Advanced Earth electromagnetism Homework-3

Build an internal geomagnetic using observatory data
Wu Xianshu 12132710

Methods and data

According to the Gauss Theory of geomagnetism, we can use the spherical harmonic analysis to describe the earth’s main magnetic field. Suppose the magnetic potential of the geomagnetic field obeys Laplace’s equation at the surface of the Earth as well as outside of the Earth, the model can be written as:

The only uncertainty in the formula is $g_n^m$ and $h_n^m$, which called Gauss Coefficient, and when we have the model and the data, we can get the parameters value by inversion methods. The problem becomes solving the following system of linear algebraic equations (the observations are taken at a spherical surface of the Earth’s mean radius, $r=a$ ):

The data in $\mathbf{d}$ are download from INTERMAGNET, the website is https://www.intermagnet.org/data-donnee/download-eng.php. I selected the definitive minute data on 2020-03-05 (Table 1), all available observation stations are included, Figure 1 is a map showing the observatories I used. the 24-hour data of each station are averaged to obtain the observed values of the day, after eliminating the outliers. I wrote a MATLAB script to implement the above process, and built a model up to spherical harmonic degree 4.

globalmagnetic-stations

Fig 1. Map showing the observatories I used

Tab 1. A table of data

Index Name Latitude Longitude X Y Z Date Index Name Latitude Longitude X Y Z Date
1 abg 18.62 72.87 38255.32861 290.4123611 20393.29507 20200305 42 kdu -12.686 132.472 35457.00736 1754.704167 -29582.17056 20200305
2 abk 68.358 18.823 11263.29438 1862.256042 52149.73104 20200305 43 kep -54.282 323.507 15328.23479 -1902.493542 -22927.075 20200305
3 aia -65.25 295.73 19743.62451 5529.188194 -31781.59625 20200305 44 kmh -26.541 18.11 10625.24368 -3290.684375 -24317.20576 20200305
4 asp -23.761 133.883 30118.9534 2384.582361 -43746.81958 20200305 45 kny 31.424 130.88 32523.31208 -3937.392708 33352.49993 20200305
5 bdv 49.08 14.015 20408.24278 1476.498472 44384.11257 20200305 46 kou 5.21 307.269 26566.90834 -8649.159903 6534.03419 20200305
6 bel 51.836 20.789 18910.50938 2155.445069 46753.62417 20200305 47 lrm -22.222 114.101 30377.57938 113.0718056 -43262.77563 20200305
7 bfo 48.331 8.325 21000.85431 971.8170139 43518.68674 20200305 48 lyc 64.612 18.748 12954.21903 1788.108194 50824.92569 20200305
8 bmt 40.3 116.2 27901.35056 -4109.268472 47580.71354 20200305 49 mab 50.298 5.683 20026.52454 676.6008535 44643.7574 20200305
9 box 58.07 38.23 15149.76771 3423.754653 50605.48924 20200305 50 maw -67.6 62.88 6637.389861 -17331.47576 -45610.43313 20200305
10 brd 49.87 260.026 15215.71021 1349.588194 54525.18188 20200305 51 mcq -54.5 158.95 10702.30986 6865.926042 -62729.04792 20200305
11 brw 71.32 203.38 8896.940417 2040.051944 56567.42097 20200305 52 mea 54.616 246.653 13676.0725 3502.193194 55141.94083 20200305
12 bsl 30.35 270.364 23897.89611 -534.3659028 40619.165 20200305 53 mmb 43.91 144.189 25860.92479 -4160.602986 42601.24806 20200305
13 cbb 69.123 254.969 5052.641181 515.9493056 58112.67063 20200305 54 ngk 52.072 12.675 18870.10889 1331.096806 45797.97604 20200305
14 cki -12.187 96.834 34897.57944 -1267.207778 -32411.88188 20200305 55 nur 60.508 24.655 14746.53667 2287.914931 50328.62708 20200305
15 clf 48.025 2.26 21268.24681 374.1634028 43038.32201 20200305 56 nvs 54.85 83.23 15977.18264 2267.85125 57794.77472 20200305
16 cmo 64.871 212.139 12071.0441 3662.327361 55060.79042 20200305 57 ott 45.403 284.448 18006.06139 -4293.210347 50757.78646 20200305
17 cnb -35.314 149.363 23121.84965 5192.365069 -52948.43965 20200305 58 pag 42.515 24.177 23724.57785 2052.341181 41095.60694 20200305
18 cpl 17.293 78.92 39453.13201 -510.5802778 17435.07875 20200305 59 pet 52.971 158.248 21521.23347 -2453.807986 47410.99722 20200305
19 csy -66.28 110.53 -1183.919167 -8905.680139 -63455.59493 20200305 60 pil -31.667 296.119 18525.13889 -1971.772847 -12926.00313 20200305
20 cta -20.09 146.264 31415.04941 4032.899791 -37523.00592 20200305 61 pst -51.704 302.107 17998.51828 816.7563025 -21589.41492 20200305
21 dou 50.1 4.599 20166.69694 564.5751389 44377.19028 20200305 62 res 74.69 265.105 2821.225278 -966.5449306 57412.04722 20200305
22 dur 41.65 14.467 24664.39583 1661.169444 39635.57229 20200305 63 sba -77.85 166.762 -10322.08271 5515.313542 -65094.66458 20200305
23 ebr 40.957 0.333 25322.75201 348.5455556 37632.16313 20200305 64 sbl 43.932 299.991 20099.86778 -6159.527222 45830.59819 20200305
24 frd 38.201 282.63 21211.1253 -3948.038707 45969.93919 20200305 65 sfs 36.667 354.055 27692.98333 -437.525625 33061.71813 20200305
25 frn 37.091 240.281 22610.76882 5106.934306 41905.74014 20200305 66 she -15.961 354.253 16010.89917 -3784.684444 -25849.75875 20200305
26 fur 48.165 11.277 20993.37243 1266.938681 43689.14313 20200305 67 sit 57.058 224.675 15003.98639 5020.095972 52959.09139 20200305
27 gan -0.695 73.154 38064.10701 -2720.955278 -12852.37549 20200305 68 sjg 18.111 293.85 26371.23667 -6125.861667 24778.29694 20200305
28 gck 44.63 20.77 22705.50861 1968.7925 42322.95653 20200305 69 spg 60.542 29.716 14451.77243 2766.171111 50554.4159 20200305
29 gng -31.356 115.715 24154.41201 -695.7534028 -52657.5475 20200305 70 spt 39.547 355.651 26157.41243 -212.2654861 35974.31007 20200305
30 gui 28.321 343.559 27788.94125 -3297.265486 22532.55771 20200305 71 tam 22.792 5.53 33853.35556 317.1153472 17125.37632 20200305
31 had 50.995 355.516 19815.59375 -487.1833333 44458.94479 20200305 72 tdc -37.067 347.684 9158.353472 -3520.939514 -22389.23111 20200305
32 hbk -25.883 27.707 12484.80071 -4183.221111 -24963.03222 20200305 73 thl 77.47 290.77 3048.201319 -2836.9075 56138.82771 20200305
33 her -34.425 19.225 9607.833819 -4713.809861 -22995.42292 20200305 74 thy 46.9 17.893 21518.14722 1798.217431 43480.7241 20200305
34 hlp 54.603 18.811 17482.39139 1693.542639 47405.90799 20200305 75 tsu -19.202 17.584 14031.38007 -2157.920833 -25488.01035 20200305
35 hrb 47.875 18.19 20998.77708 1814.178403 44071.71535 20200305 76 tuc 32.174 249.266 23874.2234 3757.009861 40291.74611 20200305
36 hyb 17.42 78.55 39586.70764 -409.4116667 18254.32278 20200305 77 ups 59.903 17.353 15093.9141 1687.224931 49301.94847 20200305
37 iqa 63.756 291.49 8624.755702 -3943.983658 56107.03804 20200305 78 vic 48.517 236.583 18091.75582 5186.817181 49871.91367 20200305
38 irt 52.17 104.45 18339.39819 -1321.375208 57631.08368 20200305 79 vna -70.683 351.718 18066.2466 -4484.883403 -33255.14722 20200305
39 izn 40.5 29.72 25075.98528 2459.532708 40544.37868 20200305 80 vss -22.4 316.35 16425.94375 -6916.265278 -14993.05438 20200305
40 jco 70.356 211.201 8698.91 2596.038472 56590.54382 20200305 81 wic 47.928 15.862 21013.83049 1646.833889 43916.92826 20200305
41 kak 36.232 140.186 29794.62132 -4036.528194 35869.73299 20200305 82 wng 53.725 9.053 18186.19222 1002.707153 46352.84486 20200305

Results

The 24 Gaussian coefficients I have calculated are in Table 2

Tab 2. Gauss Coefficients my obtained (using quasi-Schmidt normalization)

Coefficent Degree(n) Order(m) Value Coefficent Degree(n) Order(m) Value
g 1 0 -29017.523 h 3 1 -42.65915408
g 1 1 -1858.68841 h 3 2 600.217349
h 1 1 4903.10927 h 3 3 -633.5583365
g 2 0 -2497.135416 g 4 0 1132.882537
g 2 1 2617.981296 g 4 1 1296.172272
g 2 2 2178.890222 g 4 2 -2.933163263
h 2 1 -3201.763686 g 4 3 -357.3671095
h 2 2 -923.4214529 g 4 4 120.6542625
g 3 0 805.1006385 h 4 1 486.9660327
g 3 1 -2166.619627 h 4 2 -230.1865231
g 3 2 1281.757376 h 4 3 195.544086
g 3 3 90.1221627 h 4 4 -207.4272451

In order to evaluate the fitting effect of the model on the data, the prediction data on the original observation point were obtained by using $\mathbf{E} = \mathbf{A}\mathbf{m}$, and the error between the observation value and the prediction value was calculated, as shown in Figure 2, 3:

globalmagnetic-est_obs

Fig 2. Maps of observation value and prediction value

globalmagnetic-errors

Fig 3. Maps of errors at each observational stations

Global model from inverse Gauss Coefficient

The inverse Gauss coefficient can be used to calculate the global magnetic field distribution, which has been shown in Figure 4-11

globalmagnetic-F-c

Fig 4. Contour-maps of the total field intensity F

globalmagnetic-Br-c

Fig 5. Contour-maps of the radial component Br

globalmagnetic-D-c

Fig 6. Contour-maps of the Declination D

globalmagnetic-I-c

Fig 7. Contour-maps of the Inclination I

globalmagnetic-F-p

Fig 8. Contourf-maps of the total field intensity F

globalmagnetic-f9

Fig 9. Contourf-maps of the radial component Br

globalmagnetic-D-p

Fig 10. Contourf-maps of the Declination D

globalmagnetic-I-p

Fig 11. Contourf-maps of the Inclination I

It is obvious that the South Atlantic magnetic anomaly is consistent with the known characteristics of the geomagnetic field, indicating that our model can reflect the characteristics of the geomagnetic field to a certain extent.

Comparison with IGRF-13

Compute the spectrum in SH degree by:

and compare them with the results of IGRF-13 (Figure 12) shown that the inversed Gauss coefficients are in good agreement with IGRF-13 model. The characteristics of the main earth’s magnetic field are almost the same, except for a small deviation when the degree is high.

globalmagnetic-spectrum

Fig 12. A plot comparing the spectra of my model with that of IGRF-13

Discussion

Through the inversion of the observed data, I got 24 Gauss coefficients, and used them to establish the global magnetic field model, draw the magnetic field distribution on the surface of earth. Compared with the IGRF-13 model, it has good consistency, but there are still subtle differences. The reason is that when we sift through the data, the distribution of observation stations is not well-distributed enough, and we can not achieve uniform distribution on a global scale. Therefore, the inversed model is greatly affected by the local data, even if we increase the degree of the spherical harmonic coefficient will not help, there will be even greater difference. If our model is to be more accurate, it will need to include satellite data that can orbit the Earth around, which will also take into account the effects of external fields on the geomagnetic field, making the model more accurate.

Code

The code can be attained at my github repository

Reference

[1] 聂琳娟, 许智铭. 利用SWARM星群磁测数据反演地球主磁场模型WHU-MM01[J]. 测绘地理信息, 2020, 45(05): 16–19.

[2] 徐文耀, 区加明, 杜爱民. 地磁场全球建模和局域建模[J]. 地球物理学进展, 地球物理学进展, 2011, 26(2): 398–415.

[3] Monika Korte, Catherine Constable. Continuous global geomagnetic field models for the past 3000 years[J]. Physics of the Earth and Planetary Interiors, 2003, 140(1): 73–89.

[4] N. W. Peddie. USGS model coefficients for continental U.S. and Hawaii (1985)[J]. Planetary and Space Science, 1992, 40(4): 561.

[5] An Homage to Gauss and His Model of the Earth’s Magnetic Field[J]. CMS Notes, .

[6] Davis J. Mathematical Modeling of Earth’s Magnetic Field[J]. .

[7] 白春华, 徐文耀, 康国发. 地球主磁场模型[J]. 地球物理学进展, 2008(4): 1045–1057.

[8] 冯春. Matlab实现IGRF国际地磁参考场模型的计算[J]. 内蒙古石油化工, 2014, 40(12): 43–46.

[9] 安振昌. 地磁场区域模型与全球模型的比较和讨论[J]. 物探与化探, 1991(04): 248–254.

[10] 刘元元, 王仕成, 张金生, 等. 最新国际地磁参考场模型IGRF11研究[J]. 地震学报, 2013, 35(1): 125-131+133-134+138.

[11] International Geomagnetic Reference Field (IGRF-13)[EB/OL]. /2022-05-28. https://wdc.kugi.kyoto-u.ac.jp/igrf/index.html.

[12] Alken P, Thébault E, Beggan C D, 等. International Geomagnetic Reference Field: the thirteenth generation[J]. Earth, Planets and Space, 2021, 73(1): 49.

[13] Finlay C C, Kloss C, Olsen N, 等. The CHAOS-7 geomagnetic field model and observed changes in the South Atlantic Anomaly[J]. Earth, Planets and Space, 2020, 72(1): 156.