IGRF-12 GFZ candidates

Transcription

1 IGRF-12 GFZ candidates V. Lesur 1, M. Rother 1, I. Wardinski 1, R. Schachtschneider 1, M. Hamoudi 2, A. Chambodut 3 October 1, Helmholtz Centre Potsdam, GFZ German Research centre for Geosciences, Telegrafenberg, Potsdam, Germany. 2 Université des sciences et de la technologie, Bab Ezzouar, El-Alia Alger, Algeria 3 Institut de Physique du Globe de Strasbourg (UMR 7516-CNRS, Université de Strasbourg/EOST), Strasbourg, France 1 Introduction This short report describes technical aspects of the derivation of the GFZ candidates for the IGRF-12. There are three dedicated sections for the DGRF-2010, IGRF-2015 and SV , respectively. We have conducted tests to estimate the robustness of our candidates, but these are not included in this report. 2 DGRF 2010 technical description 2.1 Data Our magnetic field model is built from CHAMP level-3 satellite data from to , and observatory hourly means over the same period. We used hourly means as prepared by Macmillan and Olsen (2013). 2.2 Selection The selection criteria used for these data are similar to those used in GRIMM series of models Lesur et al. (2008); Lesur et al. (2010); Mandea et al. (2012). The satellite and observatory vector data are selected in the SM coordinate system between ±55 o magnetic latitude for magnetically quiet times according to the following criteria: - Positive value of the z-component of the interplanetary magnetic field (IMF-B z ). - Sampling points are separated by 20 seconds at minimum. 1

2 - Data are selected at local time between 23:00 and 05:00, with the sun below the horizon at 100 km above the Earth s reference radius (a = km). - Dst values should be in ±30nT and their time derivatives less than 100nT/day. - Quality flags set to have accurate satellite positioning and two star cameras operating. At high latitudes, i.e. polewards of ±55 o magnetic latitude, the three component vector magnetic data are used in North, East, Centre (NEC) system of coordinates. Their selection criteria differ from those listed above only for the local time: - Data are selected at all local times, and independently of the sun position. 2.3 Weights The weight w j used for a given data value d j, a set of model parameters g and a sensitivity matrix A, are given in equation 1. These weights are modified versions of the Huber weights where the prior data standard deviation σ j, and the scalar k j and a j are given in the table 1. Note that if g is not available, then we use : w j = 1 σ j. 1 σ j for d j A j g k j, a w j = ] 1 j 1 k 2 j for d j A j g > k j, σ j d j A j g Obvious outliers have been removed from the satellite data set. These outliers have been identified, after a few iterations of the modelling process, as the data associated with residuals larger than ±50nT at midlatitudes and larger than ±200nT at high latitudes. (1) 2.4 Model Away from its sources, the magnetic field can be described as the negative gradient of potentials associated with sources of internal and external origin: B = {V i (θ, ϕ, r, t) + V e (θ, ϕ, r, t)} V i (θ, ϕ, r, t) = a L i l l=1 m= l ( a r )l+1 gl m (t)yl m (θ, ϕ) V e (θ, ϕ, r, t) = a L e l l=1 m= l ( r a )l ql m (t)yl m (θ, ϕ) (2) where Yl m (θ, ϕ) are the Schmidt semi-normalized spherical harmonics (SHs). θ, ϕ, r and a are the colatitude, longitude, satellite altitude and model reference radius, respectively, in geocentric coordinates. We use the convention that negative orders, m < 0, are associated with sin( m ϕ) terms whereas null or positive orders, m 0, are associated with cos(mϕ) terms. For the largest wavelengths of the field generated in the core and lithosphere (here, assumed up to SH degree L i = 18), the reference radius used in equation (2) is a = 3485 km. The Gauss coefficients are parameterized in time from to , using order six B-splines ψi 6 (t), with half-year time interval between spline nodes. The time dependence of the Gauss coefficients is therefore given by: N t gl m (t) = glj m ψj 6 (t), (3) j=1 where N t = 9. For the core and lithospheric field of SH degree greater than 18, the reference radius is set to a = km. The maximum SH degree used for modelling the field of internal origin is 30, although 2

3 a constant field covering all SH degrees from 25 to 80 is subtracted from the data so that only very small contributions from the lithospheric field remain unmodelled. The remaining parts of the internal field are the induced fields that are modelled using only one coefficient, for = 4 different 6-month time intervals, scaling the internal part of the Dst index i.e. the Dst i. The time dependence of the Gauss coefficient g 0 1(t) is therefore modified to: N t g1(t) 0 = g1j 0 ψj 6 (t) + g1j 0Dst H j (Dst i ), (4) j=1 where the function H j (X) takes the value X in the time interval t j : t j+1 ] and is zero otherwise. For observatory data we also co-estimate crustal offsets. The external field parameterisation also consists of independent parts. A slowly varying part of the external field model is parameterized over each 6-month time interval by a degree l = 1 order m = 0 coefficient in the Geocentric Solar Magnetic (GSM) system of coordinates, and two coefficients of SH degree l = 1, with orders m = 0 and m = 1 in a Solar Magnetic (SM) system of coordinates. The rapidly varying part of the external field is controlled using the external part of the Dst index i.e. the Dst e, and the IMF B y time series. Here again 6-month time intervals are used. Four scaling coefficients for the Dst e are introduced in each interval: three for SH degree l = 1 and orders m = 1, 0, 1 and one for SH degree l = 2 and order m = 0. One scaling coefficients for the IMF B y is introduced in each time interval for SH degree l = 1 and orders m = 1 in SM system of coordinates. Overall, the parameterisation of the external field is: j=1 B e (θ, ϕ, r, t) = R GSM r ] j=1 { q0gsm 1j Y1 0 (θ, ϕ)} H j (1) R SM r ] j=1 { q0sm 1j Y1 0 (θ, ϕ) + q 1SM 1j Y1 1 (θ, ϕ)} H j (1) r ] j=1 { 1 m= 1 qmdst 1j Y1 m (θ, ϕ) + ( r a )q0dst 2j Y2 0 (θ, ϕ)}h j (Dst e ) R SM r ] j=1 { q 1IMF 1j Y1 1 (θ, ϕ)} H j (IMF B y ) (5) where R GSM and R SM are matrices rotating vectors defined in GSM and SM reference frames into the geocentric Earth fixed reference frame, respectively. We used independent external field parameterisations for the satellite and observatory data. In the latter, we impose that q1j 0SM is set to zero to avoid co-linearities with the observatory crustal offsets. 2.5 Process The solution is obtained after several iterations of an Iterative Reweighted Least-Squares scheme where the square root vlaues of the weights are given by equation 1. As in all the GRIMM series, the Z SM component is not used to evaluate the Gauss coefficients of the internal field. The external coefficients, that are varying rapidly in time, are determined exclusively by the mid- and low-latitude data. 2.6 Euler Angles The rotation angles between magnetic field vectors in the sensor reference frame and the satellite coordinated system are estimated on 30 days time intervals as presented in Rother et al. (2013). 3

4 Table 1: Satellite and observatory data weights, parameters and misfits. The first three rows for satellite and observatory data are mid- and low-latitude data, whereas the last three are high-latitude data. Nb is the number of data values, σ the prior standard deviation in nt, k and a are the parameters for the weights defined in equation (1), and rms are the root mean squares of the residuals to the data, with their mean in brackets (all in nt). Nb σ k a rms Satellite X SM (0.07) Y SM (0.02) Z SM (-0.38) X HL (0.81) Y HL (-0.29) Z HL (0.05) Observatories X SM (0.22) ) Y SM (-0.13) Z SM (-0.03) X HL (-3.46) Y HL (0.56) Z HL (0.27) 2.7 Regularisation Only the core field temporal variations are regularized in this model. The integral over the sphere of the squared radial component of the core field second time derivative is minimized over the time interval, while the integral over the sphere of the squared radial component of the core field first derivative is minimized in and Data misfit and residuals Data weights and misfits are provided in table 1. The misfits and residual means are unweighted estimates, and therefore can be strongly affected by outliers. That is less likely to be the case for the candidate model since our estimation process uses a modified version of the Huber norm. 2.9 DGRF candidate The DGRF candidate is the parent model snapshot for IGRF-2015 technical description 3.1 Data Our magnetic field model was built from the three Swarm L1b Baseline 0301/0302 satellite data and observatory hourly means as prepared by Macmillan and Olsen (2013). The time period is limited from to (MJD days from 4749 to 5479). Over this time span Swarm satellite data were available for: 4

5 Satellite Time Type Start End A MJD B MJD C MJD The observatory data at time of selection were available only up to MJD(2000) Selection and data weights Data were selected, and their weights estimated, in the same way as for the DGRF (see sections 2.2, 2.3). The outliers detection range was set to smaller values: ±20nT at mid-latitudes and ±150nT at high laitudes. 3.3 ASM/VFM residuals The selected Swarm satellite vector data have been corrected for their ASM/VFM differences. A correction vector is modelled as function of the sun position in the VFM sensor system for three different time windows. The models are given as SH coefficients up to SH degree 30 for each satellite and each component in the VFM system. The three different time windows are: 1. before (MJD ), 2. between and (MJD to 5308) 3. after (MJD ). The scalar misfit between ASM total intensity data, and the intensity computed from the VFM data are: Satellite Before correction After correction Id σ (nt) σ (nt) A B C Model, Process and Regularisation The model is similar to the DGRF parent model (see section 2.4), the only difference is the spline knot positions that are for the IGRF parent model spanning to The processing steps and the regularisation are the same as in the DGRF setup. 3.5 Euler Angle adaption The rotation angles between magnetic field vectors in the sensor reference frame and the satellite coordinated system are estimated on 100 day time intervals as proposed in Rother et al. (2013). Relatively large time intervals are used due to the known problems of data calibration. Further, it is clear that the stability of the Swarm VMF/star camera system does not require an evaluation of the Euler angles on short time-scales. The estimated Euler angle corrections do not show an obvious trend or pattern, the standard deviation for the set of all 27 Euler Angles is 8.4 arcsecs. 3.6 Data misfit and residuals Data weights and misfits are provided in the table 2. 5

6 Table 2: Satellite and observatory data weights, parameters and misfits. The first three rows for satellite and observatory data are mid- and low-latitude data, whereas the last three are high latitude data. For satellites, a data set is selected specifically for Euler angle estimation, with very tight selection criteria. Their misfit is given in the three intermediate rows. Nb is the number of data values, σ the prior standard deviation in nt, k and a are the parameters for the weights defined in equation (1), and rms are the root mean squares of the residuals to the data, with their means in brackets (all in nt) Nb σ(a/b/c) k a rms Satellites X SM /3.3/ (0.05) Y SM /3.9/ (0.16) Z SM /5.5/ (-0.57) X SM,Euleronly /3.3/ (0.28) Y SM,Euleronly /3.9/ (-0.07) Z SM,Euleronly /5.5/ (-0.04) X HL /10.0/ (1.00) Y HL /12.0/ (0.67) Z HL / 7.8/ (-0.07) Observatories X SM (0.23) Y SM (-0.28) Z SM (-0.32) X HL (-3.92) Y HL (1.16) Z HL (-0.15) 3.7 IGRF 2015 candidate The IGRF 2015 candidate has been estimated through the following processing: A snapshot model has been derived from the parent model for An average SV has been estimated from the parent model for the to time interval. The snapshot model for has been linearly extrapolated to , using the averaged SV model, to generate the IGRF candidate. Several approaches have been tested to estimate the IGRF candidate from the parent model. All give Gauss coefficient values inside a ±1 nt range for the first few Gauss coefficients, and much smaller ranges for higher SH degree coefficients. 4 IGRF-SV for Several radically different approaches have been used for estimating the SV for from available data. In particular we have employed Multi-channel singular value analysis of flow coefficients, where these coefficients have been derived in a joint inversion for field and flow at the core surface from geomagnetic observatory data of the period 1957 to ( similar to Wardinski and Lesur (2012). Thereby, we also tested the coestimation of the field and the flow at the core mantle boundary using different constraints for the flow. Upon the results of Multi-channel singular value analysis, predictions of the secular variation have been constructed by forecasts of the flow variation and then by forward modeling of the diffusion-less induction equation. 6

7 After a comparison of the different models obtained, and a comparison with SV derived from satellite observations, we decided to simply propose as IGRF-SV candidate the time averaged SV model obtained from the IGRF parent model between and (see section 3.7). References Lesur, V., I. Wardinski, M. Rother, and M. Mandea (2008), GRIMM - The GFZ Reference Internal Magnetic Model based on vector satellite and observatory data, Geophys. J. Int., 173, doi: /j x x. Lesur, V., N. Olsen, and A. Thomson (2010), Geomagnetic core field models in the satellite era, in Geomagnetic Observations and Models, IAGA book series, vol. 5, edited by M. Mandea and M. Korte, chap. 11, pp , Springer, doi: / Macmillan, S., and N. Olsen (2013), Observatory data and the swarm mission, Earth, Planets and Space, 65(11), , doi: /eps Mandea, M., I. Panet, V. Lesur, O. De Viron, M. Diament, and J. L. Le Mouël (2012), The earth s fluid core: recent changes derived from space observations of geopotential fields, PNAS, doi: /pnas Rother, M., V. Lesur, and R. Schachtschneider (2013), An algorithm for deriving core magnetic field models from swarm data set., Earth, Planets and Space, 65, , doi: /eps Wardinski, I., and V. Lesur (2012), An extended version of the C 3 FM geomagnetic field model - application of a continuous frozen-flux constraint, Geophys. J. Int., 189, , doi: /j x x. 7