def forward_born(model, src_coords, wavelet, rec_coords, space_order=8, nb=40, isic=False, dt=None, free_surface=False):
clear_cache()
# Parameters
nt = wavelet.shape[0]
if dt is None:
dt = model.critical_dt
m, rho, dm, damp = model.m, model.rho, model.dm, model.damp
# Create the forward and linearized wavefield
u = TimeFunction(name="u", grid=model.grid, time_order=2, space_order=space_order)
du = TimeFunction(name="du", grid=model.grid, time_order=2, space_order=space_order)
if len(model.shape) == 2:
x,y = u.space_dimensions
else:
x,y,z = u.space_dimensions
# Set up PDEs and rearrange
ulaplace, rho = acoustic_laplacian(u, rho)
dulaplace, _ = acoustic_laplacian(du, rho)
if isic:
# Sum ((u.dx * d, / rho).dx for x in dimensions)
# space_order//2 so that u.dx.dx has the same radius as u.laplace
du_aux = sum([first_derivative(first_derivative(u, dim=d, fd_order=space_order//2) * dm / rho, fd_order=space_order//2, dim=d)
for d in u.space_dimensions])
lin_source = dm /rho * u.dt2 * m - du_aux
else:
lin_source = dm / rho * u.dt2
stencil_u = damp * (2.0 * u - damp * u.backward + dt**2 * rho / m * ulaplace)
stencil_du = damp * (2.0 * du - damp * du.backward + dt**2 * rho / m * (dulaplace - lin_source))
expression_u = [Eq(u.forward, stencil_u)]
expression_du = [Eq(du.forward, stencil_du)]
# Define source symbol with wavelet
src = PointSource(name='src', grid=model.grid, ntime=nt, coordinates=src_coords)
src.data[:] = wavelet[:]
src_term = src.inject(field=u.forward, expr=src * rho * dt**2 / m)
# Define receiver symbol
rec = Receiver(name='rec', grid=model.grid, ntime=nt, coordinates=rec_coords)
rec_term = rec.interpolate(expr=du)
expression = expression_u + expression_du + src_term + rec_term
# Free surface
if free_surface is True:
expression += freesurface(u, space_order//2, model.nbpml)
expression += freesurface(du, space_order//2, model.nbpml)
# Create operator and run
set_log_level('ERROR')
subs = model.spacing_map
subs[u.grid.time_dim.spacing] = dt
op = Operator(expression, subs=subs, dse='advanced', dle='advanced')
op()
return rec.data
def forward_freq_modeling(model, src_coords, wavelet, rec_coords, freq, space_order=8, dt=None, factor=None, free_surface=False):
# Forward modeling with on-the-fly DFT of forward wavefields
clear_cache()
# Parameters
nt = wavelet.shape[0]
if dt is None:
dt = model.critical_dt
m, rho, damp = model.m, model.rho, model.damp
freq_dim = Dimension(name='freq_dim')
time = model.grid.time_dim
if factor is None:
factor = int(1 / (dt*4*np.max(freq)))
tsave = ConditionalDimension(name='tsave', parent=model.grid.time_dim, factor=factor)
if factor==1:
tsave = time
else:
tsave = ConditionalDimension(name='tsave', parent=model.grid.time_dim, factor=factor)
print("DFT subsampling factor: ", factor)
# Create wavefields
nfreq = freq.shape[0]
u = TimeFunction(name='u', grid=model.grid, time_order=2, space_order=space_order)
f = Function(name='f', dimensions=(freq_dim,), shape=(nfreq,))
f.data[:] = freq[:]
ufr = Function(name='ufr', dimensions=(freq_dim,) + u.indices[1:], shape=(nfreq,) + model.shape_domain)
ufi = Function(name='ufi', dimensions=(freq_dim,) + u.indices[1:], shape=(nfreq,) + model.shape_domain)
ulaplace, rho = acoustic_laplacian(u, rho)
# Set up PDE and rearrange
stencil = damp * (2.0 * u - damp * u.backward + dt**2 * rho / m * ulaplace)
expression = [Eq(u.forward, stencil)]
expression += [Eq(ufr, ufr + factor*u*cos(2*np.pi*f*tsave*factor*dt))]
expression += [Eq(ufi, ufi - factor*u*sin(2*np.pi*f*tsave*factor*dt))]
# Source symbol with input wavelet
src = PointSource(name='src', grid=model.grid, ntime=nt, coordinates=src_coords)
src.data[:] = wavelet[:]
src_term = src.inject(field=u.forward, expr=src * dt**2 / m)
# Data is sampled at receiver locations
rec = Receiver(name='rec', grid=model.grid, ntime=nt, coordinates=rec_coords)
rec_term = rec.interpolate(expr=u)
# Create operator and run
expression += src_term + rec_term
# Free surface
if free_surface is True:
expression += freesurface(u, space_order//2, model.nbpml)
subs = model.spacing_map
subs[u.grid.time_dim.spacing] = dt
op = Operator(expression, subs=subs, dse='advanced', dle='advanced')
cf = op.cfunction
op()
return rec.data, ufr, ufi
def adjoint_modeling(model, src_coords, rec_coords, rec_data, space_order=8, free_surface=False, dt=None):
clear_cache()
# If wavelet is file, read it
if isinstance(rec_data, str):
rec_data = np.load(rec_data)
# Parameters
nt = rec_data.shape[0]
if dt is None:
dt = model.critical_dt
m, rho, damp = model.m, model.rho, model.damp
# Create the adjoint wavefield
if src_coords is not None:
v = TimeFunction(name="v", grid=model.grid, time_order=2, space_order=space_order)
else:
v = TimeFunction(name="v", grid=model.grid, time_order=2, space_order=space_order, save=nt)
# Set up PDE and rearrange
vlaplace, rho = acoustic_laplacian(v, rho)
# Input data is wavefield
full_q = 0
if isinstance(rec_data, TimeFunction):
wf_rec = TimeFunction(name='wf_rec', grid=model.grid, time_order=2, space_order=space_order, save=nt)
wf_rec._data = rec_data._data
full_q = wf_rec
stencil = damp * (2.0 * v - damp * v.forward + dt**2 * rho / m * (vlaplace + full_q))
expression = [Eq(v.backward, stencil)]
# Free surface
if free_surface is True:
expression += freesurface(v, space_order//2, model.nbpml, forward=False)
# Adjoint source is injected at receiver locations
if rec_coords is not None:
rec = Receiver(name='rec', grid=model.grid, ntime=nt, coordinates=rec_coords)
rec.data[:] = rec_data[:]
adj_src = rec.inject(field=v.backward, expr=rec * rho * dt**2 / m)
expression += adj_src
# Data is sampled at source locations
if src_coords is not None:
src = PointSource(name='src', grid=model.grid, ntime=nt, coordinates=src_coords)
adj_rec = src.interpolate(expr=v)
expression += adj_rec
# Create operator and run
subs = model.spacing_map
subs[v.grid.time_dim.spacing] = dt
op = Operator(expression, subs=subs, dse='advanced', dle='advanced')
cf = op.cfunction
summary = op.apply()
if src_coords is None:
return v, summary
else:
return src.data, summary
def forward_modeling(model, src_coords, wavelet, rec_coords, save=False, space_order=8, nb=40, free_surface=False, op_return=False, u_return=False, dt=None, tsub_factor=1):
clear_cache()
# If wavelet is file, read it
if isinstance(wavelet, str):
wavelet = np.load(wavelet)
# Parameters
nt = wavelet.shape[0]
if dt is None:
dt = model.critical_dt
m, rho, damp = model.m, model.rho, model.damp
# Create the forward wavefield
if save is False and rec_coords is not None:
u = TimeFunction(name='u', grid=model.grid, time_order=2, space_order=space_order)
eqsave = []
elif save is True and tsub_factor > 1:
u = TimeFunction(name='u', grid=model.grid, time_order=2, space_order=space_order)
time_subsampled = ConditionalDimension(name='t_sub', parent=u.grid.time_dim, factor=tsub_factor)
nsave = (nt-1)//tsub_factor + 2
usave = TimeFunction(name='us', grid=model.grid, time_order=2, space_order=space_order, time_dim=time_subsampled, save=nsave)
eqsave = [Eq(usave.forward, u.forward)]
else:
u = TimeFunction(name='u', grid=model.grid, time_order=2, space_order=space_order, save=nt)
eqsave = []
# Set up PDE
ulaplace, rho = acoustic_laplacian(u, rho)
stencil = damp * ( 2.0 * u - damp * u.backward + dt**2 * rho / m * ulaplace)
# Input source is wavefield
if isinstance(wavelet, TimeFunction):
wf_src = TimeFunction(name='wf_src', grid=model.grid, time_order=2, space_order=space_order, save=nt)
wf_src._data = wavelet._data
stencil -= wf_src
# Rearrange expression
expression = [Eq(u.forward, stencil)]
# Data is sampled at receiver locations
if rec_coords is not None:
rec = Receiver(name='rec', grid=model.grid, ntime=nt, coordinates=rec_coords)
rec_term = rec.interpolate(expr=u)
expression += rec_term
# Create operator and run
if save:
expression += eqsave
# Free surface
kwargs = dict()
if free_surface is True:
expression += freesurface(u, space_order//2, model.nbpml)
# Source symbol with input wavelet
if src_coords is not None:
src = PointSource(name='src', grid=model.grid, ntime=nt, coordinates=src_coords)
src.data[:] = wavelet[:]
src_term = src.inject(field=u.forward, expr=src * rho * dt**2 / m)
expression += src_term
# Create operator and run
set_log_level('ERROR')
subs = model.spacing_map
subs[u.grid.time_dim.spacing] = dt
op = Operator(expression, subs=subs, dse='advanced', dle='advanced')
# Return data and wavefields
if op_return is False:
op()
if save is True and tsub_factor > 1:
if rec_coords is None:
return usave
else:
return rec.data, usave
else:
if rec_coords is None:
return u
else:
return rec.data, u
# For optimal checkpointing, return operator only
else:
return op
def adjoint_born(model, rec_coords, rec_data, u=None, op_forward=None, is_residual=False, space_order=8, nb=40, isic=False, dt=None, n_checkpoints=None, maxmem=None, free_surface=False, tsub_factor=1,):
clear_cache()
# Parameters
nt = rec_data.shape[0]
if dt is None:
dt = model.critical_dt
m, rho, damp = model.m, model.rho, model.damp
# Create adjoint wavefield and gradient
v = TimeFunction(name='v', grid=model.grid, time_order=2, space_order=space_order)
gradient = Function(name='gradient', grid=model.grid)
# Set up PDE and rearrange
vlaplace, rho = acoustic_laplacian(v, rho)
stencil = damp * (2.0 * v - damp * v.forward + dt**2 * rho / m * vlaplace)
expression = [Eq(v.backward, stencil)]
# Data at receiver locations as adjoint source
rec_g = Receiver(name='rec_g', grid=model.grid, ntime=nt, coordinates=rec_coords)
if op_forward is None:
rec_g.data[:] = rec_data[:]
adj_src = rec_g.inject(field=v.backward, expr=rec_g * rho * dt**2 / m)
# Gradient update
if u is None:
u = TimeFunction(name='u', grid=model.grid, time_order=2, space_order=space_order)
if isic is not True:
gradient_update = [Inc(gradient, - dt * u.dt2 / rho * v)]
else:
# summodel0.dm = dm u.dx * v.dx fo x in dimensions.
# space_order//2
diff_u_v = sum([first_derivative(u, dim=d, fd_order=space_order//2)*
first_derivative(v, dim=d, fd_order=space_order//2)
for d in u.space_dimensions])
gradient_update = [Inc(gradient, - tsub_factor * dt * (u * v.dt2 * m + diff_u_v) / rho)]
# Free surface
if free_surface is True:
expression += freesurface(v, space_order//2, model.nbpml, forward=False)
# Create operator and run
set_log_level('ERROR')
expression += gradient_update + adj_src
subs = model.spacing_map
subs[u.grid.time_dim.spacing] = dt
op = Operator(expression, subs=subs, dse='advanced', dle='advanced')
# Optimal checkpointing
if op_forward is not None:
rec = Receiver(name='rec', grid=model.grid, ntime=nt, coordinates=rec_coords)
cp = DevitoCheckpoint([u])
if maxmem is not None:
n_checkpoints = int(np.floor(maxmem * 10**6 / (cp.size * u.data.itemsize)))
wrap_fw = CheckpointOperator(op_forward, u=u, m=model.m, rec=rec)
wrap_rev = CheckpointOperator(op, u=u, v=v, m=model.m, rec_g=rec_g)
# Run forward
wrp = Revolver(cp, wrap_fw, wrap_rev, n_checkpoints, nt-2)
wrp.apply_forward()
# Residual and gradient
if is_residual is True: # input data is already the residual
rec_g.data[:] = rec_data[:]
else:
rec_g.data[:] = rec.data[:] - rec_data[:] # input is observed data
fval = .5*np.dot(rec_g.data[:].flatten(), rec_g.data[:].flatten()) * dt
wrp.apply_reverse()
else:
op()
clear_cache()
if op_forward is not None and is_residual is not True:
return fval, gradient.data
else:
return gradient.data
def adjoint_freq_born(model, rec_coords, rec_data, freq, ufr, ufi, space_order=8, dt=None, isic=False, factor=None,
free_surface=False):
clear_cache()
# Parameters
nt = rec_data.shape[0]
if dt is None:
dt = model.critical_dt
m, rho, damp = model.m, model.rho, model.damp
nfreq = ufr.shape[0]
time = model.grid.time_dim
if factor is None:
factor = int(1 / (dt*4*np.max(freq)))
tsave = ConditionalDimension(name='tsave', parent=model.grid.time_dim, factor=factor)
if factor==1:
tsave = time
else:
tsave = ConditionalDimension(name='tsave', parent=model.grid.time_dim, factor=factor)
dtf = factor * dt
ntf = factor / nt
print("DFT subsampling factor: ", factor)
# Create the forward and adjoint wavefield
v = TimeFunction(name='v', grid=model.grid, time_order=2, space_order=space_order)
f = Function(name='f', dimensions=(ufr.indices[0],), shape=(nfreq,))
f.data[:] = freq[:]
gradient = Function(name="gradient", grid=model.grid)
vlaplace, rho = acoustic_laplacian(v, rho)
# Set up PDE and rearrange
stencil = damp * (2.0 * v - damp * v.forward + dt**2 * rho / m * vlaplace)
expression = [Eq(v.backward, stencil)]
# Data at receiver locations as adjoint source
rec = Receiver(name='rec', grid=model.grid, ntime=nt, coordinates=rec_coords)
rec.data[:] = rec_data[:]
adj_src = rec.inject(field=v.backward, expr=rec * dt**2 / m)
# Gradient update
if isic is True:
if len(model.shape) == 2:
gradient_update = [Eq(gradient, gradient + (2*np.pi*f)**2*ntf*(ufr*cos(2*np.pi*f*tsave*dtf) - ufi*sin(2*np.pi*f*tsave*dtf))*v*model.m -
(ufr.dx*cos(2*np.pi*f*tsave*dtf) - ufi.dx*sin(2*np.pi*f*tsave*dtf))*v.dx*ntf -
(ufr.dy*cos(2*np.pi*f*tsave*dtf) - ufi.dy*sin(2*np.pi*f*tsave*dtf))*v.dy*ntf)]
else:
gradient_update = [Eq(gradient, gradient + (2*np.pi*f)**2*ntf*(ufr*cos(2*np.pi*f*tsave*dtf) - ufi*sin(2*np.pi*f*tsave*dtf))*v*model.m -
(ufr.dx*cos(2*np.pi*f*tsave*dtf) - ufi.dx*sin(2*np.pi*f*tsave*dtf))*v.dx*ntf -
(ufr.dy*cos(2*np.pi*f*tsave*dtf) - ufi.dy*sin(2*np.pi*f*tsave*dtf))*v.dy*ntf -
(ufr.dz*cos(2*np.pi*f*tsave*dtf) - ufi.dz*sin(2*np.pi*f*tsave*dtf))*v.dz*ntf)]
else:
gradient_update = [Eq(gradient, gradient + (2*np.pi*f)**2/nt*(ufr*cos(2*np.pi*f*tsave*dtf) - ufi*sin(2*np.pi*f*tsave*dtf))*v)]
# Create operator and run
# Free surface
if free_surface is True:
expression += freesurface(v, space_order//2, model.nbpml, forward=False)
expression += adj_src + gradient_update
subs = model.spacing_map
subs[v.grid.time_dim.spacing] = dt
op = Operator(expression, subs=subs, dse='advanced', dle='advanced')
cf = op.cfunction
op()
clear_cache()
return gradient.data