Comparison between the magnetic field produced by a prolate ellipsoid and a sphere

Import the required modules and functions



In [6]:

    
%matplotlib inline
import numpy as np
from matplotlib import pyplot as plt
from matplotlib.colors import BoundaryNorm
from matplotlib.ticker import MaxNLocator
from fatiando import gridder, utils
from fatiando.gravmag import sphere
from fatiando.mesher import Sphere
import prolate_ellipsoid
from mesher import ProlateEllipsoid



In [2]:

    
# Set some plot parameters
from matplotlib import rcParams
rcParams['figure.dpi'] = 300.
rcParams['font.size'] = 6
rcParams['xtick.labelsize'] = 'medium'
rcParams['ytick.labelsize'] = 'medium'
rcParams['axes.labelsize'] = 'large'
rcParams['legend.fontsize'] = 'medium'
rcParams['savefig.dpi'] = 300.

Set some parameters for modelling



In [3]:

    
# The local-geomagnetic field
F, inc, dec = 60000, 50, 20

# Create a regular grid at z = 0 m
shape = (50, 50)
area = [-5000, 5000, -4000, 6000]
xp, yp, zp = gridder.regular(area, shape, z=0)

Prolate ellipsoid versus sphere

This test compares the total-field anomalies produced by a prolate ellipsoid with that produced by a sphere. The ellipsoid has semi-axes $a$ and $b$ equal to 500.1 m and 499.9 m, respectively, and the sphere has a radius equal to 500 m. Both bodies are centered at the point (0, 0, 1000) and have the same magnetization.

Triaxial ellipsoid



In [4]:

    
ellipsoid = ProlateEllipsoid(0, 0, 1000, 500.1, 499.9, 40, -60, 180,
                             {'principal susceptibilities': [0.01, 0.01, 0.01],
                              'susceptibility angles': [-40, 90, 7],
                              'remanent magnetization': [0.7, -7, 10]})



In [7]:

    
magnetization = prolate_ellipsoid.magnetization(ellipsoid, F, inc, dec, demag=True)



In [8]:

    
magnetization









    Out[8]:





array([ 0.96939588,  0.2248678 ,  0.27951886])

Sphere



In [10]:

    
spherical_body = Sphere(ellipsoid.x, ellipsoid.y, ellipsoid.z,
                        0.5*(ellipsoid.large_axis + ellipsoid.small_axis),
                        {'magnetization': magnetization})



In [11]:

    
spherical_body.props['magnetization']









    Out[11]:





array([ 0.96939588,  0.2248678 ,  0.27951886])

Total-field anomalies



In [12]:

    
# total-field anomaly produced by the ellipsoid (in nT)
tf_t = prolate_ellipsoid.tf(xp, yp, zp, [ellipsoid],
                             F, inc, dec)

# total-field anomaly produced by the sphere (in nT)
tf_s = sphere.tf(xp, yp, zp, [spherical_body], inc, dec)

# residuals
tf_r = tf_t - tf_s



In [13]:

    
plt.figure(figsize=(3.15, 7))
plt.axis('scaled')

ranges = np.max(np.abs([np.min(tf_t), np.max(tf_t),
                        np.min(tf_s), np.max(tf_s)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,1)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             tf_t.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
cbar = plt.colorbar()
plt.annotate(s='(a)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.subplot(3,1,2)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             tf_s.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(b)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

ranges = np.max(np.abs([np.min(tf_r), np.max(tf_r)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,3)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             tf_r.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlabel('y (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(c)', xy=(0.88,0.92), 
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.tight_layout()

plt.show()

Field components



In [14]:

    
# field components produced by the ellipsoid (in nT)
bx_t = prolate_ellipsoid.bx(xp, yp, zp, [ellipsoid],
                            F, inc, dec)
by_t = prolate_ellipsoid.by(xp, yp, zp, [ellipsoid],
                            F, inc, dec)
bz_t = prolate_ellipsoid.bz(xp, yp, zp, [ellipsoid],
                            F, inc, dec)

bt = [bx_t, by_t, bz_t]

# field components produced by the sphere (in nT)
bx_s = sphere.bx(xp, yp, zp, [spherical_body])
by_s = sphere.by(xp, yp, zp, [spherical_body])
bz_s = sphere.bz(xp, yp, zp, [spherical_body])

bs = [bx_s, by_s, bz_s]

# residuals
bx_r = bx_t - bx_s
by_r = by_t - by_s
bz_r = bz_t - bz_s

br = [bx_r, by_r, bz_r]



In [15]:

    
plt.figure(figsize=(3.15, 7))
plt.axis('scaled')

ranges = np.max(np.abs([np.min(bx_t), np.max(bx_t),
                        np.min(bx_s), np.max(bx_s)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,1)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             bx_t.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
cbar = plt.colorbar()
plt.annotate(s='(a)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.subplot(3,1,2)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             bx_s.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(b)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

ranges = np.max(np.abs([np.min(bx_r), np.max(bx_r)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,3)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             bx_r.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlabel('y (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(c)', xy=(0.88,0.92), 
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.tight_layout()

plt.show()



In [16]:

    
plt.figure(figsize=(3.15, 7))
plt.axis('scaled')

ranges = np.max(np.abs([np.min(by_t), np.max(by_t),
                        np.min(by_s), np.max(by_s)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,1)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             by_t.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
cbar = plt.colorbar()
plt.annotate(s='(a)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.subplot(3,1,2)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             by_s.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(b)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

ranges = np.max(np.abs([np.min(by_r), np.max(by_r)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,3)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             by_r.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlabel('y (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(c)', xy=(0.88,0.92), 
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.tight_layout()

plt.show()



In [17]:

    
plt.figure(figsize=(3.15, 7))
plt.axis('scaled')

ranges = np.max(np.abs([np.min(bz_t), np.max(bz_t),
                        np.min(bz_s), np.max(bz_s)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,1)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             bz_t.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
cbar = plt.colorbar()
plt.annotate(s='(a)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.subplot(3,1,2)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             bz_s.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(b)', xy=(0.88,0.92),
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

ranges = np.max(np.abs([np.min(bz_r), np.max(bz_r)]))
levels = MaxNLocator(nbins=20).tick_values(-ranges, ranges)
cmap = plt.get_cmap('RdBu_r')
norm = BoundaryNorm(levels, ncolors=cmap.N, clip=True)

plt.subplot(3,1,3)
plt.contourf(0.001*yp.reshape(shape), 0.001*xp.reshape(shape),
             bz_r.reshape(shape), levels=levels,
             cmap = cmap, norm=norm)
plt.ylabel('x (km)')
plt.xlabel('y (km)')
plt.xlim(0.001*np.min(yp), 0.001*np.max(yp))
plt.ylim(0.001*np.min(xp), 0.001*np.max(xp))
plt.colorbar()
plt.annotate(s='(c)', xy=(0.88,0.92), 
             xycoords = 'axes fraction', color='k',
             fontsize = 10)

plt.tight_layout()

plt.show()



In [ ]: