 Article
 Open Access
Prediction of geomagnetic field with data assimilation: a candidate secular variation model for IGRF11
 Weijia Kuang^{1}Email author,
 Zigang Wei^{2},
 Richard Holme^{3} and
 Andrew Tangborn^{2}
https://doi.org/10.5047/eps.2010.07.008
© 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. 2010
 Received: 24 February 2010
 Accepted: 6 July 2010
 Published: 31 December 2010
Abstract
Data assimilation has been used in meteorology and oceanography to combine dynamical models and observations to predict changes in state variables. Along similar lines of development, we have created a geomagnetic data assimilation system, MoSSTDAS, which includes a numerical geodynamo model, a suite of geomagnetic and paleomagnetic field models dating back to 5000 BCE, and a data assimilation component using a sequential assimilation algorithm. To reduce systematic errors arising from the geodynamo model, a predictioncorrection iterative algorithm is applied for more accurate forecasts. This system and the new algorithm are tested with 7year geomagnetic forecasts. The results are compared independently with CHAOS and IGRF field models, and they agree very well. Utilizing the geomagnetic field models up to 2009, we provide our prediction of 5year mean secular variation (SV) for the period 2010–2015 up to degree L = 8. Our prediction is submitted to IGRF11 as a candidate SV model.
Key words
 Geodynamo
 geomagnetism
 data assimilation
 secular variation
 IGRF
1. Introduction
There is a long history of geomagnetic measurements on and near the Earth’s surface. Accurate measurements of the geomagnetic field for navigational use have been made for many hundreds of years with reliable records dating back the late 16^{th} century (e.g. Jackson et al., 2000). Measurements at dedicated ground observatories began to be recorded in 1832. Since the early 1960s, satellite measurements have enabled the Earth’s magnetic environment to be monitored from space.
Of the nearEarth measured geomagnetic signals, 97% (in terms of the magnetic energy) are of the internal origin (e.g. Langel and Hinze, 1998). This part of the field, called the core field, is generated and maintained by the geodynamo in the Earth’s fluid outer core. In particular, slow time variations of the core field (the secular variation) are the manifestations of the dynamical processes in the outer core. The remainder arises from various sources, including crustal and oceanic inductions, and electromagnetic processes in the ionosphere and magnetosphere.
These geomagnetic measurements have been used by many research groups around the world to construct global field maps (geomagnetic field modeling). A recent parallel effort is to utilize indirect magnetic measurements, e.g. paleomagnetic and archeomagnetic data for global field modeling. For more details on the global field models, we refer the reader to, e.g. Barraclough (1974), Jackson et al. (2000), Jonkers et al. (2003), Sabaka et al. (2004), Korte and Constable (2005), Korte et al. (2009), Olsen et al. (2009).
The global geomagnetic field can be modeled mathematically by a spherical harmonic expansion. The expansion coefficients, called the Gauss coefficients, can be determined using a leastsquare inversion from surface and nearsurface geomagnetic measurements. The core field can be described by the coefficients up to degree L_{obs} ≤ 13 (Langel and Estes, 1982; Cain et al., 1989). At higher degrees, it is masked by the crustal magnetic field. Recent studies suggest that it may be possible using current and future satellite measurements (e.g. SWARM mission), to deduce the core field secular variation beyond degree L_{obs} = 13 (Sabaka and Olsen, 2006; Olsen et al., 2009).
The geodynamo theory can be traced back to Larmor’s (1919) seminal work on the origin of solar magnetism, and is now widely accepted to generate and maintain the core field that varies spatially and temporally. Because of the extremely complex dynamics of rapidly rotating magnetohydrodynamic fluids, analytical dynamo solutions are nearly impossible (except perhaps for unrealistic simple systems). Since the first successful numerical simulation by Glatzmaier and Roberts (1995), we have made substantial progress in modeling the geodynamo. In particular, magnetic fields obtained from various numerical dynamo models with various boundary conditions have displayed properties qualitatively similar to those from surface geomagnetic observations (e.g. Kuang and Bloxham, 1997; Gubbins et al., 2007; Aubert et al., 2008; Sakuraba and Roberts, 2009). For a (not too) recent review on geomagnetism and the geodynamo, we refer the reader to Kono and Roberts (2002).
But these models cannot be directly used to predict geomagnetic secular variations. The numerical magnetic fields have substantial differences with the observations, in particular the nondipolar components (Kuang et al., 2008, 2009). These differences are due to many factors, including approximations and assumptions used in the models, and numerical parameter mismatches. For example. the numerical Ekman number E (measuring the fluid viscous effect) is many orders of magnitude larger than that appropriate for the Earth’s outer core. This vast parameter gap will be narrowed in time, but unlikely to disappear completely in the next 50 years (estimated based on the past development): over the 12year period from 1997 to 2009, numerical E is reduced approximately by 1.5 orders of magnitude, from 2 × 10^{−5} to 5 × 10^{−7} (e.g. Kuang and Bloxham, 1997; Sakuraba and Roberts, 2009). At the current hardware upgrade rate (doubling computing power every 1.5 years), we would need more than 50 years to reach the Earthlike value E≈5 × 10^{−14}.
Even if the numerical parameter values were appropriate, and the approximations and assumptions were validated, pure numerical simulation will still produce magnetic fields different from those observed, simply because convection in the Earth’s core is very turbulent, and thus the time variation of the core state is very chaotic. This implies that small initial differences in numerical solutions will grow rapidly in time. This situation is reminiscent of modeling atmospheric and oceanic circulations.
However, as we have learned from atmospheric and oceanic studies, assimilation of data and dynamical models can bring numerical solutions closer to the real geophysical state. Results of observing system simulation experiments (OSSEs) on simple magnetohydrodynamic system (Sun et al., 2007) and on full dynamo models (Liu et al., 2007) have shown that by assimilating magnetic data into the model output, the predicted solution can be drawn closer to the true state in a dynamically consistent manner. Using the geomagnetic field model output and the geomagnetic data assimilation system MoSSTDAS, Kuang et al. (2008, 2009) also found that the predicted field is similar to the observations. Similar results are obtained with assimilation based on simpler systems with, e.g., a twodimensional, quasigeostrophic flow (Canet et al., 2009), or a steady flow (Beggan and Whaler, 2009).
Among others, these results suggest two significant geophysical applications: using observations to constrain numerical geodynamo models, and using the improved model output to predict geomagnetic secular variation (SV).
This paper focuses on the second application, SV prediction, with an explicit goal to provide an SV candidate model to the IGRF system. This application demands not only a theoretical understanding of the numerical dynamo model and geomagnetic data, but also practical techniques to improve the prediction accuracy. Both require proper assessment of model and observational errors.
In our past work, we have assumed that errors from observations (actually the errors from geomagnetic field models) are negligible compared to dynamo model errors (Kuang et al., 2008, 2009). This is reasonable for assessing the impact of observations on dynamo model solutions, but is expected to be insufficient for providing SV forecast to IGRF.
For this purpose, we introduce in our geomagnetic data assimilation system, MoSSTDAS, a “bootstrap” type of predictioncorrection analysis algorithm to reduce forecast errors, and we also use time series of geomagnetic field models to estimate observational errors. This approach is benchmarked with independent field models. And it is the basis for our SV candidate model to IGRF for the period from 2010–2015.
This paper is organized as follows: mathematical formulation of our geomagnetic data assimilation system is summarized in Section 2. Testing of the forecast with observations is described in Section 3. The details of the SV forecast for the period 2010–2015 are provided in Section 4. Discussion is given in Section 5.
2. Geomagnetic Data Assimilation System
In numerical dynamo simulation, the truncation orders L_{ m } ≫ L_{obs}. It should be pointed out that the spectral coefficients in (7) are for the toroidal magnetic field, and are therefore not observable at the surface. In addition to these, the velocity field and the density are also among the unobservable state variables.
In the dynamo model, the velocity field v is described similarly as (6) and (7), and the density anomaly Δρ is also approximated with a truncated spherical harmonic expansion. All spherical harmonic coefficients are defined at discrete radial grid points (e.g. Kuang and Bloxham, 1999). Therefore, the dynamo state, i.e. the numerical model output, at any time can be described by an array x of all spectral coefficients on all radial grid points.
3. PredictionCorrection Iterative Algorithm
The model output x^{ f } (t) at any time t in an analysis period t_{ a } < t < t_{a+}1 is the forecast based on the observations made up to time t_{ a } . The difference between the forecast x^{ f } and the state x from a pure dynamo simulation (4) describes the correction due to the assimilation of the observations on the dynamo model solutions.
Could this error be reduced for improved predictions? To find an answer, we need to understand first the properties of the errors.
The error ε will be reduced if there are sufficient data to bring the dynamo model solutions closer to the true state in the outer core. But it can not vanish completely since the model is not perfect. However, for the purpose of forecasting, it could be reduced via a “bootstrap” approach, provided that the time scales of the error growth is different from the forecast length.
4. SV Prediction: Experiment and Assessment
To test the applicability of the predictioncorrection approach (19) and (20), we use the following procedure: first, prior to the application of (19), an assimilation run is made with the longest available observational record, i.e. the Gauss coefficients from paleomagnetic and archeomagnetic data (Korte and Constable, 2005), historical magnetic data (Jackson et al., 2000), ground observatory and satellite data (Sabaka et al., 2004). Combined, they provide up to 7000 years of observations.
In our experiments, the maximum degree L_{obs} of the Gauss coefficients used in the analysis (9) changes from L_{obs} = 6 before 1620 (the paleomagnetic field model output), and L_{obs} = 8 from 1620 to the present day (historical, observatory and satellite records). Thus we have used the maximum number of observations in order to achieve our best estimates of the core state.
Next, we carry out the predictioncorrection approach (19) and (20) for the past decade, because continuous satellite measurement during this period provides by far the best global description of the core field, in particular its SV. In addition, two sets of field models are used for the validation analysis: one is CM4 (Sabaka et al., 2004) used to produce sequences of assimilation results; and the other is CHAOS2s model (Olsen et al., 2009) used for comparing the forecast with observations.
The CM4 is a field model of the quiettime, nearEarth magnetic field from 1960 to 2002.5. It is derived with a comprehensive approach, using POGO, Magsat, ørsted and CHAMP satellite data. In this approach, all magnetic sources are coestimated, resulting in optimal separation of the fields (Sabaka and Olsen, 2006). CM4 uses the minimum mean square estimator with the assumptions that the estimator is a linear function of the data, and that the errors are isotropically and independently distributed. This approach is the socalled best, linear, unbiased estimator (BLUE) (Sorenson, 1980). In this model, the core and lithospheric fields are described by a spherical harmonic expansion up to degree and order 65. The SV part is represented by cubic Bsplines (with a 2.5year knot spacing) through degree and order 13.
The CHAOS2s model is a stateoftheart model of the geomagnetic field for the last decade, constructed with data from the three satellite missions CHAMP, ørsted and SACC, with the internal field secular variation parameterized with order 5 Bsplines, lithospheric field to degree 60 and a model of the largescale magnetospheric field (plus Earthinduced counterpart) allowing for components modeled both in a fixedEarth and a fixedSun reference frame.
In our assimilation, the Gauss coefficients from the both field models are treated as indistinguishable from the “true field” at any particular epoch away from the edges of the models (i.e. the errors in the field models are negligibly small), consistent with the assumption made in the analysis (9) that the errors from current numerical geodynamo models are assumed substantially larger than those from the field models (Kuang et al., 2008, 2009).
We have carried out two forecast experiments: the first covers the period from 1994 to 2001, and the second from 2000 to 2007. The first is to demonstrate the improvement of the forecasts, while the second is used to compare the accuracies of the forecasts with previous IGRF effort (e.g. Mandea and Macmillan, 2000; Macmillan and Maus, 2005).
In both experiments, a 7year forecast from MoSST_DAS is compared with the observations of the same period to assess the improvement of the predictioncorrection approach (19) and (20). The 7year forecast period chosen in the experiments directly aims at producing IGRF SV candidate models. The IGRF provides forecast only for the 5year period after the given epoch (e.g. for the period from 2010 to 2015 for IGRF11 defined in the year 2010). But in practice, the actual analysis could only utilize observations at most up to 0.5 year prior to the epoch. In addition, the Bsplines used in geomagnetic field modeling generate endpoint effects, which are expected to affect the coefficients at least in the last knot interval of the splines (private communication with Sabaka and Olsen). From these considerations, a 7year forecast experiment would suffice for the IGRF candidate model objective. But, in terms of general geomagnetic data assimilation, we can provide forecasts for much longer periods (Kuang et al., 2008, 2009). Of course the forecast errors increase with time (eventually reaching the values defined by the differences between the unconstrained dynamo simulation results and the observations).
Before the experiments, we need to determine the time scale τ_{ ε } of the model error ε variation in the assimilation, which will then be used to select the reanalysis time interval ta. For this purpose, 4 sequences of 5year forecasts are carried out: the first starting from 1993; and each following sequence starting one year later than the previous sequence. This implies that the reanalysis time interval is one year if they are used for the predictioncorrection process.
The scaled forecast errors at the top of the D″layer of the 5year forecasts from the analysis time t_{ a } = 1993, 1994, 1995 and 1996, respectively. The errors from a given analysis time are in the same row, and are listed from the year 1 to year 5 of the forecast period.
Analysis time  Year1  Year2  Year3  Year4  Year5 

1993  0.0012669  0.0024343  0.0035950  0.0047763  0.0059601 
1994  0.0012198  0.0024068  0.0036105  0.0048152  0.0059684 
1995  0.0012376  0.0024680  0.0036957  0.0048684  0.0059650 
1996  0.0012824  0.0025370  0.0037306  0.0048449  0.0059168 
The above experiment is not complete: the assessment is not made independently since all utilized observational information is provided by a single source (CM4). Therefore, we carry out the second experiment: we produce a 7year forecast for the period from 2000 to 2007 based on CM4 field coefficients up to year 2000 (CM4 provides the coefficients up to 2002.5). The process here is similar to that in the first experiment, but with and t_{ a } = 2000. The corrected field and SV forecasts are then compared with those from CHAOS2s, IGRF8 and IGRF9.
Selection of both IGRF8 and IGRF9 in this second experiment is intended for examining the lower and upper bounds of the MoSST_DAS forecast accuracies. IGRF8 was determined by a task Force at the end of 1999 so that as many ørsted data could be incorporated as possible (Lowes, 2000; Olsen et al, 2000). In terms of data utilization, IGRF8 SV model for 2000–2005 is the true 5year forecast, thus the closest to our forecast experiment setting. On the other hand, IGRF9 was the revision of IGRF8 in 2003, and was produced by the participating members of IAGA working group V8 (IAGA Working Group report, 2003). In this revision, observations up to later 2002 were utilized to produce SV from 2000 to 2005. From this point of view, IGRF9 produced actually a 2.5year forecast. We expect therefore that its SV forecast is more accurate than that of IGRF8, and perhaps more accurate than our 7year SV forecast. The latter implies that IGRF9 can be a good test case for the upper bound of MoSST_DAS SV forecast accuracy.
In the figure, the solid line is the CHAOS2s (observation) field model spectrum. The remaining curves are the rms differences between the observation and the forecasts of IGRF8 (dotted line with upward triangles), IGRF9 (dotted line with stars), MoSSTDAS CAF (dashed line with circles), and MoSSTDAS UAF (dashed line with diamonds). As shown in the figure, CAF and UAF are nearly identical. This is not surprising, since the forecasted field is derived from the SV forecast via (20). We can also observe from the figure that, as expected, the MoSSTDAS field forecasts are more accurate than IGRF8, but less accurate than IGRF9. This simply reflects the fact that IGRF9 actually provides a shorter period forecast with more data.
These independent assessments, in particular with respect to previous IGRF models, provide us confidence in our ability to accurately predict geomagnetic field, in particular SV, with our geomagnetic data assimilation system MoSST_DAS.
5. Predicted SV Model for IGRF
IGRF provides predictive averaged SV over a 5year period starting from a given epoch. Specifically, IGRF11 will provide the averaged SV for the period from 2010 to 2015. Therefore (19) will be used for our candidate model to IGRF.
In this part of the forecast, CHAOS2s is used to provide observations from 2002 to 2010. A mathematical smooth transition is used to migrate the Gauss coefficients of CM4 to those of CHAOS2s in the time period from 2000 to 2002. Since both models agree very well in this overlapping period, the transition is introduced only as a precaution.
The Gauss coefficients of the MoSST_DAS 5year SV forecast model for the period 2010–2015.
n  m 





1  0  11.94  0.00  0.08  0.00 
1  1  16.17  −27.14  2.15  1.86 
2  0  −11.58  0.00  0.74  0.00 
2  1  −4.13  −21.69  0.38  0.20 
2  2  2.13  −12.87  1.84  0.56 
3  0  0.63  0.00  0.15  0.00 
3  1  −3.88  7.83  0.42  0.67 
3  2  −3.14  −2.91  0.83  0.50 
3  3  −7.03  −1.68  0.49  1.22 
4  0  −1.51  0.00  0.20  0.00 
4  1  2.14  0.60  0.08  0.38 
4  2  −8.39  2.94  0.56  0.39 
4  3  4.28  3.44  0.12  0.41 
4  4  −1.81  −0.44  0.29  0.22 
5  0  −0.74  0.00  0.30  0.00 
5  1  0.58  0.44  0.01  0.18 
5  2  −1.45  1.67  0.34  0.09 
5  3  −0.97  0.94  0.05  0.26 
5  4  1.16  3.65  0.24  0.09 
5  5  1.32  −0.61  0.51  0.04 
6  0  −0.08  0.00  0.09  0.00 
6  1  −0.13  −0.14  0.13  0.12 
6  2  −0.10  −2.00  0.15  0.13 
6  3  1.82  −0.45  0.06  0.03 
6  4  −1.59  −0.48  0.02  0.06 
6  5  −0.28  0.60  0.03  0.18 
6  6  1.73  0.59  0.25  0.14 
7  0  0.12  0.00  0.01  0.00 
7  1  −0.10  0.62  0.00  0.02 
7  2  −0.50  0.29  0.07  0.01 
7  3  1.12  −0.04  0.09  0.01 
7  4  0.42  −0.03  0.05  0.06 
7  5  0.21  −0.75  0.03  0.02 
7  6  −0.67  −0.17  0.08  0.04 
7  7  0.60  0.14  0.01  0.03 
8  0  −0.01  0.00  0.02  0.00 
8  1  0.14  −0.08  0.00  0.00 
8  2  −0.47  0.13  0.02  0.01 
8  3  0.20  0.29  0.00  0.03 
8  4  −0.22  0.39  0.00  0.02 
8  5  0.25  0.11  0.01  0.00 
8  6  0.32  −0.19  0.01  0.02 
8  7  −0.53  0.45  0.02  0.00 
8  8  0.33  0.29  0.01  0.01 
6. Discussion
Advances in numerical geodynamo modeling over the past 15 years, and the long history of global geomagnetic field observations have made geomagnetic data assimilation possible. Assimilation of the geomagnetic observations with numerical geodynamo models will help validate and improve the approximations and assumptions made in the models, thus help to better understand the dynamo solutions and the dynamical states in the Earth’s outer core. The improved numerical geodynamo models, on the other hand, will help us better interpret geomagnetic observations, and predict geomagnetic secular variations. This paper provides the first attempt at predicting SV using our geomagnetic data assimilation system MoSST_DAS.
In MoSST_DAS, over 7000 years of the geomagnetic/historical/paleomagnetic records are assimilated into the core dynamics model (MoSST) via a sequential assimilation algorithm (9): at each analysis time t_{ a }, the model forecasts are modified with the data and are then used as the initial states for future forecasts.
To improve the forecast accuracy, we introduce a “bootstrap” type predictioncorrection technique (19)–(20) to reduce the model error. The idea behind this technique is very simple: we utilize two sequences of the forecast with slightly different analysis time t_{ a } and . The difference between the two sequences will then remove much of the model errors, i.e. ε in (14), varying on time scales much longer than . This difference will then be used to improve the secular variation (19), and then the field forecasts (20).
This technique has been tested in two cases. In the first case, observations prior to year 1994 are used for the forecast for the period from 1994 to 2002. As shown in Fig. 3, the technique reduces the forecast error ε by nearly an order of magnitude. In the second case, the last analysis time is set at t_{ a } = 2000. And the forecast is made for the following 7 years. The forecast is then compared with CHAOS2s, IGRF8 and IGRF9. As shown in Figs. 4 and 5, the forecasted field and SV from MoSST_DAS agree very well with the observations (CHAOS_2s field model). In addition, the MoSST_DAS field forecast accuracies are at least comparable to IGRF forecast (better than IGRF8 but slightly worse than IGRF9 which utilizes more observations for shorter period forecast). However, among all SV forecasts, those from MoSSTDAS are the closest to the observations (Fig. 5).
Using this technique, we produce a MoSST_DAS candidate model for IGRF11: a 5year geomagnetic secular variation model for the period from 2010 to 2015, as shown in Fig. 9 and table 2.
This is the first ever attempt of using geomagnetic data assimilation to forecast SV. In this approach, the dynamics of the Earth’s core that we know so far are utilized to make predictions of future changes in the core field. This is very different from techniques used in the past IGRF SV forecasts. Obviously there are many improvements that can be made to this approach.
In addition to improvements to numerical dynamo models and assimilation algorithms (which depend on further understanding of the assimilation solutions, model responses to observational constraints, dynamo properties, etc), we should also consider more direct integration of geomagnetic data into MoSST_DAS. Currently, we use the Gauss coefficients from various field models as the “real observations”. But these coefficients are derived products. Therefore, in the MoSST_DAS, we also inherit all improvements as well as limitations of these field models. One concern is the assessment of the observational errors. They are neglected under the assumption (still largely true) that the model errors are far greater. However, a better forecast needs inclusion of the data errors as well. But it is difficult to assess these errors in the field model Gauss coefficients. For example, any discrepancy between two field models (e.g. CM4 and CHAOS2s) in an overlapping observation period, different types of data used in the field models, endpoint effects (private communication with Jackson, Finlay, Olsen and Sabaka) will bring further complications on this issue. Direct assimilation of the data might help avoid some of these complications. But, any effort will be evaluated by the forecast accuracies.
Declarations
Acknowledgments
This work is supported by NASA Earth Surface and Interior Program (W. Kuang and Z. Wei), NSF Collaborative Mathematical Geophysics program under the grant EAR0327875 and NSF Mathematical Geosciences program under the grant EAR0757880 (W. Kuang and A. tangborn), NERC grant NER/O/S/2003/00675 (R. Holme). We thank A. Jackson, C. Finlay, C. Constable, M. Korte, T. Sabaka and N. Olsen to provide past global geomagnetic and paleomagnetic field models used in this research. We also thank NASA Advanced Supercomputing (NAS) division for computing resources.
Authors’ Affiliations
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