DTU candidate field models for IGRF12 and the CHAOS5 geomagnetic field model
 Christopher C Finlay^{1}Email author,
 Nils Olsen^{1} and
 Lars TøffnerClausen^{1}
Received: 10 February 2015
Accepted: 15 June 2015
Published: 22 July 2015
Abstract
We present DTU’s candidate field models for IGRF12 and the parent field model from which they were derived, CHAOS5. Ten months of magnetic field observations from ESA’s Swarm mission, together with uptodate ground observatory monthly means, were used to supplement the data sources previously used to construct CHAOS4. The internal field part of CHAOS5, from which our IGRF12 candidate models were extracted, is timedependent up to spherical harmonic degree 20 and involves sixthorder splines with a 0.5 year knot spacing. In CHAOS5, compared with CHAOS4, we update only the lowdegree internal field model (degrees 1 to 24) and the associated external field model. The highdegree internal field (degrees 25 to 90) is taken from the same model CHAOS4h, based on lowaltitude CHAMP data, which was used in CHAOS4.
We find that CHAOS5 is able to consistently fit magnetic field data from six independent low Earth orbit satellites: Ørsted, CHAMP, SACC and the three Swarm satellites (A, B and C). It also adequately describes the secular variation measured at ground observatories. CHAOS5 thus contributes to an initial validation of the quality of the Swarm magnetic data, in particular demonstrating that Huber weighted rms model residuals to Swarm vector field data are lower than those to Ørsted and CHAMP vector data (when either one or two star cameras were operating). CHAOS5 shows three pulses of secular acceleration at the core surface over the past decade; the 2006 and 2009 pulses have previously been documented, but the 2013 pulse has only recently been identified. The spatial signature of the 2013 pulse at the core surface, under the Atlantic sector where it is strongest, is well correlated with the 2006 pulse, but anticorrelated with the 2009 pulse.
Keywords
Background
In May 2014, the IAGA task force responsible for IGRF12 requested candidate geomagnetic reference field models [main field (MF) for epochs 2010.0, 2015.0 and predictive secular variation (SV) for 2015.0–2020.0] to be submitted by 1 October 2014. This article describes in detail the candidate models submitted by DTU Space and the timedependent parent model from which they were derived, called CHAOS5.
Geomagnetic field modellers producing candidate models for IGRF12 were in the fortunate position that the European Space Agency (ESA) launched the Swarm satellite constellation, whose aim is to carry out the best ever survey of the Earth’s magnetic field, in November 2013. In parallel with ongoing calibration and validation efforts, ESA promptly released L1b magnetic field data to the scientific community by May 2014. Swarm data were crucial to the DTU candidate models presented below. We therefore describe the selection, processing and modelling of the Swarm data in some detail. In addition to data from Swarm, we used data from previous satellite missions (Ørsted, CHAMP and SACC), along with ground observatory data kindly provided and checked by the British Geological Survey (Macmillan and Olsen 2013).
CHAOS5, the parent model for the IGRF12 candidates reported here, is the latest update of the CHAOS field model series (Olsen et al. 2006; Olsen et al. 2009; 2010; Olsen et al. 2014). The crucial aspects of this model are a timedependent model of the largescale internal field, a static model of the smallerscale internal field, a parameterization of the largescale external field in both solar magnetic (SM) coordinates (with timedependence parameterized by a disturbance index) and geocentric solar magnetospheric (GSM) coordinates, and a coestimation of the Euler angles used for the rotation of the threecomponent vector field from the magnetometer frame to the star camera frame.
The main improvement of CHAOS5 over CHAOS4 is its use of 10 months of Swarm data, as well as more recent ground observatory data. The modelling technique and data selection closely follow those previously described by Olsen et al. (2014). CHAOS5 is similar to the IGRF parent models produced by a number of other teams (for example Maus et al. 2010; Rother et al. 2013; Thomson et al. 2010) in not explicitly modelling the ionospheric field, in contrast to the more sophisticated comprehensive modelling approach (Sabaka et al. 2015; Thébault et al. 2015). Instead, data selection for CHAOS5 is limited to darkregion data from geomagnetically quiet times (when ionospheric currents are weak, at least at nonpolar latitudes), in an effort to isolate as best as possible the field of internal origin.
Comparison of the CHAOS4 and CHAOS5 geomagnetic field models
CHAOS4  CHAOS5  

Data sources  
Observatory monthly means  June 1997  June 2013  June 1997  Sept 2014 
Ørsted vector  March 1999  Dec 2004  March 1999  Dec 2004 
Ørsted scalar  March 1999  June 2013  March 1999  June 2013 
SACC scalar  Jan 2001  Dec 2004  Jan 2001  Dec 2004 
CHAMP vector and scalar  Aug 2000  Sept 2010  Aug 2000  Sept 2010 
Swarm A vector and scalar  –  Nov 2013  Sept 2014 
Swarm B vector and scalar  –  Nov 2013  Sept 2014 
Swarm C vector and scalar  –  Nov 2013  Sept 2014 
Timedependent internal field  
Model time span  1997.0–2013.5  1997.0–2015.0 
Spherical harmonic degree  n=1–20  n=1–20 
Spline basis  6th order, 0.5 year knots  6th order, 0.5 year knots 
Based on  CHAOS4l  CHAOS5l 
Static internal field  
Spherical harmonic degree  n=21–90  n=21–90 
Based on  CHAOS4l (n=21–24)  CHAOS5l (n=21–24) 
and CHAOS4h (n=25–90)  and CHAOS4h (n=25–90)  
External field  
SM  n=1: 1 h, RC int + ext  n=1: 1 h, RC int + ext 
5 day \(\Delta {q^{0}_{1}}\), 30 day \(\Delta {q^{1}_{1}}\), \(\Delta {s^{1}_{1}}\)  5 day \(\Delta {q^{0}_{1}}\), 30 day \(\Delta {q^{1}_{1}}\), \(\Delta {s^{1}_{1}}\)  
n=2: static  n=2: static  
GSM  n=1–2, m=0: static  n=1–2, m=0: static 
Euler angles  
Ørsted  Before and after Jan 24 2000  Before and after Jan 24 2000 
CHAMP  10 day bins  10 day bins 
Swarm  –  10 day bins 
Regularization  
Spatial  Static field n>85, \(<{B_{r}^{2}}>\)  Static field n>85, \(<{B_{r}^{2}}>\) 
λ _{0}=1 nT^{−2}  λ _{0}=1 nT^{−2}  
Temporal, interior  \(<(d{B^{3}_{r}}/dt^{3})^{2}>\)  \(<(d{B^{3}_{r}}/dt^{3})^{2}>\) 
λ _{3}=0.33 (nT/year^{−3})^{−2}  λ _{3}=0.33 (nT/year^{−3})^{−2}  
except \({g^{0}_{1}}\), λ _{3}=10 (nT/year^{−3})^{−2}  except m=0, λ _{3}=100 (nT/year ^{−3})^{−2}  
Temporal, endpoints  \(<(d{B^{2}_{r}}/dt^{2})^{2}>\)  \(<(d{B^{2}_{r}}/dt^{2})^{2}>\) 
λ _{2}=10 (nT/year^{−2})^{−2}  λ _{2}=100 (nT/year ^{−2})^{−2} 
Data
Satellite data

Dark regions only (sun at least 10° below the horizon).

Strength of the magnetospheric ring current, estimated using the RC index (Olsen et al. 2014), was required to change by at most 2 nT/h.

Three vector components of the magnetic field were taken for quasidipole (QD) latitudes equatorward of ±55°, while scalar field (intensity) data only were used for higher QD latitudes or when attitude data were not available.

Geomagnetic activity at nonpolar latitudes (equatorward of ±55° QD latitude) was sufficiently low, such that the index Kp≤2^{0}.

Poleward of ±55° QD latitude, scalar data were only selected when the merging electric field at the magnetopause \(E_{m} = 0.33 v^{4/3} B_{t}^{2/3} \sin ^{8/3} \left (\Theta  / 2 \right)\), where v is the solar wind speed, \(B_{t}=\sqrt {{B_{y}^{2}} + {B_{z}^{2}}}\) is the magnitude of the interplanetary magnetic field in the yz plane in GSM coordinates and Θ= arctan(B _{ y }/B _{ z }) (Newell et al. 2007), was sufficiently small. More precisely, the weighted average over the preceding 1 h, E _{ m,12}≤0.8 mV/m.
Observatory data
Methods
Model parameterization
The parametrization of the CHAOS5 field model follows closely that of previous versions in the CHAOS model series (Olsen et al. 2006, 2009, 2010, 2014). We assume measurements take place in a region free from electric currents, in which case the vector magnetic field B may be described by a potential such that B=−∇V. The magnetic scalar potential V=V ^{int}+V ^{ext} consists of V ^{int}, describing internal (core and lithospheric) sources, and V ^{ext}, describing external (mainly magnetospheric) sources and their Earthinduced counterparts. Both internal and external parts are expanded in spherical harmonics. The CHAOS5 model thus consists of spherical harmonic coefficients together with sets of Euler angles needed to rotate the satellite vector field readings from the magnetometer frame to the star camera frame.
where a=6371.2 km is a reference radius, (r,θ,ϕ) are geographic spherical polar coordinates, \({P_{n}^{m}}\left (\cos \theta \right)\) are the Schmidt seminormalized associated Legendre functions, \(\left \{{g_{n}^{m}},{h_{n}^{m}}\right \} \) are the Gauss coefficients describing internal sources, and N _{int} is the maximum degree and order of the internal expansion. The internal coefficients \(\{{g_{n}^{m}}(t),{h_{n}^{m}}(t)\}\) up to n=20 are timedependent; this dependence is described by order 6 Bsplines (De Boor 2001) with a 6month knot separation and fivefold knots at the endpoints t=1997.0 and t=2015.0. Internal coefficients for degrees 21 and above are static, and a maximum degree of 80 was used during the derivation of the new model for the low degree field (CHAOS5l, where ‘l’ denotes low degrees) described here.
where θ _{ d } and T _{ d } are dipole colatitude and dipole local time. The degree1 coefficients in SM coordinates are timedependent and are further expanded as
where the terms in brackets describe the contributions from the magnetospheric ring current and its Earthinduced counterpart as estimated by the RC index (Olsen et al. 2014), RC(t)=ε(t)+ι(t). We coestimate the timeindependent regression factors \(\hat {q}_{1}^{0}, \hat {q}_{1}^{1}, \hat {s}_{1}^{1}\) and the timevarying ‘RC baseline corrections’ \(\Delta {q_{1}^{0}}, \Delta {q_{1}^{1}}\) and \(\Delta {s_{1}^{1}}\) in bins of 5 days (for \(\Delta {q_{1}^{0}}\)) and 30 days (for \(\Delta {q_{1}^{1}}, \Delta {s_{1}^{1}}\)), respectively. These allow for differences between the groundbased estimate of the degree 1 external magnetic signal (the RC index) and that inferred from low Earth orbit satellites.
In addition to the above spherical harmonic coefficients, we coestimate the Euler angles describing the rotation between the vector magnetometer frame and the star camera frame. For Ørsted, this yields two sets of Euler angles (one for the period before 24 January 2000 when the onboard software of the star camera was updated and one for the period after that date), while for CHAMP and each Swarm satellite, we solve for Euler angles in bins of 10 days.
The new model described here, derived specifically to produce candidate models for IGRF12, is essentially an update of the model CHAOS4l including 10 months of Swarm data and the latest annual differences of observatory revised month means. We refer to this new parent model as CHAOS5l. It involves timedependent terms (for degrees n=1–20, 18,040 coefficients) and static terms (for n=21–80, 6120 coefficients) together resulting in a total of 24,160 internal Gauss coefficients. The total number of external field parameters is 1301, which is the sum of 5 SM terms (\({q_{2}^{m}}, {s_{2}^{m}}\) for n=2), 3 RC regression coefficients \(\tilde {q}_{1}^{0}, \tilde {q}_{1}^{1}, \tilde {s}_{1}^{1}\), 2 GSM coefficients \(\left (q_{n}^{1, \text {GSM}}, q_{n}^{2, \text {GSM}}\right)\), 949 baseline corrections \(\Delta {q_{1}^{0}}\) and 2×171 baseline corrections \(\Delta {q_{1}^{1}}, \Delta {s_{1}^{1}}\). Considering the Euler angles for the Ørsted, CHAMP and the Swarm satellites yields an additional 3×(2+366+94)=1386 model parameters. This finally results in a total of 24,160+1301+1386=26,847 model parameters to be estimated.
Model estimation and regularization
The model parameters described above for CHAOS5l were estimated from 753,996 scalar data and 3×741,440 vector data by means of a regularized iteratively reweighted leastsquares algorithm using Huber weights, minimizing the cost function
where m is the model vector, the residual vector e=d _{obs}−d _{mod} is the difference between the vector of observations d _{obs} and the vector of model predictions d _{mod}, and \(\underline {\underline {C}}\) is the data error covariance matrix.
In the data error covariance matrix \(\underline {\underline {C}}\), anisotropic errors due to attitude uncertainty (Holme and Bloxham 1996) are considered for the vector field satellite data. A priori data error variances for the scalar field were assumed to be 2.5 nT for Ørsted and 2.2 nT for CHAMP and Swarm, while the attitude uncertainties were allocated as in CHAOS4 (Olsen et al. 2014), but with a pointing uncertainty of 10 arc sec for Swarm vector field data.
\(\underline {\underline {\Lambda }}_{3}\) and \(\underline {\underline {\Lambda }}_{2}\) are block diagonal regularization matrices penalizing the squared values of the third and second, respectively, time derivatives of the radial field B _{ r } at the core surface. \(\underline {\underline {\Lambda }}_{3}\) involves integration over the full timespan of the model, while \(\underline {\underline {\Lambda }}_{2}\) involves evaluating the second time derivative only at the model endpoints t=1997.0 and 2015.0. The parameters λ _{3} and λ _{2} control the strength of the regularization applied to the model time dependence during the entire modelled interval and at the endpoints, respectively. We tested several values for these parameters and finally selected λ _{3}=0.33 (nT/year ^{3})^{−2} (the same as used in CHAOS4l) and λ _{2}=100 (nT/year ^{2})^{−2} (a stronger endpoint constraint than used in CHAOS4l). In addition, all zonal terms were treated separately (in CHAOS4l, only the axial dipole was treated separately), with λ _{3} increased to 100 (nT/year ^{3})^{−2}, since we found these internal field components were more strongly perturbed by (i) unmodelled external field fluctuations and (ii) shortcomings in the data coverage due to lack of data in the summer polar region. The regularization parameters were chosen following a series of experiments, primarily relying on comparisons to the SV recorded at ground observatories.
Since both scalar data and Huber weights are involved, the cost function depends nonlinearly on the model parameters. The solution to the minimization problem was therefore obtained iteratively using a Newtontype algorithm. The starting model was a single epoch model with linear SV centred on 2010.0. The final model was obtained after six iterations, by which point sufficient convergence was obtained with the rms misfit converging to better than 0.01 nT and the Euclidean norm of the model change in the final iteration less than 0.005 % that of the model itself.
The complete CHAOS5 field model was obtained in a final step by combining the spherical harmonic coefficients of new model CHAOS5l with the previous CHAOS4h model (Olsen et al. 2014), which in September 2014 was our best model for the highdegree lithospheric field. The transition between these models was implemented at n=24 as for CHAOS4. The various differences between CHAOS5 and CHAOS4 are collected for reference in Table 1. Note that the model statistics reported below are those for CHAOS5l, the parent model from which our IGRF12 candidate models were extracted.
Derivation of candidate models for IGRF12

DGRF, epoch 2010.0
The parent model CHAOS5l, with its splinebased time dependence, was evaluated at epoch 2010.0, and the internal spherical harmonic coefficients up to degree and order 13 output to 0.01 nT.

IGRF, epoch 2015.0
The parent model CHAOS5l, with its splinebased time dependence was evaluated at epoch 2014.75, the end of the month when the last input satellite data were available to constrain the model. The resulting coefficients were then propagated forward to epoch 2015.0, using the linear SV evaluated from CHAOS5l in epoch 2014.0 (as in our SV candidate, to avoid spline model endeffects) as follows:$$ {}{g^{m}_{n}}(t=2015.0)={g^{m}_{n}}(t=2014.75) + 0.25\cdot \dot{g}^{m}_{n}(t=2014.0) $$(5)Here \({g^{m}_{n}}\) represents each of the Gauss coefficients \(\{{g_{n}^{m}},{h_{n}^{m}}\}\), while \(\dot {g}^{m}_{n}\) represents the SV coefficients \(\{\dot {g}_{n}^{m},\dot {h}_{n}^{m}\}\) in nT/year. The resulting internal spherical harmonic coefficients for the internal field in epoch 2015.0 up to degree and order 13 were output to 0.01 nT.

Predicted average SV, 2015.0 to 2020.0
Since there can be spline model endeffects in the secular acceleration (SA), we evaluated the SV from CHAOS5l at epoch 2014.0, rather than in 2015.0, and did not attempt any extrapolation. These endeffects are essentially due to the lack of ‘future’ data for constraining the SV and SA at the model endpoint, and because SV estimates based on annual differences of ground observatory monthly means are available only up to 6 months before the latest available ground observatory data. It should also be noted that the SV in a splinebased model such CHAOS5l at a particular epoch is not the true instantaneous SV, but a weighted time average, with the amount of time averaging varying with spherical harmonic degree according to the imposed regularization.
The SV spherical harmonic coefficients (first time derivative of the spline model) for the internal field in epoch 2014.0, up to degree and order 8, were then output to 0.01 nT/year. We also provided SV predictions to degree and order 13 as a test secular variation model.
No uncertainty estimates were provided with our candidate models, since we are unable to calculate satisfactory estimates. The largest errors are likely biases caused by unmodelled sources (Sabaka et al. 2015) which cannot be assessed using a formal model error covariance matrix, or by constructing models using the same technique from independent datasets.
Results and discussion
Fit to satellite data
Number of data points N and the Huber weighted mean and rms misfits (in nT for the satellite data and in nT/year for the ground observatory data) of the data to the CHAOS5l parent field model
CHAOS5l  

Data  Component  N  Mean  rms 
Ørsted  F _{polar}  121,293  0.46  3.44 
F _{nonpolar}+B _{ B }  367,713  0.16  2.37  
B _{⊥}  87,672  −0.05  7.37  
B _{3}  87,672  0.15  3.35  
B _{ r }  87,672  0.13  4.47  
B _{ θ }  87,672  0.23  5.36  
B _{ ϕ }  87,672  0.00  5.03  
CHAMP  F _{polar}  188,015  −0.37  4.90 
F _{nonpolar}+B _{ B }  497,394  −0.09  2.07  
B _{⊥}  497,394  −0.02  3.30  
B _{3}  497,394  0.07  3.42  
B _{ r }  497,394  0.02  2.77  
B _{ θ }  497,394  0.10  3.56  
B _{ ϕ }  497,394  −0.01  2.71  
SACC  F _{polar}  26,118  0.43  3.78 
F _{nonpolar}  86,603  0.40  2.72  
Swarm A  F _{polar}  17,485  −0.03  3.80 
F _{nonpolar}+B _{ B }  53,137  −0.01  2.09  
B _{⊥}  53,137  −0.05  2.79  
B _{3}  53,137  0.05  2.72  
B _{ r }  53,137  −0.01  1.83  
B _{ θ }  53,137  0.18  2.95  
B _{ ϕ }  53,137  −0.16  2.69  
Swarm B  F _{polar}  17,774  0.15  3.65 
F _{nonpolar}+B _{ B }  53,253  −0.06  2.07  
B _{⊥}  53,253  −0.03  2.80  
B _{3}  53,253  0.08  2.84  
B _{ r }  53,253  −0.02  1.99  
B _{ θ }  53,253  0.22  3.00  
B _{ ϕ }  53,253  −0.13  2.71  
Swarm C  F _{polar}  16,697  0.13  3.82 
F _{nonpolar}+B _{ B }  49,984  0.05  2.09  
B _{⊥}  49,984  −0.05  2.80  
B _{3}  49,984  0.04  2.80  
B _{ r }  49,984  0.02  1.93  
B _{ θ }  49,984  0.11  3.00  
B _{ ϕ }  49,984  −0.15  2.71  
Observatory  d B _{ r }/d t  21,733  0.13  3.91 
d B _{ θ }/d t  21,733  −0.02  3.83  
d B _{ ϕ }/d t  21,733  −0.00  3.12 
Fit to observatory monthly means
Time dependence of secular variation coefficients
Spectral properties of DTU IGRF12 candidate models
Rationale for choice of SV candidate
The construction and evaluation of SV candidates have long been considered the most challenging aspects of producing a new IGRF generation (Lowes 2000). Here, we derived our IGRF12 SV candidate taking the position that it is not yet possible to reliably predict future SA events (for example related to geomagnetic jerks) since prognostic forward models capturing the relevant core physics on short time scales are not yet available. We therefore take our estimate of the current SV to be our prediction of the SV for 2015.0 to 2020.0, essentially assuming no average SA or equivalently that the SA will average to zero over the upcoming 5 years. As discussed above, we take the SV from 2014.0 in our spline model as our estimate of the present SV, to avoid problems related to spline model endeffects.
Secular acceleration pulses in 2006, 2009 and 2013
A striking example of the oscillatory core surface SV that now requires an explanation is that the strongest feature in the radial SA under the eastern edge of Brazil was negative in 2006, positive in 2009 and negative again in 2013. Gillet et al. (2015) have proposed that such events can be explained by oscillations in the nonzonal (i.e. nonaxisymmetric) part of the azimuthal (eastwest) quasigeostrophic core flow at low latitudes. Chulliat et al. (2015) suggest an alternative idea that fast equatorial MHD waves in a stratified layer at the top of the core may be responsible. The identification of the 2013 pulse in CHAOS5 opens the door to further detailed study of such hypotheses. The occurrence of SA pulses in 2006.2, 2009.2 and 2013.9 also leads us to wonder whether the next pulse, expected to have the same polarity as the 2009 event, might occur around 2016, before the end of the nominal Swarm mission. Since Swarm should be providing highquality magnetic field measurements with unprecedented spacetime coverage throughout this period, it promises to be an exciting opportunity to characterize a SA pulse in great detail.
Conclusions
We have presented the CHAOS5 geomagnetic field model, including the parent model CHAOS5l from which DTU’s candidate field models for IGRF12 were derived. Details of the magnetic data used to construct CHAOS5 (including their selection and processing) have been documented, with a focus on data from ESA’s Swarm satellite constellation. The CHAOS5 model parameterization and estimation scheme has been reported, and details given concerning how the candidate field models for IGRF12 were extracted.
We find acceptable misfits of CHAOS5 to both ground observatory and Swarm data in 2014, and no evidence of unreasonable model oscillations or spurious trends. CHAOS5 thus provides a consistent representation of magnetic data from six independent satellites (Ørsted, CHAMP, SACC and Swarm A, B, C), as well as ground observatory data, between 1999 and 2015. The Huber weighted rms misfit of the CHAOS5 model to the Swarm vector field data is found to be lower than the Huber weighted rms misfit to the Ørsted and CHAMP vector field data (where either 1 or 2 star cameras were operating), for example considering the radial field component, Huber weighted rms misfits of 1.83, 1.99 and 1.93 nT to Swarm A, B, C data were obtained, compared to 2.77 nT for CHAMP. Overall, the Swarm data seems very well suited for geomagnetic field modelling, and we had no hesitation in using field models based on Swarm L1b magnetic field data, version 0301/0302, to construct our IGRF12 candidate models.
CHAOS5 provides evidence of a secular acceleration pulse around 2013 at the core surface. The amplitude of this new 2013 pulse appears to be larger than the 2009 pulse, and in the Atlantic sector of the core surface, its spatial pattern is well correlated to the 2006 pulse and anticorrelated to 2009 pulse (see also Chulliat et al. 2015). If another pulse happens around 2016, then Swarm will be ideally placed to provide a much more detailed characterization of this presently poorly understood phenomenon.
The CHAOS5 model, as well as the Matlab software to evaluate it, is available from www.spacecenter.dk/files/magneticmodels/CHAOS5/.
Declarations
Acknowledgements
We wish to thank Benoit Langlais, an anonymous reviewer, and the guest editor Erwan Thébault for constructive comments that helped us to improve the manuscript. ESA is thanked for providing prompt access to the Swarm L1b data. The staff of the geomagnetic observatories and INTERMAGNET are thanked for supplying highquality observatory data, and BGS is thanked for providing us with checked and corrected observatory hourly mean values. The support of the CHAMP mission by the German Aerospace Center (DLR) and the Federal Ministry of Education and Research is gratefully acknowledged. The Ørsted Project was made possible by extensive support from the Danish Government, NASA, ESA, CNES, DARA and the Thomas B. Thriges Foundation. Support for CCF by the Research Council of Norway through the Petromaks programme, by ConocoPhillips and Lundin Norway, and by the Technical University of Denmark is highly appreciated.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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