def from_scratch_gradient_test(shape=(70, 70), kernel='OT2', space_order=6):
spacing = tuple(10. for _ in shape)
wave = setup(shape=shape,
spacing=spacing,
dtype=np.float64,
kernel=kernel,
space_order=space_order,
nbl=40)
v0 = Function(name='v0', grid=wave.model.grid, space_order=space_order)
smooth(v0, wave.model.vp)
v = wave.model.vp.data
dm = np.float64(v**(-2) - v0.data**(-2))
# Compute receiver data for the true velocity
rec, _, _ = wave.forward()
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0, _ = wave.forward(vp=v0, save=True)
# Objective function value
F0 = .5 * linalg.norm(rec0.data - rec.data)**2
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec',
grid=wave.model.grid,
data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
gradient, _ = wave.jacobian_adjoint(residual, u0, vp=v0)
v0 = v0.data
basic_gradient_test(wave, space_order, v0, v, rec, F0, gradient, dm)
def test_gradientJ(self, shape, kernel, space_order):
"""
This test ensures that the Jacobian computed with devito
satisfies the Taylor expansion property:
.. math::
F(m0 + h dm) = F(m0) + \O(h) \\
F(m0 + h dm) = F(m0) + J dm + \O(h^2) \\
with F the Forward modelling operator.
:param dimensions: size of the domain in all dimensions
in number of grid points
:param time_order: order of the time discretization scheme
:param space_order: order of the spacial discretization scheme
:return: assertion that the Taylor properties are satisfied
"""
spacing = tuple(15. for _ in shape)
wave = setup(shape=shape, spacing=spacing, dtype=np.float64,
kernel=kernel, space_order=space_order,
tn=1000., nbpml=10+space_order/2)
m0 = Function(name='m0', grid=wave.model.grid, space_order=space_order)
smooth(m0, wave.model.m)
dm = np.float64(wave.model.m.data - m0.data)
linrec = Receiver(name='rec', grid=wave.model.grid,
time_range=wave.receiver.time_range,
coordinates=wave.receiver.coordinates.data)
# Compute receiver data and full wavefield for the smooth velocity
rec, u0, _ = wave.forward(m=m0, save=False)
# Gradient: J dm
Jdm, _, _, _ = wave.born(dm, rec=linrec, m=m0)
# FWI Gradient test
H = [0.5, 0.25, .125, 0.0625, 0.0312, 0.015625, 0.0078125]
error1 = np.zeros(7)
error2 = np.zeros(7)
for i in range(0, 7):
# Add the perturbation to the model
def initializer(data):
data[:] = m0.data + H[i] * dm
mloc = Function(name='mloc', grid=wave.model.m.grid, space_order=space_order,
initializer=initializer)
# Data for the new model
d = wave.forward(m=mloc)[0]
delta_d = (d.data - rec.data).reshape(-1)
# First order error F(m0 + hdm) - F(m0)
error1[i] = np.linalg.norm(delta_d, 1)
# Second order term F(m0 + hdm) - F(m0) - J dm
error2[i] = np.linalg.norm(delta_d - H[i] * Jdm.data.reshape(-1), 1)
# Test slope of the tests
p1 = np.polyfit(np.log10(H), np.log10(error1), 1)
p2 = np.polyfit(np.log10(H), np.log10(error2), 1)
info('1st order error, Phi(m0+dm)-Phi(m0) with slope: %s compared to 1' % (p1[0]))
info('2nd order error, Phi(m0+dm)-Phi(m0) - <J(m0)^T \delta d, dm>with slope:'
' %s comapred to 2' % (p2[0]))
assert np.isclose(p1[0], 1.0, rtol=0.1)
assert np.isclose(p2[0], 2.0, rtol=0.1)
def test_gradientJ(self, shape, kernel, space_order):
r"""
This test ensures that the Jacobian computed with devito
satisfies the Taylor expansion property:
.. math::
F(m0 + h dm) = F(m0) + \O(h) \\
F(m0 + h dm) = F(m0) + J dm + \O(h^2) \\
with F the Forward modelling operator.
"""
spacing = tuple(15. for _ in shape)
wave = setup(shape=shape, spacing=spacing, dtype=np.float64,
kernel=kernel, space_order=space_order,
tn=1000., nbpml=10+space_order/2)
m0 = Function(name='m0', grid=wave.model.grid, space_order=space_order)
smooth(m0, wave.model.m)
dm = np.float64(wave.model.m.data - m0.data)
linrec = Receiver(name='rec', grid=wave.model.grid,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
# Compute receiver data and full wavefield for the smooth velocity
rec, u0, _ = wave.forward(m=m0, save=False)
# Gradient: J dm
Jdm, _, _, _ = wave.born(dm, rec=linrec, m=m0)
# FWI Gradient test
H = [0.5, 0.25, .125, 0.0625, 0.0312, 0.015625, 0.0078125]
error1 = np.zeros(7)
error2 = np.zeros(7)
for i in range(0, 7):
# Add the perturbation to the model
def initializer(data):
data[:] = m0.data + H[i] * dm
mloc = Function(name='mloc', grid=wave.model.m.grid, space_order=space_order,
initializer=initializer)
# Data for the new model
d = wave.forward(m=mloc)[0]
delta_d = (d.data - rec.data).reshape(-1)
# First order error F(m0 + hdm) - F(m0)
error1[i] = np.linalg.norm(delta_d, 1)
# Second order term F(m0 + hdm) - F(m0) - J dm
error2[i] = np.linalg.norm(delta_d - H[i] * Jdm.data.reshape(-1), 1)
# Test slope of the tests
p1 = np.polyfit(np.log10(H), np.log10(error1), 1)
p2 = np.polyfit(np.log10(H), np.log10(error2), 1)
info('1st order error, Phi(m0+dm)-Phi(m0) with slope: %s compared to 1' % (p1[0]))
info(r'2nd order error, Phi(m0+dm)-Phi(m0) - <J(m0)^T \delta d, dm>with slope:'
' %s comapred to 2' % (p2[0]))
assert np.isclose(p1[0], 1.0, rtol=0.1)
assert np.isclose(p2[0], 2.0, rtol=0.1)
def test_gradient_checkpointing(self, dtype, opt, space_order, kernel,
shape, spacing, setup_func, time_order):
"""
This test ensures that the FWI gradient computed with checkpointing matches
the one without checkpointing. Note that this test fails with dynamic openmp
scheduling enabled so this test disables it.
"""
wave = setup_func(shape=shape,
spacing=spacing,
dtype=dtype,
kernel=kernel,
space_order=space_order,
nbl=40,
opt=opt,
time_order=time_order)
v0 = Function(name='v0',
grid=wave.model.grid,
space_order=space_order,
dtype=dtype)
smooth(v0, wave.model.vp)
# Compute receiver data for the true velocity
rec = wave.forward()[0]
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0 = wave.forward(vp=v0, save=True)[0:2]
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec',
grid=wave.model.grid,
data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions,
dtype=dtype)
grad = Function(name='grad', grid=wave.model.grid, dtype=dtype)
gradient, _ = wave.jacobian_adjoint(residual,
u0,
vp=v0,
checkpointing=True,
grad=grad)
grad = Function(name='grad', grid=wave.model.grid, dtype=dtype)
gradient2, _ = wave.jacobian_adjoint(residual,
u0,
vp=v0,
checkpointing=False,
grad=grad)
assert np.allclose(gradient.data, gradient2.data, atol=0, rtol=0)
def test_gradient_checkpointing(self, shape, kernel, space_order):
r"""
This test ensures that the FWI gradient computed with devito
satisfies the Taylor expansion property:
.. math::
\Phi(m0 + h dm) = \Phi(m0) + \O(h) \\
\Phi(m0 + h dm) = \Phi(m0) + h \nabla \Phi(m0) + \O(h^2) \\
\Phi(m0) = .5* || F(m0 + h dm) - D ||_2^2
where
.. math::
\nabla \Phi(m0) = <J^T \delta d, dm> \\
\delta d = F(m0+ h dm) - D \\
with F the Forward modelling operator.
"""
spacing = tuple(10. for _ in shape)
wave = setup(shape=shape,
spacing=spacing,
dtype=np.float64,
kernel=kernel,
space_order=space_order,
nbl=40)
v0 = Function(name='v0', grid=wave.model.grid, space_order=space_order)
smooth(v0, wave.model.vp)
# Compute receiver data for the true velocity
rec, u, _ = wave.forward()
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0, _ = wave.forward(vp=v0, save=True)
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec',
grid=wave.model.grid,
data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
gradient, _ = wave.jacobian_adjoint(residual,
u0,
vp=v0,
checkpointing=True)
gradient2, _ = wave.jacobian_adjoint(residual,
u0,
vp=v0,
checkpointing=False)
assert np.allclose(gradient.data, gradient2.data)
def test_gradient_checkpointing(self, shape, kernel, space_order):
"""
This test ensures that the FWI gradient computed with devito
satisfies the Taylor expansion property:
.. math::
\Phi(m0 + h dm) = \Phi(m0) + \O(h) \\
\Phi(m0 + h dm) = \Phi(m0) + h \nabla \Phi(m0) + \O(h^2) \\
\Phi(m0) = .5* || F(m0 + h dm) - D ||_2^2
where
.. math::
\nabla \Phi(m0) = <J^T \delta d, dm> \\
\delta d = F(m0+ h dm) - D \\
with F the Forward modelling operator.
:param dimensions: size of the domain in all dimensions
in number of grid points
:param time_order: order of the time discretization scheme
:param space_order: order of the spacial discretization scheme
:return: assertion that the Taylor properties are satisfied
"""
spacing = tuple(10. for _ in shape)
wave = setup(shape=shape,
spacing=spacing,
dtype=np.float64,
kernel=kernel,
space_order=space_order,
nbpml=40)
m0 = Function(name='m0', grid=wave.model.grid, space_order=space_order)
smooth(m0, wave.model.m)
# Compute receiver data for the true velocity
rec, u, _ = wave.forward()
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0, _ = wave.forward(m=m0, save=True)
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec',
grid=wave.model.grid,
data=rec0.data - rec.data,
time_range=rec.time_range,
coordinates=rec0.coordinates.data)
gradient, _ = wave.gradient(residual, u0, m=m0, checkpointing=True)
gradient2, _ = wave.gradient(residual, u0, m=m0, checkpointing=False)
assert np.allclose(gradient.data, gradient2.data)
def test_gradient_checkpointing(self, shape, kernel, space_order):
r"""
This test ensures that the FWI gradient computed with devito
satisfies the Taylor expansion property:
.. math::
\Phi(m0 + h dm) = \Phi(m0) + \O(h) \\
\Phi(m0 + h dm) = \Phi(m0) + h \nabla \Phi(m0) + \O(h^2) \\
\Phi(m0) = .5* || F(m0 + h dm) - D ||_2^2
where
.. math::
\nabla \Phi(m0) = <J^T \delta d, dm> \\
\delta d = F(m0+ h dm) - D \\
with F the Forward modelling operator.
"""
spacing = tuple(10. for _ in shape)
wave = setup(shape=shape, spacing=spacing, dtype=np.float64,
kernel=kernel, space_order=space_order,
nbpml=40)
m0 = Function(name='m0', grid=wave.model.grid, space_order=space_order)
smooth(m0, wave.model.m)
# Compute receiver data for the true velocity
rec, u, _ = wave.forward()
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0, _ = wave.forward(m=m0, save=True)
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec', grid=wave.model.grid, data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
gradient, _ = wave.gradient(residual, u0, m=m0, checkpointing=True)
gradient2, _ = wave.gradient(residual, u0, m=m0, checkpointing=False)
assert np.allclose(gradient.data, gradient2.data)
def test_gradientFWI(self, shape, kernel, space_order, checkpointing):
r"""
This test ensures that the FWI gradient computed with devito
satisfies the Taylor expansion property:
.. math::
\Phi(m0 + h dm) = \Phi(m0) + \O(h) \\
\Phi(m0 + h dm) = \Phi(m0) + h \nabla \Phi(m0) + \O(h^2) \\
\Phi(m0) = .5* || F(m0 + h dm) - D ||_2^2
where
.. math::
\nabla \Phi(m0) = <J^T \delta d, dm> \\
\delta d = F(m0+ h dm) - D \\
with F the Forward modelling operator.
"""
spacing = tuple(10. for _ in shape)
wave = setup(shape=shape,
spacing=spacing,
dtype=np.float64,
kernel=kernel,
space_order=space_order,
nbpml=40)
m0 = Function(name='m0', grid=wave.model.grid, space_order=space_order)
smooth(m0, wave.model.m)
dm = np.float32(wave.model.m.data[:] - m0.data[:])
# Compute receiver data for the true velocity
rec, u, _ = wave.forward()
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0, _ = wave.forward(m=m0, save=True)
# Objective function value
F0 = .5 * linalg.norm(rec0.data - rec.data)**2
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec',
grid=wave.model.grid,
data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
gradient, _ = wave.gradient(residual,
u0,
m=m0,
checkpointing=checkpointing)
G = np.dot(gradient.data.reshape(-1), dm.reshape(-1))
# FWI Gradient test
H = [0.5, 0.25, .125, 0.0625, 0.0312, 0.015625, 0.0078125]
error1 = np.zeros(7)
error2 = np.zeros(7)
for i in range(0, 7):
# Add the perturbation to the model
def initializer(data):
data[:] = m0.data + H[i] * dm
mloc = Function(name='mloc',
grid=wave.model.m.grid,
space_order=space_order,
initializer=initializer)
# Data for the new model
d = wave.forward(m=mloc)[0]
# First order error Phi(m0+dm) - Phi(m0)
F_i = .5 * linalg.norm((d.data - rec.data).reshape(-1))**2
error1[i] = np.absolute(F_i - F0)
# Second order term r Phi(m0+dm) - Phi(m0) - <J(m0)^T \delta d, dm>
error2[i] = np.absolute(F_i - F0 - H[i] * G)
# Test slope of the tests
p1 = np.polyfit(np.log10(H), np.log10(error1), 1)
p2 = np.polyfit(np.log10(H), np.log10(error2), 1)
info('1st order error, Phi(m0+dm)-Phi(m0): %s' % (p1))
info(
r'2nd order error, Phi(m0+dm)-Phi(m0) - <J(m0)^T \delta d, dm>: %s'
% (p2))
assert np.isclose(p1[0], 1.0, rtol=0.1)
assert np.isclose(p2[0], 2.0, rtol=0.1)
def test_gradient_equivalence(self, shape, kernel, space_order, preset,
nbl, dtype, tolerance, spacing, tn):
""" This test asserts that the gradient calculated through the following three
expressions should match within floating-point precision:
- grad = sum(-u.dt2 * v)
- grad = sum(-u * v.dt2)
- grad = sum(-u.dt * v.dt)
The computation has the following number of operations:
u.dt2 (5 ops) * v = 6ops * 500 (nt) ~ 3000 ops ~ 1e4 ops
Hence tolerances are eps * ops = 1e-4 (sp) and 1e-13 (dp)
"""
model = demo_model(preset,
space_order=space_order,
shape=shape,
nbl=nbl,
dtype=dtype,
spacing=spacing)
m = model.m
v_true = model.vp
geometry = setup_geometry(model, tn)
dt = model.critical_dt
src = geometry.src
rec = geometry.rec
rec_true = geometry.rec
rec0 = geometry.rec
s = model.grid.stepping_dim.spacing
u = TimeFunction(name='u',
grid=model.grid,
time_order=2,
space_order=space_order,
save=geometry.nt)
eqn_fwd = iso_stencil(u, model, kernel)
src_term = src.inject(field=u.forward, expr=src * s**2 / m)
rec_term = rec.interpolate(expr=u)
fwd_op = Operator(eqn_fwd + src_term + rec_term,
subs=model.spacing_map,
name='Forward')
v0 = Function(name='v0',
grid=model.grid,
space_order=space_order,
dtype=dtype)
smooth(v0, model.vp)
grad_u = Function(name='gradu', grid=model.grid)
grad_v = Function(name='gradv', grid=model.grid)
grad_uv = Function(name='graduv', grid=model.grid)
v = TimeFunction(name='v',
grid=model.grid,
save=None,
time_order=2,
space_order=space_order)
s = model.grid.stepping_dim.spacing
eqn_adj = iso_stencil(v, model, kernel, forward=False)
receivers = rec.inject(field=v.backward, expr=rec * s**2 / m)
gradient_update_v = Eq(grad_v, grad_v - u * v.dt2)
grad_op_v = Operator(eqn_adj + receivers + [gradient_update_v],
subs=model.spacing_map,
name='GradientV')
gradient_update_u = Eq(grad_u, grad_u - u.dt2 * v)
grad_op_u = Operator(eqn_adj + receivers + [gradient_update_u],
subs=model.spacing_map,
name='GradientU')
gradient_update_uv = Eq(grad_uv, grad_uv + u.dt * v.dt)
grad_op_uv = Operator(eqn_adj + receivers + [gradient_update_uv],
subs=model.spacing_map,
name='GradientUV')
fwd_op.apply(dt=dt, vp=v_true, rec=rec_true)
fwd_op.apply(dt=dt, vp=v0, rec=rec0)
residual = Receiver(name='rec',
grid=model.grid,
data=(rec0.data - rec_true.data),
time_range=geometry.time_axis,
coordinates=geometry.rec_positions,
dtype=dtype)
grad_op_u.apply(dt=dt, vp=v0, rec=residual)
# Reset v before calling the second operator since the object is shared
v.data[:] = 0.
grad_op_v.apply(dt=dt, vp=v0, rec=residual)
v.data[:] = 0.
grad_op_uv.apply(dt=dt, vp=v0, rec=residual)
assert (np.allclose(grad_u.data,
grad_v.data,
rtol=tolerance,
atol=tolerance))
assert (np.allclose(grad_u.data,
grad_uv.data,
rtol=tolerance,
atol=tolerance))
def test_gradientJ(self, dtype, space_order, kernel, shape, spacing,
time_order, setup_func):
"""
This test ensures that the Jacobian computed with devito
satisfies the Taylor expansion property:
.. math::
F(m0 + h dm) = F(m0) + \O(h) \\
F(m0 + h dm) = F(m0) + J dm + \O(h^2) \\
with F the Forward modelling operator.
"""
wave = setup_func(shape=shape,
spacing=spacing,
dtype=dtype,
kernel=kernel,
space_order=space_order,
time_order=time_order,
tn=1000.,
nbl=10 + space_order / 2)
v0 = Function(name='v0', grid=wave.model.grid, space_order=space_order)
smooth(v0, wave.model.vp)
v = wave.model.vp.data
dm = dtype(wave.model.vp.data**(-2) - v0.data**(-2))
# Compute receiver data and full wavefield for the smooth velocity
rec = wave.forward(vp=v0, save=False)[0]
# Gradient: J dm
Jdm = wave.jacobian(dm, vp=v0)[0]
# FWI Gradient test
H = [0.5, 0.25, .125, 0.0625, 0.0312, 0.015625, 0.0078125]
error1 = np.zeros(7)
error2 = np.zeros(7)
for i in range(0, 7):
# Add the perturbation to the model
def initializer(data):
data[:] = np.sqrt(v0.data**2 * v**2 /
((1 - H[i]) * v**2 + H[i] * v0.data**2))
vloc = Function(name='vloc',
grid=wave.model.grid,
space_order=space_order,
initializer=initializer)
# Data for the new model
d = wave.forward(vp=vloc)[0]
delta_d = (d.data - rec.data).reshape(-1)
# First order error F(m0 + hdm) - F(m0)
error1[i] = np.linalg.norm(delta_d, 1)
# Second order term F(m0 + hdm) - F(m0) - J dm
error2[i] = np.linalg.norm(delta_d - H[i] * Jdm.data.reshape(-1),
1)
# Test slope of the tests
p1 = np.polyfit(np.log10(H), np.log10(error1), 1)
p2 = np.polyfit(np.log10(H), np.log10(error2), 1)
info('1st order error, Phi(m0+dm)-Phi(m0) with slope: %s compared to 1'
% (p1[0]))
info(
r'2nd order error, Phi(m0+dm)-Phi(m0) - <J(m0)^T \delta d, dm>with slope:'
' %s compared to 2' % (p2[0]))
assert np.isclose(p1[0], 1.0, rtol=0.1)
assert np.isclose(p2[0], 2.0, rtol=0.1)
def test_gradientFWI(self, dtype, space_order, kernel, shape, ckp,
setup_func, time_order):
"""
This test ensures that the FWI gradient computed with devito
satisfies the Taylor expansion property:
.. math::
\Phi(m0 + h dm) = \Phi(m0) + \O(h) \\
\Phi(m0 + h dm) = \Phi(m0) + h \nabla \Phi(m0) + \O(h^2) \\
\Phi(m0) = .5* || F(m0 + h dm) - D ||_2^2
where
.. math::
\nabla \Phi(m0) = <J^T \delta d, dm> \\
\delta d = F(m0+ h dm) - D \\
with F the Forward modelling operator.
"""
spacing = tuple(10. for _ in shape)
wave = setup_func(shape=shape,
spacing=spacing,
dtype=dtype,
kernel=kernel,
tn=400.0,
space_order=space_order,
nbl=40,
time_order=time_order)
vel0 = Function(name='vel0',
grid=wave.model.grid,
space_order=space_order)
smooth(vel0, wave.model.vp)
v = wave.model.vp.data
dm = dtype(wave.model.vp.data**(-2) - vel0.data**(-2))
# Compute receiver data for the true velocity
rec = wave.forward()[0]
# Compute receiver data and full wavefield for the smooth velocity
if setup_func is tti_setup:
rec0, u0, v0, _ = wave.forward(vp=vel0, save=True)
else:
rec0, u0 = wave.forward(vp=vel0, save=True)[0:2]
# Objective function value
F0 = .5 * linalg.norm(rec0.data - rec.data)**2
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec',
grid=wave.model.grid,
data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
if setup_func is tti_setup:
gradient, _ = wave.jacobian_adjoint(residual,
u0,
v0,
vp=vel0,
checkpointing=ckp)
else:
gradient, _ = wave.jacobian_adjoint(residual,
u0,
vp=vel0,
checkpointing=ckp)
G = np.dot(gradient.data.reshape(-1), dm.reshape(-1))
# FWI Gradient test
H = [0.5, 0.25, .125, 0.0625, 0.0312, 0.015625, 0.0078125]
error1 = np.zeros(7)
error2 = np.zeros(7)
for i in range(0, 7):
# Add the perturbation to the model
def initializer(data):
data[:] = np.sqrt(vel0.data**2 * v**2 /
((1 - H[i]) * v**2 + H[i] * vel0.data**2))
vloc = Function(name='vloc',
grid=wave.model.grid,
space_order=space_order,
initializer=initializer)
# Data for the new model
d = wave.forward(vp=vloc)[0]
# First order error Phi(m0+dm) - Phi(m0)
F_i = .5 * linalg.norm((d.data - rec.data).reshape(-1))**2
error1[i] = np.absolute(F_i - F0)
# Second order term r Phi(m0+dm) - Phi(m0) - <J(m0)^T \delta d, dm>
error2[i] = np.absolute(F_i - F0 - H[i] * G)
# Test slope of the tests
p1 = np.polyfit(np.log10(H), np.log10(error1), 1)
p2 = np.polyfit(np.log10(H), np.log10(error2), 1)
info('1st order error, Phi(m0+dm)-Phi(m0): %s' % (p1))
info(
r'2nd order error, Phi(m0+dm)-Phi(m0) - <J(m0)^T \delta d, dm>: %s'
% (p2))
assert np.isclose(p1[0], 1.0, rtol=0.1)
assert np.isclose(p2[0], 2.0, rtol=0.1)
def test_gradientFWI(self, shape, kernel, space_order, checkpointing):
r"""
This test ensures that the FWI gradient computed with devito
satisfies the Taylor expansion property:
.. math::
\Phi(m0 + h dm) = \Phi(m0) + \O(h) \\
\Phi(m0 + h dm) = \Phi(m0) + h \nabla \Phi(m0) + \O(h^2) \\
\Phi(m0) = .5* || F(m0 + h dm) - D ||_2^2
where
.. math::
\nabla \Phi(m0) = <J^T \delta d, dm> \\
\delta d = F(m0+ h dm) - D \\
with F the Forward modelling operator.
"""
spacing = tuple(10. for _ in shape)
wave = setup(shape=shape, spacing=spacing, dtype=np.float64,
kernel=kernel, space_order=space_order,
nbpml=40)
m0 = Function(name='m0', grid=wave.model.grid, space_order=space_order)
smooth(m0, wave.model.m)
dm = np.float32(wave.model.m.data[:] - m0.data[:])
# Compute receiver data for the true velocity
rec, u, _ = wave.forward()
# Compute receiver data and full wavefield for the smooth velocity
rec0, u0, _ = wave.forward(m=m0, save=True)
# Objective function value
F0 = .5*linalg.norm(rec0.data - rec.data)**2
# Gradient: <J^T \delta d, dm>
residual = Receiver(name='rec', grid=wave.model.grid, data=rec0.data - rec.data,
time_range=wave.geometry.time_axis,
coordinates=wave.geometry.rec_positions)
gradient, _ = wave.gradient(residual, u0, m=m0, checkpointing=checkpointing)
G = np.dot(gradient.data.reshape(-1), dm.reshape(-1))
# FWI Gradient test
H = [0.5, 0.25, .125, 0.0625, 0.0312, 0.015625, 0.0078125]
error1 = np.zeros(7)
error2 = np.zeros(7)
for i in range(0, 7):
# Add the perturbation to the model
def initializer(data):
data[:] = m0.data + H[i] * dm
mloc = Function(name='mloc', grid=wave.model.m.grid, space_order=space_order,
initializer=initializer)
# Data for the new model
d = wave.forward(m=mloc)[0]
# First order error Phi(m0+dm) - Phi(m0)
F_i = .5*linalg.norm((d.data - rec.data).reshape(-1))**2
error1[i] = np.absolute(F_i - F0)
# Second order term r Phi(m0+dm) - Phi(m0) - <J(m0)^T \delta d, dm>
error2[i] = np.absolute(F_i - F0 - H[i] * G)
# Test slope of the tests
p1 = np.polyfit(np.log10(H), np.log10(error1), 1)
p2 = np.polyfit(np.log10(H), np.log10(error2), 1)
info('1st order error, Phi(m0+dm)-Phi(m0): %s' % (p1))
info(r'2nd order error, Phi(m0+dm)-Phi(m0) - <J(m0)^T \delta d, dm>: %s' % (p2))
assert np.isclose(p1[0], 1.0, rtol=0.1)
assert np.isclose(p2[0], 2.0, rtol=0.1)
