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POGO satellite orbit corrections: an opportunity to improve the quality of the geomagnetic field measurements?
 Reto Stockmann^{1}Email author,
 Freddy Christiansen^{2},
 Nils Olsen^{2} and
 Andrew Jackson^{1}Email author
https://doi.org/10.1186/s4062301502547
© Stockmann et al.; licensee Springer. 2015
 Received: 1 December 2014
 Accepted: 20 May 2015
 Published: 27 June 2015
Abstract
We present an attempt to improve the quality of the geomagnetic field measurements from the Polar Orbiting Geophysical Observatory (POGO) satellite missions in the late 1960s. Inaccurate satellite positions are believed to be a major source of errors for using the magnetic observations for field modelling. To improve the data, we use an iterative approach consisting of two main parts: one is a main field modelling process to obtain the radial field gradient to perturb the orbits and the other is the stateoftheart GPS orbit modelling software BERNESE to calculate new physical orbits. We report results based on a singleday approach showing a clear increase of the data quality. That singleday approach leads, however, to undesirable orbital jumps at midnight. Furthermore, we report results obtained for a much larger data set comprising almost all of the data from the three missions. With this approach, we eliminate the orbit discontinuities at midnight but only tiny quality improvements could be achieved for geomagnetically quiet data. We believe that improvements to the data are probably still possible, but it would require the original tracking observations to be found.
Keywords
 Inverse theory
 Satellite magnetics
 Satellite orbits
Background
Measurements of the magnetic field during the previous century are of importance for characterising the secular variation. Starting with Cosmos49 in 1964, satellite data has played an increasingly important role in defining the magnetic field, culminating in the presentday Swarm mission beginning in 2013 (Olsen and The Scarf team 2013). Here we report on data collected by the series of Orbiting Geophysical Observatory (OGO) satellites from the 1960s. These data quite likely suffer from imprecision in their geographical positions as a result of poor tracking abilities and rudimentary gravitational models. We feel that there is an opportunity to improve these heritage data by using uptodate methods and gravity fields. As we shall see, the original tracking data were lost, so our efforts represent a compromise at what could be done if the original data were to be found.
Mission summary for the OGO programme
Spacecraft  Launch date  Orbit (km)  Inclination (°) 

OGO2  October 14, 1965  410–1510  87.3 
OGO4  July 28, 1967  410–910  86.0 
OGO6  June 5, 1969  400–1100  82.0 
Following Gleghorn and Wiggins (1965), the tracking functions and command of the OGO were designed to be compatible with the National Aeronautics and Space Administration’s Minitrack Network as well as prime data acquisition stations located at Rosman, North Carolina; Fairbanks, Alaska; Australia; Johannesburg, South Africa and Quito, Ecuador. Orbits of most satellites before the OGO programme had been determined by the use of the worldwide satellite network, formerly known as the Minitrack Network. This network of tracking stations and the necessary computational techniques were well established and were also used in the OGO programme. The use of two ground tracking stations simultaneously permitted high accuracy trilateration of the satellite. This system was supplemented by a range and rangerate system which was expected to permit the more accurate computation of the orbit parameters in a shorter time, especially in the case of the highly eccentric orbit in which the satellite spent a large fraction of its time at large distances from the Earth where the angular rates are very low. The overall goal of the tracking programme was to be able to determine the position of the satellite at all times within a sphere of uncertainty having a radius of 1 km or less at radial distances of less than 1000 km and of 100 km at radial distances of 17 Earth radii (Ludwig 1963).
Intensity error Δ F at 400 km altitude due to errors in position
Positioning error  Δ F max (nT/100 m)  Δ F rms (nT/100 m) 

Δ r  2.62  1.13 
r Δ θ  0.49  0.19 
r sin(θ)Δ ϕ  0.22  0.08 
POGO orbits were determined using simple gravity field models up to spherical harmonic degree 7 and order 6, plus three higher order resonance terms (Taylor et al. 1981). Thus, there are two sources contributing to positional error: i) imprecision in determination of actual spot position, through range and rangerate determination; and ii) imprecision in reduction of the data to produce valid orbits, i.e. determination of observations when only timing information was available. Unfortunately, the unprocessed original data are not available any more and we are left only with processed data which are available from the Goddard Space Flight Center (Greenbelt, Maryland, USA). Therefore, a reprocessing of the orbits with BERNESE, using a stateoftheart gravity field and tidal models, may reduce the position errors and hence increase the data quality. To remove the potential bias from the old orbit in an objective way, we radially perturb the orbits according to the maximum radial gradient of B, reducing the residual to a magnetic potential field. For more detail about our approach, see the ‘Data compilation’ section.
The data compilation that we used is described in the next section. In the ‘Results and discussion’ section, we report the results obtained by processing a single day only, namely August 8, 1969, together with the results obtained by using the combinedday approach. A discussion of the results obtained is given in the ‘Conclusions’ section.
Methods
Data compilation
where F _{mod} is the total field intensity used for the modelling, F _{obs} is the observed total field intensity, Δ F is the portion of the magnetospheric field to the observed total field intensity and B _{main} is the CM4 main field at corresponding epoch. We refer to that data set as POGO_mod. Justification of the use of the xCHAOS magnetospheric model derives from the following rationale: Of the three external constituent parts, the first two are assumed to be stationary in the SM and GSM frames, respectively. This means they are assumed to be independent of the solar cycle phase and thus identical in the 1960s and the years 2000–2010, which is the time span from which the model coefficients have been derived. The time variation of those two parts contains daily and seasonal variations, given by the ‘wobble’ of the GSM and SM frames wrt a fixed location on Earth. The third part depends explicitly on the D _{ st }index (more concretely: on the decomposition of D _{ st } into its external part E _{ st } and its induced part I _{ st }) and therefore has an explicit time dependence (given by that of D _{ st }). The estimated regression coefficients are assumed to be solar cycle independent, but we used the actual values of D _{ st }=E _{ st }+I _{ st } of the POGO time instant to correct the POGO magnetic observations for magnetospheric field contributions. The xCHAOS model coefficients have been derived using the Ørsted and CHAMP data including solar maximum and minimum conditions. The model fits the observations equally well for both conditions, indicating that the assumed solar cycle independence is justified.
We note that the use of CM4 for removing outliers is slightly circular, since CM4 itself was built using POGO data; however, the tolerence level of 30 nT is sufficiently large that we do not expect that this has a substantial effect. The error budgets for the modelling were set very conservatively to 7 nT independent of the measurement location, even though it can be assumed that the real errors of the original data are around 5.6 nT (Langel 1974). The first POGO_mod data set contains about 696,997 observations.
BERNESE GPS software
The BERNESE GPS software was developed by the Astronomical Institute of the University of Bern, AIUB, Switzerland. It includes a reduced dynamics orbit generator. The force model in the orbit generator includes i) Earth’s gravitational potential to selected order and degree, here 120; ii) gravitational effects from Sun, Moon, Jupiter, Venus, and Mars; and iii) elastic Earth tidal corrections, pole tide and ocean tides. Further, the solar radiation pressure and air drag are estimated and applied as pseudostochastic pulses (instantaneous velocity changes at specified epochs) in order to make the best fit between the input orbit and the reduced dynamics orbit. However, it must be said that the BERNESE software originally is intended to determine orbits for the GPS satellites and it could not be guaranteed that it also will work properly for our purpose because the POGO time was earlier than the actual GPS invention (personal communication of R. Dach, head GPS research and BERNESE development group at AIUB). Further, note that BERNESE is not able to generate unique orbits, it rather verifies if the input orbit points come from a physical orbit or not.
For a singleday input orbit, the new reduced dynamics orbit can easily be generated within one arc. However, by processing all days as single days, it is certainly sure that the new orbits are not continuous at midnight which of course is the case for the real satellite orbit. Because of memory and runtime limitations, it was not possible to generate all the new orbits for one POGO mission into one arc, which would have been the optimal satellite solution. We tackled the continuity problem in such a way that we used a moving window fit (similar to a moving average) consisting of 5 days fitting within three arcs. That leads to a middle arc which is overlapping into the neighbouring days by 8 h. The individual discontinuities between the three arcs are negligibly small. For some days, BERNESE was unable to produce valid orbits with that setup; in these cases, we used either a 3day fitting window or the arcs from neighbouring days (±2 days). In some rare cases, we were forced to use a singleday fit only. The new orbits modified all three coordinates (radius, colatitude and longitude) of the original orbit.
The new orbit coordinates are generated on a regular time interval. We were using a 1s time interval which was on one side dense enough for the chosen data (see the ‘Data compilation’ section for more information about the used data sets) and on the other side small enough to be able to handle the output files in a sensible manner. Since the original data were not measured on the same time grid, linear interpolation was used within the time intervals to obtain the new positions for the time steps of the original data.
Main field modelling
with a similar expression for the \({h_{l}^{m}}\). The chosen time span is 1966–1971 using a knot spacing of 0.5 years.
For the inverse problem, we use a robust L _{1}norm measure. Model estimation methods using such an L _{1}norm measure of misfit have been found to perform well in geomagnetic field modelling applications (Lesur et al. 2008; Thomson and Lesur 2007; Walker and Jackson 2000). We implemented the L _{1}norm using an iteratively reweighted least squares (IRWLS) algorithm (Scales et al. 1998; Walker and Jackson 2000).
where d are the field observations, m are the model parameters and A is the forward functional matrix, W _{ k } is a weighting matrix derived from the misfit of each datum in the previous (kth) model iteration (Walker and Jackson 2000) and C _{ e } is the data covariance matrix containing information concerning estimated errors: we take it to be diagonal, consistent with the assumption of independence of errors. We do not apply any weighting to account for variations in data density over the sphere, meaning that we aim to fit each datum equally well.
Note that when W _{ k }=I, where I is the identity matrix, this scheme reduces to that for the conventional L _{2}norm measure of misfit. Iteration is required to find a solution because W _{ k } depends on m _{ k } and because we use scalar (intensity) data rather than vector data. Note that for the singleday results presented in the ‘Results and discussion’ section, the timeindependent version of (2) and (6) was used for the field modelling.
Orbit perturbation
where δ F is the residual F _{obs}(r)−F(r).
However, to obtain reasonable orbit changes with BERNESE (reducing the number of critical days producing errors), a threshold ρ is set for dr, so that the maximum radial perturbation per iteration is less than that threshold, d r≤ρ. For the singleday analysis, ρ was set to 300 m whereas for the combinedday approach, ρ was reduced to 150 m after the first three iterations.
Processing

Use the original data as the first input orbits for the BERNESE software;

The new orbits are used for the main field modelling;

For each data point, find the residual to that model, calculate the maximum gradient of B with respect to the radius (see Eq. 9), and perturb the altitude according to that;

These new points, which are now on inadmissible orbits, build the new input orbits for the next run with the BERNESE software;

That process is thought to be iterative to find a convergence between the introduced perturbation due the magnetic field gradient and the orbit changes in BERNESE. That means that the corrections induced due to the residuals from the magnetic field model are back corrected by BERNESE.
Note that the orbit perturbations are applied to both quiet and noisy data whereas for the field modelling, only quiet data was used. That is because all the data should be used for BERNESE since magnetic quiet data do not always cover long enough arcs as it can be seen, for example, in the red section of the arc in Fig. 4 for the case of August 8, 1968. Therefore, longer arcs are preferable due to the nonuniqueness of the new orbits created by BERNESE. In the following section, we present the results obtained from the single day as well as with combining all the data described in the ‘Data compilation’ section.
Results and discussion
Residual statistics of the revised satellite positions used in the singleday approach
N  min_{res} (nT)  max_{res} (nT)  μ _{res} (nT)  σ _{res} (nT)  

Original data  7911  −90.337  64.600  −0.773  15.050 
Starting data  7907  −85.292  63.558  −0.311  14.319 
1. Iteration  7919  −79.485  59.828  0.002  13.730 
2. Iteration  7926  −73.672  57.002  0.200  13.196 
3. iteration  7932  −67.552  52.559  0.370  12.616 
4. Iteration  7931  −63.245  49.945  0.473  12.240 
5. Iteration  7932  −58.522  46.838  0.604  11.821 
6. Iteration  7934  −52.706  48.095  0.746  11.493 
7. Iteration  7936  −47.413  48.373  0.826  11.239 
8. Iteration  7935  −42.317  49.143  0.925  11.040 
9. Iteration  7934  −38.428  49.794  1.058  10.926 
10. Iteration  7930  −38.127  49.749  1.119  10.851 
11. Iteration  7925  −37.869  49.991  1.222  10.858 
12. Iteration  7920  −37.882  50.546  1.254  10.939 
The reader may ask themselves whether one should worry about the socalled the Backus effect when making field models from purely intensity data. It is certainly the fact that the models derived are nonunique, but here our purposes are different; we seek to look at whether the data are fitted to an adequate level. The problem of nonuniqueness is not one that bears on this latter point.
Conclusions
From the analysis of the residuals in Table 3 and Fig. 5, it appears that we were able to improve the data quality for a single day applying our iterative approach. However, our iterative scheme (see the ‘Processing’ subsection) did not fully converge even though the residual values settled down after 10 iterations. Of course, it remains a question if the rms of the magnetic residuals is a good indicator for data quality or not—we think so. The lower rms value is however not the only indication for an improvement of the data. We believe that the comprehensive CM4 model (Sabaka et al. 2004) is able to represent the magnetic field for the time period of the POGO mission well. Therefore, an other possible proof for the data improvement is the fact that the number of observations with an residual less than 30 nT to the CM4 model could be increased. The results show that our iterative approach is feasible to tackle the problem of improving the POGO data quality by altering the satellite orbit.
In contrast to the singleday results, the results obtained from the combined days were somewhat disappointing. The residual values (see Fig. 6) for the modelling data sets POGO_mod could not really be reduced with the iteration process. It seems that the maximum residual was already saturated after two iterations even though we were also using a radial correction threshold ρ of ±300 m up to the third iteration. The rms of the residuals could be reduced but only by a very tiny amount compared to the singleday approach. A reason for the little improvement is certainly the fact that we were using many fewer iterations compared to the singleday case. The limited number of iterations was a direct consequence of the increasing numbers of days causing errors in the BERNESE step, which we were unfortunately not able to overcome. The rather weak results obtained using the combined days might also be a result of the field modelling process, namely the rather conservative error budget and the use of no temporal and spatial damping.
From the analysis of the introduced orbital changes (not reported here in the paper), one can see that the changes are in a consistent way over the whole day, especially for the radial component (the component we perturbed in our process). Therefore, we do not have to expect large orbit jumps at midnight. Further, the rather small improvements achieved in the data quality probably will not introduce a significant change in already existing magnetic field models obtained by using the original data.
Future authors who are interested in improving the position corresponding to the POGO magnetic field measurements may wish to use more sophisticated magnetic field models and include observatory data, which will allow to solve additionally for the remaining external field component. In general, our method for satellite orbit corrections presented here could in principle be used for further work.
Declarations
Acknowledgements
We would like to thank Markus Rothacher and Rolf Dach for the helpful discussions regarding the BERNESE GPS software. We thank Mike Purucker and Joe Cain for the assistance with POGOrelated questions. This work was partially supported by NERC grant O/S/2001/01227 to the Geospace consortium. We also thank Andrey Sheyko for performing some initial calculations. The paper benefitted from reviews by R. Holme and an anonymous reviewer.
Authors’ Affiliations
References
 Aster, R, Borchers B, Thurber C (2005) Parameter estimation and inverse problems. Elsevier Academic Press, Amsterdam.Google Scholar
 Bloxham, J, Jackson A (1992) Timedependent mapping of the magnetic field at the coremantle boundary. J Geophys Res 97: 19537–19563.View ArticleGoogle Scholar
 Cain, JC, Langel RA (1971) Geomagnetic survey by the polarorbiting geophysical observatories. In: Zmuda AJ (ed)World Magnetic Survey. IAGA Bull. 28, 65–75.. Int. Ass Geomagn. Aeron., Paris.Google Scholar
 Farthing, WH, Folz WC (1967) Rubidium vapor magnetometer for near earth orbiting spacecraft. Rev Sci Instrum 38: 1023–1030.View ArticleGoogle Scholar
 Gleghorn, GJ, Wiggins ET (1965) Design and development of the Orbiting Geophysical Observatory. Ann N Y Acad Sci 134: 205–233.View ArticleGoogle Scholar
 Gubbins, D (2004) Time series analysis and inverse theory for geophysicists. Cambridge, UK.View ArticleGoogle Scholar
 Jackson, A, Olsen N (2003) Possibilities of reanalysis of old satellite data In: 4th Oersted International Science Team Conference.. DMI, Copenhagen.Google Scholar
 Langel, R (1987) The main field, in geomagnetism, Vol. I (Jacobs JA, ed.). Academic Press, London.Google Scholar
 Langel, RA (1974) Nearearth magnetic disturbance in total field at high latitudes 1. Summary of data from OGO 2, 4, and 6. J Geophys Res 79: 2363–2371.View ArticleGoogle Scholar
 Lesur, V, Wardinski I, Rother M, Mandea M (2008) GRIMM: the GFZ reference internal magnetic model based on vector satellite and observatory data. Geophys J Int 173: 382–394.View ArticleGoogle Scholar
 Ludwig, GH (1963) The Orbiting Geophysical Observatories. Space Sci Rev 2: 175–218.View ArticleGoogle Scholar
 Luenberger, DG (1969) Optimization by vector space methods. John Wiley & Sons,Inc., New York.Google Scholar
 Olsen, N, The Scarf team (2013) The Swarm satellite constellation application and research facility (SCARF) and Swarm data products. Earth Planets Space 65: 1189–1200.View ArticleGoogle Scholar
 Olsen, N, Mandea M (2008) Rapidly changing flows in the earth’s core. Nat Geosci 1: 390–394.View ArticleGoogle Scholar
 Sabaka, T, Olsen N, Purucker M (2004) Extending comprehensive models of the Earth’s magnetic field with Ørsted and CHAMP data. Geophys J Int 159: 521–547.View ArticleGoogle Scholar
 Scales, J, Gersztenkorn A, Treital S (1998) Fast lp solution of large, sparse, linear systems: application to seismic travel time tomography. J Comp Phys 75. pages=314–333,Google Scholar
 Tarantola, A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, Society for Industrial and Applied Mathematics, Philadelphia.View ArticleGoogle Scholar
 Taylor, PT, Schanzle A, Jones T, Langel R, Kahn W (1981) Influence of gravity field uncertainties on the results from POGO and MAGSAT geomagnetic surveys. Geophys Res Lett 8: 1246–1248.View ArticleGoogle Scholar
 Thomson, A, Lesur V (2007) An improved geomagnetic data selection algorithm for global geomagnetic field modelling. Geophys J Int 169: 951–963.View ArticleGoogle Scholar
 Walker, M, Jackson A (2000) Robust modelling of the Earth’s magnetic field. Geophys J Int 143: 799–808.View ArticleGoogle Scholar
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