def isic_freq(u, v, model, **kwargs):
"""
Inverse scattering imaging condition
Parameters
----------
u: TimeFunction or Tuple
Forward wavefield (tuple of fields for TTI or dft)
v: TimeFunction or Tuple
Adjoint wavefield (tuple of fields for TTI)
model: Model
Model structure
"""
freq = kwargs.get('freq')
# Subsampled dft time axis
time = model.grid.time_dim
dt = time.spacing
tsave, factor = sub_time(time, kwargs.get('factor'))
expr = 0
for uu, vv in zip(u, v):
ufr, ufi = uu
# Frequencies
nfreq = freq.shape[0]
f = Function(name='f',
dimensions=(ufr.dimensions[0], ),
shape=(nfreq, ))
f.data[:] = freq[:]
omega_t = 2 * np.pi * f * tsave * factor * dt
# Gradient weighting is (2*np.pi*f)**2/nt
w = (2 * np.pi * f)**2 / time.symbolic_max
expr += (w * (ufr * cos(omega_t) - ufi * sin(omega_t)) * vv * model.m -
factor / time.symbolic_max *
(grads(ufr * cos(omega_t) - ufi * sin(omega_t), so_fact=2).T *
grads(vv, so_fact=2)))
return expr
def inner_grad(u, v):
"""
Inner product of the gradient of two Function.
Parameters
----------
u: TimeFunction or Function
First wavefield
v: TimeFunction or Function
Second wavefield
"""
return sum([a*b for a, b in zip(grads(u, so_fact=2), grads(v, so_fact=2))])
def isic_time(u, v, model, **kwargs):
"""
Inverse scattering imaging condition
Parameters
----------
u: TimeFunction or Tuple
Forward wavefield (tuple of fields for TTI or dft)
v: TimeFunction or Tuple
Adjoint wavefield (tuple of fields for TTI)
model: Model
Model structure
"""
w = -u[0].indices[0].spacing * model.irho
return w * sum(
uu * vv.dt2 * model.m + grads(uu, so_fact=2).T * grads(vv, so_fact=2)
for uu, vv in zip(u, v))
def isic_src(model, u, **kwargs):
"""
ISIC source for linearized modeling
Parameters
----------
u: TimeFunction or Tuple
Forward wavefield (tuple of fields for TTI or dft)
model: Model
Model containing the perturbation dm
"""
m, dm, irho = model.m, model.dm, model.irho
dus = []
for uu in u:
du_aux = divs(dm * irho * grads(uu, so_fact=2), so_fact=2)
dus.append(dm * irho * uu.dt2 * m - du_aux)
if model.is_tti:
return (-dus[0], -dus[1])
return -dus[0]