percent=True,
return_stddev=True)
# Make the solver and run the inversion using damping regularization
# (assumes known thicknesses of the layers)
solver = (LayeredStraight(tts, zp, thickness) +
0.1 * Damping(len(thickness))).fit()
velocity_ = solver.estimate_
# Plot the results
mpl.figure(figsize=(12, 5))
mpl.subplot(1, 2, 1)
mpl.grid()
mpl.title("Vertical seismic profile")
mpl.plot(tts, zp, 'ok', label='Observed')
mpl.plot(solver[0].predicted(), zp, '-r', linewidth=3, label='Predicted')
mpl.legend(loc='upper right', numpoints=1)
mpl.xlabel("Travel-time (s)")
mpl.ylabel("Z (m)")
mpl.ylim(sum(thickness), 0)
mpl.subplot(1, 2, 2)
mpl.grid()
mpl.title("Velocity profile")
mpl.layers(thickness, velocity_, 'o-k', linewidth=2, label='Estimated')
mpl.layers(thickness, velocity, '--b', linewidth=2, label='True')
mpl.ylim(zmax, zmin)
mpl.xlim(vmin, vmax)
leg = mpl.legend(loc='upper right', numpoints=1)
leg.get_frame().set_alpha(0.5)
mpl.xlabel("Velocity (m/s)")
mpl.ylabel("Z (m)")
mpl.show()
# Calculate and print the standard deviation of the residuals
# Should be close to the data error if the inversion was able to fit the data
residuals = tomo.residuals()
print "Assumed error: %g" % (error)
print "Standard deviation of residuals: %g" % (np.std(residuals))
mpl.figure(figsize=(14, 5))
mpl.subplot(1, 2, 1)
mpl.axis('scaled')
mpl.title('Vp model')
mpl.squaremesh(model, prop='vp', cmap=mpl.cm.seismic)
cb = mpl.colorbar()
cb.set_label('Velocity')
mpl.points(src_loc, '*y', label="Sources")
mpl.points(rec_loc, '^r', label="Receivers")
mpl.legend(loc='lower left', shadow=True, numpoints=1, prop={'size': 10})
mpl.m2km()
mpl.subplot(1, 2, 2)
mpl.axis('scaled')
mpl.title('Tomography result')
mpl.squaremesh(mesh, prop='vp', vmin=4000, vmax=10000,
cmap=mpl.cm.seismic)
cb = mpl.colorbar()
cb.set_label('Velocity')
mpl.m2km()
mpl.figure()
mpl.grid()
mpl.title('Residuals (data with %.4f s error)' % (error))
mpl.hist(residuals, color='gray', bins=10)
mpl.xlabel("seconds")
mpl.show()
zp = numpy.arange(0, 100, 1)
temp, error = utils.contaminate(climsig.abrupt(amp, age, zp),
0.02,
percent=True,
return_stddev=True)
# Preparing for the inversion
p, residuals = climsig.iabrupt(temp, zp)
est_amp, est_age = p
mpl.figure(figsize=(12, 5))
mpl.subplot(1, 2, 1)
mpl.title("Climate signal (abrupt)")
mpl.plot(temp, zp, 'ok', label='Observed')
mpl.plot(temp - residuals, zp, '--r', linewidth=3, label='Predicted')
mpl.legend(loc='lower right', numpoints=1)
mpl.xlabel("Temperature (C)")
mpl.ylabel("Z")
mpl.ylim(100, 0)
ax = mpl.subplot(1, 2, 2)
ax2 = mpl.twinx()
mpl.title("Age and amplitude")
width = 0.3
ax.bar([1 - width], [age], width, color='b', label="True")
ax.bar([1], [est_age], width, color='r', label="Estimate")
ax2.bar([2 - width], [amp], width, color='b')
ax2.bar([2], [est_amp], width, color='r')
ax.legend(loc='upper center', numpoints=1)
ax.set_ylabel("Age (years)")
ax2.set_ylabel("Amplitude (C)")
ax.set_xticks([1, 2])
relief.addprop('density', (2670 for i in xrange(relief.size)))
gridarea = (-80, 80, -220, 220)
gridshape = (100, 100)
xp, yp, zp = gridder.regular(gridarea, gridshape, z=-200)
gz = prism.gz(xp, yp, zp, relief)
mpl.figure(figsize=(10, 7))
mpl.subplot(1, 2, 1)
mpl.title("Synthetic topography")
mpl.axis('scaled')
mpl.pcolor(x, y, height, shape)
cb = mpl.colorbar()
cb.set_label("meters")
mpl.square(gridarea, label='Computation grid')
mpl.legend()
mpl.subplot(1, 2, 2)
mpl.title("Topographic effect")
mpl.axis('scaled')
mpl.pcolor(xp, yp, gz, gridshape)
cb = mpl.colorbar()
cb.set_label("mGal")
mpl.show()
myv.figure()
myv.prisms(relief, prop='density')
axes = myv.axes(myv.outline())
myv.wall_bottom(axes.axes.bounds, opacity=0.2)
myv.wall_north(axes.axes.bounds)
myv.show()
mpl.points(src, '*y')
initial = mpl.pick_points(area, mpl.gca(), marker='*', color='b')
if len(initial) > 1:
print "Don't be greedy! Pick only one point"
sys.exit()
estimate = solver.config('levmarq', initial=initial[0]).fit().estimate_
mpl.figure(figsize=(10, 4))
mpl.subplot(1, 2, 1)
mpl.title('Epicenter + %d recording stations' % (len(recs)))
mpl.axis('scaled')
mpl.points(src, '*y', label="True")
mpl.points(recs, '^r', label="Stations")
mpl.points(initial, '*b', label="Initial")
mpl.points([estimate], '*g', label="Estimate")
mpl.set_area(area)
mpl.legend(loc='lower right', shadow=True, numpoints=1, prop={'size': 12})
mpl.xlabel("X")
mpl.ylabel("Y")
ax = mpl.subplot(1, 2, 2)
mpl.title('Travel-time residuals + error bars')
s = numpy.arange(len(traveltime)) + 1
width = 0.3
mpl.bar(s - width, traveltime, width, color='g', label="Observed", yerr=error)
mpl.bar(s, solver.predicted(), width, color='r', label="Predicted")
ax.set_xticks(s)
mpl.legend(loc='upper right', shadow=True, prop={'size': 12})
mpl.xlabel("Station number")
mpl.ylabel("Travel-time residual")
mpl.show()
log.info("Generating synthetic data")
verts = [(10000, 1.), (90000, 1.), (90000, 7000), (10000, 3330)]
model = mesher.Polygon(verts, {'density':-100})
xp = numpy.arange(0., 100000., 1000.)
zp = numpy.zeros_like(xp)
gz = utils.contaminate(gravmag.talwani.gz(xp, zp, [model]), 0.5)
log.info("Preparing for the inversion")
solver = inversion.gradient.levmarq(initial=(9000, 500))
estimate, residuals = gravmag.basin2d.trapezoidal(xp, zp, gz, verts[0:2], -100,
solver)
mpl.figure()
mpl.subplot(2, 1, 1)
mpl.title("Gravity anomaly")
mpl.plot(xp, gz, 'ok', label='Observed')
mpl.plot(xp, gz - residuals, '-r', linewidth=2, label='Predicted')
mpl.legend(loc='lower left', numpoints=1)
mpl.ylabel("mGal")
mpl.xlim(0, 100000)
mpl.subplot(2, 1, 2)
mpl.polygon(estimate, 'o-r', linewidth=2, fill='r', alpha=0.3,
label='Estimated')
mpl.polygon(model, '--k', linewidth=2, label='True')
mpl.legend(loc='lower left', numpoints=1)
mpl.xlabel("X")
mpl.ylabel("Z")
mpl.set_area((0, 100000, 10000, -500))
mpl.show()
0.02, percent=True, return_stddev=True)
# Assume that the thicknesses are unknown. In this case, use a mesh of many
# thin layers and invert for each slowness
thick = 10.
mesh = [thick]*int(sum(thickness)/thick)
solver = (LayeredStraight(tts, zp, mesh) +
5*Smoothness1D(len(mesh))).fit()
velocity_ = solver.estimate_
mpl.figure(figsize=(12,5))
mpl.subplot(1, 2, 1)
mpl.grid()
mpl.title("Vertical seismic profile")
mpl.plot(tts, zp, 'ok', label='Observed')
mpl.plot(solver.predicted(), zp, '-r', linewidth=3, label='Predicted')
mpl.legend(loc='upper right', numpoints=1)
mpl.xlabel("Travel-time (s)")
mpl.ylabel("Z (m)")
mpl.ylim(sum(mesh), 0)
mpl.subplot(1, 2, 2)
mpl.grid()
mpl.title("True velocity + smooth estimate")
mpl.layers(mesh, velocity_, '.-k', linewidth=2, label='Estimated')
mpl.layers(thickness, velocity, '--b', linewidth=2, label='True')
mpl.ylim(sum(mesh), 0)
mpl.xlim(0, 10000)
mpl.legend(loc='upper right', numpoints=1)
mpl.xlabel("Velocity (m/s)")
mpl.ylabel("Z (m)")
mpl.show()
initial = mpl.pick_points(area, mpl.gca(), marker='*', color='b')
if len(initial) > 1:
print "Don't be greedy! Pick only one point"
sys.exit()
estimate = solver.config('levmarq', initial=initial[0]).fit().estimate_
mpl.figure(figsize=(10,4))
mpl.subplot(1, 2, 1)
mpl.title('Epicenter + %d recording stations' % (len(recs)))
mpl.axis('scaled')
mpl.points(src, '*y', label="True")
mpl.points(recs, '^r', label="Stations")
mpl.points(initial, '*b', label="Initial")
mpl.points([estimate], '*g', label="Estimate")
mpl.set_area(area)
mpl.legend(loc='lower right', shadow=True, numpoints=1, prop={'size':12})
mpl.xlabel("X")
mpl.ylabel("Y")
ax = mpl.subplot(1, 2, 2)
mpl.title('Travel-time residuals + error bars')
s = numpy.arange(len(traveltime)) + 1
width = 0.3
mpl.bar(s - width, traveltime, width, color='g', label="Observed",
yerr=error)
mpl.bar(s, solver.predicted(), width, color='r', label="Predicted")
ax.set_xticks(s)
mpl.legend(loc='upper right', shadow=True, prop={'size':12})
mpl.xlabel("Station number")
mpl.ylabel("Travel-time residual")
mpl.show()
mpl.pcolor(layer.y, layer.x, layer.props['density'], layer.shape)
mpl.colorbar().set_label(r'Density $kg.m^{-3}$')
mpl.m2km()
mpl.subplot(1, 3, 2)
mpl.axis('scaled')
mpl.title('Fit gz (mGal)')
levels = mpl.contour(y1, x1, gz, shape, 15, color='k', interp=True)
mpl.contour(y1,
x1,
solver.predicted()[0],
shape,
levels,
color='r',
interp=True)
mpl.plot(y1, x1, 'xk', label='Data points')
mpl.legend()
mpl.m2km()
mpl.subplot(1, 3, 3)
mpl.axis('scaled')
mpl.title('Fit gzz (Eotvos)')
levels = mpl.contour(y2, x2, gzz, shape, 10, color='k', interp=True)
mpl.contour(y2,
x2,
solver.predicted()[1],
shape,
levels,
color='r',
interp=True)
mpl.plot(y2, x2, 'xk', label='Data points')
mpl.legend()
mpl.m2km()
z = data(x, y)
shape = (100, 100)
# First, we need to know the real data at the grid points
grdx, grdy = gridder.regular(area, shape)
grdz = data(grdx, grdy)
mpl.figure()
mpl.subplot(2, 2, 1)
mpl.axis("scaled")
mpl.title("True grid data")
mpl.plot(x, y, ".k", label="Data points")
mpl.contourf(grdx, grdy, grdz, shape, 50)
mpl.colorbar()
mpl.legend(loc="lower right", numpoints=1)
# Use the default interpolation (cubic)
grdx, grdy, grdz = gridder.interp(x, y, z, shape)
mpl.subplot(2, 2, 2)
mpl.axis("scaled")
mpl.title("Interpolated using cubic minimum-curvature")
mpl.plot(x, y, ".k", label="Data points")
mpl.contourf(grdx, grdy, grdz, shape, 50)
mpl.colorbar()
mpl.legend(loc="lower right", numpoints=1)
# Use the nearest neighbors interpolation
grdx, grdy, grdz = gridder.interp(x, y, z, shape, algorithm="nearest")
mpl.subplot(2, 2, 3)
mpl.axis("scaled")
"""
Gridding: Cut a section from a grid
"""
from fatiando import gridder, utils
from fatiando.vis import mpl
# Generate some synthetic data on a regular grid
x, y = gridder.regular((-10, 10, -10, 10), (100,100))
# Using a 2D Gaussian
z = utils.gaussian2d(x, y, 1, 1)
subarea = [-2, 2, -3, 3]
subx, suby, subscalar = gridder.cut(x, y, [z], subarea)
mpl.figure(figsize=(12, 5))
mpl.subplot(1, 2, 1)
mpl.title("Whole grid")
mpl.axis('scaled')
mpl.pcolor(x, y, z, (100,100))
mpl.square(subarea, 'k', linewidth=2, label='Cut this region')
mpl.legend(loc='lower left')
mpl.subplot(1, 2, 2)
mpl.title("Cut grid")
mpl.axis('scaled')
mpl.pcolor(subx, suby, subscalar[0], (40,60), interp=True)
mpl.show()
x, y = gridder.scatter((-2, 2, -2, 2), n=300, seed=1)
# And calculate 2D Gaussians on these points as sample data
def data(x, y):
return (utils.gaussian2d(x, y, -0.6, -1) -
utils.gaussian2d(x, y, 1.5, 1.5))
d = data(x, y)
# Extract a profile along the diagonal
p1, p2 = [-1.5, 0], [1.5, 1.5]
xp, yp, distance, dp = gridder.profile(x, y, d, p1, p2, 100)
dp_true = data(xp, yp)
mpl.figure()
mpl.subplot(2, 1, 2)
mpl.title("Irregular grid")
mpl.plot(xp, yp, '-k', label='Profile', linewidth=2)
mpl.contourf(x, y, d, (100, 100), 50, interp=True)
mpl.colorbar(orientation='horizontal')
mpl.legend(loc='lower right')
mpl.subplot(2, 1, 1)
mpl.title('Profile')
mpl.plot(distance, dp, '.b', label='Extracted')
mpl.plot(distance, dp_true, '-k', label='True')
mpl.xlim(distance.min(), distance.max())
mpl.legend(loc='lower right')
mpl.show()
"""
Gridding: Cut a section from a grid
"""
from fatiando import gridder, utils
from fatiando.vis import mpl
# Generate some synthetic data on a regular grid
x, y = gridder.regular((-10, 10, -10, 10), (100, 100))
# Using a 2D Gaussian
z = utils.gaussian2d(x, y, 1, 1)
subarea = [-2, 2, -3, 3]
subx, suby, subscalar = gridder.cut(x, y, [z], subarea)
mpl.figure(figsize=(12, 5))
mpl.subplot(1, 2, 1)
mpl.title("Whole grid")
mpl.axis('scaled')
mpl.pcolor(x, y, z, (100, 100))
mpl.square(subarea, 'k', linewidth=2, label='Cut this region')
mpl.legend(loc='lower left')
mpl.subplot(1, 2, 2)
mpl.title("Cut grid")
mpl.axis('scaled')
mpl.pcolor(subx, suby, subscalar[0], (40, 60), interp=True)
mpl.show()
from fatiando import utils
from fatiando.geothermal.climsig import linear, SingleChange
from fatiando.vis import mpl
# Generating synthetic data
amp = 5.43
age = 78.2
# along a well at these depths
zp = numpy.arange(0, 100, 1)
temp, error = utils.contaminate(linear(amp, age, zp), 0.02,
percent=True, return_stddev=True)
# Preparing for the inversion
data = SingleChange(temp, zp, mode='linear').config('levmarq', initial=[1, 1])
amp_, age_ = data.fit().estimate_
print "Linear change in temperature"
print " true: amp=%.3f age=%.3f" % (amp, age)
print " estimated: amp=%.3f age=%.3f" % (amp_, age_)
mpl.figure(figsize=(4, 5))
mpl.title("Residual well temperature")
mpl.plot(temp, zp, 'ok', label='Observed')
mpl.plot(data.predicted(), zp, '--r', linewidth=3, label='Predicted')
mpl.legend(loc='lower right', numpoints=1)
mpl.xlabel("Temperature (C)")
mpl.ylabel("Z (m)")
mpl.ylim(100, 0)
mpl.show()
verts = [(10000, 1.), (90000, 1.), (80000, 5000)]
model = mesher.Polygon(verts, {'density': -100})
xp = numpy.arange(0., 100000., 1000.)
zp = numpy.zeros_like(xp)
gz = utils.contaminate(gravmag.talwani.gz(xp, zp, [model]), 1)
solver = inversion.gradient.levmarq(initial=(10000, 1000))
estimate, residuals = gravmag.basin2d.triangular(xp, zp, gz, verts[0:2], -100,
solver)
mpl.figure()
mpl.subplot(2, 1, 1)
mpl.title("Gravity anomaly")
mpl.plot(xp, gz, 'ok', label='Observed')
mpl.plot(xp, gz - residuals, '-r', linewidth=2, label='Predicted')
mpl.legend(loc='lower left')
mpl.ylabel("mGal")
mpl.xlim(0, 100000)
mpl.subplot(2, 1, 2)
mpl.polygon(estimate,
'o-r',
linewidth=2,
fill='r',
alpha=0.3,
label='Estimated')
mpl.polygon(model, '--k', linewidth=2, label='True')
mpl.legend(loc='lower left', numpoints=1)
mpl.xlabel("X")
mpl.ylabel("Z")
mpl.set_area((0, 100000, 10000, -500))
mpl.show()
# Print the estimated and true dipole monents, inclinations and declinations
print 'Estimated magnetization (intensity, inclination, declination)'
for e in solver.estimate_:
print e
# Plot the fit and the normalized histogram of the residuals
mpl.figure(figsize=(14, 5))
mpl.subplot(1, 2, 1)
mpl.title("Total Field Anomaly (nT)", fontsize=14)
mpl.axis('scaled')
nlevels = mpl.contour(y, x, tf, (50, 50), 15, interp=True, color='r',
label='Observed', linewidth=2.0)
mpl.contour(y, x, solver.predicted(), (50, 50), nlevels, interp=True,
color='b', label='Predicted', style='dashed', linewidth=2.0)
mpl.legend(loc='upper left', shadow=True, prop={'size': 13})
mpl.xlabel('East y (m)', fontsize=14)
mpl.ylabel('North x (m)', fontsize=14)
mpl.subplot(1, 2, 2)
residuals_mean = numpy.mean(solver.residuals())
residuals_std = numpy.std(solver.residuals())
# Each residual is subtracted from the mean and the resulting
# difference is divided by the standard deviation
s = (solver.residuals() - residuals_mean) / residuals_std
mpl.hist(s, bins=21, range=None, normed=True, weights=None,
cumulative=False, bottom=None, histtype='bar', align='mid',
orientation='vertical', rwidth=None, log=False,
color=None, label=None)
mpl.xlim(-4, 4)
mpl.title("mean = %.3f std = %.3f" % (residuals_mean, residuals_std),
fontsize=14)
# Generate random points
x, y = gridder.scatter((-2, 2, -2, 2), n=300, seed=1)
# And calculate 2D Gaussians on these points as sample data
def data(x, y):
return (utils.gaussian2d(x, y, -0.6, -1)
- utils.gaussian2d(x, y, 1.5, 1.5))
d = data(x, y)
# Extract a profile along the diagonal
p1, p2 = [-1.5, 0], [1.5, 1.5]
xp, yp, distance, dp = gridder.profile(x, y, d, p1, p2, 100)
dp_true = data(xp, yp)
mpl.figure()
mpl.subplot(2, 1, 2)
mpl.title("Irregular grid")
mpl.plot(xp, yp, '-k', label='Profile', linewidth=2)
mpl.contourf(x, y, d, (100, 100), 50, interp=True)
mpl.colorbar(orientation='horizontal')
mpl.legend(loc='lower right')
mpl.subplot(2, 1, 1)
mpl.title('Profile')
mpl.plot(distance, dp, '.b', label='Extracted')
mpl.plot(distance, dp_true, '-k', label='True')
mpl.xlim(distance.min(), distance.max())
mpl.legend(loc='lower right')
mpl.show()
# Calculate and print the standard deviation of the residuals
# Should be close to the data error if the inversion was able to fit the data
residuals = tomo.residuals()
print "Assumed error: %g" % (error)
print "Standard deviation of residuals: %g" % (np.std(residuals))
mpl.figure(figsize=(14, 5))
mpl.subplot(1, 2, 1)
mpl.axis('scaled')
mpl.title('Vp model')
mpl.squaremesh(model, prop='vp', cmap=mpl.cm.seismic)
cb = mpl.colorbar()
cb.set_label('Velocity')
mpl.points(src_loc, '*y', label="Sources")
mpl.points(rec_loc, '^r', label="Receivers")
mpl.legend(loc='lower left', shadow=True, numpoints=1, prop={'size': 10})
mpl.m2km()
mpl.subplot(1, 2, 2)
mpl.axis('scaled')
mpl.title('Tomography result')
mpl.squaremesh(mesh, prop='vp', vmin=4000, vmax=10000, cmap=mpl.cm.seismic)
cb = mpl.colorbar()
cb.set_label('Velocity')
mpl.m2km()
mpl.figure()
mpl.grid()
mpl.title('Residuals (data with %.4f s error)' % (error))
mpl.hist(residuals, color='gray', bins=10)
mpl.xlabel("seconds")
mpl.show()
tf, (50, 50),
15,
interp=True,
color='r',
label='Observed',
linewidth=2.0)
mpl.contour(y,
x,
solver.predicted(), (50, 50),
nlevels,
interp=True,
color='b',
label='Predicted',
style='dashed',
linewidth=2.0)
mpl.legend(loc='upper left', shadow=True, prop={'size': 13})
mpl.xlabel('East y (m)', fontsize=14)
mpl.ylabel('North x (m)', fontsize=14)
mpl.subplot(1, 2, 2)
residuals_mean = numpy.mean(solver.residuals())
residuals_std = numpy.std(solver.residuals())
# Each residual is subtracted from the mean and the resulting
# difference is divided by the standard deviation
s = (solver.residuals() - residuals_mean) / residuals_std
mpl.hist(s,
bins=21,
range=None,
normed=True,
weights=None,
cumulative=False,
bottom=None,
