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.
The parameterization for the advanced algorithm is listed in Table 2 and is similar to the basic algorithm in the core field except for higher time resolving splines that are higher in order and knot density. The lithosphere is now split into two parameter types, “nominal” and “nuisance”, that will be discussed in the next section. The magnetosphere is the same as in the basic algorithm while the ionosphere is different in that its a priori conductivity model now has 3D structure (Kuvshinov, 2011). If ϵ(ω) and ι(ω) are the vectors of SH coefficients for the inducing and induced ionospheric fields, respectively, at frequencyωthen the a priori coupling via the conductivity model is manifested in the relationship ι(ω) = Q (ω)ϵ(ω), where Q(ω) is the coupling matrix at frequencyωIn a 1D treatment, as in Sabaka et al (2002, 2004) and Sabaka and Olsen (2006), Q(ω) is diagonal and square and its elements are only dependent upon SH degree. In the full 3D treatment, Q(ω) is a dense, generally rectangular matrix allowing for very complicated induced structure to result from relatively smooth inducing structure. Therefore, the change from 1D to 3D comes from simply using a different set of Q(ω). In addition, toroidal magnetic fields due to meridional currents that exist within the satellite sampling shells are also modeled in the advanced CI algorithm. These follow the parameterization of CM4 (Sabaka et al., 2004). Finally, because OHM data are now processed in the advanced algorithm, static vector biases are now included in the parameter set for each observatory in order to absorb effects such as local crustal anomalies (Sabaka et al., 2002, 2004).
The truly new parameters are those that describe the magnetometer alignment, that is, the rotation of the vector magnetometer measurement BVFM in the VFM frame to CRF
For Swarm, this rotation is parameterized by a set of 3 positive counter-clockwise Euler angles of type (XYZ) for each satellite such that (Olsen et al., 2013)
In the advanced CI algorithm the observation equations involving vector magnetometer measurements are expressed in the CRF. If the model parameter vector at the k-th GN step xk is split into two subsets, the “geophysical” parameters in vector z
and the Euler parameters for a particular satellite in vector e
, then for the i-th vector measurement of that satellite, the observation equation is
is the error vector, and g
) and a
(xk) are the geophysical and total model vectors in the CRF, respectively. The reason for solving in the CRF rather than the VFM frame is to decouple the product that exists in the latter system, thus decreasing the level of non-linearity in the estimation process. Recalling Eq. (6), it can be seen that Eq. (51) is in a form that is equivalent to having d
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
One of the great advantages of the Swarm constellation is that the low satellite pair have orbits that differ only by 1.4° in the values of their Right Ascension of the Ascending Node (RAAN), thus allowing for east-west gradiome-try to be carried out at low-mid latitudes. Let the Swarm low pair, denoted “A” and “B”, be at positions (r, θ, ϕ) and (r, θ, ϕ + ϕ), respectively, where r, Ϙ, and ϕ are the radius, colatitude and longitude, respectively, and Δø is a longitude increment. Assume that they provide vector measurements BECEF( r, θ, ϕ) and BECEF( r, θ, ϕ + Δϕ) that have been rotated into the Earth Centered Earth Fixed (ECEF) frame, where the z axis points to the north geographic pole, the x axis points along the prime meridian, and the y axis completes the right-handed system. If these vectors are further rotated into the local spherical NEC frame at the midpoint of the satellite positions, then for small Δϕ certain components of their difference behave as a negative gradient of a potential function whose SH coefficients are multiplied by a gain factor of approximately
as compared to the potential coefficients leading to the individual field measurements. These components correspond to the direction of the ECEF z axis in the NEC frame and the direction of the average of the two measurement vectors in the NEC frame. If these two directions are coincident, then all components will exhibit this gain enhancement. It can be seen that and that within the range 0 ≤ m ≤ 150, the maximum gain for vector difference measurements is found at approximately m = 129. Conversely, certain components of their sum behave as a negative gradient of a potential function whose SH coefficients are multiplied by a gain factor of approximately
These components correspond to the direction of the ECEF z axis in the NEC frame and the direction of the difference of the two measurement vectors in the NEC frame. Again, if these two directions are coincident, then all components will exhibit this gain. Note that these gain factors are out of phase such that . The gain factors are shown in Fig. 2 for the order range of the core and crustal fields and are derived in Appendix B.
If one were only interested in recovering high degree/order lithospheric signals, then based on the gain factors one might naively exclude the vector summation data and focus only on the vector differences. However, the summation data is critical for determining broad-scale, highly time varying fields such as the magnetospheric and the high-frequency induced fields, which if not properly modeled can cause spurious signals that mimic lithospheric signal. This strongly suggests using the SIVW mechanism in order to preserve the vector summation data, but account for systematic bias in its high degree/order lithospheric signal that must certainly exist, especially given its low gain factors at high orders. Because the CRFs of Swarm A and B (CRFA and CRFB, respectively) cannot be considered the same, the CI algorithm first rotates the observation equations for each satellite to the local NEC coordinate system at the mid-point between the two satellites before adding and subtracting. If and be the rotations from CRFa and CRFb to the mid-point, respectively, then a given pair of vector measurements from Swarm A and B are transformed to differences and sums via the following orthogonal transformation
The covariances are similarly transformed as
where the notation C(∙) is now used to indicate auto or cross-covariance.
At this point, the observation equations and covariance of Eqs. (54) and (55) could be formally introduced into the LSLE-GN framework of Eq. (6), with the modification of an additional infinite variance term in the covariance to account for high degree/order lithospheric systematic bias in the vector summations. In practice, however, it is much more feasible to perform this through co-estimation of nuisance parameters as shown in Section 3. This means that while Eq. (6) is strictly followed, Eqs. (7) and (8) are modified to include the crustal nuisance parameters. This essentially modifies the Ak matrix in the previous equations and renders the linearized observation equations at GN step k to be
where the subscript “k” has been suppressed, the subscripts “−”, “∋”, “C”, and “OHM” indicate vector differences, summations, Swarm satellite C (the high satellite), and the ground observatories, respectively, and the superscripts “h” and “r” indicate the high degree/order lithospheric field parameters containing systematic bias in the summation data, and the remainder of the parameters, respectively. Likewise, Δxh, Δnh, and Δxr are the vector adjustments to the nominal and nuisance high degree/order lithospheric fields, and the remaining nominal parameters, respectively. Note that because of its high altitude, the measurements from Swarm C are assumed to have a low SNR in the high degree/order lithospheric field, and are therefore eliminated from the nominal model. In the case of the OHM measurements, the static vector biases that are solved for effectively decouple this data from the static lithosphere and so it is not affected by either the nominal or nuisance lithospheric parameters, at least for n > 20. The C matrix in Eqs. (7) and (8) is now that in Eq. (55). However, in the current implementation of the CI algorithm, C−+ is ignored. Again, it should be stated that when solving LSLE-GN, only the nominal parameters are used to calculate Δx
and are the only parameters updated. The nuisance parameters are only included to expedite the use of the dense SIVW covariance matrix. In this study, the high degree/order lithospheric nuisance field is defined to be in the range n, m > 20, as shown by the green vertical line in Fig. 2, and so does not include the time varying part of the internal field.
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
), 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.
If the linearized model residuals are Gaussian distributed, then a weighted least-squares estimate, which minimizes ℓ2 norm of a vector, as does LSLE-GN, would provide the maximum-likelihood estimate. Since this is rarely the case in real-world problems, the estimator can suffer deleterious effects due to excessive influence of outliers. To combat this, a robust estimation procedure know as “Iteratively Reweighted Least-Squares” (IRLS) with Huber weighting will be employed here (Constable, 1988). This method has been used successfully in such models as CM4 (Sabaka et al., 2004) where details may be found. What is important here is that the IRLS formulation is defined for uncorrelated, scalar measurements. IRLS assigns Huber weights to the i-th measurement at the k-th GN iteration as a function of its standard deviation σ
and current residual eii, k according to
where the underlying Huber distribution is defined as having a Gaussian core for ǀe
ǀ ≤ cσ
, and Laplacian tails (Constable, 1988). A value of c = 1.5 is used by the CI algorithm. All measurement types conform to the structure of Eq. (57) except the vector differences and summations. To rectify this, the Huber weighting is applied to the principle components of the covariance in Eq. (55), which follows readily from the eigendecompositions of CAA and CBB.
In Appendix A it is shown that Euu = 0. It follows from Eq. (35) that , which means that exists in a 2-dimensional subspace that is spanned by the columns of a 3 × 2 matrix that are orthogonal to . The eigendecomposition of CAA is then given by
where is the unit vector in the direction of is a 3 × 2 matrix whose columns span the range of , and is the eigendecomposition of where the 2×2 matrix Ua is orthogonal and the 2×2 matrix Λa is diagonal with positive eigenvalue entries. Of course a similar development leading to Eq. (58) applies to Cbb . Thus, using Eq. (55) and Eq. (58) the residuals are rotated into the principle axes of the covariance matrix in Eq. (55) where and supply the σ
needed for determining the Huber weighting in Eq. (57).
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
= 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.