def crosscorr_freq(u, v, model, freq=None, dft_sub=None, **kwargs):
"""
Standard cross-correlation imaging condition with on-th-fly-dft
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: Array
Array of frequencies for on-the-fly DFT
factor: int
Subsampling factor for DFT
"""
# Subsampled dft time axis
time = model.grid.time_dim
dt = time.spacing
tsave, factor = sub_time(time, dft_sub)
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
return expr
def crosscorr_freq(u, v, model, freq=None, dft_sub=None, **kwargs):
"""
Standard cross-correlation imaging condition with on-th-fly-dft
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: Array
Array of frequencies for on-the-fly DFT
factor: int
Subsampling factor for DFT
"""
# Subsampled dft time axis
time = model.grid.time_dim
dt = time.spacing
tsave, factor = sub_time(time, dft_sub)
expr = 0
for uu, vv in zip(u, v):
ufr, ufi = uu
# Frequencies
nfreq = np.shape(freq)[0]
fdim = ufr.dimensions[0]
omega_t = lambda f: 2*np.pi*f*tsave*factor*dt
# Gradient weighting is (2*np.pi*f)**2/nt
w = lambda f: -(2*np.pi*f)**2/time.symbolic_max
expr += sum(w(freq[ff])*(ufr._subs(fdim, ff)*cos(omega_t(freq[ff])) -
ufi._subs(fdim, ff)*sin(omega_t(freq[ff])))
for ff in range(nfreq))*vv
return expr
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 * (grad(ufr * cos(omega_t) -
ufi * sin(omega_t)).T * grad(vv)))
return expr
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 = np.shape(freq)[0]
fdim = ufr.dimensions[0]
omega_t = lambda f: 2*np.pi*f*tsave*factor*dt
# Gradient weighting is (2*np.pi*f)**2/nt
w = lambda f: -(2*np.pi*f)**2/time.symbolic_max
w2 = factor / time.symbolic_max
for ff in range(nfreq):
cwt, swt = cos(omega_t(freq[ff])), sin(omega_t(freq[ff]))
ufrf, ufif = ufr._subs(fdim, ff), ufi._subs(fdim, ff)
idftu = (ufrf * cwt - ufif * swt)
expr += w(freq[ff]) * idftu * vv * model.m - w2 * inner_grad(idftu, vv)
return expr
