def doSlice(v):
if vType == "CC":
return getIndSlice(self.r(v, "CC", "CC", "M"))
elif vType == "CCv":
assert view == "vec", "Other types for CCv not supported"
else:
# Now just deal with 'F' and 'E' (x,y,z, maybe...)
aveOp = "ave" + vType + ("2CCV" if view == "vec" else "2CC")
Av = getattr(self, aveOp)
if v.size == Av.shape[1]:
v = Av * v
else:
v = self.r(v, vType[0], vType) # get specific component
v = Av * v
# we should now be averaged to cell centers (might be a vector)
v = self.r(v.reshape((self.nC, -1), order="F"), "CC", "CC", "M")
if view == "vec":
outSlice = []
if "X" not in normal:
outSlice.append(getIndSlice(v[0]))
if "Y" not in normal:
outSlice.append(getIndSlice(v[1]))
if "Z" not in normal:
outSlice.append(getIndSlice(v[2]))
return np.r_[mkvc(outSlice[0]), mkvc(outSlice[1])]
else:
return getIndSlice(self.r(v, "CC", "CC", "M"))
def getJtJdiag(self, m, W=None):
"""
Return the diagonal of JtJ
"""
dmudm = self.chiMap.deriv(m)
self._dSdm = None
self._dfdm = None
self.model = m
if (self.gtgdiag is None) and (self.modelType != 'amplitude'):
if W is None:
w = np.ones(self.G.shape[1])
else:
w = W.diagonal()
self.gtgdiag = np.zeros(dmudm.shape[1])
for ii in range(self.G.shape[0]):
self.gtgdiag += (w[ii]*self.G[ii, :]*dmudm)**2.
if self.coordinate_system == 'cartesian':
if self.modelType == 'amplitude':
return np.sum((W * self.dfdm * self.G * dmudm)**2., axis=0)
else:
return self.gtgdiag
else: # spherical
if self.modelType == 'amplitude':
return np.sum(((W * self.dfdm) * self.G * (self.dSdm * dmudm))**2., axis=0)
else:
Japprox = sdiag(mkvc(self.gtgdiag)**0.5*dmudm.T) * (self.dSdm * dmudm)
return mkvc(np.sum(Japprox.power(2), axis=0))
def gridDrappedXYZ(width, height, spacing, x0=(0, 0, 0), topo=None):
"""
grid_survey(width)
Generate concentric grid surveys centered at the origin
:grid: List of grid spacing
:width: Grid width
return: rxLoc an n-by-3 receiver locations
"""
nCx = int(width/spacing)
nCy = int(height/spacing)
dx = width/nCx
dy = height/nCy
rxVec = -width/2. + dx/2. + np.asarray(range(nCx))*dx
ryVec = -height/2. + dy/2. + np.asarray(range(nCy))*dy
rxGridx, rxGridy = np.meshgrid(rxVec, ryVec)
rxGridx += x0[0]
rxGridy += x0[1]
if topo is not None:
rxGridz = sp.interpolate.griddata(topo[:, :2], topo[:, 2],
(rxGridx, rxGridy),
method='linear') + x0[2]
else:
rxGridz = np.zeros_like(rxGridx) + x0[2]
return np.c_[mkvc(rxGridx), mkvc(rxGridy), mkvc(rxGridz)]
def gridCC(self):
if getattr(self, '_gridCC', None) is None:
X, Y = np.meshgrid(self.hx, self.hy)
self._gridCC = np.c_[mkvc(X), mkvc(Y)]
return self._gridCC
def refine_box(mesh):
# Refine for sphere
xp, yp, zp = np.meshgrid([-55., 50.], [-50., 50.], [-40., 20.])
xyz = np.c_[mkvc(xp), mkvc(yp), mkvc(zp)]
mesh = refine_tree_xyz(mesh,
xyz,
octree_levels=[2],
method='box',
finalize=False)
return mesh
def refine_topography(mesh):
# Define topography and refine
[xx, yy] = np.meshgrid(mesh.vectorNx, mesh.vectorNy)
zz = -3 * np.exp((xx**2 + yy**2) / 60**2) + 45.
topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)]
mesh = refine_tree_xyz(mesh,
topo,
octree_levels=[3, 2],
method='surface',
finalize=False)
return mesh
def Jtvec(self, m, v, u=None):
"""
:param numpy.array m: Conductivity model
:param numpy.ndarray,SimPEG.Survey.Data v: vector (data object)
:param simpegEM.TDEM.FieldsTDEM u: Fields resulting from m
:rtype: numpy.ndarray
:return: w (model object)
Multiplying \\\(\\\mathbf{J}^\\\\top\\\) onto a vector can be broken into three steps
* Compute \\\(\\\\vec{p} = \\\mathbf{Q}^\\\\top \\\\vec{v}\\\)
* Solve \\\(\\\hat{\\\mathbf{A}}^\\\\top \\\\vec{y} = \\\\vec{p}\\\)
* Compute \\\(\\\\vec{w} = -\\\mathbf{G}^\\\\top y\\\)
"""
if self.verbose: print '%s\nCalculating J^T(v)\n%s'%('*'*50,'*'*50)
self.curModel = m
if u is None:
u = self.fields(m)
if not isinstance(v, self.dataPair):
v = self.dataPair(self.survey, v)
p = self.survey.projectFieldsDeriv(u, v=v, adjoint=True)
y = self.solveAht(m, p)
w = self.Gtvec(m, y, u)
if self.verbose: print '%s\nDone calculating J^T(v)\n%s'%('*'*50,'*'*50)
return - mkvc(w)
def Jvec(self, m, v, f=None):
"""
:param numpy.array m: Conductivity model
:param numpy.ndarray v: vector (model object)
:param FieldsTDEM f: Fields resulting from m
:rtype: numpy.ndarray
:return: w (data object)
Multiplying \\\(\\\mathbf{J}\\\) onto a vector can be broken into three steps
* Compute \\\(\\\\vec{p} = \\\mathbf{G}v\\\)
* Solve \\\(\\\hat{\\\mathbf{A}} \\\\vec{y} = \\\\vec{p}\\\)
* Compute \\\(\\\\vec{w} = -\\\mathbf{Q} \\\\vec{y}\\\)
"""
if self.verbose:
print '{0!s}\nCalculating J(v)\n{1!s}'.format('*' * 50, '*' * 50)
self.curModel = m
if f is None:
f = self.fields(m)
p = self.Gvec(m, v, f)
y = self.solveAh(m, p)
Jv = self.survey.evalDeriv(f, v=y)
if self.verbose:
print '{0!s}\nDone calculating J(v)\n{1!s}'.format(
'*' * 50, '*' * 50)
return -mkvc(Jv)
def calculate(self):
self.nD = self.rxLoc.shape[0]
if self.parallelized:
if self.n_cpu is None:
# By default take half the cores, turns out be faster
# than running full threads
self.n_cpu = int(multiprocessing.cpu_count()/2)
pool = multiprocessing.Pool(self.n_cpu)
result = pool.map(self.calcTrow, [self.rxLoc[ii, :] for ii in range(self.nD)])
pool.close()
pool.join()
else:
result = []
for ii in range(self.nD):
result += [self.calcTrow(self.rxLoc[ii, :])]
self.progress(ii, self.nD)
if self.forwardOnly:
return mkvc(np.vstack(result))
else:
return np.vstack(result)
def upwardContinuation(self, z=0):
"""
Function to calculate upward continued data
"""
self.heightUC = z
upFact = -(
np.sqrt(self.Kx**2. + self.Ky**2.) *
z /
np.sqrt(self.dx**2. + self.dy**2.)
)
FzUpw = self.gridFFT*np.exp(upFact)
zUpw_pad = np.fft.ifft2(FzUpw)
zUpw = np.real(
zUpw_pad[self.npady:-self.npady, self.npadx:-self.npadx])
self._valuesFilledUC = zUpw.copy()
if self.indNan is not None:
zUpw = mkvc(zUpw)
zUpw[self.indNan] = np.nan
zUpw = zUpw.reshape(self.values.shape, order='F')
return zUpw
def setRTP(self, isRTP):
if isRTP:
if np.isnan(self.inc):
print("Attibute 'inc' needs to be set")
if np.isnan(self.dec):
print("Attibute 'dec' needs to be set")
h0_xyz = MathUtils.dipazm_2_xyz(self.inc, self.dec)
Frtp = self.gridFFT / (
(h0_xyz[2] + 1j*(self.Kx*h0_xyz[0] + self.Ky*h0_xyz[1]))**2.
)
rtp_pad = np.fft.ifft2(Frtp)
rtp = np.real(
rtp_pad[self.npady:-self.npady, self.npadx:-self.npadx])
if self.indNan is not None:
rtp = mkvc(rtp)
rtp[self.indNan] = np.nan
rtp = rtp.reshape(self.values.shape, order='F')
self._RTP = rtp
for prop in gridProps:
setattr(self, '_{}'.format(prop), None)
else:
self._RTP = None
for prop in gridProps:
setattr(self, '_{}'.format(prop), None)
def test_zero(self):
z = Zero()
assert z == 0
assert not (z < 0)
assert z <= 0
assert not (z > 0)
assert z >= 0
assert +z == z
assert -z == z
assert z + 1 == 1
assert z + 3 +z == 3
assert z - 3 == -3
assert z - 3 -z == -3
assert 3*z == 0
assert z*3 == 0
assert z/3 == 0
a = 1
a += z
assert a == 1
a = 1
a += z
assert a == 1
self.assertRaises(ZeroDivisionError, lambda:3/z)
assert mkvc(z) == 0
assert sdiag(z)*a == 0
assert z.T == 0
assert z.transpose == 0
def Jtvec(self, m, v, f=None):
"""
:param numpy.array m: Conductivity model
:param numpy.ndarray v: vector (or a :class:`SimPEG.Survey.Data` object)
:param FieldsTDEM u: Fields resulting from m
:rtype: numpy.ndarray
:return: w (model object)
Multiplying \\\(\\\mathbf{J}^\\\\top\\\) onto a vector can be broken into three steps
* Compute \\\(\\\\vec{p} = \\\mathbf{Q}^\\\\top \\\\vec{v}\\\)
* Solve \\\(\\\hat{\\\mathbf{A}}^\\\\top \\\\vec{y} = \\\\vec{p}\\\)
* Compute \\\(\\\\vec{w} = -\\\mathbf{G}^\\\\top y\\\)
"""
if self.verbose:
print '{0!s}\nCalculating J^T(v)\n{1!s}'.format('*' * 50, '*' * 50)
self.curModel = m
if f is None:
f = self.fields(m)
if not isinstance(v, self.dataPair):
v = self.dataPair(self.survey, v)
p = self.survey.evalDeriv(f, v=v, adjoint=True)
y = self.solveAht(m, p)
w = self.Gtvec(m, y, f)
if self.verbose:
print '{0!s}\nDone calculating J^T(v)\n{1!s}'.format(
'*' * 50, '*' * 50)
return -mkvc(w)
def calculate(self):
self.nD = self.rxLoc.shape[0]
if self.parallelized:
if self.n_cpu is None:
# By default take half the cores, turns out be faster
# than running full threads
self.n_cpu = int(multiprocessing.cpu_count()/2)
pool = multiprocessing.Pool(self.n_cpu)
result = pool.map(self.calcTrow, [self.rxLoc[ii, :] for ii in range(self.nD)])
pool.close()
pool.join()
else:
result = []
for ii in range(self.nD):
result += [self.calcTrow(self.rxLoc[ii, :])]
self.progress(ii, self.nD)
if self.forwardOnly:
return mkvc(np.vstack(result))
else:
return np.vstack(result)
def test_zero(self):
z = Zero()
assert z == 0
assert not (z < 0)
assert z <= 0
assert not (z > 0)
assert z >= 0
assert +z == z
assert -z == z
assert z + 1 == 1
assert z + 3 +z == 3
assert z - 3 == -3
assert z - 3 -z == -3
assert 3*z == 0
assert z*3 == 0
assert z/3 == 0
a = 1
a += z
assert a == 1
a = 1
a += z
assert a == 1
self.assertRaises(ZeroDivisionError, lambda:3/z)
assert mkvc(z) == 0
assert sdiag(z)*a == 0
assert z.T == 0
assert z.transpose == 0
def eval(self, prob):
if prob._formulation == 'HJ':
inds = closestPoints(prob.mesh, self.loc)
q = np.zeros(prob.mesh.nC)
q[inds] = self.current * np.r_[1.]
elif prob._formulation == 'EB':
q = prob.mesh.getInterpolationMat(self.loc, locType='N').todense()
q = self.current * mkvc(q)
return q
def eval(self, prob):
if prob._formulation == 'HJ':
inds = closestPoints(prob.mesh, self.loc)
q = np.zeros(prob.mesh.nC)
q[inds] = self.current * np.r_[1.]
elif prob._formulation == 'EB':
q = prob.mesh.getInterpolationMat(self.loc, locType='N').todense()
q = self.current * mkvc(q)
return q
def eval(self, prob):
if prob._formulation == 'HJ':
inds = closestPoints(prob.mesh, self.loc, gridLoc='CC')
q = np.zeros(prob.mesh.nC)
q[inds] = self.current * np.r_[1., -1.]
elif prob._formulation == 'EB':
qa = prob.mesh.getInterpolationMat(self.loc[0], locType='N').todense()
qb = -prob.mesh.getInterpolationMat(self.loc[1], locType='N').todense()
q = self.current * mkvc(qa+qb)
return q
def eval(self, prob):
if prob._formulation == 'HJ':
inds = closestPoints(prob.mesh, self.loc, gridLoc='CC')
q = np.zeros(prob.mesh.nC)
q[inds] = self.current * np.r_[1., -1.]
elif prob._formulation == 'EB':
qa = prob.mesh.getInterpolationMat(self.loc[0],
locType='N').todense()
qb = -prob.mesh.getInterpolationMat(self.loc[1],
locType='N').todense()
q = self.current * mkvc(qa + qb)
return q
def GravSphereFreeSpace(x, y, z, R, xc, yc, zc, rho):
"""
Computing the induced response of magnetic sphere in free-space.
>> Input
x, y, z: Observation locations
R: radius of the sphere
xc, yc, zc: Location of the sphere
rho: Density of sphere
By convention, z-component is positive downward for a
positive density anomaly
"""
if (~np.size(x) == np.size(y) == np.size(z)):
print("Specify same size of x, y, z")
return
unit_conv = 1e+8 # Unit conversion from SI to (mgal*g/cc)
x = mkvc(x)
y = mkvc(y)
z = mkvc(z)
nobs = len(x)
M = R**3. * 4. / 3. * np.pi * rho
rx = (x - xc)
ry = (y - yc)
rz = (zc - z)
rvec = np.c_[rx, ry, rz]
r = np.sqrt((rx)**2 + (ry)**2 + (rz)**2)
g = -G * (1. / r**2) * M * unit_conv
gx = g * (rx / r)
gy = g * (ry / r)
gz = g * (rz / r)
return gx, gy, gz
def GravSphereFreeSpace(x, y, z, R, xc, yc, zc, rho):
"""
Computing the induced response of gravity sphere in free-space.
>> Input
x, y, z: Observation locations
R: radius of the sphere
xc, yc, zc: Location of the sphere
rho: Density of sphere
By convention, z-component is positive downward for a
positive density anomaly
"""
if (~np.size(x) == np.size(y) == np.size(z)):
print("Specify same size of x, y, z")
return
unit_conv = 1e+8 # Unit conversion from SI to (mgal*g/cc)
x = mkvc(x)
y = mkvc(y)
z = mkvc(z)
nobs = len(x)
M = R**3. * 4. / 3. * np.pi * rho
rx = (x - xc)
ry = (y - yc)
rz = (zc - z)
rvec = np.c_[rx, ry, rz]
r = np.sqrt((rx)**2 + (ry)**2 + (rz)**2)
g = -G * (1. / r**2) * M * unit_conv
gx = g * (rx / r)
gy = g * (ry / r)
gz = g * (rz / r)
return gx, gy, gz
def run(plotIt=True, nx=5, ny=5):
# 2D mesh
mesh = Mesh.TensorMesh([nx, ny], x0='CC')
xtopo = mesh.vectorNx
# define a topographic surface
topo = 0.4*np.sin(xtopo*5)
# make it an array
Topo = np.hstack([Utils.mkvc(xtopo, 2), Utils.mkvc(topo, 2)])
# Compare the different options
indtopoCC_near = surface2ind_topo(mesh, Topo, gridLoc='CC', method='nearest')
indtopoN_near = surface2ind_topo(mesh, Topo, gridLoc='N', method='nearest')
indtopoCC_linear = surface2ind_topo(mesh, Topo, gridLoc='CC', method='linear')
indtopoN_linear = surface2ind_topo(mesh, Topo, gridLoc='N', method='linear')
indtopoCC_cubic = surface2ind_topo(mesh, Topo, gridLoc='CC', method='cubic')
indtopoN_cubic = surface2ind_topo(mesh, Topo, gridLoc='N', method='cubic')
if plotIt:
fig, ax = plt.subplots(2, 3, figsize=(9, 6))
ax = mkvc(ax)
xinterpolate = np.linspace(mesh.gridN[:, 0].min(), mesh.gridN[:, 0].max(),100)
listindex = [indtopoCC_near,indtopoN_near,indtopoCC_linear,indtopoN_linear,indtopoCC_cubic,indtopoN_cubic]
listmethod = ['nearest','nearest', 'linear', 'linear', 'cubic', 'cubic']
for i in range(6):
mesh.plotGrid(ax=ax[i], nodes=True, centers=True)
mesh.plotImage(listindex[i], ax=ax[i], pcolorOpts = {"alpha":0.5, "cmap":plt.cm.gray})
ax[i].scatter(Topo[:,0], Topo[:,1], color = 'black', marker = 'o',s = 50)
ax[i].plot(
xinterpolate,
interp1d(Topo[:, 0], Topo[:, 1], kind=listmethod[i])(xinterpolate),
'--k',
linewidth=3
)
ax[i].xaxis.set_ticklabels([])
ax[i].yaxis.set_ticklabels([])
ax[i].set_aspect('equal')
ax[i].set_xlabel('')
ax[i].set_ylabel('')
ax[0].set_xlabel('Nearest Interpolation', fontsize=16)
ax[2].set_xlabel('Linear Interpolation', fontsize=16)
ax[4].set_xlabel('Cubic Interpolation', fontsize=16)
ax[0].set_ylabel('Cells Center \n based selection', fontsize=16)
ax[1].set_ylabel('Nodes \n based selection', fontsize=16)
plt.tight_layout()
def NMOstack(data, xorig, time, v):
if np.isscalar(v):
v = np.ones_like(time) * v
Time = (time.reshape([1, -1])).repeat(data.shape[0], axis=0)
singletrace = np.zeros(data.shape[1])
for i in range(time.size):
toffset = np.sqrt(xorig**2 / v[i]**2 + time[i]**2)
Time = (time.reshape([1, -1])).repeat(data.shape[0], axis=0)
Toffset = (toffset.reshape([-1, 1])).repeat(data.shape[1], axis=1)
indmin = np.argmin(abs(Time - Toffset), axis=1)
singletrace[i] = (mkvc(data)[sub2ind(
data.shape, np.c_[np.arange(data.shape[0]), indmin])]).sum()
return singletrace
def fields(self, m):
self.model = self.rhoMap*m
if self.forwardOnly:
# Compute the linear operation without forming the full dense G
fields = self.Intrgl_Fwr_Op()
return mkvc(fields)
else:
vec = np.dot(self.G, (self.model).astype(np.float32))
return vec.astype(np.float64)
def fields(self, m):
model = self.rhoMap*m
if self.forwardOnly:
# Compute the linear operation without forming the full dense G
fields = self.Intrgl_Fwr_Op(m=m)
return mkvc(fields)
else:
vec = np.dot(self.G, model.astype(np.float32))
return vec.astype(np.float64)
def doSlice(v):
if vType == 'CC':
return getIndSlice(self.r(v, 'CC', 'CC', 'M'))
elif vType == 'CCv':
assert view == 'vec', 'Other types for CCv not supported'
else:
# Now just deal with 'F' and 'E' (x,y,z, maybe...)
aveOp = 'ave' + vType + ('2CCV' if view == 'vec' else '2CC')
Av = getattr(self, aveOp)
if v.size == Av.shape[1]:
v = Av * v
else:
v = self.r(v, vType[0], vType) # get specific component
v = Av * v
# we should now be averaged to cell centers (might be a vector)
v = self.r(v.reshape((self.nC, -1), order='F'), 'CC', 'CC', 'M')
if view == 'vec':
outSlice = []
if 'X' not in normal: outSlice.append(getIndSlice(v[0]))
if 'Y' not in normal: outSlice.append(getIndSlice(v[1]))
if 'Z' not in normal: outSlice.append(getIndSlice(v[2]))
return np.r_[mkvc(outSlice[0]), mkvc(outSlice[1])]
else:
return getIndSlice(self.r(v, 'CC', 'CC', 'M'))
def doSlice(v):
if vType == 'CC':
return getIndSlice(self.r(v,'CC','CC','M'))
elif vType == 'CCv':
assert view == 'vec', 'Other types for CCv not supported'
else:
# Now just deal with 'F' and 'E' (x,y,z, maybe...)
aveOp = 'ave' + vType + ('2CCV' if view == 'vec' else '2CC')
Av = getattr(self,aveOp)
if v.size == Av.shape[1]:
v = Av * v
else:
v = self.r(v,vType[0],vType) # get specific component
v = Av * v
# we should now be averaged to cell centers (might be a vector)
v = self.r(v.reshape((self.nC,-1),order='F'),'CC','CC','M')
if view == 'vec':
outSlice = []
if 'X' not in normal: outSlice.append(getIndSlice(v[0]))
if 'Y' not in normal: outSlice.append(getIndSlice(v[1]))
if 'Z' not in normal: outSlice.append(getIndSlice(v[2]))
return np.r_[mkvc(outSlice[0]), mkvc(outSlice[1])]
else:
return getIndSlice(self.r(v,'CC','CC','M'))
def firstVertical(self):
if getattr(self, '_firstVertical', None) is None:
FHzD = self.gridFFT*np.sqrt(self.Kx**2. + self.Ky**2.)
fhzd_pad = np.fft.ifft2(FHzD)
firstVD = np.real(
fhzd_pad[self.npady:-self.npady, self.npadx:-self.npadx])
if self.indNan is not None:
firstVD = mkvc(firstVD)
firstVD[self.indNan] = np.nan
firstVD = firstVD.reshape(self.values.shape, order='F')
self._firstVertical = firstVD
return self._firstVertical
def derivativeX(self):
if getattr(self, '_derivativeX', None) is None:
FHxD = (self.Kx*1j)*self.gridFFT
fhxd_pad = np.fft.ifft2(FHxD)
derivX = np.real(
fhxd_pad[self.npady:-self.npady, self.npadx:-self.npadx])
if self.indNan is not None:
derivX = mkvc(derivX)
derivX[self.indNan] = np.nan
derivX = derivX.reshape(self.values.shape, order='F')
self._derivativeX = derivX
return self._derivativeX
def dfdm(self):
if self.model is None:
self.model = np.zeros(self.G.shape[1])
if getattr(self, '_dfdm', None) is None:
Bxyz = self.Bxyz_a(self.chiMap * self.model)
# Bx = sp.spdiags(Bxyz[:, 0], 0, self.nD, self.nD)
# By = sp.spdiags(Bxyz[:, 1], 0, self.nD, self.nD)
# Bz = sp.spdiags(Bxyz[:, 2], 0, self.nD, self.nD)
ii = np.kron(np.asarray(range(self.survey.nD), dtype='int'), np.ones(3))
jj = np.asarray(range(3*self.survey.nD), dtype='int')
# (data, (row, col)), shape=(3, 3))
# P = s
self._dfdm = sp.csr_matrix(( mkvc(Bxyz), (ii,jj)), shape=(self.survey.nD, 3*self.survey.nD))
return self._dfdm
def valuesFilled(self):
if getattr(self, '_valuesFilled', None) is None:
values = mkvc(self.values)
indVal = np.where(~self.indNan)[0]
tree = cKDTree(self.gridCC[indVal, :])
dists, indexes = tree.query(self.gridCC[self.indNan, :])
uInd = np.unique(indVal[indexes])
xyz = self.gridCC[uInd, :]
_, values[self.indNan] = MathUtils.minCurvatureInterp(
xyz, values[uInd], xyzOut=self.gridCC[self.indNan, :])
self._valuesFilled = values.reshape(self.values.shape, order='F')
return self._valuesFilled
def rotatePointsFromNormals(XYZ,n0,n1,x0=np.r_[0.,0.,0.]):
"""
rotates a grid so that the vector n0 is aligned with the vector n1
:param numpy.array n0: vector of length 3, should have norm 1
:param numpy.array n1: vector of length 3, should have norm 1
:param numpy.array x0: vector of length 3, point about which we perform the rotation
:rtype: numpy.array, 3x3
:return: rotation matrix which rotates the frame so that n0 is aligned with n1
"""
R = rotationMatrixFromNormals(n0, n1)
assert XYZ.shape[1] == 3, "Grid XYZ should be 3 wide"
assert len(x0) == 3, "x0 should have length 3"
X0 = np.ones([XYZ.shape[0],1])*mkvc(x0)
return (XYZ - X0).dot(R.T) + X0 # equivalent to (R*(XYZ - X0)).T + X0
def derivativeY(self):
if getattr(self, '_derivativeY', None) is None:
FHyD = (self.Ky*1j)*self.gridFFT
fhyd_pad = np.fft.ifft2(FHyD)
derivY = np.real(
fhyd_pad[self.npady:-self.npady, self.npadx:-self.npadx])
if self.indNan is not None:
derivY = mkvc(derivY)
derivY[self.indNan] = np.nan
derivY = derivY.reshape(self.values.shape, order='F')
self._derivativeY = derivY
return self._derivativeY
def gridPadded(self):
if getattr(self, '_gridPadded', None) is None:
if getattr(self, '_valuesFilledUC', None) is not None:
# if ~np.array_equal(
# self.valuesFilled[~np.isnan(self.valuesFilled)],
# self.values[~np.isnan(self.indNan)]
# ):
# self._valuesFilled = None
grid = self.valuesFilledUC
else:
if np.any(np.isnan(self.values)):
self.indNan = np.isnan(mkvc(self.values))
grid = self.valuesFilled
else:
grid = self.values
# Add paddings
dpad = np.c_[
np.fliplr(grid[:, 0:self.npadx]),
grid,
np.fliplr(grid[:, -self.npadx:])
]
dpad = np.r_[
np.flipud(dpad[0:self.npady, :]),
dpad,
np.flipud(dpad[-self.npady:, :])
]
# Tapper the paddings
rampx = -np.cos(np.pi*np.asarray(range(self.npadx))/self.npadx)
rampx = np.r_[rampx, np.ones(grid.shape[1]), -rampx]/2. + 0.5
rampy = -np.cos(np.pi*np.asarray(range(self.npady))/self.npady)
rampy = np.r_[rampy, np.ones(grid.shape[0]), -rampy]/2. + 0.5
tapperx, tappery = np.meshgrid(rampx, rampy)
self._gridPadded = tapperx*tappery*dpad
return self._gridPadded
def rotatePointsFromNormals(XYZ, n0, n1, x0=np.r_[0., 0., 0.]):
"""
rotates a grid so that the vector n0 is aligned with the vector n1
:param numpy.array n0: vector of length 3, should have norm 1
:param numpy.array n1: vector of length 3, should have norm 1
:param numpy.array x0: vector of length 3, point about which we perform the rotation
:rtype: numpy.array, 3x3
:return: rotation matrix which rotates the frame so that n0 is aligned with n1
"""
R = rotationMatrixFromNormals(n0, n1)
assert XYZ.shape[1] == 3, "Grid XYZ should be 3 wide"
assert len(x0) == 3, "x0 should have length 3"
X0 = np.ones([XYZ.shape[0], 1]) * mkvc(x0)
return (XYZ - X0).dot(R.T) + X0 # equivalent to (R*(XYZ - X0)).T + X0
def Jvec(self, m, v, u=None):
"""
:param numpy.array m: Conductivity model
:param numpy.ndarray v: vector (model object)
:param simpegEM.TDEM.FieldsTDEM u: Fields resulting from m
:rtype: numpy.ndarray
:return: w (data object)
Multiplying \\\(\\\mathbf{J}\\\) onto a vector can be broken into three steps
* Compute \\\(\\\\vec{p} = \\\mathbf{G}v\\\)
* Solve \\\(\\\hat{\\\mathbf{A}} \\\\vec{y} = \\\\vec{p}\\\)
* Compute \\\(\\\\vec{w} = -\\\mathbf{Q} \\\\vec{y}\\\)
"""
if self.verbose: print '%s\nCalculating J(v)\n%s'%('*'*50,'*'*50)
self.curModel = m
if u is None:
u = self.fields(m)
p = self.Gvec(m, v, u)
y = self.solveAh(m, p)
Jv = self.survey.projectFieldsDeriv(u, v=y)
if self.verbose: print '%s\nDone calculating J(v)\n%s'%('*'*50,'*'*50)
return - mkvc(Jv)
def plotGrid(self, ax=None, nodes=False, faces=False, centers=False, edges=False, lines=True, showIt=False):
"""Plot the nodal, cell-centered and staggered grids for 1,2 and 3 dimensions.
.. plot::
:include-source:
from SimPEG import Mesh, Utils
X, Y = Utils.exampleLrmGrid([3,3],'rotate')
M = Mesh.CurvilinearMesh([X, Y])
M.plotGrid(showIt=True)
"""
import matplotlib.pyplot as plt
import matplotlib
from mpl_toolkits.mplot3d import Axes3D
axOpts = {'projection':'3d'} if self.dim == 3 else {}
if ax is None: ax = plt.subplot(111, **axOpts)
NN = self.r(self.gridN, 'N', 'N', 'M')
if self.dim == 2:
if lines:
X1 = np.c_[mkvc(NN[0][:-1, :]), mkvc(NN[0][1:, :]), mkvc(NN[0][:-1, :])*np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :]), mkvc(NN[1][1:, :]), mkvc(NN[1][:-1, :])*np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1]), mkvc(NN[0][:, 1:]), mkvc(NN[0][:, :-1])*np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1]), mkvc(NN[1][:, 1:]), mkvc(NN[1][:, :-1])*np.nan].flatten()
X = np.r_[X1, X2]
Y = np.r_[Y1, Y2]
ax.plot(X, Y, 'b-')
if centers:
ax.plot(self.gridCC[:,0],self.gridCC[:,1],'ro')
# Nx = self.r(self.normals, 'F', 'Fx', 'V')
# Ny = self.r(self.normals, 'F', 'Fy', 'V')
# Tx = self.r(self.tangents, 'E', 'Ex', 'V')
# Ty = self.r(self.tangents, 'E', 'Ey', 'V')
# ax.plot(self.gridN[:, 0], self.gridN[:, 1], 'bo')
# nX = np.c_[self.gridFx[:, 0], self.gridFx[:, 0] + Nx[0]*length, self.gridFx[:, 0]*np.nan].flatten()
# nY = np.c_[self.gridFx[:, 1], self.gridFx[:, 1] + Nx[1]*length, self.gridFx[:, 1]*np.nan].flatten()
# ax.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'rs')
# ax.plot(nX, nY, 'r-')
# nX = np.c_[self.gridFy[:, 0], self.gridFy[:, 0] + Ny[0]*length, self.gridFy[:, 0]*np.nan].flatten()
# nY = np.c_[self.gridFy[:, 1], self.gridFy[:, 1] + Ny[1]*length, self.gridFy[:, 1]*np.nan].flatten()
# #ax.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'gs')
# ax.plot(nX, nY, 'g-')
# tX = np.c_[self.gridEx[:, 0], self.gridEx[:, 0] + Tx[0]*length, self.gridEx[:, 0]*np.nan].flatten()
# tY = np.c_[self.gridEx[:, 1], self.gridEx[:, 1] + Tx[1]*length, self.gridEx[:, 1]*np.nan].flatten()
# ax.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'r^')
# ax.plot(tX, tY, 'r-')
# nX = np.c_[self.gridEy[:, 0], self.gridEy[:, 0] + Ty[0]*length, self.gridEy[:, 0]*np.nan].flatten()
# nY = np.c_[self.gridEy[:, 1], self.gridEy[:, 1] + Ty[1]*length, self.gridEy[:, 1]*np.nan].flatten()
# #ax.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'g^')
# ax.plot(nX, nY, 'g-')
elif self.dim == 3:
X1 = np.c_[mkvc(NN[0][:-1, :, :]), mkvc(NN[0][1:, :, :]), mkvc(NN[0][:-1, :, :])*np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :, :]), mkvc(NN[1][1:, :, :]), mkvc(NN[1][:-1, :, :])*np.nan].flatten()
Z1 = np.c_[mkvc(NN[2][:-1, :, :]), mkvc(NN[2][1:, :, :]), mkvc(NN[2][:-1, :, :])*np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1, :]), mkvc(NN[0][:, 1:, :]), mkvc(NN[0][:, :-1, :])*np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1, :]), mkvc(NN[1][:, 1:, :]), mkvc(NN[1][:, :-1, :])*np.nan].flatten()
Z2 = np.c_[mkvc(NN[2][:, :-1, :]), mkvc(NN[2][:, 1:, :]), mkvc(NN[2][:, :-1, :])*np.nan].flatten()
X3 = np.c_[mkvc(NN[0][:, :, :-1]), mkvc(NN[0][:, :, 1:]), mkvc(NN[0][:, :, :-1])*np.nan].flatten()
Y3 = np.c_[mkvc(NN[1][:, :, :-1]), mkvc(NN[1][:, :, 1:]), mkvc(NN[1][:, :, :-1])*np.nan].flatten()
Z3 = np.c_[mkvc(NN[2][:, :, :-1]), mkvc(NN[2][:, :, 1:]), mkvc(NN[2][:, :, :-1])*np.nan].flatten()
X = np.r_[X1, X2, X3]
Y = np.r_[Y1, Y2, Y3]
Z = np.r_[Z1, Z2, Z3]
ax.plot(X, Y, 'b', zs=Z)
ax.set_zlabel('x3')
ax.grid(True)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
if showIt:
plt.show()
def _plotCylTensorMesh(self, plotType, *args, **kwargs):
if not self.isSymmetric:
raise Exception('We have not yet implemented this type of view.')
assert plotType in ['plotImage', 'plotGrid']
# Hackity Hack:
# Just create a TM and use its view.
from SimPEG.Mesh import TensorMesh
vType = kwargs.pop('vType', None)
if vType is not None:
if vType.upper() != 'CCV':
if vType.upper() == 'F':
val = mkvc(self.aveF2CCV * args[0])
kwargs['vType'] = 'CCv' # now the vector is cell centered
if vType.upper() == 'E':
val = mkvc(self.aveE2CCV * args[0])
args = (val,) + args[1:]
mirror = kwargs.pop('mirror', None)
if mirror is True:
# create a mirrored mesh
hx = np.hstack([np.flipud(self.hx), self.hx])
x00 = self.x0[0] - self.hx.sum()
M = TensorMesh([hx, self.hz], x0=[x00, self.x0[2]])
# mirror the data
if len(args) > 0:
val = args[0]
if len(val) == self.nC: # only a single value at cell centers
val = val.reshape(self.vnC[0], self.vnC[2], order='F')
val = mkvc(np.vstack([np.flipud(val), val]))
elif len(val) == 2*self.nC:
val_x = val[:self.nC]
val_z = val[self.nC:]
val_x = val_x.reshape(self.vnC[0], self.vnC[2], order='F')
val_x = mkvc(np.vstack([-1.*np.flipud(val_x), val_x])) # by symmetry
val_z = val_z.reshape(self.vnC[0], self.vnC[2], order='F')
val_z = mkvc(np.vstack([np.flipud(val_z), val_z]))
val = np.hstack([val_x, val_z])
args = (val,) + args[1:]
else:
M = TensorMesh([self.hx, self.hz], x0=[self.x0[0], self.x0[2]])
ax = kwargs.get('ax', None)
if ax is None:
fig = plt.figure()
ax = plt.subplot(111)
kwargs['ax'] = ax
else:
assert isinstance(ax, matplotlib.axes.Axes), "ax must be an matplotlib.axes.Axes"
fig = ax.figure
# Don't show things in the TM.plotImage
showIt = kwargs.get('showIt', False)
kwargs['showIt'] = False
out = getattr(M, plotType)(*args, **kwargs)
ax.set_xlabel('x')
ax.set_ylabel('z')
if showIt:
plt.show()
return out
def Intrgl_Fwr_Op(self, m=None, magType='H0', rx_type='tmi'):
"""
Magnetic forward operator in integral form
magType = 'H0' | 'x' | 'y' | 'z'
rx_type = 'tmi' | 'x' | 'y' | 'z'
Return
_G = Linear forward operator | (forwardOnly)=data
"""
if m is not None:
self.model = self.chiMap*m
# Find non-zero cells
if getattr(self, 'actInd', None) is not None:
if self.actInd.dtype == 'bool':
inds = np.where(self.actInd)[0]
else:
inds = self.actInd
else:
inds = np.asarray(range(self.mesh.nC))
nC = len(inds)
# Create active cell projector
P = sp.csr_matrix((np.ones(nC), (inds, range(nC))),
shape=(self.mesh.nC, nC))
# Create vectors of nodal location
# (lower and upper coners for each cell)
if isinstance(self.mesh, Mesh.TreeMesh):
# Get upper and lower corners of each cell
bsw = (self.mesh.gridCC - self.mesh.h_gridded/2.)
tne = (self.mesh.gridCC + self.mesh.h_gridded/2.)
xn1, xn2 = bsw[:, 0], tne[:, 0]
yn1, yn2 = bsw[:, 1], tne[:, 1]
zn1, zn2 = bsw[:, 2], tne[:, 2]
else:
xn = self.mesh.vectorNx
yn = self.mesh.vectorNy
zn = self.mesh.vectorNz
yn2, xn2, zn2 = np.meshgrid(yn[1:], xn[1:], zn[1:])
yn1, xn1, zn1 = np.meshgrid(yn[:-1], xn[:-1], zn[:-1])
# If equivalent source, use semi-infite prism
if self.equiSourceLayer:
zn1 -= 1000.
self.Yn = P.T*np.c_[mkvc(yn1), mkvc(yn2)]
self.Xn = P.T*np.c_[mkvc(xn1), mkvc(xn2)]
self.Zn = P.T*np.c_[mkvc(zn1), mkvc(zn2)]
# survey = self.survey
self.rxLoc = self.survey.srcField.rxList[0].locs
if magType == 'H0':
if getattr(self, 'M', None) is None:
self.M = matutils.dip_azimuth2cartesian(np.ones(nC) * self.survey.srcField.param[1],
np.ones(nC) * self.survey.srcField.param[2])
Mx = sdiag(self.M[:, 0] * self.survey.srcField.param[0])
My = sdiag(self.M[:, 1] * self.survey.srcField.param[0])
Mz = sdiag(self.M[:, 2] * self.survey.srcField.param[0])
self.Mxyz = sp.vstack((Mx, My, Mz))
elif magType == 'full':
self.Mxyz = sp.identity(3*nC) * self.survey.srcField.param[0]
else:
raise Exception('magType must be: "H0" or "full"')
# Loop through all observations and create forward operator (nD-by-nC)
print("Begin forward: M=" + magType + ", Rx type= " + self.rx_type)
# Switch to determine if the process has to be run in parallel
job = Forward(
rxLoc=self.rxLoc, Xn=self.Xn, Yn=self.Yn, Zn=self.Zn,
n_cpu=self.n_cpu, forwardOnly=self.forwardOnly,
model=self.model, rx_type=self.rx_type, Mxyz=self.Mxyz,
P=self.ProjTMI, parallelized=self.parallelized
)
G = job.calculate()
return G
def Jvec(self, m, v, u=None):
"""
Computing Jacobian multiplied by vector
By setting our problem as
.. math ::
\mathbf{C}(\mathbf{m}, \mathbf{u}) = \mathbf{A}\mathbf{u} - \mathbf{rhs} = 0
And taking derivative w.r.t m
.. math ::
\\nabla \mathbf{C}(\mathbf{m}, \mathbf{u}) = \\nabla_m \mathbf{C}(\mathbf{m}) \delta \mathbf{m} +
\\nabla_u \mathbf{C}(\mathbf{u}) \delta \mathbf{u} = 0
\\frac{\delta \mathbf{u}}{\delta \mathbf{m}} = - [\\nabla_u \mathbf{C}(\mathbf{u})]^{-1}\\nabla_m \mathbf{C}(\mathbf{m})
With some linear algebra we can have
.. math ::
\\nabla_u \mathbf{C}(\mathbf{u}) = \mathbf{A}
\\nabla_m \mathbf{C}(\mathbf{m}) =
\\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} - \\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}}
.. math ::
\\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} =
\\frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[\Div \diag (\Div^T \mathbf{u}) \dMfMuI \\right]
\dMfMuI = \diag(\MfMui)^{-1}_{vec} \mathbf{Av}_{F2CC}^T\diag(\mathbf{v})\diag(\\frac{1}{\mu^2})
\\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} = \\frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[
\Div \diag(\M^f_{\mu_{0}^{-1}}\mathbf{B}_0) \dMfMuI \\right] - \diag(\mathbf{v})\mathbf{D} \mathbf{P}_{out}^T\\frac{\partial B_{sBC}}{\partial \mathbf{m}}
In the end,
.. math ::
\\frac{\delta \mathbf{u}}{\delta \mathbf{m}} =
- [ \mathbf{A} ]^{-1}\left[ \\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u}
- \\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} \\right]
A little tricky point here is we are not interested in potential (u), but interested in magnetic flux (B).
Thus, we need sensitivity for B. Now we take derivative of B w.r.t m and have
.. math ::
\\frac{\delta \mathbf{B}} {\delta \mathbf{m}} = \\frac{\partial \mathbf{\mu} } {\partial \mathbf{m} }
\left[
\diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \\
- \diag (\Div^T\mathbf{u})\dMfMuI
\\right ]
- (\MfMui)^{-1}\Div^T\\frac{\delta\mathbf{u}}{\delta \mathbf{m}}
Finally we evaluate the above, but we should remember that
.. note ::
We only want to evalute
.. math ::
\mathbf{J}\mathbf{v} = \\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}}\mathbf{v}
Since forming sensitivity matrix is very expensive in that this monster is "big" and "dense" matrix!!
"""
if u is None:
u = self.fields(m)
B, u = u['B'], u['u']
mu = self.muMap * (m)
dmudm = self.muDeriv
# dchidmu = sdiag(1 / mu_0 * np.ones(self.mesh.nC))
vol = self.mesh.vol
Div = self._Div
Dface = self.mesh.faceDiv
P = self.survey.projectFieldsDeriv(B) # Projection matrix
B0 = self.getB0()
MfMuIvec = 1 / self.MfMui.diagonal()
dMfMuI = sdiag(MfMuIvec**2) * \
self.mesh.aveF2CC.T * sdiag(vol * 1. / mu**2)
# A = self._Div*self.MfMuI*self._Div.T
# RHS = Div*MfMuI*MfMu0*B0 - Div*B0 + Mc*Dface*Pout.T*Bbc
# C(m,u) = A*m-rhs
# dudm = -(dCdu)^(-1)dCdm
dCdu = self.getA(m)
dCdm_A = Div * (sdiag(Div.T * u) * dMfMuI * dmudm)
dCdm_RHS1 = Div * (sdiag(self.MfMu0 * B0) * dMfMuI)
# temp1 = (Dface * (self._Pout.T * self.Bbc_const * self.Bbc))
# dCdm_RHS2v = (sdiag(vol) * temp1) * \
# np.inner(vol, dchidmu * dmudm * v)
# dCdm_RHSv = dCdm_RHS1*(dmudm*v) + dCdm_RHS2v
dCdm_RHSv = dCdm_RHS1 * (dmudm * v)
dCdm_v = dCdm_A * v - dCdm_RHSv
m1 = sp.linalg.interface.aslinearoperator(
sdiag(1 / dCdu.diagonal())
)
sol, info = sp.linalg.bicgstab(dCdu, dCdm_v,
tol=1e-6, maxiter=1000, M=m1)
if info > 0:
print("Iterative solver did not work well (Jvec)")
# raise Exception ("Iterative solver did not work well")
# B = self.MfMuI*self.MfMu0*B0-B0-self.MfMuI*self._Div.T*u
# dBdm = d\mudm*dBd\mu
dudm = -sol
dBdmv = (
sdiag(self.MfMu0 * B0) * (dMfMuI * (dmudm * v))
- sdiag(Div.T * u) * (dMfMuI * (dmudm * v))
- self.MfMuI * (Div.T * (dudm))
)
return mkvc(P * dBdmv)
def plotGrid(self, ax=None, nodes=False, faces=False, centers=False, edges=False, lines=True, showIt=False):
"""Plot the nodal, cell-centered and staggered grids for 1,2 and 3 dimensions.
.. plot::
:include-source:
from SimPEG import Mesh, Utils
X, Y = Utils.exampleLrmGrid([3,3],'rotate')
M = Mesh.CurvilinearMesh([X, Y])
M.plotGrid(showIt=True)
"""
import matplotlib.pyplot as plt
import matplotlib
from mpl_toolkits.mplot3d import Axes3D
axOpts = {"projection": "3d"} if self.dim == 3 else {}
if ax is None:
ax = plt.subplot(111, **axOpts)
NN = self.r(self.gridN, "N", "N", "M")
if self.dim == 2:
if lines:
X1 = np.c_[mkvc(NN[0][:-1, :]), mkvc(NN[0][1:, :]), mkvc(NN[0][:-1, :]) * np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :]), mkvc(NN[1][1:, :]), mkvc(NN[1][:-1, :]) * np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1]), mkvc(NN[0][:, 1:]), mkvc(NN[0][:, :-1]) * np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1]), mkvc(NN[1][:, 1:]), mkvc(NN[1][:, :-1]) * np.nan].flatten()
X = np.r_[X1, X2]
Y = np.r_[Y1, Y2]
ax.plot(X, Y, "b-")
if centers:
ax.plot(self.gridCC[:, 0], self.gridCC[:, 1], "ro")
# Nx = self.r(self.normals, 'F', 'Fx', 'V')
# Ny = self.r(self.normals, 'F', 'Fy', 'V')
# Tx = self.r(self.tangents, 'E', 'Ex', 'V')
# Ty = self.r(self.tangents, 'E', 'Ey', 'V')
# ax.plot(self.gridN[:, 0], self.gridN[:, 1], 'bo')
# nX = np.c_[self.gridFx[:, 0], self.gridFx[:, 0] + Nx[0]*length, self.gridFx[:, 0]*np.nan].flatten()
# nY = np.c_[self.gridFx[:, 1], self.gridFx[:, 1] + Nx[1]*length, self.gridFx[:, 1]*np.nan].flatten()
# ax.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'rs')
# ax.plot(nX, nY, 'r-')
# nX = np.c_[self.gridFy[:, 0], self.gridFy[:, 0] + Ny[0]*length, self.gridFy[:, 0]*np.nan].flatten()
# nY = np.c_[self.gridFy[:, 1], self.gridFy[:, 1] + Ny[1]*length, self.gridFy[:, 1]*np.nan].flatten()
# #ax.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'gs')
# ax.plot(nX, nY, 'g-')
# tX = np.c_[self.gridEx[:, 0], self.gridEx[:, 0] + Tx[0]*length, self.gridEx[:, 0]*np.nan].flatten()
# tY = np.c_[self.gridEx[:, 1], self.gridEx[:, 1] + Tx[1]*length, self.gridEx[:, 1]*np.nan].flatten()
# ax.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'r^')
# ax.plot(tX, tY, 'r-')
# nX = np.c_[self.gridEy[:, 0], self.gridEy[:, 0] + Ty[0]*length, self.gridEy[:, 0]*np.nan].flatten()
# nY = np.c_[self.gridEy[:, 1], self.gridEy[:, 1] + Ty[1]*length, self.gridEy[:, 1]*np.nan].flatten()
# #ax.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'g^')
# ax.plot(nX, nY, 'g-')
elif self.dim == 3:
X1 = np.c_[mkvc(NN[0][:-1, :, :]), mkvc(NN[0][1:, :, :]), mkvc(NN[0][:-1, :, :]) * np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :, :]), mkvc(NN[1][1:, :, :]), mkvc(NN[1][:-1, :, :]) * np.nan].flatten()
Z1 = np.c_[mkvc(NN[2][:-1, :, :]), mkvc(NN[2][1:, :, :]), mkvc(NN[2][:-1, :, :]) * np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1, :]), mkvc(NN[0][:, 1:, :]), mkvc(NN[0][:, :-1, :]) * np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1, :]), mkvc(NN[1][:, 1:, :]), mkvc(NN[1][:, :-1, :]) * np.nan].flatten()
Z2 = np.c_[mkvc(NN[2][:, :-1, :]), mkvc(NN[2][:, 1:, :]), mkvc(NN[2][:, :-1, :]) * np.nan].flatten()
X3 = np.c_[mkvc(NN[0][:, :, :-1]), mkvc(NN[0][:, :, 1:]), mkvc(NN[0][:, :, :-1]) * np.nan].flatten()
Y3 = np.c_[mkvc(NN[1][:, :, :-1]), mkvc(NN[1][:, :, 1:]), mkvc(NN[1][:, :, :-1]) * np.nan].flatten()
Z3 = np.c_[mkvc(NN[2][:, :, :-1]), mkvc(NN[2][:, :, 1:]), mkvc(NN[2][:, :, :-1]) * np.nan].flatten()
X = np.r_[X1, X2, X3]
Y = np.r_[Y1, Y2, Y3]
Z = np.r_[Z1, Z2, Z3]
ax.plot(X, Y, "b", zs=Z)
ax.set_zlabel("x3")
ax.grid(True)
ax.set_xlabel("x1")
ax.set_ylabel("x2")
if showIt:
plt.show()
# Get extent of points
F = NearestNDInterpolator(topo[:, :2], topo[:, 2])
zOffset = 0
# Cycle through the first 3 octree levels
for ii in range(3):
dx = mesh.hx.min() * 2**ii
nCx = int((limx[0] - limx[1]) / dx)
nCy = int((limy[0] - limy[1]) / dx)
# Create a grid at the octree level in xy
CCx, CCy = np.meshgrid(np.linspace(limx[1], limx[0], nCx),
np.linspace(limy[1], limy[0], nCy))
z = F(mkvc(CCx), mkvc(CCy))
# level means number of layers in current OcTree level
for level in range(int(nCpad[ii])):
mesh.insert_cells(np.c_[mkvc(CCx), mkvc(CCy), z - zOffset],
np.ones_like(z) * maxLevel - ii,
finalize=False)
zOffset += dx
mesh.finalize()
# Define an active cells from topo
actv = Utils.surface2ind_topo(mesh, topo)
nC = int(actv.sum())
def test_mkvc3(self):
x = mkvc(self.a, 3)
self.assertTrue(x.shape, (3, 1, 1))
def plotGrid(self, length=0.05, showIt=False):
"""Plot the nodal, cell-centered and staggered grids for 1,2 and 3 dimensions.
.. plot::
:include-source:
from SimPEG import Mesh, Utils
X, Y = Utils.exampleCurvGird([3,3],'rotate')
M = Mesh.CurvilinearMesh([X, Y])
M.plotGrid(showIt=True)
"""
NN = self.r(self.gridN, 'N', 'N', 'M')
if self.dim == 2:
fig = plt.figure(2)
fig.clf()
ax = plt.subplot(111)
X1 = np.c_[mkvc(NN[0][:-1, :]),
mkvc(NN[0][1:, :]),
mkvc(NN[0][:-1, :]) * np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :]),
mkvc(NN[1][1:, :]),
mkvc(NN[1][:-1, :]) * np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1]),
mkvc(NN[0][:, 1:]),
mkvc(NN[0][:, :-1]) * np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1]),
mkvc(NN[1][:, 1:]),
mkvc(NN[1][:, :-1]) * np.nan].flatten()
X = np.r_[X1, X2]
Y = np.r_[Y1, Y2]
plt.plot(X, Y)
plt.hold(True)
Nx = self.r(self.normals, 'F', 'Fx', 'V')
Ny = self.r(self.normals, 'F', 'Fy', 'V')
Tx = self.r(self.tangents, 'E', 'Ex', 'V')
Ty = self.r(self.tangents, 'E', 'Ey', 'V')
plt.plot(self.gridN[:, 0], self.gridN[:, 1], 'bo')
nX = np.c_[self.gridFx[:, 0], self.gridFx[:, 0] + Nx[0] * length,
self.gridFx[:, 0] * np.nan].flatten()
nY = np.c_[self.gridFx[:, 1], self.gridFx[:, 1] + Nx[1] * length,
self.gridFx[:, 1] * np.nan].flatten()
plt.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'rs')
plt.plot(nX, nY, 'r-')
nX = np.c_[self.gridFy[:, 0], self.gridFy[:, 0] + Ny[0] * length,
self.gridFy[:, 0] * np.nan].flatten()
nY = np.c_[self.gridFy[:, 1], self.gridFy[:, 1] + Ny[1] * length,
self.gridFy[:, 1] * np.nan].flatten()
#plt.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'gs')
plt.plot(nX, nY, 'g-')
tX = np.c_[self.gridEx[:, 0], self.gridEx[:, 0] + Tx[0] * length,
self.gridEx[:, 0] * np.nan].flatten()
tY = np.c_[self.gridEx[:, 1], self.gridEx[:, 1] + Tx[1] * length,
self.gridEx[:, 1] * np.nan].flatten()
plt.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'r^')
plt.plot(tX, tY, 'r-')
nX = np.c_[self.gridEy[:, 0], self.gridEy[:, 0] + Ty[0] * length,
self.gridEy[:, 0] * np.nan].flatten()
nY = np.c_[self.gridEy[:, 1], self.gridEy[:, 1] + Ty[1] * length,
self.gridEy[:, 1] * np.nan].flatten()
#plt.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'g^')
plt.plot(nX, nY, 'g-')
plt.axis('equal')
elif self.dim == 3:
fig = plt.figure(3)
fig.clf()
ax = fig.add_subplot(111, projection='3d')
X1 = np.c_[mkvc(NN[0][:-1, :, :]),
mkvc(NN[0][1:, :, :]),
mkvc(NN[0][:-1, :, :]) * np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :, :]),
mkvc(NN[1][1:, :, :]),
mkvc(NN[1][:-1, :, :]) * np.nan].flatten()
Z1 = np.c_[mkvc(NN[2][:-1, :, :]),
mkvc(NN[2][1:, :, :]),
mkvc(NN[2][:-1, :, :]) * np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1, :]),
mkvc(NN[0][:, 1:, :]),
mkvc(NN[0][:, :-1, :]) * np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1, :]),
mkvc(NN[1][:, 1:, :]),
mkvc(NN[1][:, :-1, :]) * np.nan].flatten()
Z2 = np.c_[mkvc(NN[2][:, :-1, :]),
mkvc(NN[2][:, 1:, :]),
mkvc(NN[2][:, :-1, :]) * np.nan].flatten()
X3 = np.c_[mkvc(NN[0][:, :, :-1]),
mkvc(NN[0][:, :, 1:]),
mkvc(NN[0][:, :, :-1]) * np.nan].flatten()
Y3 = np.c_[mkvc(NN[1][:, :, :-1]),
mkvc(NN[1][:, :, 1:]),
mkvc(NN[1][:, :, :-1]) * np.nan].flatten()
Z3 = np.c_[mkvc(NN[2][:, :, :-1]),
mkvc(NN[2][:, :, 1:]),
mkvc(NN[2][:, :, :-1]) * np.nan].flatten()
X = np.r_[X1, X2, X3]
Y = np.r_[Y1, Y2, Y3]
Z = np.r_[Z1, Z2, Z3]
plt.plot(X, Y, 'b', zs=Z)
ax.set_zlabel('x3')
ax.grid(True)
ax.hold(False)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
if showIt: plt.show()
def _plotImage2D(self,
v,
vType='CC',
grid=False,
view='real',
ax=None,
clim=None,
showIt=False,
pcolorOpts={},
streamOpts={'color': 'k'},
gridOpts={'color': 'k'}):
vTypeOptsCC = ['N', 'CC', 'Fx', 'Fy', 'Ex', 'Ey']
vTypeOptsV = ['CCv', 'F', 'E']
vTypeOpts = vTypeOptsCC + vTypeOptsV
if view == 'vec':
assert vType in vTypeOptsV, "vType must be in ['%s'] when view='vec'" % "','".join(
vTypeOptsV)
assert vType in vTypeOpts, "vType must be in ['%s']" % "','".join(
vTypeOpts)
viewOpts = ['real', 'imag', 'abs', 'vec']
assert view in viewOpts, "view must be in ['%s']" % "','".join(
viewOpts)
if ax is None:
fig = plt.figure()
ax = plt.subplot(111)
else:
assert isinstance(
ax, matplotlib.axes.Axes), "ax must be an matplotlib.axes.Axes"
fig = ax.figure
# Reshape to a cell centered variable
if vType == 'CC':
pass
elif vType == 'CCv':
assert view == 'vec', 'Other types for CCv not supported'
elif vType in ['F', 'E', 'N']:
aveOp = 'ave' + vType + ('2CCV' if view == 'vec' else '2CC')
v = getattr(
self, aveOp) * v # average to cell centers (might be a vector)
elif vType in ['Fx', 'Fy', 'Ex', 'Ey']:
aveOp = 'ave' + vType[0] + '2CCV'
v = getattr(
self, aveOp) * v # average to cell centers (might be a vector)
xORy = {'x': 0, 'y': 1}[vType[1]]
v = v.reshape((self.nC, -1), order='F')[:, xORy]
out = ()
if view in ['real', 'imag', 'abs']:
v = self.r(v, 'CC', 'CC', 'M')
v = getattr(np, view)(v) # e.g. np.real(v)
if clim is None:
clim = [v.min(), v.max()]
v = np.ma.masked_where(np.isnan(v), v)
out += (ax.pcolormesh(self.vectorNx,
self.vectorNy,
v.T,
vmin=clim[0],
vmax=clim[1],
**pcolorOpts), )
elif view in ['vec']:
U, V = self.r(v.reshape((self.nC, -1), order='F'), 'CC', 'CC', 'M')
if clim is None:
uv = np.sqrt(U**2 + V**2)
clim = [uv.min(), uv.max()]
# Matplotlib seems to not support irregular
# spaced vectors at the moment. So we will
# Interpolate down to a regular mesh at the
# smallest mesh size in this 2D slice.
nxi = int(self.hx.sum() / self.hx.min())
nyi = int(self.hy.sum() / self.hy.min())
tMi = self.__class__([
np.ones(nxi) * self.hx.sum() / nxi,
np.ones(nyi) * self.hy.sum() / nyi
], self.x0)
P = self.getInterpolationMat(tMi.gridCC, 'CC', zerosOutside=True)
Ui = tMi.r(P * mkvc(U), 'CC', 'CC', 'M')
Vi = tMi.r(P * mkvc(V), 'CC', 'CC', 'M')
# End Interpolation
out += (ax.pcolormesh(self.vectorNx,
self.vectorNy,
np.sqrt(U**2 + V**2).T,
vmin=clim[0],
vmax=clim[1],
**pcolorOpts), )
out += (ax.streamplot(tMi.vectorCCx, tMi.vectorCCy, Ui.T, Vi.T,
**streamOpts), )
if grid:
xXGrid = np.c_[self.vectorNx, self.vectorNx,
np.nan * np.ones(self.nNx)].flatten()
xYGrid = np.c_[self.vectorNy[0] * np.ones(self.nNx),
self.vectorNy[-1] * np.ones(self.nNx),
np.nan * np.ones(self.nNx)].flatten()
yXGrid = np.c_[self.vectorNx[0] * np.ones(self.nNy),
self.vectorNx[-1] * np.ones(self.nNy),
np.nan * np.ones(self.nNy)].flatten()
yYGrid = np.c_[self.vectorNy, self.vectorNy,
np.nan * np.ones(self.nNy)].flatten()
out += (ax.plot(np.r_[xXGrid, yXGrid], np.r_[xYGrid, yYGrid],
**gridOpts)[0], )
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_xlim(*self.vectorNx[[0, -1]])
ax.set_ylim(*self.vectorNy[[0, -1]])
if showIt: plt.show()
return out
def calcTrow(self, xyzLoc):
"""
Load in the active nodes of a tensor mesh and computes the gravity tensor
for a given observation location xyzLoc[obsx, obsy, obsz]
INPUT:
Xn, Yn, Zn: Node location matrix for the lower and upper most corners of
all cells in the mesh shape[nC,2]
M
OUTPUT:
Tx = [Txx Txy Txz]
Ty = [Tyx Tyy Tyz]
Tz = [Tzx Tzy Tzz]
where each elements have dimension 1-by-nC.
Only the upper half 5 elements have to be computed since symetric.
Currently done as for-loops but will eventually be changed to vector
indexing, once the topography has been figured out.
"""
NewtG = constants.G*1e+8 # Convertion from mGal (1e-5) and g/cc (1e-3)
eps = 1e-8 # add a small value to the locations to avoid
# Pre-allocate space for 1D array
row = np.zeros((1, self.Xn.shape[0]))
dz = xyzLoc[2] - self.Zn
dy = self.Yn - xyzLoc[1]
dx = self.Xn - xyzLoc[0]
# Compute contribution from each corners
for aa in range(2):
for bb in range(2):
for cc in range(2):
r = (
mkvc(dx[:, aa]) ** 2 +
mkvc(dy[:, bb]) ** 2 +
mkvc(dz[:, cc]) ** 2
) ** (0.50)
if self.rx_type == 'x':
row -= NewtG * (-1) ** aa * (-1) ** bb * (-1) ** cc * (
dy[:, bb] * np.log(dz[:, cc] + r + eps) +
dz[:, cc] * np.log(dy[:, bb] + r + eps) -
dx[:, aa] * np.arctan(dy[:, bb] * dz[:, cc] /
(dx[:, aa] * r + eps)))
elif self.rx_type == 'y':
row -= NewtG * (-1) ** aa * (-1) ** bb * (-1) ** cc * (
dx[:, aa] * np.log(dz[:, cc] + r + eps) +
dz[:, cc] * np.log(dx[:, aa] + r + eps) -
dy[:, bb] * np.arctan(dx[:, aa] * dz[:, cc] /
(dy[:, bb] * r + eps)))
else:
row -= NewtG * (-1) ** aa * (-1) ** bb * (-1) ** cc * (
dx[:, aa] * np.log(dy[:, bb] + r + eps) +
dy[:, bb] * np.log(dx[:, aa] + r + eps) -
dz[:, cc] * np.arctan(dx[:, aa] * dy[:, bb] /
(dz[:, cc] * r + eps)))
if self.forwardOnly:
return np.dot(row, self.model)
else:
return row
def test_mkvc1(self):
x = mkvc(self.a)
self.assertTrue(x.shape, (3,))
def plotGrid(self, length=0.05, showIt=False):
"""Plot the nodal, cell-centered and staggered grids for 1,2 and 3 dimensions.
.. plot::
:include-source:
from SimPEG import Mesh, Utils
X, Y = Utils.exampleCurvGird([3,3],'rotate')
M = Mesh.CurvilinearMesh([X, Y])
M.plotGrid(showIt=True)
"""
NN = self.r(self.gridN, 'N', 'N', 'M')
if self.dim == 2:
fig = plt.figure(2)
fig.clf()
ax = plt.subplot(111)
X1 = np.c_[mkvc(NN[0][:-1, :]), mkvc(NN[0][1:, :]), mkvc(NN[0][:-1, :])*np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :]), mkvc(NN[1][1:, :]), mkvc(NN[1][:-1, :])*np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1]), mkvc(NN[0][:, 1:]), mkvc(NN[0][:, :-1])*np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1]), mkvc(NN[1][:, 1:]), mkvc(NN[1][:, :-1])*np.nan].flatten()
X = np.r_[X1, X2]
Y = np.r_[Y1, Y2]
plt.plot(X, Y)
plt.hold(True)
Nx = self.r(self.normals, 'F', 'Fx', 'V')
Ny = self.r(self.normals, 'F', 'Fy', 'V')
Tx = self.r(self.tangents, 'E', 'Ex', 'V')
Ty = self.r(self.tangents, 'E', 'Ey', 'V')
plt.plot(self.gridN[:, 0], self.gridN[:, 1], 'bo')
nX = np.c_[self.gridFx[:, 0], self.gridFx[:, 0] + Nx[0]*length, self.gridFx[:, 0]*np.nan].flatten()
nY = np.c_[self.gridFx[:, 1], self.gridFx[:, 1] + Nx[1]*length, self.gridFx[:, 1]*np.nan].flatten()
plt.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'rs')
plt.plot(nX, nY, 'r-')
nX = np.c_[self.gridFy[:, 0], self.gridFy[:, 0] + Ny[0]*length, self.gridFy[:, 0]*np.nan].flatten()
nY = np.c_[self.gridFy[:, 1], self.gridFy[:, 1] + Ny[1]*length, self.gridFy[:, 1]*np.nan].flatten()
#plt.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'gs')
plt.plot(nX, nY, 'g-')
tX = np.c_[self.gridEx[:, 0], self.gridEx[:, 0] + Tx[0]*length, self.gridEx[:, 0]*np.nan].flatten()
tY = np.c_[self.gridEx[:, 1], self.gridEx[:, 1] + Tx[1]*length, self.gridEx[:, 1]*np.nan].flatten()
plt.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'r^')
plt.plot(tX, tY, 'r-')
nX = np.c_[self.gridEy[:, 0], self.gridEy[:, 0] + Ty[0]*length, self.gridEy[:, 0]*np.nan].flatten()
nY = np.c_[self.gridEy[:, 1], self.gridEy[:, 1] + Ty[1]*length, self.gridEy[:, 1]*np.nan].flatten()
#plt.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'g^')
plt.plot(nX, nY, 'g-')
plt.axis('equal')
elif self.dim == 3:
fig = plt.figure(3)
fig.clf()
ax = fig.add_subplot(111, projection='3d')
X1 = np.c_[mkvc(NN[0][:-1, :, :]), mkvc(NN[0][1:, :, :]), mkvc(NN[0][:-1, :, :])*np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][:-1, :, :]), mkvc(NN[1][1:, :, :]), mkvc(NN[1][:-1, :, :])*np.nan].flatten()
Z1 = np.c_[mkvc(NN[2][:-1, :, :]), mkvc(NN[2][1:, :, :]), mkvc(NN[2][:-1, :, :])*np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, :-1, :]), mkvc(NN[0][:, 1:, :]), mkvc(NN[0][:, :-1, :])*np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, :-1, :]), mkvc(NN[1][:, 1:, :]), mkvc(NN[1][:, :-1, :])*np.nan].flatten()
Z2 = np.c_[mkvc(NN[2][:, :-1, :]), mkvc(NN[2][:, 1:, :]), mkvc(NN[2][:, :-1, :])*np.nan].flatten()
X3 = np.c_[mkvc(NN[0][:, :, :-1]), mkvc(NN[0][:, :, 1:]), mkvc(NN[0][:, :, :-1])*np.nan].flatten()
Y3 = np.c_[mkvc(NN[1][:, :, :-1]), mkvc(NN[1][:, :, 1:]), mkvc(NN[1][:, :, :-1])*np.nan].flatten()
Z3 = np.c_[mkvc(NN[2][:, :, :-1]), mkvc(NN[2][:, :, 1:]), mkvc(NN[2][:, :, :-1])*np.nan].flatten()
X = np.r_[X1, X2, X3]
Y = np.r_[Y1, Y2, Y3]
Z = np.r_[Z1, Z2, Z3]
plt.plot(X, Y, 'b', zs=Z)
ax.set_zlabel('x3')
ax.grid(True)
ax.hold(False)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
if showIt: plt.show()
def test_mkvc2(self):
x = mkvc(self.a, 2)
self.assertTrue(x.shape, (3, 1))
def _plotImage2D(
self,
v,
vType="CC",
grid=False,
view="real",
ax=None,
clim=None,
showIt=False,
pcolorOpts=None,
streamOpts=None,
gridOpts=None,
):
if pcolorOpts is None:
pcolorOpts = {}
if streamOpts is None:
streamOpts = {"color": "k"}
if gridOpts is None:
gridOpts = {"color": "k"}
vTypeOptsCC = ["N", "CC", "Fx", "Fy", "Ex", "Ey"]
vTypeOptsV = ["CCv", "F", "E"]
vTypeOpts = vTypeOptsCC + vTypeOptsV
if view == "vec":
assert vType in vTypeOptsV, "vType must be in ['{0!s}'] when view='vec'".format("','".join(vTypeOptsV))
assert vType in vTypeOpts, "vType must be in ['{0!s}']".format("','".join(vTypeOpts))
viewOpts = ["real", "imag", "abs", "vec"]
assert view in viewOpts, "view must be in ['{0!s}']".format("','".join(viewOpts))
if ax is None:
fig = plt.figure()
ax = plt.subplot(111)
else:
assert isinstance(ax, matplotlib.axes.Axes), "ax must be an matplotlib.axes.Axes"
fig = ax.figure
# Reshape to a cell centered variable
if vType == "CC":
pass
elif vType == "CCv":
assert view == "vec", "Other types for CCv not supported"
elif vType in ["F", "E", "N"]:
aveOp = "ave" + vType + ("2CCV" if view == "vec" else "2CC")
v = getattr(self, aveOp) * v # average to cell centers (might be a vector)
elif vType in ["Fx", "Fy", "Ex", "Ey"]:
aveOp = "ave" + vType[0] + "2CCV"
v = getattr(self, aveOp) * v # average to cell centers (might be a vector)
xORy = {"x": 0, "y": 1}[vType[1]]
v = v.reshape((self.nC, -1), order="F")[:, xORy]
out = ()
if view in ["real", "imag", "abs"]:
v = self.r(v, "CC", "CC", "M")
v = getattr(np, view)(v) # e.g. np.real(v)
if clim is None:
clim = [v.min(), v.max()]
v = np.ma.masked_where(np.isnan(v), v)
out += (ax.pcolormesh(self.vectorNx, self.vectorNy, v.T, vmin=clim[0], vmax=clim[1], **pcolorOpts),)
elif view in ["vec"]:
U, V = self.r(v.reshape((self.nC, -1), order="F"), "CC", "CC", "M")
if clim is None:
uv = np.sqrt(U ** 2 + V ** 2)
clim = [uv.min(), uv.max()]
# Matplotlib seems to not support irregular
# spaced vectors at the moment. So we will
# Interpolate down to a regular mesh at the
# smallest mesh size in this 2D slice.
nxi = int(self.hx.sum() / self.hx.min())
nyi = int(self.hy.sum() / self.hy.min())
tMi = self.__class__([np.ones(nxi) * self.hx.sum() / nxi, np.ones(nyi) * self.hy.sum() / nyi], self.x0)
P = self.getInterpolationMat(tMi.gridCC, "CC", zerosOutside=True)
Ui = tMi.r(P * mkvc(U), "CC", "CC", "M")
Vi = tMi.r(P * mkvc(V), "CC", "CC", "M")
# End Interpolation
out += (
ax.pcolormesh(
self.vectorNx, self.vectorNy, np.sqrt(U ** 2 + V ** 2).T, vmin=clim[0], vmax=clim[1], **pcolorOpts
),
)
out += (ax.streamplot(tMi.vectorCCx, tMi.vectorCCy, Ui.T, Vi.T, **streamOpts),)
if grid:
xXGrid = np.c_[self.vectorNx, self.vectorNx, np.nan * np.ones(self.nNx)].flatten()
xYGrid = np.c_[
self.vectorNy[0] * np.ones(self.nNx), self.vectorNy[-1] * np.ones(self.nNx), np.nan * np.ones(self.nNx)
].flatten()
yXGrid = np.c_[
self.vectorNx[0] * np.ones(self.nNy), self.vectorNx[-1] * np.ones(self.nNy), np.nan * np.ones(self.nNy)
].flatten()
yYGrid = np.c_[self.vectorNy, self.vectorNy, np.nan * np.ones(self.nNy)].flatten()
out += (ax.plot(np.r_[xXGrid, yXGrid], np.r_[xYGrid, yYGrid], **gridOpts)[0],)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_xlim(*self.vectorNx[[0, -1]])
ax.set_ylim(*self.vectorNy[[0, -1]])
if showIt:
plt.show()
return out
def _plotImage2D(self, v, vType='CC', grid=False, view='real',
ax=None, clim=None, showIt=False,
pcolorOpts={},
streamOpts={'color':'k'},
gridOpts={'color':'k'}
):
vTypeOptsCC = ['N','CC','Fx','Fy','Ex','Ey']
vTypeOptsV = ['CCv','F','E']
vTypeOpts = vTypeOptsCC + vTypeOptsV
if view == 'vec':
assert vType in vTypeOptsV, "vType must be in ['%s'] when view='vec'" % "','".join(vTypeOptsV)
assert vType in vTypeOpts, "vType must be in ['%s']" % "','".join(vTypeOpts)
viewOpts = ['real','imag','abs','vec']
assert view in viewOpts, "view must be in ['%s']" % "','".join(viewOpts)
if ax is None:
fig = plt.figure()
ax = plt.subplot(111)
else:
assert isinstance(ax, matplotlib.axes.Axes), "ax must be an matplotlib.axes.Axes"
fig = ax.figure
# Reshape to a cell centered variable
if vType == 'CC':
pass
elif vType == 'CCv':
assert view == 'vec', 'Other types for CCv not supported'
elif vType in ['F', 'E', 'N']:
aveOp = 'ave' + vType + ('2CCV' if view == 'vec' else '2CC')
v = getattr(self,aveOp)*v # average to cell centers (might be a vector)
elif vType in ['Fx','Fy','Ex','Ey']:
aveOp = 'ave' + vType[0] + '2CCV'
v = getattr(self,aveOp)*v # average to cell centers (might be a vector)
xORy = {'x':0,'y':1}[vType[1]]
v = v.reshape((self.nC,-1), order='F')[:,xORy]
out = ()
if view in ['real','imag','abs']:
v = self.r(v, 'CC', 'CC', 'M')
v = getattr(np,view)(v) # e.g. np.real(v)
if clim is None:
clim = [v.min(),v.max()]
v = np.ma.masked_where(np.isnan(v), v)
out += (ax.pcolormesh(self.vectorNx, self.vectorNy, v.T, vmin=clim[0], vmax=clim[1], **pcolorOpts),)
elif view in ['vec']:
U, V = self.r(v.reshape((self.nC,-1), order='F'), 'CC', 'CC', 'M')
if clim is None:
uv = np.sqrt(U**2 + V**2)
clim = [uv.min(),uv.max()]
# Matplotlib seems to not support irregular
# spaced vectors at the moment. So we will
# Interpolate down to a regular mesh at the
# smallest mesh size in this 2D slice.
nxi = int(self.hx.sum()/self.hx.min())
nyi = int(self.hy.sum()/self.hy.min())
tMi = self.__class__([np.ones(nxi)*self.hx.sum()/nxi,
np.ones(nyi)*self.hy.sum()/nyi], self.x0)
P = self.getInterpolationMat(tMi.gridCC,'CC',zerosOutside=True)
Ui = tMi.r(P*mkvc(U), 'CC', 'CC', 'M')
Vi = tMi.r(P*mkvc(V), 'CC', 'CC', 'M')
# End Interpolation
out += (ax.pcolormesh(self.vectorNx, self.vectorNy, np.sqrt(U**2+V**2).T, vmin=clim[0], vmax=clim[1], **pcolorOpts),)
out += (ax.streamplot(tMi.vectorCCx, tMi.vectorCCy, Ui.T, Vi.T, **streamOpts),)
if grid:
xXGrid = np.c_[self.vectorNx,self.vectorNx,np.nan*np.ones(self.nNx)].flatten()
xYGrid = np.c_[self.vectorNy[0]*np.ones(self.nNx),self.vectorNy[-1]*np.ones(self.nNx),np.nan*np.ones(self.nNx)].flatten()
yXGrid = np.c_[self.vectorNx[0]*np.ones(self.nNy),self.vectorNx[-1]*np.ones(self.nNy),np.nan*np.ones(self.nNy)].flatten()
yYGrid = np.c_[self.vectorNy,self.vectorNy,np.nan*np.ones(self.nNy)].flatten()
out += (ax.plot(np.r_[xXGrid,yXGrid],np.r_[xYGrid,yYGrid],**gridOpts)[0],)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_xlim(*self.vectorNx[[0,-1]])
ax.set_ylim(*self.vectorNy[[0,-1]])
if showIt: plt.show()
return out
def plotGrid(self, ax=None, nodes=False, faces=False, centers=False, edges=False, lines=True, showIt=False):
"""Plot the nodal, cell-centered and staggered grids for 1,2 and 3 dimensions.
:param bool nodes: plot nodes
:param bool faces: plot faces
:param bool centers: plot centers
:param bool edges: plot edges
:param bool lines: plot lines connecting nodes
:param bool showIt: call plt.show()
.. plot::
:include-source:
from SimPEG import Mesh, np
h1 = np.linspace(.1,.5,3)
h2 = np.linspace(.1,.5,5)
mesh = Mesh.TensorMesh([h1, h2])
mesh.plotGrid(nodes=True, faces=True, centers=True, lines=True, showIt=True)
.. plot::
:include-source:
from SimPEG import Mesh, np
h1 = np.linspace(.1,.5,3)
h2 = np.linspace(.1,.5,5)
h3 = np.linspace(.1,.5,3)
mesh = Mesh.TensorMesh([h1,h2,h3])
mesh.plotGrid(nodes=True, faces=True, centers=True, lines=True, showIt=True)
"""
axOpts = {'projection':'3d'} if self.dim == 3 else {}
if ax is None:
fig = plt.figure()
ax = plt.subplot(111, **axOpts)
else:
assert isinstance(ax, matplotlib.axes.Axes), "ax must be an matplotlib.axes.Axes"
fig = ax.figure
if self.dim == 1:
if nodes:
ax.plot(self.gridN, np.ones(self.nN), 'bs')
if centers:
ax.plot(self.gridCC, np.ones(self.nC), 'ro')
if lines:
ax.plot(self.gridN, np.ones(self.nN), 'b.-')
ax.set_xlabel('x1')
elif self.dim == 2:
if nodes:
ax.plot(self.gridN[:, 0], self.gridN[:, 1], 'bs')
if centers:
ax.plot(self.gridCC[:, 0], self.gridCC[:, 1], 'ro')
if faces:
ax.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'g>')
ax.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'g^')
if edges:
ax.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'c>')
ax.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'c^')
# Plot the grid lines
if lines:
NN = self.r(self.gridN, 'N', 'N', 'M')
X1 = np.c_[mkvc(NN[0][0, :]), mkvc(NN[0][self.nCx, :]), mkvc(NN[0][0, :])*np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][0, :]), mkvc(NN[1][self.nCx, :]), mkvc(NN[1][0, :])*np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, 0]), mkvc(NN[0][:, self.nCy]), mkvc(NN[0][:, 0])*np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, 0]), mkvc(NN[1][:, self.nCy]), mkvc(NN[1][:, 0])*np.nan].flatten()
X = np.r_[X1, X2]
Y = np.r_[Y1, Y2]
ax.plot(X, Y, 'b-')
ax.set_xlabel('x1')
ax.set_ylabel('x2')
elif self.dim == 3:
if nodes:
ax.plot(self.gridN[:, 0], self.gridN[:, 1], 'bs', zs=self.gridN[:, 2])
if centers:
ax.plot(self.gridCC[:, 0], self.gridCC[:, 1], 'ro', zs=self.gridCC[:, 2])
if faces:
ax.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'g>', zs=self.gridFx[:, 2])
ax.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'g<', zs=self.gridFy[:, 2])
ax.plot(self.gridFz[:, 0], self.gridFz[:, 1], 'g^', zs=self.gridFz[:, 2])
if edges:
ax.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'k>', zs=self.gridEx[:, 2])
ax.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'k<', zs=self.gridEy[:, 2])
ax.plot(self.gridEz[:, 0], self.gridEz[:, 1], 'k^', zs=self.gridEz[:, 2])
# Plot the grid lines
if lines:
NN = self.r(self.gridN, 'N', 'N', 'M')
X1 = np.c_[mkvc(NN[0][0, :, :]), mkvc(NN[0][self.nCx, :, :]), mkvc(NN[0][0, :, :])*np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][0, :, :]), mkvc(NN[1][self.nCx, :, :]), mkvc(NN[1][0, :, :])*np.nan].flatten()
Z1 = np.c_[mkvc(NN[2][0, :, :]), mkvc(NN[2][self.nCx, :, :]), mkvc(NN[2][0, :, :])*np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, 0, :]), mkvc(NN[0][:, self.nCy, :]), mkvc(NN[0][:, 0, :])*np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, 0, :]), mkvc(NN[1][:, self.nCy, :]), mkvc(NN[1][:, 0, :])*np.nan].flatten()
Z2 = np.c_[mkvc(NN[2][:, 0, :]), mkvc(NN[2][:, self.nCy, :]), mkvc(NN[2][:, 0, :])*np.nan].flatten()
X3 = np.c_[mkvc(NN[0][:, :, 0]), mkvc(NN[0][:, :, self.nCz]), mkvc(NN[0][:, :, 0])*np.nan].flatten()
Y3 = np.c_[mkvc(NN[1][:, :, 0]), mkvc(NN[1][:, :, self.nCz]), mkvc(NN[1][:, :, 0])*np.nan].flatten()
Z3 = np.c_[mkvc(NN[2][:, :, 0]), mkvc(NN[2][:, :, self.nCz]), mkvc(NN[2][:, :, 0])*np.nan].flatten()
X = np.r_[X1, X2, X3]
Y = np.r_[Y1, Y2, Y3]
Z = np.r_[Z1, Z2, Z3]
ax.plot(X, Y, 'b-', zs=Z)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('x3')
ax.grid(True)
if showIt: plt.show()
zOffset = 0
# Cycle through the first 3 octree levels
for ii in range(3):
dx = mesh.hx.min()*2**ii
nCx = int((limx[0]-limx[1]) / dx)
nCy = int((limy[0]-limy[1]) / dx)
# Create a grid at the octree level in xy
CCx, CCy = np.meshgrid(
np.linspace(limx[1], limx[0], nCx),
np.linspace(limy[1], limy[0], nCy)
)
z = F(mkvc(CCx), mkvc(CCy))
# level means number of layers in current OcTree level
for level in range(int(nCpad[ii])):
mesh.insert_cells(
np.c_[mkvc(CCx), mkvc(CCy), z-zOffset],
np.ones_like(z)*maxLevel-ii,
finalize=False
)
zOffset += dx
mesh.finalize()
# Define an active cells from topo
def get_dist_wgt(mesh, rxLoc, actv, R, R0):
"""
get_dist_wgt(xn,yn,zn,rxLoc,R,R0)
Function creating a distance weighting function required for the magnetic
inverse problem.
INPUT
xn, yn, zn : Node location
rxLoc : Observation locations [obsx, obsy, obsz]
actv : Active cell vector [0:air , 1: ground]
R : Decay factor (mag=3, grav =2)
R0 : Small factor added (default=dx/4)
OUTPUT
wr : [nC] Vector of distance weighting
Created on Dec, 20th 2015
@author: dominiquef
"""
# Find non-zero cells
if actv.dtype == 'bool':
inds = np.where(actv)[0]
else:
inds = actv
nC = len(inds)
# Create active cell projector
P = sp.csr_matrix((np.ones(nC), (inds, range(nC))),
shape=(mesh.nC, nC))
# Geometrical constant
p = 1 / np.sqrt(3)
# Create cell center location
Ym, Xm, Zm = np.meshgrid(mesh.vectorCCy, mesh.vectorCCx, mesh.vectorCCz)
hY, hX, hZ = np.meshgrid(mesh.hy, mesh.hx, mesh.hz)
# Remove air cells
Xm = P.T * mkvc(Xm)
Ym = P.T * mkvc(Ym)
Zm = P.T * mkvc(Zm)
hX = P.T * mkvc(hX)
hY = P.T * mkvc(hY)
hZ = P.T * mkvc(hZ)
V = P.T * mkvc(mesh.vol)
wr = np.zeros(nC)
ndata = rxLoc.shape[0]
count = -1
print("Begin calculation of distance weighting for R= " + str(R))
for dd in range(ndata):
nx1 = (Xm - hX * p - rxLoc[dd, 0])**2
nx2 = (Xm + hX * p - rxLoc[dd, 0])**2
ny1 = (Ym - hY * p - rxLoc[dd, 1])**2
ny2 = (Ym + hY * p - rxLoc[dd, 1])**2
nz1 = (Zm - hZ * p - rxLoc[dd, 2])**2
nz2 = (Zm + hZ * p - rxLoc[dd, 2])**2
R1 = np.sqrt(nx1 + ny1 + nz1)
R2 = np.sqrt(nx1 + ny1 + nz2)
R3 = np.sqrt(nx2 + ny1 + nz1)
R4 = np.sqrt(nx2 + ny1 + nz2)
R5 = np.sqrt(nx1 + ny2 + nz1)
R6 = np.sqrt(nx1 + ny2 + nz2)
R7 = np.sqrt(nx2 + ny2 + nz1)
R8 = np.sqrt(nx2 + ny2 + nz2)
temp = (R1 + R0)**-R + (R2 + R0)**-R + (R3 + R0)**-R + \
(R4 + R0)**-R + (R5 + R0)**-R + (R6 + R0)**-R + \
(R7 + R0)**-R + (R8 + R0)**-R
wr = wr + (V * temp / 8.)**2.
count = progress(dd, count, ndata)
wr = np.sqrt(wr) / V
wr = mkvc(wr)
wr = np.sqrt(wr / (np.max(wr)))
print("Done 100% ...distance weighting completed!!\n")
return wr
def plotGrid(self,
ax=None,
nodes=False,
faces=False,
centers=False,
edges=False,
lines=True,
showIt=False):
"""Plot the nodal, cell-centered and staggered grids for 1,2 and 3 dimensions.
:param bool nodes: plot nodes
:param bool faces: plot faces
:param bool centers: plot centers
:param bool edges: plot edges
:param bool lines: plot lines connecting nodes
:param bool showIt: call plt.show()
.. plot::
:include-source:
from SimPEG import Mesh, np
h1 = np.linspace(.1,.5,3)
h2 = np.linspace(.1,.5,5)
mesh = Mesh.TensorMesh([h1, h2])
mesh.plotGrid(nodes=True, faces=True, centers=True, lines=True, showIt=True)
.. plot::
:include-source:
from SimPEG import Mesh, np
h1 = np.linspace(.1,.5,3)
h2 = np.linspace(.1,.5,5)
h3 = np.linspace(.1,.5,3)
mesh = Mesh.TensorMesh([h1,h2,h3])
mesh.plotGrid(nodes=True, faces=True, centers=True, lines=True, showIt=True)
"""
axOpts = {'projection': '3d'} if self.dim == 3 else {}
if ax is None:
fig = plt.figure()
ax = plt.subplot(111, **axOpts)
else:
assert isinstance(
ax, matplotlib.axes.Axes), "ax must be an matplotlib.axes.Axes"
fig = ax.figure
if self.dim == 1:
if nodes:
ax.plot(self.gridN, np.ones(self.nN), 'bs')
if centers:
ax.plot(self.gridCC, np.ones(self.nC), 'ro')
if lines:
ax.plot(self.gridN, np.ones(self.nN), 'b.-')
ax.set_xlabel('x1')
elif self.dim == 2:
if nodes:
ax.plot(self.gridN[:, 0], self.gridN[:, 1], 'bs')
if centers:
ax.plot(self.gridCC[:, 0], self.gridCC[:, 1], 'ro')
if faces:
ax.plot(self.gridFx[:, 0], self.gridFx[:, 1], 'g>')
ax.plot(self.gridFy[:, 0], self.gridFy[:, 1], 'g^')
if edges:
ax.plot(self.gridEx[:, 0], self.gridEx[:, 1], 'c>')
ax.plot(self.gridEy[:, 0], self.gridEy[:, 1], 'c^')
# Plot the grid lines
if lines:
NN = self.r(self.gridN, 'N', 'N', 'M')
X1 = np.c_[mkvc(NN[0][0, :]),
mkvc(NN[0][self.nCx, :]),
mkvc(NN[0][0, :]) * np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][0, :]),
mkvc(NN[1][self.nCx, :]),
mkvc(NN[1][0, :]) * np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, 0]),
mkvc(NN[0][:, self.nCy]),
mkvc(NN[0][:, 0]) * np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, 0]),
mkvc(NN[1][:, self.nCy]),
mkvc(NN[1][:, 0]) * np.nan].flatten()
X = np.r_[X1, X2]
Y = np.r_[Y1, Y2]
ax.plot(X, Y, 'b-')
ax.set_xlabel('x1')
ax.set_ylabel('x2')
elif self.dim == 3:
if nodes:
ax.plot(self.gridN[:, 0],
self.gridN[:, 1],
'bs',
zs=self.gridN[:, 2])
if centers:
ax.plot(self.gridCC[:, 0],
self.gridCC[:, 1],
'ro',
zs=self.gridCC[:, 2])
if faces:
ax.plot(self.gridFx[:, 0],
self.gridFx[:, 1],
'g>',
zs=self.gridFx[:, 2])
ax.plot(self.gridFy[:, 0],
self.gridFy[:, 1],
'g<',
zs=self.gridFy[:, 2])
ax.plot(self.gridFz[:, 0],
self.gridFz[:, 1],
'g^',
zs=self.gridFz[:, 2])
if edges:
ax.plot(self.gridEx[:, 0],
self.gridEx[:, 1],
'k>',
zs=self.gridEx[:, 2])
ax.plot(self.gridEy[:, 0],
self.gridEy[:, 1],
'k<',
zs=self.gridEy[:, 2])
ax.plot(self.gridEz[:, 0],
self.gridEz[:, 1],
'k^',
zs=self.gridEz[:, 2])
# Plot the grid lines
if lines:
NN = self.r(self.gridN, 'N', 'N', 'M')
X1 = np.c_[mkvc(NN[0][0, :, :]),
mkvc(NN[0][self.nCx, :, :]),
mkvc(NN[0][0, :, :]) * np.nan].flatten()
Y1 = np.c_[mkvc(NN[1][0, :, :]),
mkvc(NN[1][self.nCx, :, :]),
mkvc(NN[1][0, :, :]) * np.nan].flatten()
Z1 = np.c_[mkvc(NN[2][0, :, :]),
mkvc(NN[2][self.nCx, :, :]),
mkvc(NN[2][0, :, :]) * np.nan].flatten()
X2 = np.c_[mkvc(NN[0][:, 0, :]),
mkvc(NN[0][:, self.nCy, :]),
mkvc(NN[0][:, 0, :]) * np.nan].flatten()
Y2 = np.c_[mkvc(NN[1][:, 0, :]),
mkvc(NN[1][:, self.nCy, :]),
mkvc(NN[1][:, 0, :]) * np.nan].flatten()
Z2 = np.c_[mkvc(NN[2][:, 0, :]),
mkvc(NN[2][:, self.nCy, :]),
mkvc(NN[2][:, 0, :]) * np.nan].flatten()
X3 = np.c_[mkvc(NN[0][:, :, 0]),
mkvc(NN[0][:, :, self.nCz]),
mkvc(NN[0][:, :, 0]) * np.nan].flatten()
Y3 = np.c_[mkvc(NN[1][:, :, 0]),
mkvc(NN[1][:, :, self.nCz]),
mkvc(NN[1][:, :, 0]) * np.nan].flatten()
Z3 = np.c_[mkvc(NN[2][:, :, 0]),
mkvc(NN[2][:, :, self.nCz]),
mkvc(NN[2][:, :, 0]) * np.nan].flatten()
X = np.r_[X1, X2, X3]
Y = np.r_[Y1, Y2, Y3]
Z = np.r_[Z1, Z2, Z3]
ax.plot(X, Y, 'b-', zs=Z)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('x3')
ax.grid(True)
if showIt: plt.show()
vtem = TDEM.Src.VTEMWaveform()
trapezoid = TDEM.Src.TrapezoidWaveform(
ramp_on=np.r_[0., 1.5e-3], ramp_off=max_t-np.r_[1.5e-3, 0]
)
quarter_sine = TDEM.Src.QuarterSineRampOnWaveform(
ramp_on=np.r_[0., 1.5e-3], ramp_off=max_t-np.r_[1.5e-3, 0]
)
waveforms = dict(zip(
[
'RampOffWaveform', 'VTEMWaveform', 'TrapezoidWaveform',
'QuarterSineRampOnWaveform'
],
[ramp_off, vtem, trapezoid, quarter_sine]
))
# plot the waveforms
fig, ax = plt.subplots(2, 2, figsize=(7, 7))
ax = mkvc(ax)
for a, key in zip(ax, waveforms):
wave = waveforms[key]
wave_plt = [wave.eval(t) for t in times]
a.plot(times, wave_plt)
a.set_title(key)
a.set_xlabel('time (s)')
plt.tight_layout()
plt.show()
def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None, expectedOrder=2, tolerance=0.85, eps=1e-10, ax=None):
"""
Basic derivative check
Compares error decay of 0th and 1st order Taylor approximation at point
x0 for a randomized search direction.
:param lambda fctn: function handle
:param numpy.array x0: point at which to check derivative
:param int num: number of times to reduce step length, h
:param bool plotIt: if you would like to plot
:param numpy.array dx: step direction
:param int expectedOrder: The order that you expect the derivative to yield.
:param float tolerance: The tolerance on the expected order.
:param float eps: What is zero?
:rtype: bool
:return: did you pass the test?!
.. plot::
:include-source:
from SimPEG import Tests, Utils, np
def simplePass(x):
return np.sin(x), Utils.sdiag(np.cos(x))
Tests.checkDerivative(simplePass, np.random.randn(5))
"""
print "%s checkDerivative %s" % ('='*20, '='*20)
print "iter h |ft-f0| |ft-f0-h*J0*dx| Order\n%s" % ('-'*57)
f0, J0 = fctn(x0)
x0 = mkvc(x0)
if dx is None:
dx = np.random.randn(len(x0))
h = np.logspace(-1, -num, num)
E0 = np.ones(h.shape)
E1 = np.ones(h.shape)
def l2norm(x):
# because np.norm breaks if they are scalars?
return np.sqrt(np.real(np.vdot(x, x)))
for i in range(num):
# Evaluate at test point
ft, Jt = fctn( x0 + h[i]*dx )
# 0th order Taylor
E0[i] = l2norm( ft - f0 )
# 1st order Taylor
if inspect.isfunction(J0):
E1[i] = l2norm( ft - f0 - h[i]*J0(dx) )
else:
# We assume it is a numpy.ndarray
E1[i] = l2norm( ft - f0 - h[i]*J0.dot(dx) )
order0 = np.log10(E0[:-1]/E0[1:])
order1 = np.log10(E1[:-1]/E1[1:])
print " %d %1.2e %1.3e %1.3e %1.3f" % (i, h[i], E0[i], E1[i], np.nan if i == 0 else order1[i-1])
# Ensure we are about precision
order0 = order0[E0[1:] > eps]
order1 = order1[E1[1:] > eps]
belowTol = (order1.size == 0 and order0.size >= 0)
# Make sure we get the correct order
correctOrder = order1.size > 0 and np.mean(order1) > tolerance * expectedOrder
passTest = belowTol or correctOrder
if passTest:
print "%s PASS! %s" % ('='*25, '='*25)
print happiness[np.random.randint(len(happiness))]+'\n'
else:
print "%s\n%s FAIL! %s\n%s" % ('*'*57, '<'*25, '>'*25, '*'*57)
print sadness[np.random.randint(len(sadness))]+'\n'
if plotIt:
ax = ax or plt.subplot(111)
ax.loglog(h, E0, 'b')
ax.loglog(h, E1, 'g--')
ax.set_title('Check Derivative - %s' % ('PASSED :)' if passTest else 'FAILED :('))
ax.set_xlabel('h')
ax.set_ylabel('Error')
leg = ax.legend(['$\mathcal{O}(h)$', '$\mathcal{O}(h^2)$'], loc='best',
title="$f(x + h\Delta x) - f(x) - h g(x) \Delta x - \mathcal{O}(h^2) = 0$",
frameon=False)
plt.setp(leg.get_title(),fontsize=15)
plt.show()
return passTest
def Jtvec(self, m, v, u=None):
"""
Computing Jacobian^T multiplied by vector.
.. math ::
(\\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}})^{T} = \left[ \mathbf{P}_{deriv}\\frac{\partial \mathbf{\mu} } {\partial \mathbf{m} }
\left[
\diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \\
- \diag (\Div^T\mathbf{u})\dMfMuI
\\right ]\\right]^{T}
- \left[\mathbf{P}_{deriv}(\MfMui)^{-1}\Div^T\\frac{\delta\mathbf{u}}{\delta \mathbf{m}} \\right]^{T}
where
.. math ::
\mathbf{P}_{derv} = \\frac{\partial \mathbf{P}}{\partial\mathbf{B}}
.. note ::
Here we only want to compute
.. math ::
\mathbf{J}^{T}\mathbf{v} = (\\frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}})^{T} \mathbf{v}
"""
if u is None:
u = self.fields(m)
B, u = u['B'], u['u']
mu = self.mapping * (m)
dmudm = self.mapping.deriv(m)
# dchidmu = sdiag(1 / mu_0 * np.ones(self.mesh.nC))
vol = self.mesh.vol
Div = self._Div
Dface = self.mesh.faceDiv
P = self.survey.projectFieldsDeriv(
B) # Projection matrix
B0 = self.getB0()
MfMuIvec = 1 / self.MfMui.diagonal()
dMfMuI = sdiag(MfMuIvec**2) * \
self.mesh.aveF2CC.T * sdiag(vol * 1. / mu**2)
# A = self._Div*self.MfMuI*self._Div.T
# RHS = Div*MfMuI*MfMu0*B0 - Div*B0 + Mc*Dface*Pout.T*Bbc
# C(m,u) = A*m-rhs
# dudm = -(dCdu)^(-1)dCdm
dCdu = self.getA(m)
s = Div * (self.MfMuI.T * (P.T * v))
m1 = sp.linalg.interface.aslinearoperator(
sdiag(1 / (dCdu.T).diagonal())
)
sol, info = sp.linalg.bicgstab(dCdu.T, s, tol=1e-6, maxiter=1000, M=m1)
if info > 0:
print("Iterative solver did not work well (Jtvec)")
# raise Exception ("Iterative solver did not work well")
# dCdm_A = Div * ( sdiag( Div.T * u )* dMfMuI *dmudm )
# dCdm_Atsol = ( dMfMuI.T*( sdiag( Div.T * u ) * (Div.T * dmudm)) ) * sol
dCdm_Atsol = (dmudm.T * dMfMuI.T *
(sdiag(Div.T * u) * Div.T)) * sol
# dCdm_RHS1 = Div * (sdiag( self.MfMu0*B0 ) * dMfMuI)
# dCdm_RHS1tsol = (dMfMuI.T*( sdiag( self.MfMu0*B0 ) ) * Div.T * dmudm) * sol
dCdm_RHS1tsol = (
dmudm.T * dMfMuI.T *
(sdiag(self.MfMu0 * B0)) * Div.T
) * sol
# temp1 = (Dface*(self._Pout.T*self.Bbc_const*self.Bbc))
# temp1sol = (Dface.T * (sdiag(vol) * sol))
# temp2 = self.Bbc_const * (self._Pout.T * self.Bbc).T
# dCdm_RHS2v = (sdiag(vol)*temp1)*np.inner(vol, dchidmu*dmudm*v)
# dCdm_RHS2tsol = (dmudm.T * dchidmu.T * vol) * np.inner(temp2, temp1sol)
# dCdm_RHSv = dCdm_RHS1*(dmudm*v) + dCdm_RHS2v
# temporary fix
# dCdm_RHStsol = dCdm_RHS1tsol - dCdm_RHS2tsol
dCdm_RHStsol = dCdm_RHS1tsol
# dCdm_RHSv = dCdm_RHS1*(dmudm*v) + dCdm_RHS2v
# dCdm_v = dCdm_A*v - dCdm_RHSv
Ctv = dCdm_Atsol - dCdm_RHStsol
# B = self.MfMuI*self.MfMu0*B0-B0-self.MfMuI*self._Div.T*u
# dBdm = d\mudm*dBd\mu
# dPBdm^T*v = Atemp^T*P^T*v - Btemp^T*P^T*v - Ctv
Atemp = sdiag(self.MfMu0 * B0) * (dMfMuI * (dmudm))
Btemp = sdiag(Div.T * u) * (dMfMuI * (dmudm))
Jtv = Atemp.T * (P.T * v) - Btemp.T * (P.T * v) - Ctv
return mkvc(Jtv)