 FULL PAPER
 Open Access
Stochastic forecasting of the geomagnetic field from the COVOBS.x1 geomagnetic field model, and candidate models for IGRF12
 Nicolas Gillet^{1}Email author,
 Olivier Barrois^{1} and
 Christopher C Finlay^{2}
 Received: 15 December 2014
 Accepted: 4 April 2015
 Published: 14 May 2015
Abstract
We present the geomagnetic field model COVOBS.x1, covering 1840 to 2020, from which have been derived candidate models for the IGRF12. Towards the most recent epochs, it is primarily constrained by first differences of observatory annual means and measurements from the Oersted, Champ, and Swarm satellite missions. Stochastic information derived from the temporal spectra of geomagnetic series is used to construct the a priori model covariance matrix that complements the constraint brought by the data. This approach makes it possible the use of a posteriori model errors, for instance, to measure the ‘observations’ uncertainties in data assimilation schemes for the study of the outer core dynamics.
We also present and illustrate a stochastic algorithm designed to forecast the geomagnetic field. The radial field at the outer core surface is advected by core motions governed by an autoregressive process of order 1. This particular choice is motivated by the slope observed for the power spectral density of geomagnetic series. Accounting for timecorrelated model errors (subgrid processes associated with the unresolved magnetic field) is made possible thanks to the use of an augmented state ensemble Kalman filter algorithm. We show that the envelope of forecasts includes the observed secular variation of the geomagnetic field over 5year intervals, even in the case of rapid changes. In a purpose of testing hypotheses about the core dynamics, this prototype method could be implemented to build the ‘state zero’ of the ability to forecast the geomagnetic field, by measuring what can be predicted when no deterministic physics is incorporated into the dynamical model.
Keywords
 Geomagnetic field
 Secular variation
 Stochastic equations
 Ensemble Kalman filter
Background
The 12th generation of the IGRF has been computed from several candidates by different institutes, all submitted in September 2014 for evaluation by the IGRF working group (Thébault et al. 2015). We present here our proposed candidates for the main field (MF) DGRF model in 2010, IGRF in 2015, and the time average secular variation (SV) prediction over 2015 to 2020, together with their associated uncertainties. These are derived from the COVOBS.x1 field model and its posterior model error covariance matrix. This model, that covers the observatory era since 1840, is primarily constrained at recent epochs by first differences of observatory annual means (OAM) and by satellite observations, including data from the Swarm constellation (Olsen et al. 2013). It differs from regularized field models in the sense that the information contained into observations is complemented with a stochastic a priori information derived from temporal spectra of geomagnetic series, following Gillet et al. (2013). The a priori information integrated in the construction of COVOBStype models makes it possible to use the model errors, as estimated from the posterior covariance matrix, in data assimilation algorithms designed to reanalyze or forecast the outer core dynamics (Fournier et al. 2010). However, being derived over a long timespan, it cannot contain a detailed description of external magnetic signals (as in the comprehensive approach, see Sabaka et al. 2015). This leads to a compromise where external sources are handled in a simple way in comparison with models built exclusively from the most recent observations (e.g., Lesur et al. 2010; Olsen et al. 2014).
We also present a test SV prediction (time average over 2015 to 2020) based on a stochastic model of the flow at the coremantle boundary, using an augmented state ensemble Kalman filter (EnKF, see Evensen 2003). This prototype method is presented as a proof of concept study. It should be considered as an attempt to produce a ‘state zero’ of the ability to forecast the geomagnetic field, i.e., a measure of our ignorance when we have no deterministic knowledge about the core physics and only have access to statistical information. It may be used in the future for the purpose of validating geomagnetic data assimilation algorithms (Fournier et al. 2010). Possibilities for future improvements are listed in the ‘Conclusions’ section.
Methods
Derivation of the COVOBS.x1 field model
Data selection
Our data selection process follows closely that used to construct the original COVOBS field model. Full details can be found in Gillet et al. (2013) ; here, we only briefly describe the new or updated aspects of the datasets. Again, the most important data sources are satellite data, as measured by highprecision magnetometers on low Earthorbiting missions (in particular, CHAMP, Oersted, and Swarm) and observatory annual means.
Regarding satellite data, we used data from ESA’s Swarm satellite trio (FriisChristensen et al. 2006) and also reselected data from CHAMP, Oersted, and SACC, taking as a basis the dataset employed for a recent update of the CHAOS4 field model (CHAOS4plus_V4, Olsen et al. 2014). Most importantly for our IGRF12 candidate models, the Swarm data used were ESA Level 1B operational data from 26th November 2013 to 17th August 2014 (baseline 0302 until 4th July 2014, baseline 0301 thereafter). Only quiettime, nightside data were selected according to the CHAOS4 criteria, and vector data were used only up to 55° degrees quasidipole latitude; at high latitudes, scalar intensity data were used. Following the procedure previously introduced for the COVOBS model, the CHAOS4plus_V4 dataset was then subsampled onto a grid of 72 cells in longitude by 36 cells in cosine latitude, with each cell refilled each year using random sampling from the data falling within that cell during the year. This provided a geographical and temporal coverage sufficient for studying the main field up to degree 14 and its secular variation up to degree 8, as required for IGRF, while avoiding problems due to alongtrack correlated errors. Vector data were selected where possible (preferentially CHAMP data when two star cameras were available) or later from SwarmB (which is slightly further from the disturbing ionosphere than SwarmA or SwarmC); otherwise, scalar data was used. In particular, during the gap from the end of the CHAMP mission in September 2010 and the start of the Swarm mission in November 2013, only Oersted scalar data were available.
Errors statistics on F data for various satellites
Satellite  Period  No  σ  \(\boldsymbol {{\boldsymbol{\mathcal {M}}}^{*}}\)  μ ^{ ∗ }  μ  

Oersted  1999.2 to 2013.5  5580  5.5 to 8.5  0.95  0.03  5.58  0.10 
CHAMP  2000.6 to 2010.7  4489  4 to 10  0.95  −0.05  8.39  −0.55 
SacC  2001.1 to 2001.2  32  5.5 to 8.5  0.98  0.19  5.51  0.93 
Swarm  2013.9 to 2014.6  574  4 to 10  0.65  −0.08  5.39  −0.88 
Total  1999.2 to 2014.6  10675  0.93  −0.01  6.90  −0.23 
An update of the OAM dataset previously used to derive the COVOBS model was also carried out. This was based on the worldwide OAM database from the World Data Centre for Geomagnetism, BGS Edinburgh (pers. comm. S. Macmillan, September 2014). It involved 13 new observatories compared to the previous dataset used by Gillet et al. (2013) and comprises annual means up to 2013.5 where available. First differences of OAM thus provide constraints on the secular variation up until 2013.0 that are not sensitive to the crustal field. Uncertainty estimates were recalculated for each updated observatory time series using a generalized cross validation approach as in Bloxham and Jackson (1992), Gillet et al. (2013), and Jackson et al. (2000). No corrections for external field variations, beyond the averaging inherent in the annual means, were applied.
Parameterization and prior information for the COVOBS.x1 model
We refer to Gillet et al. (2013) for the details concerning the construction of the COVOBS field model. Here, we recall only the main points. We use a spherical harmonic expansion of the internal field up to degree N=14. A single coefficient, the axial dipole in geomagnetic dipole coordinates, is used for the spherical harmonic expansion of the external field (with a 20nT background value). All coefficients are projected onto order 4 cubic Bsplines, with knots every 2 years, spanning the period 1838 to 2022.
τ is the time lag between two epochs, \(E(\dots)\) means the statistical expectation. For each coefficient, the a priori distribution is centered on zero. We consider that coefficients of different orders or degrees are a priori independent (\({h_{n}^{m}}\) and \({g_{n}^{m}}\) are also independent). For the sake of simplicity, the \({\mathcal {C}}_{n}^{m}\) do not vary with the spherical harmonic order m. The main field variances σ _{ g }(n)^{2} are estimated from the variances of satellite MF model coefficients in 2005, averaged over all orders. Given Equation (1), the relevant timescales are by definition \(\tau _{c}(n)=\sqrt {3}\sigma _{g}(n) / \sigma _{\dot {g}}(n)\), with \(\sigma _{\dot {g}}(n)\) the secular variation variances, estimated from the variances of satellite SV model coefficients in 2005, averaged over all orders (see Gillet et al. 2013).
with ε a white noise process^{a}. This description is chosen since it is consistent with the slope close to −4 found for the power spectrum density (PSD) of observatory series at periods from 5 to 70 years (De Santis et al. 2003). The continuation of such a slope for the PSD of the internal field at periods shorter than a few years is unsure given the domination of the external signal towards high frequencies (see Figure two in Ou et al. 2015), but there is no need either for a steeper slope given the current estimate of the mantle conductivity (Velímský and Finlay 2011).
Since D,I, and F data are involved, model estimation is a nonlinear optimization problem. We solve it with a Newton algorithm, using a L2 measure of the data misfit together with a rejection criterion. We first build, starting from COVOBS, an intermediate model rejecting data for which the residual magnitude is above 10 σ _{ e } (σ _{ e } is the a priori data error). Next, from this model we obtain within six iterations (typically a few iterations are enough for convergence) the COVOBS.x1 field model together with its a posteriori model errors’ covariance matrix using a 3σ _{ e } rejection criterion.
Stochastic forecast of the geomagnetic field
We present below how we derive our test SV model for IGRF12. It results from the integration of a stochastic core flow model into an augmented state ensemble Kalman filter (Evensen 2003).
Stochastic core flow modeling
For the sake of simplicity, a single value of τ _{ u }=100 years is assumed for all the flow coefficients (see Gillet et al. 2015).
From (8), we construct the covariance matrix C _{ z } for z. The linear set of constraints (6) is transformed into matrix form, L z=0 (if z satisfies the constraints, then it is also the case for u). We generate the random vectors z by multiplying the Choleski decomposition of \(\mu {\sf L}^{T}{\sf L}+{\sf C}_{\sf z}\) (with μ a very large scalar, we use μ=10^{9}) to a random unit vector normally distributed. Then, flow models u derived from z satisfy the constraints (6) and present the variances of Equation (7).
Augmented state ensemble Kalman filter
with a white noise process. This defines the correlation function for , \(\rho _{\mathcal {E}}(\tau)=\exp \left (\tau /\tau _{\mathcal {E}}\right)\). This choice, with \(\tau _{\mathcal {E}}=10\) years, builds on the investigations of Gillet et al. (2015) who constructed flow models from COVOBS over 1940 to 2010 (see their Figure one). Variances for the SV model error coefficients, which depend only on the degree n, are also estimated from their results.
We use an EulerMaruyama scheme (e.g., Kloeden and Platen 1992) to timestep the stochastic equations. We consider here as observations snapshots of the MF and SV coefficients of the COVOBS.x1 field model, obtained from the continuous parameterization (through Bsplines) of the field model. Observations fall on a subset of the discrete timesteps involved when integrating the forward problem, so there is no need for interpolating in time the stochastic model.
with \(\left <\textbf {x}^{f}\right >\) the ensemble average of the models. Given the small ensemble size compared to the model dimension, crosscovariances in P ^{ f } are ignored to ensure a positivedefinite matrix (one may consider reduced rank filters or localization techniques to address this issue, while conserving the information carried by offdiagonal elements). All quantities are represented in the spherical harmonic domain.

we first perform an analysis of \(\textbf {x}_{1}=\overline {B}_{r}\) from K noisy realizations of the COVOBS.x1 field model, which involves \(\textbf {y}^{o}_{1}=\overline {B}_{r}^{o}\), R=R _{ MF }, and \({\sf H}_{1}(\textbf {x})=\overline {B}_{r}\) (i.e., ∇H _{1}=I _{ P } the identity matrix of size P=N(N+2));

next, we compute an analysis of \(\textbf {x}_{2}=\left [\textbf {u},\mathcal {E}\right ]\), with \(\textbf {y}^{o}_{2}=\partial _{t}\overline {B}_{r}^{o}\), R=R _{ SV } and:$$\begin{array}{@{}rcl@{}} \nabla{\sf H}_{2} = \left[ \begin{array}{cc} {\sf A}(\overline{B}_{r}^{a}) & {\sf 0} \\ {\sf 0} & {\sf I}_{P} \end{array} \right]\,, \end{array} $$
where \({\sf A}(\overline {B}_{r}^{a})\textbf {u}=\overline {\nabla _{H}\cdot \left (\textbf {u}\overline {B}_{r}^{a}\right)}\).
We perform an analysis of the observations every 5 years, starting from 1985 (except the last analysis, performed in 2014.5 instead of 2015). Several (at least two) analysis steps are required in order to eliminate the initial condition memory on the covariances that enter the EnKF.
Results and discussion
The COVOBS.x1 field model
Statistics on the COVOBS.x1 prediction errors
Errors statistics on vector data for various satellites
Satellite  No  \(\boldsymbol {{\boldsymbol{\mathcal {M}}}^{*}}\)  μ ^{ ∗ }  μ  

B _{ b }  B _{ ⊥ }  B _{ 3 }  B _{ b }  B _{ ⊥ }  B _{ 3 }  B _{ b }  B _{ ⊥ }  B _{ 3 }  B _{ b }  B _{ ⊥ }  B _{ 3 }  
Oersted  3971  1.01  1.12  0.86  0.02  −0.08  0.01  5.54  8.38  4.92  0.11  −0.61  0.08 
CHAMP  18507  1.13  0.94  1.24  0.00  0.05  −0.04  4.51  4.07  5.43  0.00  0.21  −0.18 
Swarm  3935  1.04  0.88  1.19  −0.01  0.01  −0.06  4.16  3.82  5.15  −0.03  0.05  −0.26 
Total  26413  1.10  0.96  1.18  0.00  0.02  −0.04  4.63  4.94  5.31  0.01  0.07  −0.15 
The dependence of the unweighted residuals on the geomagnetic latitude is shown in Figure 2 for various satellites. As expected, larger prediction errors on F data are observed above 60° latitude. The envelope of the residuals shows a minimum around 35° latitude and a local maximum at the equator for both F and B _{ b } data for all three satellites. This indicates some unmodelled contributions from the ring current (see the discussion of Figure two in Olsen 2002). It may at least partly be due to the inability of our model to fit highfrequency external sources due to the choice of a 2year knot spacing also for the external field. As in Olsen (2002), B _{ p } Oersted data and B _{3} Oersted and CHAMP data are associated with lower residuals towards the geomagnetic equator. This trend is less obvious with Swarm data, which present on average lower residuals for both scalar and vector data.
Description of COVOBS.x1 and its uncertainties
We obtain relatively larger error bars at higher degrees. In comparison with the period prior to 1999, where the SV is mainly constrained by OAM, we find smaller uncertainties during the satellite era. These increase after 2014.6, when the control imposed by a priori covariances gradually takes over from that of the observations. The ensemble average SV prediction after 2014.6 becomes almost flat. This behavior is in agreement with our assumed prior information. Indeed, the correlation function in Equation (1) is that of an autoregressive process of order 2 for the MF, i.e., approximately a random walk for the SV on short periods, characterized by an envelope of possible solutions \(\propto \sqrt {t}\).
Results of the MF and SV stochastic forecast
For most coefficients, observations are accounted for by one standard deviation σ ^{ f } estimated from our ensemble of forecasts. This indicates that the statistics of our stochastic model are fairly realistic, although this diagnostic is imperfect: COVOBS.x1 coefficients at two different epochs are not independent. Thus, the forecast at a given epoch is not independent from the observation at this epoch. To obtain an independent diagnostic, forecasts produced from COVOBStype field models constructed from datasets without data from the forecast era would be required. For some coefficients, there exists a systematic bias (see, for instance, the positive trend taken by the average \(\dot {h}_{2}^{1}\) forecast) : even though the observed series usually lies within the ensemble of forecast realizations, it sometimes lies in an area of lower probability, as measured by the pdf of the forecast. Initial investigations of this problem suggest it is not related to our choice of topological constraint (6) but probably to our use of stochastic models for u and that are centered around zero (we use no background model).
We also make a comparison of the EnKF forecast with the IGRF11 predictions for 2010 to 2015 (Finlay et al. 2010). We find that the IGRF11 SV predictions are accounted for by one standard deviation σ ^{ f } estimated from our ensemble of forecasts. The results presented in this section should be considered as a proof of concept study; possible future applications or modifications are presented in the ‘Conclusions’ section.
Our IGRF12 candidate models
Additionally, we presented a test IGRF SV candidate as the ensemble average of SV forecast resulting from the stochastic EnKF, averaged in time over 2015 to 2020 (its associated error bars are estimated from the dispersion relative to the ensemble average within the ensemble of SV forecast realizations).
Conclusions
Beyond the ‘state zero’ forecast of the geomagnetic field
Forecasting the geomagnetic field requires, upon accurate observations at or above the Earth’s surface, understanding the physics occurring within the Earth’s outer core over millennial to interannual timescales. The knowledge of the field behavior on long timescales can be inferred from field models derived from archeomagnetic and lake sediment data bases (e.g., Korte et al. 2011; Licht et al. 2013; Nilsson et al. 2014; PavónCarrasco et al. 2014). Since the turnover time is of the order of a few hundreds of years, the reanalysis of such models may help define a background flow above which a flow perturbation could be described (equivalent of a climatic mean in oceanic studies). This may reduce systematic biases observed in the SV predictions. One may think, for instance, of the coupledEarth dynamo scenario by Aubert et al. (2013), where a slowly evolving westward drift naturally arises from the coupling between the inner core and the mantle.
In this study, the evolution of the geomagnetic field is constrained by statistical properties inferred from the temporal PSD recovered for magnetic series. There exists dynamical interpretations associated with the observed slopes of the spectra (Tanriverdi and Tilgner 2011) (see also Buffett and Matsui 2015; Buffett et al. 2014; Olson et al. 2012, for studies focused on the analysis of the dipole moment in geodynamo simulations). Nevertheless, the method derived in the ‘Stochastic forecast of the geomagnetic field’ section does not contain any deterministic equation for advecting the magnetic field and the flow inside the outer core. As a consequence, it provides an envelope of possible states, and its average prediction can only be marginally better than those obtained from a stationary flow hypothesis (Beggan and Whaler 2009; Waddington et al. 1995). To go further, still accounting for unresolved processes, one may introduce random forcings such as those of Equations (4) and (10) into prognostic models of the core dynamics used for the reanalysis of the core state – for instance, into geodynamo (Fournier et al. 2011) or quasigeostrophic (Canet et al. 2009) equations. With this in mind, the development of smoother algorithms may also be envisioned (see Evensen and Van Leeuwen 2000).
The model presented in this study aims to produce, as far as possible, an unbiased estimate of the core state, considering SV model error covariances via a stochastic equation. To go further, we could reduce the importance of SV model errors by coestimating the unresolved field at degrees n>14 (and its associated uncertainties) together with the field at large lengthscales, following Aubert (2014) who performed such a coestimation using the crosscovariances between Gauss coefficients obtained from the forward integration of a geodynamo simulation. Implementing more sophisticated stochastic models for can also be envisioned, given that the Laplace correlation function associated with the AR1 Equation (10) is only an approximation of that found by Gillet et al. (2015). Finally, we used here the compressible QG constraint on core surface flows. Any possible topological constraint may be considered, and the algorithm presented above could be used to test the ability of each hypothesis to predict the observed SV within reanalysis studies. With the prototype algorithm illustrated above, one could, for instance, explore the hypothesis of a stratified layer at the top of the outer core, through its implications on the structure in space and time of core surface flow (Buffett 2014).
Endnotes
^{a} Note that Equation (2) differs from Equation (twelve) in Gillet et al. (2013). Both define processes with the same autocovariance function (1), but their Equation (twelve), contrary to our Equation (2), corresponds to a nonstationary process. This does not affects their results that have been computed from the covariance function and not the stochastic equation.
^{b} By replacing in Equation (4) the scalar quantity 1/τ _{ u } by a matrix homogeneous to a frequency, one could derive more complex stochastic models still compatible with the −4 slope for the PSD of observatory series.
Declarations
Acknowledgements
We thank the national institutes that support ground magnetic observatories and INTERMAGNET for promoting high standards of practice and making the data available. The support of the CHAMP mission by the German Aerospace Center (DLR) and the Federal Ministry of Education and Research is gratefully acknowledged. The Oersted Project was made possible by extensive support from the Danish Government, NASA, ESA, CNES, DARA, and the Thomas B. Thriges Foundation. We thank T. Sabaka and A. De Santis for their review of our manuscript. This work was supported in part by the French Centre National d’Etudes Spatiales (CNES) for the preparation of the Swarm mission of ESA. This work has been partially supported by the French ‘Agence Nationale de la Recherche’ under the grant ANR11BS56011. We thank the International Space Science Institute for providing support for the workshops of international team no. 241.
Authors’ Affiliations
References
 Aubert, J (2014) Earth’s core internal dynamics 1840–2010 imaged by inverse geodynamo modelling. Geophys J Int 197(3): 1321–1334.View ArticleGoogle Scholar
 Aubert, J, Finlay CC, Fournier A (2013) Bottomup control of geomagnetic secular variation by the Earth’s inner core. Nature 502: 219–223.View ArticleGoogle Scholar
 Baerenzung, J, Holschneider M, Lesur V (2014) Bayesian inversion for the filtered flow at the Earth’s core mantle boundary. J Geophys Res Solid Earth 119: 2695–2720. DOI 10.1002/2013JB010358.View ArticleGoogle Scholar
 Beggan, C, Whaler K (2009) Forecasting change of the magnetic field using core surface flows and ensemble Kalman filtering. Geophys Res Lett36: L18303. DOI 10.1029/2009GL039927.View ArticleGoogle Scholar
 Bloxham, J, Jackson A (1992) Timedependent mapping of the magnetic field at the coremantle boundary. J Geophys Res 97: 19,537–19,563.View ArticleGoogle Scholar
 Buffett, B (2014) Geomagnetic fluctuations reveal stable stratification at the top of the Earth’s core. Nature 507: 484–487.View ArticleGoogle Scholar
 Buffett, B, Matsui H (2015) A power spectrum for the geomagnetic dipole moment. Earth Planetary Sci Lett 411: 20–26.View ArticleGoogle Scholar
 Buffett, BA, King EM, Matsui H (2014) A physical interpretation of stochastic models for fluctuations in the earth’s dipole field. Geophys J Int 198(1): 597–608.View ArticleGoogle Scholar
 Canet, E, Fournier A, Jault D (2009) Forward and adjoint quasigeostrophic models of the geomagnetic secular variation. J Geophys Res 114: B11,101. doi:10.1029/2008JB006,189.View ArticleGoogle Scholar
 De Santis, A, Barraclough D, Tozzi R (2003) Spatial and temporal spectra of the geomagnetic field and their scaling properties. Phys Earth Planet Int 135(2): 125–134.View ArticleGoogle Scholar
 Evensen, G (2003) The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn 53(4): 343–367.View ArticleGoogle Scholar
 Evensen, G, Van Leeuwen PJ (2000) An ensemble Kalman smoother for nonlinear dynamics. Mon Weather Rev 128(6): 1852–1867.View ArticleGoogle Scholar
 Eymin, C, Hulot G (2005) On core surface flows inferred from satellite magnetic data. Phys Earth Planet Inter 152: 200–220.View ArticleGoogle Scholar
 Finlay, CC, Maus S, Beggan CD, Bondar TN, Chambodut A, Chernova TA, Chulliat A, Golovkov VP, Hamilton B, Hamoudi M, Holme R, Hulot G, Kuang W, Langlais B, Lesur V, Lowes FJ, Lühr H, Macmillan S, Mandea M, McLean S, Manoj C, Menvielle M, Michaelis I, Olsen N, Rauberg J, Rother M, Sabaka TJ, Tangborn A, TøffnerClausen L, Thébault E, et al. (2010) International geomagnetic reference field: the eleventh generation. Geophys J Int 183(3): 1216–1230.View ArticleGoogle Scholar
 Finlay, CC, Jackson A, Gillet N, Olsen N (2012) Core surface magnetic field evolution 2000–2010. Geophys J Int 189: 761–781. DOI 10.1111/j.1365246X.2012.05395.x.View ArticleGoogle Scholar
 Finlay, CC, Olsen N, TøffnerClausen L (2015) DTU candidate field models for IGRF12 and the CHAOS5 geomagnetic field model. Earth, Planets and Space, in press.Google Scholar
 Fournier, A, Hulot G, Jault D, Kuang W, Tangborn A, Gillet N, Canet E, Aubert J, Lhuillier F (2010) An introduction to data assimilation and predictability in geomagnetism. Space Sci Rev 155: 247–291. http://dx.doi.org/10.1007/s1121401096694.View ArticleGoogle Scholar
 Fournier, A, Aubert J, Thébault E (2011) Inference on core surface flow from observations and 3D dynamo modelling. Geophys J Int 186: 118–136.View ArticleGoogle Scholar
 FriisChristensen, E, Lühr H, Hulot G (2006) Swarm a constellation to study the Earth’s magnetic field. Earth Planet Space 58: 351–358.View ArticleGoogle Scholar
 Gillet, N, Jault D, Finlay CC, Olsen N (2013) Stochastic modeling of the earth’s magnetic field: inversion for covariances over the observatory era. Geochem Geophys Geosyst 14(4): 766–786.View ArticleGoogle Scholar
 Gillet, N, Jault D, Finlay CC (2015) Planetary gyre, timedependent eddies, torsional waves and equatorial jets at the Earth’s core surface. J. Geophys Res, in press.Google Scholar
 Holme R (2007) Large scale flow in the core. In: Olson P Schubert G (eds)Treatise in geophysics, core dynamics, 107–129.. Elsevier, chap 4.Google Scholar
 Holme, R, Bloxham J (1996) The treatment of attitude errors in satellite geomagnetic data. Phys Earth Planet Inter 98: 221–223.View ArticleGoogle Scholar
 Jackson, A, Jonkers ART, Walker MR (2000) Four centuries of geomagnetic secular variation from historical records. Phil Trans R Soc Lond A 358: 957–990.View ArticleGoogle Scholar
 Kloeden, PE, Platen E (1992) Numerical solution of stochastic differential equations, Vol. 23. SpringerVerlag, Berlin Heidelberg.View ArticleGoogle Scholar
 Korte, M, Constable C, Donadini F, Holme R (2011) Reconstructing the holocene geomagnetic field. Earth Planet Sci Lett 312: 497–505.View ArticleGoogle Scholar
 Le Mouël, JL, Gire C, Madden T (1985) Motions of the core surface in the geostrophic approximation. Phys Earth Planet Inter 39: 270–287.View ArticleGoogle Scholar
 Lesur, V, Wardinski I, Hamoudi M, Rother M (2010) The second generation of the GFZ reference internal magnetic model: GRIMM2. Earth Planets Space 62: 765–773.View ArticleGoogle Scholar
 Licht, A, Hulot G, Gallet Y, Thébault E (2013) Ensembles of low degree archeomagnetic field models for the past three millennia. Phys Earth Planetary Inter 224: 38–67.View ArticleGoogle Scholar
 Nilsson, A, Holme R, Korte M, Suttie N, Hill M (2014) Reconstructing holocene geomagnetic field variation: new methods, models and implications. Geophys J Int 198(1): 229–248.View ArticleGoogle Scholar
 Olsen, N (2002) A model of the geomagnetic field and its secular variation for epoch 2000 estimated from ørsted data. Geophys J Int 149(2): 454–462.View ArticleGoogle Scholar
 Olsen, N, FriisChristensen E, Floberghagen R, Alken P, Beggan CD, Chulliat A, Doornbos E, da Encarnaçao JT, Hamilton B, Hulot G, et al (2013) The swarm satellite constellation application and research facility (scarf) and Swarm data products. Earth Planets Space 65(11): 1189–1200.View ArticleGoogle Scholar
 Olsen, N, Luhr H, Finlay CC, Sabaka TJ, Michaelis I, Rauberg J, TøffnerClausen L (2014) The CHAOS4 geomagnetic field model. Geophys J Int 197: 815–827.View ArticleGoogle Scholar
 Olson, PL, Christensen UR, Driscoll PE (2012) From superchrons to secular variation: a broadband dynamo frequency spectrum for the geomagnetic dipole. Earth Planet Sci Lett319–320: 75–82. http://www.sciencedirect.com/science/article/pii/S0012821X11007229.View ArticleGoogle Scholar
 Ou, J, Gillet N, Du A (2015) Geomagnetic observatory monthly series, 1930 to 2010: empirical analysis and unmodeled signal estimation. Earth Planets Space 67(1): 13. DOI 10.1186/s406230140173z.View ArticleGoogle Scholar
 Pais, MA, Jault D (2008) Quasigeostrophic flows responsible for the secular variation of the Earth’s magnetic field. Geophys J Int 173: 421–443.View ArticleGoogle Scholar
 Papoulis, A, Pillai SU (2002) Probability, random variable and stochastic process. 4th ed., McGrawHill, New York.Google Scholar
 PavónCarrasco, FJ, Osete ML, Torta JM, De Santis A (2014) A geomagnetic field model for the holocene based on archaeomagnetic and lava flow data. Earth Planetary Sci Lett 388: 98–109.View ArticleGoogle Scholar
 Sabaka, TJ, Olsen N, Tyler RH, Kuvshinov A (2015) Cm5, a preswarm comprehensive geomagnetic field model derived from over 12 yr of champ, ørsted, sacc and observatory data. Geophys J Int 200(3): 1596–1626.View ArticleGoogle Scholar
 Tanriverdi, V, Tilgner A (2011) Global fluctuations in magnetohydrodynamic dynamos. New J Phys 13(3): 033,019.View ArticleGoogle Scholar
 Thébault, E, Finlay CC, Beggan C, Alken P, Aubert J, Barrois O, Bertrand F, Bondar T, Boness A, Brocco L, Canet E, Chambodut A, Chulliat A, Coïsson P, Civet F, Du A, Fournier A, Fratter I, Gillet N, Hamilton B, Hamoudi M, Hulot G, Jager J, Korte M, Kuang W, Lalanne X, Langlais B, Léger JM, Lesur V, Lowes FJ, et al. (2015) International geomagnetic reference field: the twelfth generation. Earth, Planets and Space, accepted.Google Scholar
 Velímský, J, Finlay CC, Effect of a metallic core on transient geomagnetic induction (2011). Geochem Geophys Geosyst 12(5): Q05,011. DOI 10.1029/2011GC003557.View ArticleGoogle Scholar
 Waddington, R, Gubbins D, Barber N (1995) Geomagnetic field analysis –V. Determining steady core surface flows directly from geomagnetic observations. Geophys J Int 122: 326–350.View ArticleGoogle Scholar
 Yaglom, AM (1962) An introduction to the theory of stationary random functions, PrenticeHall, Englewood Cliffs NJ.Google Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.