HB theory”. This theory has been used successfully in such models as the Oersted Initial Field Model (OIFM) (Olsen et al., 2000) and Comprehensive Model-4 (CM4) (Sabaka et al., 2004) for instance. However, it turns out that simplifying assumptions have been made in the HB theory that render the forms used in these models (equations (13) and (18) of Holme and Bloxham (1996)) less suitable for describing the actual attitude error encountered. While Holme and Bloxham (1996) provide a form that is technically always applicable (their equation (20)), the quantities involved are non-intuitive and require prior knowledge of the eigen-structure of the attitude covariance; something that is not obvious, even in isotropic error cases. In this section an attempt is made to remedy the situation by applying a slightly more generalized treatment to attitude errors in the CI algorithm.
4.1 Covariance under general, finite rotations
4.2 Covariance under HB theory assumptions
It turns out that the covariance matrix corresponding to the “no equal variances” case in Eq. (A.24) of Appendix A is a general expression that is always true and is equivalent to setting the three rotation axes and corresponding angular variances equal to the eigenvectors and corresponding eigenvalues, respectively, of the symmetric positive semi-definite matrix ACaAT in Eq. (35). The “three variances equal” case in Eq. (A.22) and “two variances equal” case in Eq. (A.23) of Appendix A are true when either all three or just two out of three eigenvalues are equal, respectively.
4.3 Inertial to Common Reference Frame transformations
Uncertainties in the STR or CRF are usually quoted in terms of errors in the angles comprising the rotation matrices, such as errors in bore-sight pointing angles δ and λ and errors in rotation angles χ about bore-sights. If these angles are finite, then one begins to see the pitfalls in using the HB theory expressions because the columns of the matrix A in Eq. (44) are rarely orthonormal. This means that one has to compute the eigen-decompositon of ACaAT in order to use the “no equal variance” or general formula of the HB theory (equation (20) of Holme and Bloxham (1996)). If two or all three of the eigenvalues are equal, then one can use the more specialized HB formulas (equations (18) and (13) of Holme and Bloxham (1996), respectively), but this is also likely a very rare event. Consequently, while the HB theory general formula is always available, it is very non-intuitive to use as one must compute eigen-decompositions to even use it. Alternatively, the general covariance formula in Eq. (35) follows directly from the statements of error that are presumed to be provided for the STR or CRF and is thus very intuitive.
4.4 Application to CHAMP transformations
5. Development of the Advanced CI Algorithm
An advanced CI algorithm will now be built upon the foundation outlined in Section 2 with improvements from SIVW in Section 3 and the revised treatment of vector magnetometer attitude errors in Section 4. The modified parameterization will be described as well as how Swarm gradient information is to be exploited for improved lithospheric recovery which entails a more sophisticated error analysis then was used in the basic algorithm.
Note that while the magnetospheric and associated induced field parameters described so far are estimated by iteratively solving LSLE-GN in Eq. (6) using Eqs. (7) and (8), they do not represent the final Level-2 product MMA_SHA_2 because they are only estimated during geomagnetic quiet times. Rather, they provide a crucial step in the generation of these products, which is elaborated upon further in Section 7.5. This is the reason for using the term “precursor” in Table 2.
5.2 Exploiting Swarm gradient information
5.3 Weighting and robust estimation
The next task is to define C in Eqs. (7) and (8) for each measurement type. For the vector differences and summations, this is commensurate to defining CAA and CBB in Eq. (55). Beginning with the simplest case, the OHM measurement noise covariance is expressed in the form , where is a function of geomagnetic latitude with polar stations having higher variance than lower latitude stations. Thus the noise is treated as isotropic and uncorrelated between vector components and other data. Likewise, satellite scalar measurements are treated as uncorrelated with all other data and the variance is denoted by . For satellite vector measurements, the formalism of Section 4 is employed to account for the CRF attitude error while an additional isotropic term is added to account for instrument noise (Holme, 2000) and is chosen here to match the scalar variance, which is assumed the same for each spacecraft fluxgate magnetometer. Therefore, and . Notice that because the attitude error (second term) is a function of Bcrf(x k ), then Caa and Cbb change at each GN iteration. Specifically, both and are in the form of Eq. (35) under the assumptions of Section 4.3. These vector measurements are also assumed uncorrelated with all other data.
What remains is to define the quadratic norms in Eq. (8) that are used to regularize the system. For the core SV and ionosphere the norms are similar to those used in the CM4 model (Sabaka et al., 2004) and earlier Swarm simulation studies (Sabaka and Olsen, 2006). A combination of the mean-squared magnitude of over the sphere at the core-mantle boundary (CMB) and at Earth’s surface were used to constrain the core SV, while the nightside ionospheric E-region currents were minimized by including a norm that measures the mean-squared magnitude of the E-region equivalent current, Jeq, flowing at 110 km over the night-time sector defined as 1100–0500 hrs local time. In addition, these currents are further smoothed by minimizing the mean-squared magnitude of the surface divergence of the diurnally varying portion of Jeq at mid-latitudes at all local times.
For this study two additional norms were employed. The first is motivated by the presence of a gap in the coverage of the satellites resulting in a polar cap of a few degrees in half-angle. Because zonal SH terms are most affected by these gaps, a norm which minimizes the square of the magnetic potential of the lithospheric field for degrees n ≥ 60 at both the north and south geographic poles was developed. The final norm minimizes the sum of square deviations of the Euler angles in each time bin with the average value over the entire mission domain as determined from the current nominal values. This is done separately for each of the three angles.
In summary, N q = 6 quadratic norms are applied in Eq. (8), four of which are similar to those used in previous studies, and two of which are experimental. It is expected that similar norms will be used for the actual mission analysis, but development is continuing on the CI algorithm and could quite possibly lead to better regularization techniques. The advanced CI algorithm has now been developed and will next be applied to the V2 simulation in Section 6.
6. The V2 Simulation
Target and threshold requirements of estimated field accuracy for V2.
MCO SHA 2
Core field, SV on ground, n = 2–16, time averaged
MLI SHA 2
Lithospheric field, accumulated error on ground
40 nT, for range n = 16–150
120 nT, for range n = 16–133
MIO SHA 2
Ionospheric field, average relative error in ǀBǀ on ground
10% equator-ward of ±55° magnetic latitude
MMA SHA 2
coh2> 0.8 coh2> 0.95 for dipole terms
coh2> 0.5 coh2> 0.75 for dipole terms
The core field is defined for SH degrees n = 1−20 and consists of snap-shots derived from order 6 spline models (Lesur et al, 2010) that run from 2003.0 to 2008.0 inclusive, but are shifted byδyears (i.e. to 1998.0 to 2003.0) in order to be compatible with the data period used for TDS-1. However, for SH degrees n = 14−20 the core field static terms have been replaced by those of the lithospheric field. The lithospheric field consists of SH degrees n = 14− 250, where n = 14–15 taken from model POMME-6.1 (Maus et al, 2010), degrees n = 16–90 are taken from model MF7 (Maus, 2010a), and degrees n = 91-250 are taken from model NGDC-720 (version 3p1) (Maus, 2010b) scaled by factor 1.1. The primary ionospheric field is based on that of CM4 (Sabaka et al, 2004), and the secondary field SH coefficient vector ι(ω) is computed from the primary vector ϵ (ω) at each frequency ω via the relationship ι(ω) = Q(ω)ϵ(ω) as discussed in Section 5.1. The coupling matrices Q(ω) come from a 3D mantle conductivity model (Kuvshinov, 2011). The magnetospheric primary field is similar to that of the E2E+ simulation (Tøffner-Clausen et al, 2010) and is based on an hour-by-hour spherical harmonic analysis of worldwide distributed observatory data. The secondary magnetospheric field is computed from the same set of coupling matrices as the ionospheric field.
Angles defining rotations between VFM and CRF systems in TDS-1.
7. Results and Discussion
The results of applying the advanced CI algorithm to the TDS-1 data set of the V2 simulation will now be briefly discussed. While these were found to be quite favorable, it should be stressed that this is a simulation based upon synthetic data containing contributions from forward models similar to those used in the CI, such as the ionospheric primary and secondary fields. Therefore, it is possible that performance could be degraded when analyzing data from the actual mission. The three metrics employed by Sabaka and Olsen (2006) will be used here, which are defined in terms of the real and imaginary parts of complex S H coefficients of a field, denoted generically as ,and respectively.
There is one additional metric that will be used to evaluate the V2 results. This is the “squared-magnitude coherence”, or coh2, whose range is 0 ≤ coh2 ≤ 1 and measures the similarity between input and output signals of a system. For constant parameter linear systems, coh2 = 1, but this can decrease due to a number of issues, particularly the presence of noise in the system. This has been used by Olsen (1998) to analyze C-responses describing electrical conductivity of the mantle beneath Europe, and details may be found therein.
7.2 MCO core field
7.3 MLI lithospheric field
In addition to assessing the CI algorithm performance with respect to V2 accuracy requirements, it was of interest to see the direct benefits of taking explicit advantage of the magnetic gradient information for determination of the small-scale lithospheric field. Therefore, two types of models were derived from exactly the same magnetic field observations. The first model, denoted as “field only”, was constructed by considered the Swarm data from three single satellites, whereas the second model, denoted as “field plus gradient”, was constructed by explicit use of the constellation aspect of Swarm by using magnetic gradient information with SIVW.
Average relative error in ǀBǀ on ground in MIO primary ionospheric field recovery.
ǀ Magnetic latitude ǀ < 50°
ǀ Magnetic latitude ǀ < 50°
ǀ Magnetic latitude ǀ < 50°
While these results suggest an optimistic outlook that the Swarm constellation is capable of accurately recovering small-scale lithospheric structure, the application to real data will be more challenging, especially at high latitudes. However, if V2 performance is any indication of real performance, then Swarm will go far in closing the gap in intermediate lithospheric wavelength coverage that exists now between satellites and aeromagnetic surveys.
7.4 MIO ionospheric field
The accuracy target requirement for the MIO primary ionospheric field is 10% average relative error in ǀBǀ on ground. Table 5 actually subdivides these numbers across local time sectors (midnight, morning, noon, and evening) and across seasons (February to April, May to July, August to October, and November to January) for magnetic latitudes equator-ward of ±50° and all latitudes. It can be seen that every subdivision is performing almost twice as well as the 10% requirement, with overall errors of 4.16% and 4.40% for the low latitude and all latitudes regions, respectively.
7.5 MMA magnetospheric/induced fields
Recall that the test data are selected for magnetically quiet times such that the CI products for core, lithosphere, and ionosphere (primary and secondary) all reflect this. However, the determination of continuous time series of the spherical harmonic expansion coefficients of magneto-spheric and corresponding induced sources requires data taken during all geomagnetic conditions, as required for Level-2 product MMA_SHA_2. Therefore, this is achieved by applying a second processing step, the MMA_SHA_2 analysis step, after CI in the following way: Predictions are made from the output CI core, lithospheric, and primary and secondary ionospheric models derived during quiet times and subtracted from each 1 min Swarm satellite measurement and OHMs from all available ground observatories. The resulting residuals (observations minus model values) are expected to contain the magnetospheric (primary and secondary) field plus errors due to improper removal of all other sources. From those residuals of each day estimates are made of the spherical harmonic expansion coefficients describing the external (magnetospheric) sources for Nmax = 3 and Mmax = 1, and coefficients describing the induced field for Nmax = Nmax = 5. This is done in bins of 1.5 hrs for the axial dipole coefficients and (which means that 16 values for each of those coefficient pairs are determined per day) and in bins of 6 hrs for the remaining 42 coefficients (resulting in 4 × 42 = 168 values per day). In total, 200 coefficients are estimated for each day using IRLS with Huber weights.
This paper has presented an advanced CI algorithm that includes many improvements over the basic algorithm described in Sabaka and Olsen (2006), and most important among these are the introduction of the SIVW mechanism for mitigating systematic, non-zero mean bias in the observations allowing for optimal combinations of data to achieve improved models. In addition, this paper has pointed out a way to improve on the handling of attitude error in vector magnetometer data put forth in the HB theory. This will hopefully allow for even better error characterization, and thus, model quality.
The performance of the advanced CI algorithm was evaluated using a synthetic test data set (TDS-1) from a full mission simulation. In general, it was found to perform well above both threshold and target accuracy requirements for core, lithospheric, ionospheric, and magnetospheric and induced. Only the recovery of induced fields at some periods longer than one month were of suspect quality. This may be due to the point-wise orthogonality condition with respect to the core field that has been imposed on this field, which of course would affect the longer periods since SV is on the order of these longer periods. Although there were no toroidal field contributions in the TDS-1 data, these fields were nonetheless co-estimated and found to be negligible, thus indicating a proper treatment in the model. All of this suggests, at least from the standpoint of V2, that the advanced CI algorithm will be quite competent in delivering high quality Level-2 products.
Although the advanced algorithm is a great improvement over the basic algorithm, certain issues should still be dealt with to further enhance performance. Recall that when using the covariance matrix describing the error in the vector differences and summations in Eq. (55) the cross-covariance matrices C−+ have been ignored. This was done to simplify the algorithm during development, but should now be instated. The increase in deviations between reference and recovered B r from the lithosphere around the geographic poles in Fig. 7 indicates that the polar gap problem should be further addressed. In addition, several issues surrounding SIVW should be explored, such as the optimal SH order m that delineates between the nominal and nuisance lithospheric fields, which could easily lead to better models. In addition, SIVW could be used to account for dayside bias when modeling the lithosphere, which would reflect the current best methods for crustal field modeling (Thomson and Lesur, 2007; Maus et al., 2007, 2008; Lesur et al., 2008, 2013; Olsen et al., 2011). Finally, SIVW could be applied to high degree SV modeling by mitigating the bias due to the poor distribution of ground observatories. It is planned to implement and test several of these ideas.
We thank Richard Holme and Vincent Lesur for fruitful reviews. The NASA Center for Climate Simulation at Goddard Space Flight Center provided computer support. Some figures were produced with GMT (Wessel and Smith, 1991).
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