Modeling the global geomagnetic field
Advanced Earth electromagnetism Homework-3
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.
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:
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
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.
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
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