def sum_gradient(self, gradient, p, residuals):
polygon = Polygon(self.verts + [p], self.prop)
at_p = talwani.gz(self.xp, self.zp, [polygon])
polygon = Polygon(self.verts + [[p[0] + self.delta, p[1]]], self.prop)
jacx = (talwani.gz(self.xp, self.zp, [polygon]) - at_p) / self.delta
polygon = Polygon(self.verts + [[p[0], p[1] + self.delta]], self.prop)
jacz = (talwani.gz(self.xp, self.zp, [polygon]) - at_p) / self.delta
self.jac_T = numpy.array([jacx, jacz])
return gradient - 2. * numpy.dot(self.jac_T, residuals)
def sum_gradient(self, gradient, p, residuals):
x1, x2 = self.verts[1][0], self.verts[0][0]
z1, z2 = p
delta = self.delta
polygon = Polygon(self.verts + [[x1, z1], [x2, z2]], self.prop)
at_p = talwani.gz(self.xp, self.zp, [polygon])
polygon = Polygon(self.verts + [[x1, z1 + delta], [x2, z2]], self.prop)
jacz1 = (talwani.gz(self.xp, self.zp, [polygon]) - at_p) / delta
polygon = Polygon(self.verts + [[x1, z1], [x2, z2 + delta]], self.prop)
jacz2 = (talwani.gz(self.xp, self.zp, [polygon]) - at_p) / delta
self.jac_T = numpy.array([jacz1, jacz2])
return gradient - 2. * numpy.dot(self.jac_T, residuals)
def trapezoidal(xp, zp, data, verts, density, solver, iterate=False):
"""
Estimate basement relief of a trapezoidal basin. The basin is modeled as a
triangle with two known vertices at the surface. The parameters estimated
are the x and z coordinates of the third vertice.
Parameters:
* xp, zp : array
Arrays with the x and z coordinates of the profile data points
* data : array
The profile gravity anomaly data
* verts : list of lists
List of the [x, z] coordinates of the two know vertices.
.. warning::
Very important that the vertices in the list be ordered from left to
right! Otherwise the forward model will give results with an
inverted sign and terrible things may happen!
* density : float
Density contrast of the basin
* solver : function
A non-linear inverse problem solver generated by a factory function
from the :mod:`fatiando.inversion.gradient` package.
* iterate : True or False
If True, will yield the current estimate at each iteration yielded by
*solver*. In Python terms, ``iterate=True`` transforms this function
into a generator function.
Returns:
* results : list = [estimate, residuals]:
* estimate : array or :class:`fatiando.mesher.Polygon`
If ``iterate==False``, will return a Polygon, else will yield
the estimated [z1, z2] coordinates of the bottom vertices.
* residuals : array
The residuals of the inversion (difference between measured and
predicted data)
"""
dms = [TrapezoidalGzDM(xp, zp, data, verts, density)]
if iterate:
return _iterator(dms, solver)
else:
estimate, residuals = _solver(dms, solver)
z1, z2 = estimate
left, right = verts
return Polygon([left, right, (right[0], z1), (left[0], z2)]), residuals
def update(self):
if self.polygons:
polys = []
for p, d in zip(self.polygons, self.densities):
polys.append(Polygon(1000. * numpy.array(p), {'density': d}))
self.predgz = utils.contaminate(
talwani.gz(self.xp, self.zp, polys), self.error)
else:
self.predgz = numpy.zeros_like(self.xp)
self.predplot.set_data(self.xp * 0.001, self.predgz)
if self.gz is not None:
ymin = min(self.predgz.min(), self.gz.min())
ymax = max(self.predgz.max(), self.gz.max())
else:
ymin = self.predgz.min()
ymax = self.predgz.max()
if ymin != ymax:
self.dcanvas.set_ylim(ymin, ymax)
self.draw()
def get_predicted(self, p):
x1, x2 = self.verts[1][0], self.verts[0][0]
z1, z2 = p
polygon = Polygon(self.verts + [[x1, z1], [x2, z2]], self.prop)
return talwani.gz(self.xp, self.zp, [polygon])
def get_predicted(self, p):
polygon = Polygon(self.verts + [p], self.prop)
return talwani.gz(self.xp, self.zp, [polygon])
def forward_oib(basedir, infile, dropturns=False, make_plots=False, diagnostics=False):
# type: (object, object, object, object) -> object
# -*- coding: utf-8 -*-
# %load_ext autoreload
# %autoreload 2
pd.options.mode.chained_assignment = None # None or 'warn' or 'raise'
pd.set_option("display.max_rows", 20)
# pd.set_option("precision",13)
pd.set_option('expand_frame_repr', False)
'''
Read in file
'''
# if os.path.isdir('/Volumes/C/'):
# basedir = '/Volumes/C/data/Antarctic/OIB/ATM/2009_AN_NASA_ATM'
# else:
# basedir = '/Volumes/BOOTCAMP/data/Antarctic/OIB/ATM/2009_AN_NASA_ATM'
# basedir = '/Users/dporter/Documents/data_local/Antarctica/OIB/'
# basedir = 'data'
outdir = os.path.join(basedir, 'integrated', 'forward')
df = pd.read_csv(infile)
'''
Preliminary Processing
'''
# Crossing Horizons
# df.ix[df['SURFACE_atm'] - df['ICEBASE'] <= 0, 'ICEBASE'] = (df['SURFACE_atm'] - 0.1)
# df.loc[df['surface_recalc'] - df['ICEBASE'] <= 0, 'ICEBASE'] = (df['surface_recalc'] - 0.1)
df.loc[df['surface_recalc'] - df['icebase_recalc'] <= 0, 'icebase_recalc'] = (df['surface_recalc'] - 0.1)
#
df = df.round({'RTOPO2_icemask': 0, 'D_gravmask': 0})
# TODO Add water block when HYDROAPPROX close to icebase_recalc OR use RTOPO icemask
if make_plots:
pdir = os.path.join('figs', str(df['DATE'].values[0]))
if not os.path.exists(pdir):
os.makedirs(pdir)
# Compute distance along transect
distance = np.zeros((np.size(df['LONG'])))
for i in range(2, np.size(df['LONG'])):
distance[i] = distance[i - 1] + haversine([df['LAT'].values[i - 1], df['LONG'].values[i - 1]],
[df['LAT'].values[i], df['LONG'].values[i]])
df['DIST'] = distance
# Drop where no gravity
if dropturns:
df = df.dropna(subset=['FAG070'])
'''
Run functions to read in each flight
'''
dflist = {}
dflst = [g for _, g in df.groupby(['LINE'])]
for dnum, dname in enumerate(dflst, start=0):
# print 'Mode of ATM is %.2f' % (mode)
print('L%s' % (str(dname['LINE'].values[0])))
dist = dname['DIST'][0:].values
fag070 = dname['FAG070'][0:].values
print('Distance of line is %i km' % (dist[-1] - dist[0]))
if np.isfinite(fag070).any() and (dist[-1] - dist[0] < 5e2) and (dname['surface_recalc'].any()):
if diagnostics:
print('Processing the line.')
'''
Extract data from DataFrame
'''
rho_i = 915
rho_w = 1005
rho_r = 2670
### Interpolate
if dname['icebase_recalc'].isnull().all():
print('No Icebase for this flight...')
dname['icebase_recalc'] = dname['surface_recalc'] - 1
# if dname['surface_recalc'].all():
# print('No surface for this flight...')
# TODO should these horizons be interpolated in previous script (read_OIB_ALL.py)?
dname['ICESFC_horizon'] = dname['surface_recalc']
dname['ICESFC_horizon'].interpolate(method='pad', inplace=True)
dname['ICEBASE_horizon'] = dname['icebase_recalc']
dname['ICEBASE_horizon'].interpolate(method='linear', limit_area='inside', axis=0, inplace=True)
# Find ice front position
firstmean = np.mean(dname['ICEBASE_horizon'].iloc[:5]) # or 'icebase_recalc', but this breaks when no bed
lastmean = np.mean(dname['ICEBASE_horizon'].iloc[-5:])
firsticepoint = dname['ICEBASE_horizon'].first_valid_index() - 1
lasticepoint = dname['ICEBASE_horizon'].last_valid_index() + 1
if np.isnan(firstmean):
if diagnostics:
print('Missing data at start of line.')
# firsticepoint = dname['icebase_recalc'].first_valid_index() - 1
if dname['ICESFC_horizon'].loc[firsticepoint] >= 120:
if diagnostics:
print('NOT floating.')
dname['ICEBASE_horizon'].interpolate(method='linear', limit_direction='backward', axis=0,
inplace=True)
else:
# make ice constant thick at end of line
if diagnostics:
print('LIKELY floating.')
dname['ICEBASE_horizon'].loc[firsticepoint] = dname['ICESFC_horizon'].loc[firsticepoint] - 1
dname['ICEBASE_horizon'].iloc[0] = dname['ICESFC_horizon'].iloc[0] - 1
else:
if diagnostics:
print('NO missing data at start of line.')
# Find ice front position
# lastmean = np.mean(dname['icebase_recalc'].iloc[-5:])
if np.isnan(lastmean):
if diagnostics:
print('Missing data at end of line.')
# lasticepoint = dname['icebase_recalc'].last_valid_index() + 1
if dname['ICESFC_horizon'].loc[lasticepoint] >= 120:
if diagnostics:
print('NOT floating.')
dname['ICEBASE_horizon'].interpolate(method='linear', limit_direction='forward', axis=0,
inplace=True)
else:
# make ice constant thick at end of line
if diagnostics:
print('LIKELY floating.')
dname['ICEBASE_horizon'].loc[lasticepoint] = dname['ICESFC_horizon'].loc[lasticepoint] - 1
dname['ICEBASE_horizon'].iloc[-1] = dname['ICESFC_horizon'].iloc[-1] - 1
else:
if diagnostics:
print('NO missing data at end of line.')
### Final INNER interpolation
dname['ICEBASE_horizon'].interpolate(method='linear', limit_area='inside', axis=0, inplace=True)
# Check for crossing horizons AGAIN?
# dname.loc[dname['ICESFC_horizon']-dname['ICEBASE_horizon'] <= 0, 'ICEBASE_horizon'] = (dname['ICESFC_horizon'] - 0.1)
# Create water base, set it to single depth below ice
dname['WATERBASE_horizon'] = dname['ICEBASE_horizon'] - 1
if np.mean(dname['ICESFC_horizon'].iloc[:5]) < 120:
print('Adding water block to START.')
dname['WATERBASE_horizon'].loc[dname['WATERBASE_horizon'].index[0]:firsticepoint] = \
dname['WATERBASE_horizon'].loc[
firsticepoint + 1]
if np.mean(dname['ICESFC_horizon'].iloc[-5:]) < 120:
print('Adding water block to END.')
dname['WATERBASE_horizon'].loc[lasticepoint:dname['WATERBASE_horizon'].index[-1]] = \
dname['WATERBASE_horizon'].loc[
lasticepoint - 1]
# if make_plots:
# pdir = os.path.join('figs', str(dname['DATE'].values[0]))
# print(pdir)
# if not os.path.exists(pdir):
# os.makedirs(pdir)
# oib_lineplot_all(dname, str(dname['DATE'].values[0])[:10] + '_L' + str(dname['LINE'].values[0]),
# os.path.join(pdir, str(dname['LINE'].values[0])+'_forward_lineplot.png'))
'''
Convert OIB data to polygon arrays
'''
icesfc = dname['ICESFC_horizon'][0:].values
icebase = dname['ICEBASE_horizon'][0:].values
# iceoutline = np.append(np.concatenate([icesfc, icebase[::-1]], axis=0), icesfc[0]) # no extension
iceoutline = make_outline(icesfc, icebase)
# plt.figure(facecolor='white'); plt.plot(iceoutline[1180:1220])
watertop = icebase
waterbase = dname['WATERBASE_horizon'][0:].values
wateroutline = make_outline(watertop, waterbase)
rocktop = waterbase
rockbase = -30000 * np.ones_like(waterbase)
# rockoutline = np.append(icebase[0], np.concatenate([icebase, z_r], axis=0), icebase[0], icebase[0])
rockoutline = make_outline(rocktop, rockbase)
### Distances
x = dist * 1000
xs = make_outline_dist(x, 1e6)
### Heights
elevation = dname['WGSHGT'][0:].values
# z = int(np.max(elevation))
# z = int(np.max(icesfc) + 1) * np.ones_like(x)
try:
z = elevation + int(np.max(icesfc) + 1)
except ValueError:
z = elevation + 50
# z = np.max(elevation) * np.ones_like(x)
# z = elevation + 50
'''
Build the Polygon
'''
props_i = {'density': rho_i}
props_r = {'density': rho_r}
props_w = {'density': rho_w}
# polygon = Polygon(np.transpose([xs, iceoutline]), props_i)
# polygons = [Polygon(np.transpose([xs, iceoutline]), props_i),
# Polygon(np.transpose([xs, rockoutline]), props_r)
# ]
polygons = [Polygon(np.transpose([xs, iceoutline]), props_i),
Polygon(np.transpose([xs, wateroutline]), props_w),
Polygon(np.transpose([xs, rockoutline]), props_r)
]
'''
Forward Model
'''
gz = talwani.gz(x, z, polygons)
gz_adj = (gz - np.nanmean(gz)) + np.nanmean(fag070)
n = len(gz_adj)
rmse = np.linalg.norm(gz_adj - fag070) / np.sqrt(n)
# print 'modeled'
'''
Make new channels
'''
# print 'make new channels'
try:
dname.loc[:, 'rmse'] = rmse
dname.loc[:, 'RESIDUAL'] = gz_adj - fag070
dname.loc[:, 'GZ'] = gz_adj
except:
dname.loc[:, 'rmse'] = 0
dname.loc[:, 'RESIDUAL'] = 0
dname.loc[:, 'GZ'] = 0
'''
Models of Intermediate Complexity
'''
# dnum = 7
# Interpolate across NaNs in compilation bedrock
dflst[dnum]['RTOPO2_bedrock'].interpolate(method='linear', axis=0, inplace=True)
# Remove Ice Mass Contribution
# Remove Water Mass Contribution
# Bouguer correction rock
# dflst[dnum]['BOUGUER_CORR_1'] = dflst[dnum]['RTOPO2_bedrock'] * 0.0419088 * (np.abs(rho_w - rho_r)) / 1000
dflst[dnum]['BOUGUER_CORR'] = np.where(dflst[dnum]['RTOPO2_bedrock'] > 0,
dflst[dnum]['RTOPO2_bedrock'] * 0.0419088 * (
np.abs(rho_r)) / 1000,
dflst[dnum]['RTOPO2_bedrock'] * 0.0419088 * (
np.abs(rho_w - rho_r)) / 1000)
# Bouguer Anamoly
dflst[dnum]['BOUGUER_ANOMALY'] = dflst[dnum]['FAG070'] - dflst[dnum]['BOUGUER_CORR']
# dflst[dnum][['BOUGUER_ANOMALY', 'BOUGUER_ANOMALY_1']].plot();
# plt.show()
# Low-pass filter (fill NaNs first)
dflst[dnum]['BOUGUER_ANOMALY'].interpolate(method='cubic', axis=0, inplace=True)
## Savitzky-Golay
# from scipy import signal
# dflst[dnum]['BOUGUER_ANOMALY_LP'] = signal.savgol_filter(dflst[dnum]['BOUGUER_ANOMALY'].values,
# window, 2, mode='nearest')
# dflst[dnum][['BOUGUER_ANOMALY', 'BOUGUER_ANOMALY_LP', 'FAG070']].plot();
# plt.show()
## Butterworth (filtfilt)
from scipy.signal import butter, filtfilt
import scipy.signal as signal
N = 1 # Filter order
# Wn = 0.4 # Cutoff frequency
lp_length_wallclock = 70
Wn = 1 / (lp_length_wallclock / np.nanmean(np.diff(dflst[dnum]['UNIX'])))
B, A = signal.butter(N, Wn, output='ba')
dflst[dnum]['BOUGUER_ANOMALY_LP'] = signal.filtfilt(B, A, dflst[dnum]['BOUGUER_ANOMALY'].values,
method="gust")
# dflst[dnum][['BOUGUER_ANOMALY', 'BOUGUER_ANOMALY_LP', 'FAG070']].plot();
# plt.show()
# Floating Ice
dflst[dnum]['D_floatice'] = np.where(dflst[dnum]['RTOPO2_icemask'] == 2, 1, np.nan)
# dflst[dnum][['RTOPO2_icemask', 'D_floatice']].plot();
# plt.show()
# Set bed beneath floating ice to -2000 AKA 'bedcomp'
# TODO samples SID and only do this to UNCONSTRAINED
dflst[dnum]['BEDCOMP'] = np.where((dflst[dnum]['D_floatice'] != 1),# & (dflst[dnum]['D_floatice'] != 1),
dflst[dnum]['ICEBASE_horizon'],
-2000)
# dflst[dnum][['RTOPO2_bedrock', 'ICESFC_horizon', 'ICEBASE_horizon', 'BEDCOMP']].plot();
# plt.show()
'''
Plot
'''
# diagnostics = True
if make_plots:
pdir = os.path.join('figs', str(dname['DATE'].values[0]))
if not os.path.exists(pdir):
os.makedirs(pdir)
if diagnostics:
fig, axes = plt.subplots(nrows=3, ncols=1, figsize=(12, 8))
axes[0].plot(xs, iceoutline, color='cyan', marker='o')
axes[1].plot(xs, wateroutline, color='blue')
axes[2].plot(xs, rockoutline, color='orange')
plt.savefig(os.path.join(pdir, str(dname['LINE'].values[0]) + '_polygonsplot.pdf'))
plt.show()
# ### Horizons
# plt.figure(num=None, figsize=(8, 5), dpi=80, facecolor='w', edgecolor='k')
# plt.plot(xs, iceoutline)
# # plt.plot(x, icebase, linewidth=1, color='orange', alpha=0.5)
# plt.plot(x, z, '-r', linewidth=2, label='Modeled (constant)')
# plt.xlim(min(x), max(x))
# # plt.legend()
### Plot model results
if make_plots:
talwani_lineplot(x, fag070, gz_adj, polygons, rmse, np.nanmin(rocktop) - 300, np.nanmax(icesfc) + 300,
str(dname['DATE'].values[0]) + '_L' + str(dname['LINE'].values[0]),
os.path.join(pdir, str(dname['LINE'].values[0]) + '_Talwani_lineplot.pdf'))
if make_plots:
if diagnostics:
fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(12, 8))
axes.fill(polygons[0].vertices[:, 0], polygons[0].vertices[:, 1], edgecolor='black', linewidth=1,
color='cyan', alpha=0.5)
# axes.fill(polygons[1], '.-k', linewidth=1, color='blue', alpha=0.5)
# axes.fill(polygons[2], '.-k', linewidth=1, color='orange', alpha=0.5)
plt.savefig(os.path.join(pdir, str(dname['LINE'].values[0]) + '_polygonsplot.pdf'))
plt.show()
elif not np.isfinite(fag070).any():
if diagnostics:
print('No FAG070 present: skipping.')
else:
if diagnostics:
print('Line over 1000 km: skipping.')
print(('\n' * 0))
'''
Merge back together
'''
dfout = {}
dfout = pd.concat(dflst, sort=True)
'''
Map
'''
### Entire flight
# if make_plots:
# oib_mapplot_flight(dfout['LONG'].where((dfout['D_gravmask'] != -1)),
# dfout['LAT'].where((dfout['D_gravmask'] != -1)),
# dfout['FLTENVIRO'].where((dfout['D_gravmask'] != -1)), '',
# 'FLTENVIRO ' + str(dfout['DATE'].values[0])[:10],
# os.path.join(pdir, str(dfout['DATE'].values[0])[:10] + '_mapplot_talwani_FLTENVIRO_ALL.png'))
#
# if make_plots:
# oib_mapplot(dfout['LONG'], dfout['LAT'], dfout['rmse'], 'm',
# 'rmse '+str(dfout['DATE'].values[0])[:10],
# os.path.join(pdir, str(dfout['DATE'].values[0])+'_mapplot_Talwani_rmse_ALL.pdf'))
if make_plots:
try:
oib_mapplot_zoom(dfout['LONG'].where((dfout['D_gravmask'] != -1)),
dfout['LAT'].where((dfout['D_gravmask'] != -1)),
dfout['BOUGUER_ANOMALY_LP'].where((dfout['D_gravmask'] != -1)), '', '',
os.path.join(pdir, str(dfout['DATE'].values[0])[:10] + '_oceanmapplot_BOUGUER_ALL.png'))
oib_mapplot_zoom(dfout['LONG'].where((dfout['D_gravmask'] != -1)),
dfout['LAT'].where((dfout['D_gravmask'] != -1)),
dfout['RESIDUAL'].where((dfout['D_gravmask'] != -1)), '', '',
os.path.join(pdir, str(dfout['DATE'].values[0])[:10] + '_oceanmapplot_RESIDUAL_ALL.png'))
except KeyError:
print("Can't make mapplot zoom for this flight. Sorry - do it yourself.")
'''
Save to CSV
'''
# TODO: rename some channels?
# Trim and rename
if not diagnostics:
print("Trimming output CSV for export")
dfout.drop(['FLT', 'FAG100', 'FAG140', 'FX', 'FY',
'BOTTOM', 'ELEVATION', 'NUMUSED', 'EOTGRAV', 'FACOR', 'ICEBASE',
'SURFACE_radar', 'TOPOGRAPHY_radar', 'INTCOR', 'QUALITY'],
axis=1, inplace=True)
else:
print("Exporting it all...")
# Change Precision of certain fields
dfout = dfout.round({'GZ': 1, 'DIST': 0, 'RESIDUAL': 2, 'rmse': 2, 'FAA': 1,
'BOUGUER_CORR': 1, 'BOUGUER_ANOMALY': 1, 'BOUGUER_ANOMALY_LP': 1,
'HYDROAPPX': 0, 'icebase_recalc': 1, 'surface_recalc': 1, 'BEDCOMP': 1,
'RTOPO2_icemask': 0, 'RTOPO2_bedrock': 0,
'ICESFC_horizon': 1, 'ICEBASE_horizon': 1, 'WATERBASE_horizon': 1})
# dfout['GZ_test'] = dfout['GZ'].map(lambda x: '%2.1f' % x)
# Write to CSV
if not os.path.exists(outdir):
os.makedirs(outdir)
dfout.to_csv(os.path.join(outdir, 'OIB_' + str(dfout['DATE'].values[0])[:10] + '_forward.csv'))
from fatiando.gravmag.basin2d import PolygonalBasinGravity
from fatiando.gravmag import talwani
from fatiando.mesher import Polygon
from fatiando.vis import mpl
from fatiando import utils
import numpy as np
# Make some synthetic data to test the inversion
# The model will be a polygon.
# Reverse x because vertices must be clockwise.
xs = np.linspace(0, 100000, 100)[::-1]
depths = (-1e-15*(xs - 50000)**4 + 8000 -
3000*np.exp(-(xs - 70000)**2/(10000**2)))
depths -= depths.min() # Reduce depths to zero
props = {'density': -300}
model = Polygon(np.transpose([xs, depths]), props)
x = np.linspace(0, 100000, 100)
z = -100*np.ones_like(x)
data = utils.contaminate(talwani.gz(x, z, [model]), 0.5, seed=0)
# Make the solver using smoothness regularization and run the inversion
misfit = PolygonalBasinGravity(x, z, data, 50, props, top=0)
regul = Smoothness1D(misfit.nparams)
solver = misfit + 1e-4*regul
# This is a non-linear problem so we need to pick an initial estimate
initial = 3000*np.ones(misfit.nparams)
solver.config('levmarq', initial=initial).fit()
mpl.figure()
mpl.subplot(2, 1, 1)
mpl.plot(x, data, 'ok', label='observed')
"""
GravMag: Simple gravity inversion for the relief of a 2D trapezoidal basin
"""
import numpy
from fatiando import utils
from fatiando.mesher import Polygon
from fatiando.gravmag import talwani, basin2d
from fatiando.vis import mpl
verts = [(10000, 1.), (90000, 1.), (90000, 7000), (10000, 3330)]
model = Polygon(verts, {'density': -100})
x = numpy.arange(0., 100000., 1000.)
z = numpy.zeros_like(x)
gz = utils.contaminate(talwani.gz(x, z, [model]), 0.5)
solver = basin2d.Trapezoidal(x, z, gz, verts[0:2],
density=-100).config('levmarq',
initial=[9000, 500]).fit()
estimate = solver.estimate_
mpl.figure()
mpl.subplot(2, 1, 1)
mpl.title("Gravity anomaly")
mpl.plot(x, gz, 'ok', label='Observed')
mpl.plot(x, solver.predicted(), '-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',