Magnetic anomaly due to a vertical-sided polygon
Calculate the magnetic anomaly due to one or more buried, vertical-sided polygons using Plouff's algorithm
Contents
Introduction
Many geologic units can be characterized by complex outlines in plan view and a thickness. A vertical-sided polygon, with many faces and horizontal top and bottom can be used to simulate such a geometry. A method is presented for calculating the components of the magnetic field strength anomaly, and the total magnetic field anomaly for a vertical-sided polygon. The polygon has many vertical faces. The top and bottom of the polygon are flat. The method developed by Plouff (1976) is used. Note that since magnetic anomalies add, many vertical-sided polygons can be added together to simulate a complex 3D shape, as proposed by Talwani (1965).
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Mathematical Model
A method for calculating the total magnetic field anomaly due to bodies of arbitrary geometry was developed by Manik Talwani (1965). Talwani was the first person to subdivide the equation for the magnetic field induced by a body, into separate geometric and magnetic components, thus, simplifying the calculation. In the most general case, consider a 3D magnetic body in an orthogonal coordinate system ($x, y, z$). Usually by convention, $x$ points North, $y$ points East, and $z$ points down.
The magnetic potential, $\Omega$, at a point located at the origin ($x=0$, $y = 0$, $z = 0$) due to a volume element in the 3D body is: $$\Omega = \frac{\vec \mu \cdot \vec R}{R^3} $$ where $\vec \mu$ is the magnetic dipole moment (amp m$^2$) of the volume element of dimension $dx, dy, dz$, and $\vec R$ is a vector of length $R$, the distance from the origin of the coordinate system to the center of the volume element. The magnetic potential is expressed in amperes. Assuming the magnetization, $\vec M$, is constant within the volume element, the magnetic dipole moment is: $$\vec \mu = \vec M dx dy dz $$ The components of magnetization are $$m_x = m_i \cos (I_m) \cos (D_m) $$ $$m_y = m_i \cos (I_m) \sin (D_m) $$ $$m_z = m_i \sin (I_m) $$ where $m_i$ is the magnitude of magnetization (amp/m), $I_m$ is the inclination and $D_m$ is the declination of the magnetization (radians). For example, the vector of magnetization is parallel to the Earth's magnetic field if the magnetization is induced, anti-parallel for reversely magnetized rocks with high remanent magnetization, or a completely different orientation if the rocks have high remanent magnetization and have been tectonically rotated.
Then: $$\Omega = \frac{m_x x + m_y y + m_z z}{R^3} \Delta x \Delta y \Delta z$$ The three components of the magnetic induction (magnetic field strength) outside the body are: $$\vec B_x = \mu_o \int \int \int - \frac{\partial \Omega}{\partial x} dx dy dz$$ $$\vec B_y = \mu_o \int \int \int - \frac{\partial \Omega}{\partial y} dx dy dz$$ $$\vec B_z = \mu_o \int \int \int - \frac{\partial \Omega}{\partial z} dx dy dz$$ where $\mu_o$ is the magnetic permeability ($4 \pi \times 10^{-7}$ T m/amp). Substituting in the expression for $\Omega$ allows for a system of equations to be developed $$\vec B_x = \mu_o(V_1 m_x + V_2 m_y + V_3 m_z) $$ $$\vec B_y = \mu_o(V_2 m_x + V_4 m_y + V_5 m_z) $$ $$\vec B_z = \mu_o(V_3 m_x + V_5 m_y + V_6 m_z) $$ where $$V_1 = \int \int \int \frac{3x^2-R^2}{R^5} dx dy dz$$ $$V_2 = \int \int \int \frac{3xy}{R^5} dx dy dz$$ $$V_3 = \int \int \int \frac{3xz}{R^5} dx dy dz$$ $$V_4 = \int \int \int \frac{3y^2-R^2}{R^5} dx dy dz$$ $$V_5 = \int \int \int \frac{3yz}{R^5} dx dy dz$$ $$V_6 = \int \int \int \frac{3z^2-R^2}{R^5} dx dy dz$$ $V_1-V_6$ are integrated over the volume of the 3D body. They are dimensionless, so it does not matter what scale (e.g., what units of $x$, $y$ and $z$) is chosen for the problem, as long as the same scale is used for each dimension. The units of magnetic field strength are Tesla, usually converted to nanoTesla (1 nT$ = 1 \times 10^{-9}$ T). The three components of magnetic field strength can be rewritten in units of nanoTesla with a proportionality constant $400 \pi = (4 \pi \times 10^{-7}) (1 \times 10^9$) (nT m/amp):
$$\vec B_x = 400 \pi(V_1 m_x + V_2 m_y + V_3 m_z) $$ $$\vec B_y = 400 \pi(V_2 m_x + V_4 m_y + V_5 m_z) $$ $$\vec B_z = 400 \pi(V_3 m_x + V_5 m_y + V_6 m_z) $$As long as the magnetic anomaly is small compared to the magnitude of the Earth's magnetic field, which is almost always the case, then the magnetic anomaly is: $$B = L B_x + M B_y + N B_z $$ where $L, M, N$ are the direction cosines of the Earth's magnetic field, or other external magnetic field. $$L = \cos (I_E) \cos (D_E) $$ $$M = \cos (I_E) \sin (D_E) $$ $$N = \sin (I_E) $$ $I_E$ is the inclination and $D_E$ is the declination of the Earth's magnetic field at the location of the anomaly (in radians). For example, they might be provided by the International Geomagnetic Reference Frame (IGRF).
Talwani's approach is extremely helpful because the volume integrals, $V_1-V_6$, are calculated independently of the vector of magnetization (Talwani, 1965). The volume integrals $V_2, V_3$ and $V_5$ produce asymmetry in the magnetic anomaly, depending on the vector of magnetization. This gives rise to the dipolar nature of magnetic anomalies. Since Talwani's development of the volume integrals, numerous computationally efficient and simplifying solutions have been proposed to estimate the volume integrals (e.g., Bhattacharya, 1964; Rao et al., 1973; Plouff, 1976; Cady, 1980; Rao and Babu, 1991, 1993).Here the geometric solution of Plouff (1976) is used to calculate the magnetic anomaly due to a vertical-sided polygon.
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For an observation point located at $P(0,0,0)$, the distances to the corners of the first face (polygon side) are:
$$r_{11} = \sqrt{x_1^2 + y_1^2 + z_1^2} $$
$$r_{12} = \sqrt{x_1^2 + y_1^2 + z_2^2} $$
$$r_{21} = \sqrt{x_2^2 + y_2^2 + z_1^2} $$
$$r_{22} = \sqrt{x_2^2 + y_2^2 + z_2^2} $$
Note the following geometric relationships:
$$ c = \frac{\Delta y}{ \Delta s} $$
$$ s = \frac{\Delta x}{\Delta s} $$
$$ p = \frac{(x_1 y_2 - x_2 y_1)}{\Delta s} $$
$$ d_1 = x_1 s + y_1 c $$
$$ d_2 = x_2 s + y_2 c $$
$$f = \ln \left [\frac{r_{22}+z_2}{r_{12}+z_2} \frac{r_{11} + z_1}{r_{21}+ z_1} \right ] $$
$$q = \ln \left [\frac{r_{22}+d_2}{r_{12}+d_1} \frac{r_{11} + d_1}{r_{21}+ d_2} \right ] $$
$$w = \tan^{-1} \left [ \frac{z_2 d_2}{p r_{22}} \right ] - \tan^{-1} \left [ \frac{z_2 d_1}{p r_{12}} \right ] - \tan^{-1} \left [ \frac{z_1 d_2}{p r_{21}} \right ] + \tan^{-1} \left [ \frac{z_1 d_1}{p r_{11}} \right ]$$
Model Geometry
Plouff(1976) formulated the volume integrals for a vertical-sided polygon like the one shown in the following figure. The model geometry is for the magnetic anomaly at point $P$ due to the vertical-sided polygon. The geometry is simplified compared to a general 3D body because the sides (faces) of the polygon are vertical and the top and bottom are flat. This allowed Plouff to formulate the solution for a polygon of any thickness. Looking down on the polygon top, the coordinates of the vertices of the faces are arranged in a clockwise order. This clockwise order is essential to solve the volume integrals, $V_1-V_6$, properly. The depth ($z$) is positive downward, the Northing ($x$) is positive North, and the Easting ($y$) is positive East. In this example. the Northing coordinate os given first, followed by the Easting, followed by the depth and $z_2 > z_1 > 0$ for the buried polygon.
In plan view, looking down on the top of the prism, the geometry between the point $P(0,0)$ and one face is:
Plouff found that with these geometric terms, the volume integrals for one face (polygon side) become: $$V_{1,i} = scf - c^2w $$ $$V_{2,i} = scw + c^2f $$ $$V_{3,i} = cq $$ $$V_{4,i} = -(scf + s^2w) $$ $$V_{5,i} = -(sq) $$ $$V_{6,i} = w $$
He used this formulation to allow the thickness of the vertical-sided polygon ($z_2 - z_1$) to be arbitrarily large. The volume integrals for all faces are summed. The summation proceeds in clockwise order, updating the geometric terms for each face. For $n$ polygon sides: $$V_1 = \sum_{i=1}^n scf - c^2w $$ $$V_2 = \sum_{i=1}^n scw + c^2f $$ $$V_3 = \sum_{i=1}^n cq $$ $$V_4 = \sum_{i=1}^n -(scf + s^2w) $$ $$V_5 = \sum_{i=1}^n -(sq) $$ $$V_6 = \sum_{i=1}^n w $$ These summations of the volume integrals are used to calculate the components of the magnetic anomaly and/or the total magnetic anomaly.
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Model assumptions
The Plouff model assumes that:- The polygon has vertical faces and a flat top and bottom.
- $\Delta s \ne 0$ - vertices cannot be in the same place and must be arranged in clockwise order (looking down on the top of the polygon).
- $z_1 > 0$ - the observation point cannot lie exactly on a polygon edge. This will result in division by zero.
- The magnetic anomaly is relatively small compared to the magnitude of the external magnetic field (e.g., Earth's field).
Model calculations
[Top]perl scripts
Example 1, simple perl script to calculate a N–S profile over a buried 4-sided polygon
In this example, the magnetic anomaly due to a 20 x 20 x 10 cube is calculated along a N–S profile. The faces of the cube are parallel to the major axes. The perl module, Math::Trig, must be installed for this script to run. Use the perl installation tool CPAN to install this module.
#calculate magnetic anomaly for
#polygonal prism
#C. B. Connor
use warnings;
use Math::Trig;
$pi = 3.14159;
#experimental 4-sided prism
#x must be the northing (north coordinate),
#y must be the easting (east coordinate),
#coordinates must be entered in clockwise order
#map units can be arbitary by must be consistent
#for example, all units in meters
$x[0] = -10;
$y[0] = -10;
$x[1] = 10;
$y[1] = -10;
$x[2] = 10;
$y[2] = 10;
$x[3] = -10;
$y[3] = 10;
#depth to top
$z1 = 10;
#depth to bottom
$z2 = 20;
$sides = 4;
#set magnetization (amp/m)
$minc = 45*$pi/180; #inclination down in rad
$mdec = 0*$pi/180; #declination east in rad
$mi = 0.9;
#calculate direction cosines for magnetiation
$ml = cos($minc) * cos($mdec);
$mm = cos($minc) * sin($mdec);
$mn = sin($minc);
#components of magnetization in x,y,z directions
$mx = $mi * $ml;
$my = $mi * $mm;
$mz = $mi * $mn;
#set earths field
$einc = 45*$pi/180; #inclination down in rad
$edec = 0*$pi/180; #declination east in rad
$el = cos($einc) * cos($edec);
$em = cos($einc) * sin($edec);
$en = sin($einc);
#proportionality constant is
# magnetic permeability * 1e9
#to convert of nT
$prop = 400*$pi;
for ($j =0; $j<400; $j++) {
#px is the northing of the observation point
#py is the easting of the observation point
$px = $j-200;
$py = 0;
#set the volume integrals to zero
#for each observation point
$v1 = 0;
$v2 = 0;
$v3 = 0;
$v4 = 0;
$v5 = 0;
$v6 = 0;
#calculate volume integrals
for ($i=0; $i<$sides; $i++) {
#if last side of polygon
if($i == $sides-1) {
$x1 = $x[$i]-$px;
$y1 = $y[$i]-$py;
$x2 = $x[0]-$px;
$y2 = $y[0]-$py;
} else {
#all other polygon sides
$x1 = $x[$i]-$px;
$y1 = $y[$i]-$py;
$x2 = $x[$i+1]-$px;
$y2 = $y[$i+1]-$py;
}
#calculate geometry
$delta_x = $x2-$x1;
$delta_y = $y2-$y1;
$delta_s = sqrt($delta_x**2 + $delta_y**2);
#avoid division by zero if delta_s = 0
# if ($delta_s == 0) {$delta_s = 0.1};
$c = $delta_y/$delta_s;
$s = $delta_x/$delta_s;
$p = ($x1*$y2 - $x2*$y1)/$delta_s;
$d1 = $x1*$s + $y1*$c;
$d2 = $x2*$s + $y2*$c;
$r1sq = $x1**2 + $y1**2;
$r2sq = $x2**2 + $y2**2;
$r11 = sqrt($r1sq + $z1**2);
$r12 = sqrt($r1sq + $z2**2);
$r21 = sqrt($r2sq + $z1**2);
$r22 = sqrt($r2sq + $z2**2);
$f = log(($r22+$z2)/($r12+$z2) * ($r11+$z1)/($r21+$z1));
$q = log(($r22+$d2)/($r12+$d1) * ($r11+$d1)/($r21+$d2));
$w = atan2(($z2*$d2),($p*$r22)) - atan2(($z2*$d1),($p*$r12)) - atan2(($z1*$d2),($p*$r21)) + atan2(($z1*$d1),($p*$r11));
$v1 += $s*$c*$f - $c*$c*$w;
$v2 += $s*$c*$w + $c*$c*$f;
$v3 += $c*$q;
$v4 += -($s*$c*$f + $s*$s*$w);
$v5 += -($s*$q);
$v6 += $w;
}
#calculate the components of the magnetic field
$bx = $prop*($mx*$v1 + $my*$v2 + $mz* $v3);
$by = $prop*($mx*$v2 + $my*$v4 + $mz* $v5);
$bz = $prop*($mx*$v3 + $my*$v5 + $mz* $v6);
#calculate total anomaly
#this calculation works for anomaly << total field strength
$b_total = $el*$bx + $em*$by + $en*$bz;
#uncomment these lines to check volume integral
# v4 = -($v1+$v6)
#$test = -($v1+$v6);
#print"$v4 $test\n";
#print results
print " $px $b_total\n";
}
Profile data, (the output from the above script, follows a 2-column format, first column is northing (m), second column is total magnetic field anomaly (nT).
-200 0.406663024321587
-199 0.413433445594862
-198 0.420353458386151
-197 0.427427180736217
-196 0.434658866425999
-195 0.442052910189912
-194 0.449613853155896
-193 0.457346388527788
-192 0.465255367520014
-191 0.473345805553159
-190 0.481622888730453
-189 0.490091980606125
-188 0.498758629255631
-187 0.507628574674287
-186 0.516707756509662
-185 0.526002322150901
-184 0.53551863519578
-183 0.545263284304229
-182 0.555243092474835
-181 0.565465126751334
-180 0.575936708391195
-179 0.58666542351715
-178 0.597659134279273
-177 0.608925990552493
-176 0.620474442201681
-175 0.632313251943488
-174 0.644451508836735
-173 0.656898642440333
-172 0.669664437665556
-171 0.682759050375848
-170 0.696193023765809
-169 0.709977305562021
-168 0.724123266102136
-167 0.738642717332928
-166 0.753547932783008
-165 0.768851668570066
-164 0.784567185493606
-163 0.800708272291569
-162 0.817289270110613
-161 0.834325098277562
-160 0.851831281442102
-159 0.869823978172532
-158 0.888320011094969
-157 0.907336898667093
-156 0.926892888690754
-155 0.947006993661937
-154 0.967699028081265
-153 0.988989647835607
-152 1.01090039179915
-151 1.03345372576707
-150 1.05667308889569
-149 1.08058294279317
-148 1.10520882343897
-147 1.13057739610603
-146 1.15671651349528
-145 1.18365527728045
-144 1.21142410329438
-143 1.24005479059867
-142 1.26958059469335
-141 1.30003630515282
-140 1.33145832797668
-139 1.36388477299386
-138 1.39735554665284
-137 1.43191245057932
-136 1.46759928630204
-135 1.50446196658272
-134 1.54254863381459
-133 1.58190978599855
-132 1.62259841083311
-131 1.66467012852173
-130 1.70818334391016
-129 1.75319940866693
-128 1.79978279422731
-127 1.84800127631691
-126 1.89792613192474
-125 1.94963234966228
-124 2.00319885454131
-123 2.05870874826711
-122 2.11624956626844
-121 2.1759135527498
-120 2.23779795520935
-119 2.30200533995272
-118 2.36864393028867
-117 2.43782796924218
-116 2.50967810877356
-115 2.58432182768151
-114 2.66189388056526
-113 2.74253678043236
-112 2.82640131778752
-111 2.91364711929135
-110 3.00444324938529
-109 3.09896885857333
-108 3.19741388243905
-107 3.29997979584303
-106 3.40688042718606
-105 3.51834283810731
-104 3.63460827451778
-103 3.75593319544982
-102 3.88259038686332
-101 4.01487016828111
-100 4.15308170091379
-99 4.29755440685084
-98 4.44863950988077
-97 4.60671170962324
-96 4.77217100188399
-95 4.94544465955612
-94 5.12698938989603
-93 5.31729368578074
-92 5.51688039045265
-91 5.72630949744948
-90 5.94618120984654
-89 6.17713928567085
-88 6.41987469941241
-87 6.6751296530288
-86 6.94370197370774
-85 7.22644994005539
-84 7.52429758330329
-83 7.83824051573184
-82 8.16935234480778
-81 8.51879173867293
-80 8.88781021672915
-79 9.27776074821629
-78 9.69010725211155
-77 10.1264351034842
-76 10.5884627649145
-75 11.0780546768745
-74 11.597235558429
-73 12.1482062895006
-72 12.7333615686617
-71 13.3553095663606
-70 14.0168938231581
-69 14.7212176765243
-68 15.4716715386177
-67 16.2719633920743
-66 17.1261529219269
-65 18.0386897604703
-64 19.0144563892207
-63 20.0588163194573
-62 21.1776682616751
-61 22.3775070963204
-60 23.6654925753453
-59 25.0495268186513
-58 26.5383418237787
-57 28.1415983840369
-56 29.8699980125348
-55 31.735409700597
-54 33.7510136020756
-53 35.9314640335487
-52 38.2930745175227
-51 40.8540279742541
-50 43.634615589416
-49 46.6575083495444
-48 49.9480657423277
-47 53.534686657088
-46 57.4492080786908
-45 61.7273577218976
-44 66.4092672649839
-43 71.5400532524097
-42 77.170472957629
-41 83.3576623972571
-40 90.1659630748853
-39 97.667842629962
-38 105.944911979059
-37 115.089037203516
-36 125.20353757275
-35 136.404450600959
-34 148.821829404473
-33 162.601014790579
-32 177.903791676281
-31 194.909292899651
-30 213.814448425608
-29 234.833688280997
-28 258.197485963257
-27 284.149167425413
-26 312.939201374335
-25 344.815925198099
-24 380.011352213961
-23 418.720375048309
-22 461.07139086703
-21 507.086260542528
-20 556.627822674055
-19 609.334325217299
-18 664.542713664495
-17 721.207443802758
-16 777.828877082893
-15 832.414875297035
-14 882.508128247042
-13 925.313651676529
-12 957.946623650439
-11 977.783944873491
-10 982.850473469794
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-8 945.690152335931
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-6 850.584560023983
-5 785.77644233467
-4 712.273820007409
-3 631.96900791194
-2 546.433435615489
-1 456.920464084973
0 364.444062306879
1 269.902987298068
2 174.226049406408
3 78.5184536483919
4 -15.8111483064653
5 -106.968347626571
6 -192.767358936884
7 -270.734874753192
8 -338.341603565774
9 -393.342987754763
10 -434.158368370679
11 -460.176585690278
12 -471.880143263663
13 -470.741598283794
14 -458.934172429473
15 -438.96076817598
16 -413.310764468808
17 -384.213755767724
18 -353.507966236789
19 -322.604831462478
20 -292.517062257445
21 -263.919176547437
22 -237.218010045535
23 -212.619858483262
24 -190.187936702334
25 -169.888304876338
26 -151.624818124702
27 -135.264704781388
28 -120.656650229444
29 -107.64314801145
30 -96.0686092341785
31 -85.7844179029608
32 -76.6518409496597
33 -68.5434684471821
34 -61.3436748298639
35 -54.9484509128899
36 -49.2648515287381
37 -44.2102269672745
38 -39.7113513519658
39 -35.703522087788
40 -32.1296772672618
41 -28.939559150795
42 -26.0889391162894
43 -23.5389110216801
44 -21.255254455303
45 -19.2078659401814
46 -17.3702541542154
47 -15.7190941669662
48 -14.2338352527515
49 -12.896356795527
50 -11.6906670004549
51 -10.6026394665044
52 -9.61978308481759
53 -8.73104116396795
54 -7.92661611712546
55 -7.19781646038895
56 -6.53692325651807
57 -5.93707348941232
58 -5.39215817061692
59 -4.89673326050648
60 -4.44594173555185
61 -4.0354453516252
62 -3.66136484455304
63 -3.32022747584704
64 -3.00892097652858
65 -2.72465306781896
66 -2.4649158464987
67 -2.22745441718762
68 -2.0102392354682
69 -1.81144169642313
70 -1.62941256427634
71 -1.4626628916107
72 -1.30984712237456
73 -1.16974811239426
74 -1.04126383534663
75 -0.923395571784186
76 -0.815237404488558
77 -0.715966865717172
78 -0.624836601241314
79 -0.54116693287074
80 -0.464339215766191
81 -0.3937898995495
82 -0.329005213294259
83 -0.269516404120798
84 -0.214895467551545
85 -0.164751315131379
86 -0.118726331251709
87 -0.0764932767461231
88 -0.037752501759944
89 -0.00222943471499182
90 0.0303276820077301
91 0.0601498357090389
92 0.0874490894393571
93 0.112420251028674
94 0.135242383310807
95 0.156080171789441
96 0.175085164202398
97 0.192396894882914
98 0.208143905422172
99 0.222444671914197
100 0.235408447968608
101 0.247136031714414
102 0.257720464160647
103 0.267247665512064
104 0.275797015363306
105 0.283441882085517
106 0.290250106188823
107 0.29628444195278
108 0.301602961196723
109 0.306259422669454
110 0.310303610210636
111 0.313781642497744
112 0.31673625696695
113 0.31920707018923
114 0.321230816817943
115 0.322841568980883
116 0.324070937850913
117 0.324948258923015
118 0.325500762431622
119 0.325753730168088
120 0.325730639865803
121 0.325453298209148
122 0.3249419634228
123 0.324215458313089
124 0.323291274558523
125 0.322185668974594
126 0.320913752405008
127 0.319489571851388
128 0.317926186387419
129 0.316235737353097
130 0.314429513296845
131 0.312518010081044
132 0.310510986531126
133 0.308417515986738
134 0.306246034066608
135 0.304004382958382
136 0.301699852476222
137 0.299339218171951
138 0.296928776686739
139 0.294474378583997
140 0.291981458840538
141 0.289455065171169
142 0.286899884358964
143 0.28432026673132
144 0.281720248923701
145 0.279103575056641
146 0.276473716443177
147 0.273833889933483
148 0.271187074995961
149 0.26853602962955
150 0.265883305183157
151 0.263231260173392
152 0.260582073160113
153 0.257937754752176
154 0.255300158808558
155 0.25267099288196
156 0.250051827969283
157 0.247444107607292
158 0.244849156366334
159 0.242268187780716
160 0.23970231175348
161 0.237152541475895
162 0.234619799889084
163 0.232104925724059
164 0.229608679147454
165 0.227131747035559
166 0.224674747909245
167 0.22223823654686
168 0.21982270829581
169 0.217428603108746
170 0.215056309318207
171 0.212706167167274
172 0.210378472111933
173 0.208073477915132
174 0.205791399537891
175 0.203532415846906
176 0.201296672151738
177 0.199084282572534
178 0.196895332266142
179 0.194729879498753
180 0.192587957597761
181 0.190469576769212
182 0.188374725806973
183 0.186303373687875
184 0.184255471064875
185 0.182230951670556
186 0.180229733623909
187 0.178251720657703
188 0.176296803266928
189 0.174364859782158
190 0.172455757375328
191 0.170569353001142
192 0.168705494275446
193 0.166864020301918
194 0.16504476243799
195 0.163247545020007
196 0.161472186032001
197 0.159718497739073
198 0.157986287272394
199 0.156275357182817
The profile data is plotted using gnuplot.
The gnuplot script assumes that the output from plouff-mag.pl is piped to a file named plouff-mag.dat.
#this file plots a function using gnuplot. Typically to run
# type: gnuplot thin-plate-mag.gnu
# on the command line
reset
set termoption dash
# Axes label
set xlabel 'Horizontal distance North from plate center (m)'
set ylabel 'magnetic anomaly (nT)'
# Axes ranges
set xrange [-200:200]
set yrange [-1000:1000]
# Axes tics
set xtics 25
set ytics 250
set style fill transparent solid 0.15
set key Left
plot "plouff-mag.dat" using 1:2 with line lt rgb "black" notitle
set term pdf enhanced dashed
set output "plouf-mag.pdf"
replot
Example 2, simple perl script that calculates magnetic anomalies on a grid
This script is a modification of the previous code.
#calculate magnetic anomaly for
#polygonal prism
#C. B. Connor
use warnings;
use Math::Trig;
$pi = 3.14159;
#experimental 4-sided prism
#x must be the northing (north coordinate),
#y must be the easting (east coordinate),
#coordinates must be entered in clockwise order
#map units can be arbitary by must be consistent
#for example, all units in meters
$x[0] = -10;
$y[0] = -10;
$x[1] = 10;
$y[1] = -10;
$x[2] = 10;
$y[2] = 10;
$x[3] = -10;
$y[3] = 10;
#depth to top
$z1 = 10;
#depth to bottom
$z2 = 20;
$sides = 4;
#set magnetization (amp/m)
$minc = 45*$pi/180; #inclination down in rad
$mdec = 0*$pi/180; #declination east in rad
$mi = 0.9;
#calculate direction cosines for magnetiation
$ml = cos($minc) * cos($mdec);
$mm = cos($minc) * sin($mdec);
$mn = sin($minc);
#components of magnetization in x,y,z directions
$mx = $mi * $ml;
$my = $mi * $mm;
$mz = $mi * $mn;
#set earths field
$einc = 45*$pi/180; #inclination down in rad
$edec = 0*$pi/180; #declination east in rad
$el = cos($einc) * cos($edec);
$em = cos($einc) * sin($edec);
$en = sin($einc);
#proportionality constant is
# magnetic permeability * 1e9
#to convert of nT
$prop = 400*$pi;
for ($j =0; $j<101; $j++) {
for ($k =0; $k<101; $k++) {
#px is the northing of the observation point
#py is the easting of the observation point
$px = $j-50;
$py = $k-50;
#set the volume integrals to zero
#for each observation point
$v1 = 0;
$v2 = 0;
$v3 = 0;
$v4 = 0;
$v5 = 0;
$v6 = 0;
#calculate volume integrals
for ($i=0; $i<$sides; $i++) {
#if last side of polygon
if($i == $sides-1) {
$x1 = $x[$i]-$px;
$y1 = $y[$i]-$py;
$x2 = $x[0]-$px;
$y2 = $y[0]-$py;
} else {
#all other polygon sides
$x1 = $x[$i]-$px;
$y1 = $y[$i]-$py;
$x2 = $x[$i+1]-$px;
$y2 = $y[$i+1]-$py;
}
#calculate geometry
$delta_x = $x2-$x1;
$delta_y = $y2-$y1;
$delta_s = sqrt($delta_x**2 + $delta_y**2);
#avoid division by zero if delta_s = 0
# if ($delta_s == 0) {$delta_s = 0.1};
$c = $delta_y/$delta_s;
$s = $delta_x/$delta_s;
$p = ($x1*$y2 - $x2*$y1)/$delta_s;
$d1 = $x1*$s + $y1*$c;
$d2 = $x2*$s + $y2*$c;
$r1sq = $x1**2 + $y1**2;
$r2sq = $x2**2 + $y2**2;
$r11 = sqrt($r1sq + $z1**2);
$r12 = sqrt($r1sq + $z2**2);
$r21 = sqrt($r2sq + $z1**2);
$r22 = sqrt($r2sq + $z2**2);
$f = log(($r22+$z2)/($r12+$z2) * ($r11+$z1)/($r21+$z1));
$q = log(($r22+$d2)/($r12+$d1) * ($r11+$d1)/($r21+$d2));
$w = atan2(($z2*$d2),($p*$r22)) - atan2(($z2*$d1),($p*$r12)) - atan2(($z1*$d2),($p*$r21)) + atan2(($z1*$d1),($p*$r11));
$v1 += $s*$c*$f - $c*$c*$w;
$v2 += $s*$c*$w + $c*$c*$f;
$v3 += $c*$q;
$v4 += -($s*$c*$f + $s*$s*$w);
$v5 += -($s*$q);
$v6 += $w;
}
#calculate the components of the magnetic field
$bx = $prop*($mx*$v1 + $my*$v2 + $mz* $v3);
$by = $prop*($mx*$v2 + $my*$v4 + $mz* $v5);
$bz = $prop*($mx*$v3 + $my*$v5 + $mz* $v6);
#calculate total anomaly
#this calculation works for anomaly << total field strength
$b_total = $el*$bx + $em*$by + $en*$bz;
#uncomment these lines to check volume integral
# v4 = -($v1+$v6)
#$test = -($v1+$v6);
#print"$v4 $test\n";
#print results
#note the format of the output is easting, northing, mag
print " $py $px $b_total\n";
}
#a blank line is
#added for gnuplot
print " \n";
}
The output data grid follows a 3-column format, first column is easting (m), second column is northing (m), third column is total magnetic field anomaly (nT).
-50 -50 2.68557651174455
-49 -50 3.06108575170311
-48 -50 3.46176306347599
-47 -50 3.88878121876644
-46 -50 4.3433146538765
-45 -50 4.82653082426421
-44 -50 5.33958027921733
-43 -50 5.88358537092491
-42 -50 6.4596275207475
-41 -50 7.06873297785578
-40 -50 7.71185702212912
-39 -50 8.38986658484951
-38 -50 9.10352128791207
-37 -50 9.85345293533436
-36 -50 10.640143530269
-35 -50 11.4639019365763
-34 -50 12.3248393564376
-33 -50 13.2228438539216
-32 -50 14.1575542185686
-31 -50 15.128333531504
-30 -50 16.1342428681724
-29 -50 17.1740156442455
-28 -50 18.2460331823023
-27 -50 19.3483021433579
-26 -50 20.4784345258532
-25 -50 21.6336309814714
-24 -50 22.8106682280436
-23 -50 24.0058913507687
-22 -50 25.2152117700753
-21 -50 26.434111614225
-20 -50 27.65765516449
-19 -50 28.880507938745
-18 -50 30.0969638452158
-17 -50 31.3009806731092
-16 -50 32.4862239938069
-15 -50 33.6461193297482
-14 -50 34.7739122145092
-13 -50 35.8627355244858
-12 -50 36.905683219098
-11 -50 37.8958893920473
-10 -50 38.8266113208727
-9 -50 39.6913150152762
-8 -50 40.4837616151366
-7 -50 41.1980928840257
-6 -50 41.8289139889304
-5 -50 42.37137175516
-4 -50 42.8212266384029
-3 -50 43.1749167626842
-2 -50 43.429612530542
-1 -50 43.5832605154032
0 -50 43.634615589416
1 -50 43.5832605154034
2 -50 43.429612530542
3 -50 43.1749167626845
4 -50 42.8212266384026
5 -50 42.3713717551598
6 -50 41.8289139889303
7 -50 41.1980928840257
8 -50 40.4837616151366
9 -50 39.6913150152762
10 -50 38.8266113208722
11 -50 37.8958893920472
12 -50 36.9056832190981
13 -50 35.8627355244856
14 -50 34.7739122145092
15 -50 33.6461193297485
16 -50 32.4862239938068
17 -50 31.3009806731093
18 -50 30.0969638452157
19 -50 28.8805079387455
20 -50 27.6576551644898
21 -50 26.4341116142247
22 -50 25.2152117700752
23 -50 24.0058913507686
24 -50 22.8106682280437
25 -50 21.6336309814716
26 -50 20.4784345258529
27 -50 19.3483021433579
28 -50 18.2460331823021
29 -50 17.174015644246
30 -50 16.134242868173
31 -50 15.1283335315041
32 -50 14.157554218568
33 -50 13.2228438539213
34 -50 12.324839356437
35 -50 11.4639019365759
36 -50 10.6401435302684
37 -50 9.85345293533539
38 -50 9.10352128791271
39 -50 8.38986658484976
40 -50 7.71185702212874
41 -50 7.06873297785502
42 -50 6.45962752074673
43 -50 5.88358537092567
44 -50 5.3395802792172
45 -50 4.82653082426446
46 -50 4.343314653877
47 -50 3.88878121876644
48 -50 3.4617630634765
49 -50 3.06108575170311
50 -50 2.68557651174405
-50 -49 2.58784333617164
-49 -49 2.97337205376975
-48 -49 3.38560730707236
-47 -49 3.8258615339692
-46 -49 4.2954565993912
-45 -49 4.79571488784079
-44 -49 5.32794896220887
-43 -49 5.89344967898105
-42 -49 6.49347265728014
-41 -49 7.12922301050499
-40 -49 7.80183826583722
-39 -49 8.51236941897505
-38 -49 9.26176009990085
-37 -49 10.0508238612694
-36 -49 10.880219644106
-35 -49 11.7504255270081
-34 -49 12.6617109245088
-33 -49 13.6141074679445
-32 -49 14.6073788773349
-31 -49 15.6409902144437
-30 -49 16.7140769937921
-29 -49 17.8254147177003
-28 -49 18.9733894906027
-27 -49 20.1559704533588
-26 -49 21.370684855907
-25 -49 22.6145966516968
-24 -49 23.8842895447337
-23 -49 25.1758554445612
-22 -49 26.484889280905
-21 -49 27.8064910933461
-20 -49 29.1352762383818
-19 -49 30.4653944439492
-18 -49 31.7905582890183
-17 -49 33.104081493895
-16 -49 34.3989271788834
-15 -49 35.6677659898732
-14 -49 36.9030437072178
-13 -49 38.0970576582519
-12 -49 39.2420409552003
-11 -49 40.3302532912002
-10 -49 41.3540767601401
-9 -49 42.3061149331402
-8 -49 43.1792932367008
-7 -49 43.9669585441096
-6 -49 44.6629758194499
-5 -49 45.2618196468654
-4 -49 45.758658538112
-3 -49 46.1494300374879
-2 -49 46.430904831377
-1 -49 46.6007383135684
0 -49 46.6575083495444
1 -49 46.6007383135686
2 -49 46.4309048313763
3 -49 46.1494300374881
4 -49 45.7586585381121
5 -49 45.2618196468652
6 -49 44.6629758194493
7 -49 43.9669585441096
8 -49 43.1792932367004
9 -49 42.3061149331403
10 -49 41.3540767601403
11 -49 40.3302532911996
12 -49 39.2420409552004
13 -49 38.0970576582519
14 -49 36.9030437072178
15 -49 35.6677659898732
16 -49 34.3989271788839
17 -49 33.1040814938946
18 -49 31.7905582890174
19 -49 30.4653944439497
20 -49 29.1352762383809
21 -49 27.8064910933465
22 -49 26.4848892809046
23 -49 25.1758554445613
24 -49 23.8842895447342
25 -49 22.6145966516972
26 -49 21.3706848559073
27 -49 20.1559704533588
28 -49 18.9733894906025
29 -49 17.8254147176998
30 -49 16.7140769937922
31 -49 15.6409902144443
32 -49 14.6073788773347
33 -49 13.6141074679446
34 -49 12.6617109245076
35 -49 11.7504255270076
36 -49 10.8802196441052
37 -49 10.0508238612694
38 -49 9.26176009990085
39 -49 8.5123694189744
40 -49 7.801838265838
41 -49 7.12922301050398
42 -49 6.49347265728103
43 -49 5.89344967898067
44 -49 5.32794896220785
45 -49 4.7957148878418
46 -49 4.29545659939081
47 -49 3.82586153396983
48 -49 3.3856073070726
49 -49 2.97337205376986
50 -49 2.58784333617189
-50 -48 2.47633335012659
-49 -48 2.87163694414163
-48 -48 3.29524958885251
-47 -48 3.7486359854546
-46 -48 4.23327947444473
-45 -48 4.75067295230353
-44 -48 5.3023081909574
-43 -48 5.88966342195856
-42 -48 6.51418905252637
-41 -48 7.17729139052195
-40 -48 7.88031427079039
-39 -48 8.62451849725524
-38 -48 9.41105904446927
-37 -48 10.2409600001503
-36 -48 11.1150872770889
-35 -48 12.0341191798073
-34 -48 12.9985149784349
-33 -48 14.0084817199407
-32 -48 15.0639395944699
-31 -48 16.1644862713102
-30 -48 17.3093607232298
-29 -48 18.4974071673085
-28 -48 19.727039861591
-27 -48 20.9962096058814
-26 -48 22.3023728967112
-25 -48 23.6424647752983
-24 -48 25.0128764766044
-23 -48 26.4094390308607
-22 -48 27.8274139792712
-21 -48 29.2614923367267
-20 -48 30.705802860956
-19 -48 32.1539305654924
-18 -48 33.5989462413447
-17 -48 35.0334475296308
-16 -48 36.4496118180485
-15 -48 37.8392609240098
-14 -48 39.193937185765
-13 -48 40.5049902216839
-12 -48 41.7636732509143
-11 -48 42.961247511393
-10 -48 44.0890929793573
-9 -48 45.1388233036894
-8 -48 46.1024026325012
-7 -48 46.9722618401142
-6 -48 47.7414115687217
-5 -48 48.4035494858114
-4 -48 48.9531592276161
-3 -48 49.3855986486173
-2 -48 49.6971752224139
-1 -48 49.88520673247
0 -48 49.9480657423277
1 -48 49.8852067324696
2 -48 49.6971752224137
3 -48 49.3855986486171
4 -48 48.9531592276165
5 -48 48.4035494858108
6 -48 47.7414115687224
7 -48 46.9722618401145
8 -48 46.1024026325015
9 -48 45.1388233036896
10 -48 44.0890929793575
11 -48 42.9612475113928
12 -48 41.7636732509141
13 -48 40.5049902216836
14 -48 39.193937185765
15 -48 37.8392609240088
16 -48 36.4496118180487
17 -48 35.033447529631
18 -48 33.5989462413451
19 -48 32.1539305654921
20 -48 30.7058028609551
21 -48 29.2614923367271
22 -48 27.8274139792712
23 -48 26.4094390308606
24 -48 25.0128764766042
25 -48 23.6424647752981
26 -48 22.3023728967108
27 -48 20.996209605881
28 -48 19.7270398615913
29 -48 18.4974071673081
30 -48 17.3093607232297
31 -48 16.1644862713103
32 -48 15.0639395944705
33 -48 14.0084817199404
34 -48 12.9985149784357
35 -48 12.0341191798068
36 -48 11.1150872770891
37 -48 10.2409600001503
38 -48 9.41105904446927
39 -48 8.62451849725446
40 -48 7.88031427079167
41 -48 7.17729139052133
42 -48 6.5141890525265
43 -48 5.88966342195906
44 -48 5.30230819095815
45 -48 4.75067295230264
46 -48 4.23327947444498
47 -48 3.74863598545459
48 -48 3.29524958885161
49 -48 2.87163694414201
50 -48 2.47633335012761
-50 -47 2.35003311088608
-49 -47 2.75478844653106
-48 -47 3.18951737541878
-47 -47 3.65585016478171
-46 -47 4.1554465628295
-45 -47 4.68998665090714
-44 -47 5.26115992672968
-43 -47 5.87065244790913
-42 -47 6.52013186741219
-41 -47 7.2112301999986
-40 -47 7.94552417223985
-39 -47 8.72451302956295
-38 -47 9.5495937031857
-37 -47 10.4220332790151
-36 -47 11.3429387607497
-35 -47 12.3132241817187
-34 -47 13.3335751949647
-33 -47 14.404411359436
-32 -47 15.5258464415675
-31 -47 16.6976471654247
-30 -47 17.9191909693415
-29 -47 19.1894234600409
-28 -47 20.5068163931491
-27 -47 21.8693271468707
-26 -47 23.2743607876338
-25 -47 24.7187359455785
-24 -47 26.1986558160614
-23 -47 27.7096856720947
-22 -47 29.2467383032623
-21 -47 30.8040687804848
-20 -47 32.3752798757089
-19 -47 33.9533393352323
-18 -47 35.5306100116853
-17 -47 37.0988936021617
-16 -47 38.6494884220157
-15 -47 40.1732612724055
-14 -47 41.6607330458341
-13 -47 43.1021772722912
-12 -47 44.4877303566577
-11 -47 45.8075118153
-10 -47 47.0517524065911
-9 -47 48.2109276862458
-8 -47 49.2758942217178
-7 -47 50.238025485548
-6 -47 51.0893443263738
-5 -47 51.8226488948356
-4 -47 52.4316289817127
-3 -47 52.9109699045433
-2 -47 53.2564413500967
-1 -47 53.4649689334023
0 -47 53.534686657088
1 -47 53.4649689334018
2 -47 53.2564413500968
3 -47 52.9109699045431
4 -47 52.4316289817131
5 -47 51.8226488948357
6 -47 51.0893443263735
7 -47 50.2380254855479
8 -47 49.2758942217183
9 -47 48.2109276862463
10 -47 47.0517524065909
11 -47 45.8075118153
12 -47 44.4877303566586
13 -47 43.1021772722913
14 -47 41.6607330458338
15 -47 40.1732612724059
16 -47 38.6494884220157
17 -47 37.0988936021614
18 -47 35.530610011685
19 -47 33.9533393352325
20 -47 32.3752798757084
21 -47 30.8040687804851
22 -47 29.246738303262
23 -47 27.7096856720954
24 -47 26.1986558160614
25 -47 24.7187359455792
26 -47 23.2743607876338
27 -47 21.8693271468709
28 -47 20.5068163931487
29 -47 19.1894234600408
30 -47 17.9191909693415
31 -47 16.6976471654251
32 -47 15.5258464415669
33 -47 14.404411359436
34 -47 13.3335751949646
35 -47 12.3132241817182
36 -47 11.34293876075
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The buried prism is outlined by the black lines.
The gnuplot script assumes that the output from plouff-mag-map.pl is saved to a file named plouff.xyz.
#this file plots a function using gnuplot. Typically to run
# type: gnuplot thin-plate-mag.gnu
# on the command line
reset
set termoption dash
set size square
set xlabel 'East'
set ylabel 'North'
set pm3d map
set xrange[-50:50]
set yrange[-50:50]
set cbrange [-500:1000]
set cblabel 'nT'
set palette rgbformulae 22,13,10
set arrow front from -10,-10 to -10,10 nohead
set arrow front from -10,10 to 10,10 nohead
set arrow front from 10,10 to 10,-10 nohead
set arrow front from 10,-10 to -10,-10 nohead
splot 'plouff.xyz' notitle
set term pdf enhanced dashed
set output "plouff-mag-map.pdf"
replot
[Top]
LaTeX and TikZ script: plouff-mag.tex
Draw the model geometry with LaTeX and TikZ
The following LaTeX and TikZ code draws the figure of the 3D model geometry presented at the top of the page.
\documentclass[tikz, border = 1mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{shapes}
\begin{document}
\tdplotsetmaincoords{60}{110}
\begin{tikzpicture}[scale=5,tdplot_main_coords]
\coordinate (P) at (0,0,0);
\coordinate (A) at (-0.9, 0.2, -0.2);
\coordinate (B) at (-0.9, 0.5, -0.2);
\coordinate (C) at (-0.75, 0.7, -0.2);
\coordinate (D) at (-0.6, 0.7, -0.2);
\coordinate (E) at (-0.35, 0.35, -0.2);
\coordinate (F) at (-0.35, 0.25, -0.2);
\coordinate (G) at (-0.4, 0.15, -0.2);
\coordinate (H) at (-0.6, 0.05, -0.2);
\coordinate (Ap) at (-0.9, 0.2, -0.75);
\coordinate (Bp) at (-0.9, 0.5, -0.75);
\coordinate (Cp) at (-0.75, 0.7, -0.75);
\coordinate (Dp) at (-0.6, 0.7, -0.75);
\coordinate (Ep) at (-0.35, 0.35, -0.75);
\coordinate (Fp) at (-0.35, 0.25, -0.75);
\coordinate (Gp) at (-0.4, 0.15, -0.75);
\coordinate (Hp) at (-0.6, 0.05, -0.75);
\draw[thick,->] (0,0,0) -- (-1,0,0) node[anchor=east]{$x$, North};
\draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$, East};
\draw[thick,->] (0,0,0) -- (0,0,-1) node[anchor=west]{$z$, Down};
\draw[] (A)--(B);
\draw[] (B)--(C);
\draw[] (C)--(D);
\draw[] (D)--(E);
\draw[] (E)--(F);
\draw[] (F)--(G);
\draw[] (G)--(H);
\draw[] (H)--(A);
\draw[dashed] (Ap)--(Bp);
\draw[dashed] (Bp)--(Cp);
\draw[] (Cp)--(Dp);
\draw[] (Dp)--(Ep);
\draw[] (Ep)--(Fp);
\draw[] (Fp)--(Gp);
\draw[] (Gp)--(Hp);
\draw[dashed] (Hp)--(Ap);
\draw[dashed] (Ap)--(A);
\draw[dashed] (Bp)--(B);
\draw[] (Cp)--(C);
\draw[] (Dp)--(D);
\draw[] (Ep)--(E);
\draw[] (Fp)--(F);
\draw[] (Gp)--(G);
\draw[] (Hp)--(H);
\draw[dashed, orange] (P) -- node[above]{$r_{11}$}(A);
\draw[dashed, orange] (P) -- node[below left]{$r_{12}$}(Ap);
\draw[dashed, orange] (P) -- node[below]{$r_{21}$}(B);
\draw[dashed, orange] (P) -- node[above]{$r_{22}$}(Bp);
\node[anchor=south] at (A) {$x_1, y_1, z_1$};
\node[anchor=south west] at (B) {$x_2, y_2, z_1$};
\node[anchor=north east, inner sep = 1pt] at (Bp) {$x_2, y_2, z_2$};
\node[anchor=south west, inner sep = 1pt] at (Ap) {$x_1, y_1, z_2$};
\node[anchor=south west] at (C) {$x_3, y_3, z_1$};
\node[anchor=south west] at (Cp) {$x_3, y_3, z_2$};
\draw[fill=black, tdplot_screen_coords] (A) circle [radius=0.01];
\draw[fill=black, tdplot_screen_coords] (B) circle [radius=0.01];
\draw[fill=black, tdplot_screen_coords] (Bp) circle [radius=0.01];
\draw[fill=black, tdplot_screen_coords] (Ap) circle [radius=0.01];
\draw[fill=black, tdplot_screen_coords] (C) circle [radius=0.01];
\draw[fill=black, tdplot_screen_coords] (Cp) circle [radius=0.01];
\draw[fill=red, tdplot_screen_coords] (P) circle [radius=0.01];
\node[red, anchor=south east] (P) {$P(x_p, y_p, 0$)};
\end{tikzpicture}
\end{document}
[Top]
Some References
- Bhattacharyya, B.K. (1964) Magnetic anomalies due to prism-shaped bodies with arbitrary polarization. Geophysics 29(4):517–531. DOI
- Cady, J.W. (1980) Calculation of gravity and magnetic anomalies of finite-length right polygonal prisms. Geophysics 45(10):1507–1512. DOI
- Plouff, D. (1976) Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections. Geophysics 41(4):727–741. DOI
- Talwani, M. (1965) Computation with the help of a digital computer of magnetic anomalies caused by bodies of arbitrary shape. Geophysics 30(5):797–817. DOI
- MacMillan, W.D. (1958) Theory of the Potential. Dover Publications, Inc., New York, 469p. ISBN 10:0486604861
- Rao, D.B. and Babu, N.R ( 1991) A rapid method for three-dimensional modeling of magnetic anomalies. Geophysics 56(11):1729–1737. DOI
- Rao, D.B. and Babu, N.R. (1993) A FORTRAN-77 computer program for three-dimensional inversion of magnetic anomalies resulting from multiple prismatic bodies. Computers & Geosciences 19(6):781–801.DOI
Citation
Connor, C. B. (2020) Perl scripts to calculate the magnetic anomaly of a vertical-sided polygon. [Source code].
This page is developed and maintained as part of the GEODES project, funded by NASA.