3.1 Analysis of IGRF-11 DGRF-2005 candidate models
Table 1 lists the seven candidates models for DGRF 2005 giving details of the teams, the major data sources used and very brief comments concerning the various modelling approaches adopted. Two candidates (C2 and E2) were resubmissions as the original candidates were withdrawn by their authors.
3.1.1 RMS vector field differences for DGRF-2005 candidate models
Rows of Table 2 present the RMS vector field differences
R in units of nT between a particular DGRF candidate model i and another candidate j. The final three columns document
R between a candidate model i and one of three possible mean models j. The mean models considered are the arithmetic mean model M, the model MnoD which is an arithmetic mean excluding candidate D, and the model MABG that is an arithmetic mean model derived only from candidates A, B and G. Note the symmetry about the diagonal entries in this table which is included as a check on the calculations. It is readily observed that model D is consistently furthest away from the other models in terms of
R; furthermore the RMS vector field differences between the other candidates and the mean are reduced when D is removed from the calculation of the mean. On the other hand candidates A, B, G are found to be extremely similar displaying the smallest RMS vector field differences between each other. Besides candidate D, candidates C2 and E2 show the next largest
R followed by F.
The final three rows of Table 2 involve the arithmetic means of the RMS vector field differences of
R of model i from the other models j. The third from last row is , the penultimate row is the same calculation excluding candidate D while the final row involves only
R from candidates A, B and G. Candidates A, B and G have the smallest and the mean of the
R becomes smaller when only candidates A, B and G are retained.
3.1.2 Spectral analysis of DGRF-2005 candidate models
Figure 1 (left) presents the Lowes-Mauersberger spectra i0R
(defined in (3)) of the DGRF-2005 candidate models as a function of spherical harmonic degree plotted at the Earth’s core-mantle boundary (r = 3480 km). The spectra of the candidate models are mostly very similar, almost completely overlapping for degrees less than 9. The most noticeable differences occur for candidate D at degree 11 (where it contains lower power than the other candidates) and for candidate E2 at degree 13 (where it contains higher power than the other candidates). Figure 1 (right) presents the degree correlation
as defined in (9) between the DGRF-2005 candidate models and the arithmetic mean model M. Candidate D displays a low degree correlation to M above degree 9. The degree correlation of candidates C2, E2 and F to M above degree 10 is slightly lower than that of A, B and G which appear similar to each other and close to M.
In Fig. 2 coefficient by coefficient analysis of the DGRF-2005 candidate models is presented. The plot on the left shows differences as defined in (1) between the candidate models and the arithmetic mean model M. The largest differences from M are found to occur for candidate D, with significant deviations also notable for candidates E2, C2 and F. The deviations associated with candidates A, B and G are smaller, so that the curves for candidates A and B are largely hidden behind those for the other candidates. The right hand plot shows the Huber weights calculated during the determination of robust mean coefficients. Notice that the coefficients of candidate D often receive the lowest weights, particularly for the coefficients associated with the highest harmonics which receive weights as low as 0.4. Candidates E2, C2 and F also receive low weights for certain coefficients; in particular E2 receives some low weights for coefficients between n = 6 and n = 9. Almost all coefficients of candidates A, B and G receive full weights of 1.0 illustrating that they are consistently closer to the robust mean, so are arguably of higher quality.
3.1.3 Spatial analysis of DGRF-2005 candidate models
A geographical investigation of the DGRF-2005 candidate models is presented in Fig. 3. This shows the differences between the vertical (Z) component of the candidates and model MABG at radius r = a. Model MABG was chosen as a suitable reference based on the earlier analyses presented in Sections 3.1.1 and 3.1.2.
Studying differences between the candidate models and a reference model in space yields insight into the geographical locations where disparities in the candidates are located. Visual inspection of Fig. 3 reveals that candidate D involves the most striking deviations from MABG that are locally as large as 50 nT. The differences are scattered over the globe and not confined to any particular geographical location, though the largest discrepancies occur in the polar regions and in the mid-Atlantic. Candidates C2 and E2 display largest deviations from A, B and G in the polar regions (particularly in the Arctic). Model E2 shows one localized anomalous region in the equatorial Pacific while model F shows rather minor differences at high latitudes and at mid-latitudes in the northern hemisphere. Candidates A, B and G exhibit only minor differences to the reference model MABG demonstrating once more that they are consistent with each other.
3.1.4 Choice of numerical precision for DGRF-2005
An important analysis for DGRF-2005 was to calculate (using (10) and (11)) the error per degree in the unweighted arithmetic means determined for sets of candidate models. Figure 4 shows the result of such a calculation using candidates A, B and G, on the assumption that all the candidates have the same per degree sample deviation s
, which is estimated from their scatter about the mean. The solid line shows the resulting error in the mean per degree for model MABG which is typically around 0.3 nT. The dashed line in Fig. 4 shows the expected uncertainty due to rounding the model coefficients to 0.1 nT, given by the expression (see, for example, Lowes, 2000). It is observed that the error due to 0.1 nT rounding dominates the error in the mean of candidates A, B and G above degree 7. Given the decision by the task force (see next section) to adopt model MABG for the DRGF-2005, this necessitates quoting DGRF-2005 to 0.01 nT rather than 0.1 nT to avoid introducing unnecessary rounding errors. Note that based on internal consistency, the total formal RMS error in the mean model MABG (which is DGRF-2005) is remarkably only 1.0 nT.
3.1.5 Discussion and summary for DGRF-2005
Based on the tests presented above, candidate D appears consistently different in both the spectral domain (with certain spherical harmonic coefficients apparently anomalous—see Fig. 2) as well as in physical space where global problems are observed. In addition candidates E2, C2 and to lesser extent F were observed to have some problems, particularly at high degrees in the spectral domain and at high latitudes in space. In contrast candidates A, B and G were very similar despite being derived using different data selection criteria and using different modelling procedures. The task force therefore voted that DGRF-2005 be derived from a simple arithmetic mean of candidates A, B and G (i.e. model MABG as discussed above).
3.2 Retrospective analysis of IGRF-10 MF candidate models for epoch 2005
Having established a new DGRF for epoch 2005 it is possible to carry out an assessment of the quality of the candidate models that contributed to the IGRF-10 provisional model for epoch 2005. Table 3 presents the RMS vector field differences
R between the various candidate models, the IGRF-2005 model (from IGRF-10) and the DGRF-2005 model (from IGRF-11). The naming convention for the candidates is that used by Maus et al. (2005). Candidate A1 was a model from DSRI/NASA/Newcastle, Candidate B3 was a model from NGDC/GFZ, Candidate C1 was a model from BGS and Candidate D1 was a candidate from IZMIRAN. Candidate A1 agrees most closely with DGRF-2005 with a global RMS vector field difference of 9.9 nT followed closely by B3 which differs by 10.9 nT. Candidate D1 does a little worse with a difference 14.0 nT and candidate C1 is furthest from DGRF-2005 with an RMS vector field difference of 18.5 nT, almost twice that of candidate A1. The IGRF-2005 (which was the arithmetic mean of candidates A1, B3 and C1) differed from DGRF-2005 by 12.0 nT.
In Fig. 5 the difference in power per degree between the IGRF-10 candidates and DGRF-2005 (
) are presented. The mean square vector field difference per degree between the final IGRF-2005 (the arithmetic mean of A1, B3 and C1) and DGRF-2005 is shown as the black dashed line. It appears that the problems with candidate D1 are predominantly at high degree (n > 7); it is better than most other candidates at the lower degrees. Candidate C1 was further from DGRF-2005 than all the other candidates at low degrees 1–7 suggesting some systematic problem with this model. It is also noticeable that candidate A1 did better than the other candidates for the dipole (n = 1) terms while candidate B3 performed best at high degrees, especially n = 12, 13.
3.3 Analysis of IGRF-11 MF candidate models for epoch 2010
Having completed the analysis of MF models for epoch 2005.0 we now move on to consider epoch 2010.0. Table 4 summarizes the candidate models submitted for IGRF-2010. Note that model C2 was a resubmission by BGS who withdrew their initial candidate. Further details are again given in the papers in this special issue focusing on the various candidate models, and their descriptions are available online at http://www. ngdc.noaa.gov/IAGA/vmod/candidatemodels.html. Models for epoch 2010.0 were submitted in October 2009; teams therefore faced the additional challenge of how to propagate their estimates forward to 2010.0; this was not an issue faced when deriving retrospective models for epoch 2005.0. A brief indication of the method used to propagate to epoch 2010.0 is provided in the final column of Table 4. Larger differences in the candidate models are expected due to this additional complication; it the IGRF-11 model for epoch 2010.0 is therefore only provisional and will be updated to a DGRF in 2014 during the IGRF-12 process.
3.3.1 RMS vector field differences for IGRF-2010 candidate models
Table 5 displays the RMS vector field differences
R between the IGRF-11 candidates for epoch 2010.0 and also between the candidates and the arithmetic mean model M and a weighted mean model M
is reported here because it was important in the final voting process; it consists of candidates A, B, C2, F and G having weight 1.0 and candidates D, E having weight 0.25 (in addition coefficients and of candidate A were disregarded following a vote by the task force). The bottom row of Table 5 shows , the mean of the differences
R (excluding the zero value for the difference between candidates and themselves—see (7)).
As anticipated, the differences between the IGRF-2010 candidates are larger than between the DGRF-2005 candidates, with the mean of the differences between the candidates and the mean model (i.e. the mean of
R) being 7.3 nT here for epoch 2010.0 compared to 4.9 nT for epoch 2005.0. Candidates D and E display the largest differences from the other candidates and to the mean models M and M
. Candidate B is most similar to M and it also agrees reasonably closely with candidates F and G (differences less than 5.5 nT) and slightly less well with candidates A and C2 (differences of less than 8.5 nT).
3.3.2 Spectral analysis of IGRF-2010 candidate models
In Fig. 6 (left) we plot the Lowes-Mauersberger spectra i,0R
from (3) of the IGRF-2010 candidates at the core-mantle boundary. Candidates E and D have noticeably higher power in degrees 11 and 13 suggesting that they may have difficulties with noise being mapped into some model coefficients at high degree.
Figure 6 (right) shows the degree correlation per degree
from (9) between the candidates and the arithmetic mean model M. Candidates E and especially D show the largest differences above degree 10; candidates C2, F and G show smaller deviations from M while candidates A and B are closest to M.
In Fig. 7 the left hand plot presents the coefficient by coefficient differences as defined in (1) between the IGRF-2010 candidates and the mean model M. It is apparent that there are some systematic problems. Candidate A possesses particularly large differences from M in coefficients and . Candidate D displays many remarkable differences from M in the sectoral harmonics while candidate E shows anomalous coefficients, particularly at degrees n = 11−13. Candidate C2 shows differences from M predominantly in the terms, most noticeably in degrees n = 3−9. The right hand plot in Fig. 7 displays the Huber weights as a function of the index of the spherical harmonic coefficient. It shows how the robust weighting scheme would in this circumstance strongly down-weight many (but not all) of the coefficients of candidate D at n > 10, as well as many of the coefficients of candidate E. The lowest Huber weight for the important axial dipole coefficient is allocated to candidate A. Aside from this exception candidates A, B, C2, F and G receive Huber weighting factors close to 1 for the majority of their coefficients.
3.3.3 Spatial analysis of IGRF-2010 candidate models
In Fig. 8 we plot at Earth’s surface the differences between the Z component of the IGRF-2010 candidate models and the weighted mean model M
in which candidates D and E are weighted by a factor 0.25 and the and coefficients of candidate A are discarded. The largest discrepancies are observed for candidates D and E. Candidate D displays major differences from M
along the dip equator, and in the high latitude Arctic region where differences as large as 50 nT are evident. Candidate E also displays prominent deviations from M
in the Arctic region, but predominantly of the opposite sign to those of candidate D; in addition it possesses low latitude anomalies linked to its anomalous sectoral harmonics. For both candidates E and D the deviations are globally distributed rather than localized. Candidate C2 has its largest differences from the other models in the polar regions. Candidates A, B, F, and G show more minor deviations from M
, the differences being largest in the polar regions in all cases. The analysis of the IGRF-2010 candidate models in geographical space highlights that the most serious differences in the candidate models occur in the polar regions and to a lesser extent along the dip equator. Future efforts towards improved field models will require better models of external and induced fields in these regions.
3.3.4 Discussion and summary for IGRF-2010
The evaluations of the IGRF-2010 candidates presented above suggest that candidates D and E have some problems, particularly at spherical harmonic degree greater than 10. Consequently the task force voted to allocate these candidates weight 0.25 while candidates A, B, C2, F, G were allocated weight 1.0 in the determination of the new IGRF-11 model for epoch 2010. In addition the task force voted to disregard coefficients and from candidate A since these were thought to be suspect. Subsequent analysis has shown that a model that includes more recent data but is otherwise similar to the parent model for candidate A results in values of and 11 that are in much better agreement with model M (Olsen et al., 2010). The final IGRF-2010 was therefore fixed to be the model discussed above as M