def test_taper2d(par):
"""Create taper wavelet and check size and values"""
tap = taper2d(par["nt"], par["nspat"][0], par["ntap"][0], par["tapertype"])
assert tap.shape == (par["nspat"][0], par["nt"])
assert_array_equal(tap[0], np.zeros(par["nt"]))
assert_array_equal(tap[-1], np.zeros(par["nt"]))
assert_array_equal(tap[par["ntap"][0] + 1], np.ones(par["nt"]))
assert_array_equal(tap[par["nspat"][0] // 2], np.ones(par["nt"]))
def Deghosting(
p,
nt,
nr,
dt,
dr,
vel,
zrec,
pd=None,
win=None,
npad=(11, 11),
ntaper=(11, 11),
restriction=None,
sptransf=None,
solver=lsqr,
dottest=False,
dtype="complex128",
**kwargs_solver
):
r"""Wavefield deghosting.
Apply seismic wavefield decomposition from single-component (pressure)
data. This process is also generally referred to as model-based deghosting.
Parameters
----------
p : :obj:`np.ndarray`
Pressure data of of size :math:`\lbrack n_{r_x}\,(\times n_{r_y})
\times n_t \rbrack` (or :math:`\lbrack n_{r_{x,\text{sub}}}\,
(\times n_{r_{y,\text{sub}}}) \times n_t \rbrack`
in case a ``restriction`` operator is provided. Note that
:math:`n_{r_{x,\text{sub}}}` (and :math:`n_{r_{y,\text{sub}}}`)
must agree with the size of the output of this operator)
nt : :obj:`int`
Number of samples along the time axis
nr : :obj:`int` or :obj:`tuple`
Number of samples along the receiver axis (or axes)
dt : :obj:`float`
Sampling along the time axis
dr : :obj:`float` or :obj:`tuple`
Sampling along the receiver array of the separated
pressure consituents
vel : :obj:`float`
Velocity along the receiver array (must be constant)
zrec : :obj:`float`
Depth of receiver array
pd : :obj:`np.ndarray`, optional
Direct arrival to be subtracted from ``p``
win : :obj:`np.ndarray`, optional
Time window to be applied to ``p`` to remove the direct arrival
(if ``pd=None``)
ntaper : :obj:`float` or :obj:`tuple`, optional
Number of samples of taper applied to propagator to avoid edge
effects
npad : :obj:`float` or :obj:`tuple`, optional
Number of samples of padding applied to propagator to avoid edge
effects
angle
restriction : :obj:`pylops.LinearOperator`, optional
Restriction operator
sptransf : :obj:`pylops.LinearOperator`, optional
Sparsifying operator
solver : :obj:`float`, optional
Function handle of solver to be used if ``kind='inverse'``
dottest : :obj:`bool`, optional
Apply dot-test
dtype : :obj:`str`, optional
Type of elements in input array. If ``None``, directly inferred
from ``p``
**kwargs_solver
Arbitrary keyword arguments for chosen ``solver``
Returns
-------
pup : :obj:`np.ndarray`
Up-going wavefield
pdown : :obj:`np.ndarray`
Down-going wavefield
Notes
-----
Up- and down-going components of seismic data :math:`p^-(x, t)`
and :math:`p^+(x, t)` can be estimated from single-component data
:math:`p(x, t)` using a ghost model.
The basic idea [1]_ is that of using a one-way propagator in the f-k domain
(also referred to as ghost model) to predict the down-going field
from the up-going one (excluded the direct arrival and its source
ghost referred here to as :math:`p_d(x, t)`):
.. math::
p^+ - p_d = e^{-j k_z 2 z_\text{rec}} p^-
where :math:`k_z` is the vertical wavenumber and :math:`z_\text{rec}` is the
depth of the array of receivers
In a matrix form we can thus write the total wavefield as:
.. math::
\mathbf{p} - \mathbf{p_d} = (\mathbf{I} + \Phi) \mathbf{p}^-
where :math:`\Phi` is one-way propagator implemented via the
:class:`pylops.waveeqprocessing.PhaseShift` operator.
.. [1] Amundsen, L., 1993, Wavenumber-based filtering of marine point-source
data: GEOPHYSICS, 58, 1335–1348.
"""
ndims = p.ndim
if ndims == 2:
dims = (nt, nr)
nrs = nr
nkx = nr + 2 * npad
kx = np.fft.ifftshift(np.fft.fftfreq(nkx, dr))
ky = None
else:
dims = (nt, nr[0], nr[1])
nrs = nr[0] * nr[1]
nkx = nr[0] + 2 * npad[0]
kx = np.fft.ifftshift(np.fft.fftfreq(nkx, dr[0]))
nky = nr[1] + 2 * npad[1]
ky = np.fft.ifftshift(np.fft.fftfreq(nky, dr))
nf = nt
freq = np.fft.rfftfreq(nf, dt)
# Phase shift operator
zprop = 2 * zrec
if ndims == 2:
taper = taper2d(nt, nr, ntaper).T
Padop = Pad(dims, ((0, 0), (npad, npad)))
else:
taper = taper3d(nt, nr, ntaper).transpose(2, 0, 1)
Padop = Pad(dims, ((0, 0), (npad[0], npad[0]), (npad[1], npad[1])))
Pop = (
-Padop.H
* PhaseShift(vel, zprop, nt, freq, kx, ky)
* Padop
* Diagonal(taper.ravel(), dtype=dtype)
)
# Decomposition operator
Dupop = Identity(nt * nrs, dtype=p.dtype) + Pop
if dottest:
Dottest(Dupop, nt * nrs, nt * nrs, verb=True)
# Add restriction
if restriction is not None:
Dupop_norestr = Dupop
Dupop = restriction * Dupop
# Add sparsify transform
if sptransf is not None:
Dupop_norestr = Dupop_norestr * sptransf
Dupop = Dupop * sptransf
# Define data
if pd is not None:
d = p - pd
else:
d = win * p
# Inversion
pup = solver(Dupop, d.ravel(), **kwargs_solver)[0]
# Apply sparse transform
if sptransf is not None:
p = Dupop_norestr * pup # reconstruct p at finely sampled spatial axes
pup = sptransf * pup
p = np.real(p).reshape(dims)
# Finalize estimates
pup = np.real(pup).reshape(dims)
pdown = p - pup
return pup, pdown
def Sliding2D(Op, dims, dimsd, nwin, nover, tapertype="hanning", design=False):
"""2D Sliding transform operator.
Apply a transform operator ``Op`` repeatedly to slices of the model
vector in forward mode and slices of the data vector in adjoint mode.
More specifically, in forward mode the model vector is divided into
slices, each slice is transformed, and slices are then recombined in a
sliding window fashion. Both model and data are internally reshaped and
interpreted as 2-dimensional arrays: each slice contains a portion
of the array in the first dimension (and the entire second dimension).
This operator can be used to perform local, overlapping transforms (e.g.,
:obj:`pylops.signalprocessing.FFT2D`
or :obj:`pylops.signalprocessing.Radon2D`) on 2-dimensional arrays.
.. note:: The shape of the model has to be consistent with
the number of windows for this operator not to return an error. As the
number of windows depends directly on the choice of ``nwin`` and
``nover``, it is recommended to use ``design=True`` if unsure about the
choice ``dims`` and use the number of windows printed on screen to
define such input parameter.
.. warning:: Depending on the choice of `nwin` and `nover` as well as the
size of the data, sliding windows may not cover the entire first dimension.
The start and end indices of each window can be displayed using
``design=True`` while defining the best sliding window approach.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Transform operator
dims : :obj:`tuple`
Shape of 2-dimensional model. Note that ``dims[0]`` should be multiple
of the model size of the transform in the first dimension
dimsd : :obj:`tuple`
Shape of 2-dimensional data
nwin : :obj:`int`
Number of samples of window
nover : :obj:`int`
Number of samples of overlapping part of window
tapertype : :obj:`str`, optional
Type of taper (``hanning``, ``cosine``, ``cosinesquare`` or ``None``)
design : :obj:`bool`, optional
Print number of sliding window (``True``) or not (``False``)
Returns
-------
Sop : :obj:`pylops.LinearOperator`
Sliding operator
Raises
------
ValueError
Identified number of windows is not consistent with provided model
shape (``dims``).
"""
# model windows
mwin_ins, mwin_ends = _slidingsteps(dims[0], Op.shape[1] // dims[1], 0)
# data windows
dwin_ins, dwin_ends = _slidingsteps(dimsd[0], nwin, nover)
nwins = len(dwin_ins)
# create tapers
if tapertype is not None:
tap = taper2d(dimsd[1], nwin, nover, tapertype=tapertype)
tapin = tap.copy()
tapin[:nover] = 1
tapend = tap.copy()
tapend[-nover:] = 1
taps = {}
taps[0] = tapin
for i in range(1, nwins - 1):
taps[i] = tap
taps[nwins - 1] = tapend
# check that identified number of windows agrees with mode size
if design:
logging.warning("%d windows required...", nwins)
logging.warning("model wins - start:%s, end:%s", str(mwin_ins),
str(mwin_ends))
logging.warning("data wins - start:%s, end:%s", str(dwin_ins),
str(dwin_ends))
if nwins * Op.shape[1] // dims[1] != dims[0]:
raise ValueError("Model shape (dims=%s) is not consistent with chosen "
"number of windows. Choose dims[0]=%d for the "
"operator to work with estimated number of windows, "
"or create the operator with design=True to find "
"out the optimal number of windows for the current "
"model size..." %
(str(dims), nwins * Op.shape[1] // dims[1]))
# transform to apply
if tapertype is None:
OOp = BlockDiag([Op for _ in range(nwins)])
else:
OOp = BlockDiag(
[Diagonal(taps[itap].ravel()) * Op for itap in range(nwins)])
combining = HStack([
Restriction(np.prod(dimsd),
range(win_in, win_end),
dims=dimsd,
dtype=Op.dtype).H
for win_in, win_end in zip(dwin_ins, dwin_ends)
])
Sop = combining * OOp
return Sop
def Patch2D(Op,
dims,
dimsd,
nwin,
nover,
nop,
tapertype="hanning",
design=False):
"""2D Patch transform operator.
Apply a transform operator ``Op`` repeatedly to patches of the model
vector in forward mode and patches of the data vector in adjoint mode.
More specifically, in forward mode the model vector is divided into
patches, each patch is transformed, and patches are then recombined
together. Both model and data are internally reshaped and
interpreted as 2-dimensional arrays: each patch contains a portion
of the array in both the first and second dimension.
This operator can be used to perform local, overlapping transforms (e.g.,
:obj:`pylops.signalprocessing.FFT2D`
or :obj:`pylops.signalprocessing.Radon2D`) on 2-dimensional arrays.
.. note:: The shape of the model has to be consistent with
the number of windows for this operator not to return an error. As the
number of windows depends directly on the choice of ``nwin`` and
``nover``, it is recommended to use ``design=True`` if unsure about the
choice ``dims`` and use the number of windows printed on screen to
define such input parameter.
.. warning:: Depending on the choice of `nwin` and `nover` as well as the
size of the data, patches may not cover the entire size of the data.
The start and end indices of each window can be displayed using
``design=True`` while defining the best patching approach.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Transform operator
dims : :obj:`tuple`
Shape of 2-dimensional model. Note that ``dims[0]`` and ``dims[1]``
should be multiple of the model size of the transform in their
respective dimensions
dimsd : :obj:`tuple`
Shape of 2-dimensional data
nwin : :obj:`tuple`
Number of samples of window
nover : :obj:`tuple`
Number of samples of overlapping part of window
nop : :obj:`tuple`
Size of model in the transformed domain
tapertype : :obj:`str`, optional
Type of taper (``hanning``, ``cosine``, ``cosinesquare`` or ``None``)
design : :obj:`bool`, optional
Print number of sliding window (``True``) or not (``False``)
Returns
-------
Sop : :obj:`pylops.LinearOperator`
Sliding operator
Raises
------
ValueError
Identified number of windows is not consistent with provided model
shape (``dims``).
See Also
--------
Sliding2d: 2D Sliding transform operator.
"""
# model windows
mwin0_ins, mwin0_ends = _slidingsteps(dims[0], nop[0], 0)
mwin1_ins, mwin1_ends = _slidingsteps(dims[1], nop[1], 0)
# data windows
dwin0_ins, dwin0_ends = _slidingsteps(dimsd[0], nwin[0], nover[0])
dwin1_ins, dwin1_ends = _slidingsteps(dimsd[1], nwin[1], nover[1])
nwins0 = len(dwin0_ins)
nwins1 = len(dwin1_ins)
nwins = nwins0 * nwins1
# create tapers
if tapertype is not None:
tap = taper2d(nwin[1], nwin[0], nover,
tapertype=tapertype).astype(Op.dtype)
taps = {itap: tap for itap in range(nwins)}
# topmost tapers
taptop = tap.copy()
taptop[:nover[0]] = tap[nwin[0] // 2]
for itap in range(0, nwins1):
taps[itap] = taptop
# bottommost tapers
tapbottom = tap.copy()
tapbottom[-nover[0]:] = tap[nwin[0] // 2]
for itap in range(nwins - nwins1, nwins):
taps[itap] = tapbottom
# leftmost tapers
tapleft = tap.copy()
tapleft[:, :nover[1]] = tap[:, nwin[1] // 2][:, np.newaxis]
for itap in range(0, nwins, nwins1):
taps[itap] = tapleft
# rightmost tapers
tapright = tap.copy()
tapright[:, -nover[1]:] = tap[:, nwin[1] // 2][:, np.newaxis]
for itap in range(nwins1 - 1, nwins, nwins1):
taps[itap] = tapright
# lefttopcorner taper
taplefttop = tap.copy()
taplefttop[:, :nover[1]] = tap[:, nwin[1] // 2][:, np.newaxis]
taplefttop[:nover[0]] = taplefttop[nwin[0] // 2]
taps[0] = taplefttop
# righttopcorner taper
taprighttop = tap.copy()
taprighttop[:, -nover[1]:] = tap[:, nwin[1] // 2][:, np.newaxis]
taprighttop[:nover[0]] = taprighttop[nwin[0] // 2]
taps[nwins1 - 1] = taprighttop
# leftbottomcorner taper
tapleftbottom = tap.copy()
tapleftbottom[:, :nover[1]] = tap[:, nwin[1] // 2][:, np.newaxis]
tapleftbottom[-nover[0]:] = tapleftbottom[nwin[0] // 2]
taps[nwins - nwins1] = tapleftbottom
# rightbottomcorner taper
taprightbottom = tap.copy()
taprightbottom[:, -nover[1]:] = tap[:, nwin[1] // 2][:, np.newaxis]
taprightbottom[-nover[0]:] = taprightbottom[nwin[0] // 2]
taps[nwins - 1] = taprightbottom
# check that identified number of windows agrees with mode size
if design:
logging.warning("%d-%d windows required...", nwins0, nwins1)
logging.warning(
"model wins - start:%s, end:%s / start:%s, end:%s",
str(mwin0_ins),
str(mwin0_ends),
str(mwin1_ins),
str(mwin1_ends),
)
logging.warning(
"data wins - start:%s, end:%s / start:%s, end:%s",
str(dwin0_ins),
str(dwin0_ends),
str(dwin1_ins),
str(dwin1_ends),
)
if nwins0 * nop[0] != dims[0] or nwins1 * nop[1] != dims[1]:
raise ValueError("Model shape (dims=%s) is not consistent with chosen "
"number of windows. Choose dims[0]=%d and "
"dims[1]=%d for the operator to work with "
"estimated number of windows, or create "
"the operator with design=True to find out the"
"optimal number of windows for the current "
"model size..." %
(str(dims), nwins0 * nop[0], nwins1 * nop[1]))
# transform to apply
if tapertype is None:
OOp = BlockDiag([Op for _ in range(nwins)])
else:
OOp = BlockDiag([
Diagonal(taps[itap].ravel(), dtype=Op.dtype) * Op
for itap in range(nwins)
])
hstack = HStack([
Restriction(
dimsd[1] * nwin[0],
range(win_in, win_end),
dims=(nwin[0], dimsd[1]),
dir=1,
dtype=Op.dtype,
).H for win_in, win_end in zip(dwin1_ins, dwin1_ends)
])
combining1 = BlockDiag([hstack] * nwins0)
combining0 = HStack([
Restriction(
np.prod(dimsd),
range(win_in, win_end),
dims=dimsd,
dir=0,
dtype=Op.dtype,
).H for win_in, win_end in zip(dwin0_ins, dwin0_ends)
])
Pop = combining0 * combining1 * OOp
return Pop