def PrestackInversion(data,
theta,
wav,
m0=None,
linearization='akirich',
explicit=False,
simultaneous=False,
epsI=None,
epsR=None,
dottest=False,
returnres=False,
epsRL1=None,
kind='centered',
**kwargs_solver):
r"""Pre-stack linearized seismic inversion.
Invert pre-stack seismic operator to retrieve a set of elastic property
profiles from band-limited seismic pre-stack data (i.e., angle gathers).
Depending on the choice of input parameters, inversion can be
trace-by-trace with explicit operator or global with either
explicit or linear operator.
Parameters
----------
data : :obj:`np.ndarray`
Band-limited seismic post-stack data of size
:math:`[(n_{lins} \times) n_{t0} \times n_{\theta} (\times n_x \times n_y)]`
theta : :obj:`np.ndarray`
Incident angles in degrees
wav : :obj:`np.ndarray`
Wavelet in time domain (must had odd number of elements
and centered to zero)
m0 : :obj:`np.ndarray`, optional
Background model of size :math:`[n_{t0} \times n_{m}
(\times n_x \times n_y)]`
linearization : :obj:`str` or :obj:`list`, optional
choice of linearization, ``akirich``: Aki-Richards, ``fatti``: Fatti
``PS``: PS or a combination of them (required only when ``m0``
is ``None``).
explicit : :obj:`bool`, optional
Create a chained linear operator (``False``, preferred for large data)
or a ``MatrixMult`` linear operator with dense matrix
(``True``, preferred for small data)
simultaneous : :obj:`bool`, optional
Simultaneously invert entire data (``True``) or invert
trace-by-trace (``False``) when using ``explicit`` operator
(note that the entire data is always inverted when working
with linear operator)
epsI : :obj:`float` or :obj:`list`, optional
Damping factor(s) for Tikhonov regularization term. If a list of
:math:`n_{m}` elements is provided, the regularization term will have
different strenght for each elastic property
epsR : :obj:`float`, optional
Damping factor for additional Laplacian regularization term
dottest : :obj:`bool`, optional
Apply dot-test
returnres : :obj:`bool`, optional
Return residuals
epsRL1 : :obj:`float`, optional
Damping factor for additional blockiness regularization term
kind : :obj:`str`, optional
Derivative kind (``forward`` or ``centered``).
**kwargs_solver
Arbitrary keyword arguments for :py:func:`scipy.linalg.lstsq`
solver (if ``explicit=True`` and ``epsR=None``)
or :py:func:`scipy.sparse.linalg.lsqr` solver (if ``explicit=False``
and/or ``epsR`` is not ``None``))
Returns
-------
minv : :obj:`np.ndarray`
Inverted model of size :math:`[n_{t0} \times n_{m}
(\times n_x \times n_y)]`
datar : :obj:`np.ndarray`
Residual data (i.e., data - background data) of
size :math:`[n_{t0} \times n_{\theta} (\times n_x \times n_y)]`
Notes
-----
The different choices of cost functions and solvers used in the
seismic pre-stack inversion module follow the same convention of the
seismic post-stack inversion module.
Refer to :py:func:`pylops.avo.poststack.PoststackInversion` for
more details.
"""
ncp = get_array_module(data)
# find out dimensions
if m0 is None and linearization is None:
raise NotImplementedError('either m0 or linearization '
'must be provided')
elif m0 is None:
if isinstance(linearization, str):
nm = _linearizations[linearization]
else:
nm = _linearizations[linearization[0]]
else:
nm = m0.shape[1]
data_shape = data.shape
data_ndim = data.ndim
n_lins = 1
multi = 0
if not isinstance(linearization, str):
n_lins = data_shape[0]
data_shape = data_shape[1:]
data_ndim -= 1
multi = 1
if data_ndim == 2:
dims = 1
nt0, ntheta = data_shape
nspat = None
nspatprod = nx = 1
elif data_ndim == 3:
dims = 2
nt0, ntheta, nx = data_shape
nspat = (nx, )
nspatprod = nx
else:
dims = 3
nt0, ntheta, nx, ny = data_shape
nspat = (nx, ny)
nspatprod = nx * ny
data = data.reshape(nt0, ntheta, nspatprod)
# check if background model and data have same shape
if m0 is not None:
if nt0 != m0.shape[0] or\
(dims >= 2 and nx != m0.shape[2]) or\
(dims == 3 and ny != m0.shape[3]):
raise ValueError('data and m0 must have same time and space axes')
# create operator
if isinstance(linearization, str):
# single operator
PPop = PrestackLinearModelling(wav,
theta,
nt0=nt0,
spatdims=nspat,
linearization=linearization,
explicit=explicit,
kind=kind)
else:
# multiple operators
if not isinstance(wav, (list, tuple)):
wav = [
wav,
] * n_lins
PPop = [
PrestackLinearModelling(w,
theta,
nt0=nt0,
spatdims=nspat,
linearization=lin,
explicit=explicit)
for w, lin in zip(wav, linearization)
]
if explicit:
PPop = MatrixMult(np.vstack([Op.A for Op in PPop]),
dims=nspat,
dtype=PPop[0].A.dtype)
else:
PPop = VStack(PPop)
if dottest:
Dottest(PPop,
n_lins * nt0 * ntheta * nspatprod,
nt0 * nm * nspatprod,
raiseerror=True,
verb=True,
backend=get_module_name(ncp))
# swap axes for explicit operator
if explicit:
data = data.swapaxes(0 + multi, 1 + multi)
if m0 is not None:
m0 = m0.swapaxes(0, 1)
# invert model
if epsR is None:
# create and remove background data from original data
datar = data.flatten() if m0 is None else \
data.flatten() - PPop * m0.flatten()
# inversion without spatial regularization
if explicit:
if epsI is None and not simultaneous:
# solve unregularized equations indipendently trace-by-trace
minv = \
get_lstsq(data)(PPop.A,
datar.reshape(n_lins*nt0*ntheta, nspatprod).squeeze(),
**kwargs_solver)[0]
elif epsI is None and simultaneous:
# solve unregularized equations simultaneously
if ncp == np:
minv = lsqr(PPop, datar, **kwargs_solver)[0]
else:
minv = cgls(PPop,
datar,
x0=ncp.zeros(int(PPop.shape[1]), PPop.dtype),
**kwargs_solver)[0]
elif epsI is not None:
# create regularized normal equations
PP = ncp.dot(PPop.A.T, PPop.A) + \
epsI * ncp.eye(nt0*nm, dtype=PPop.A.dtype)
datarn = np.dot(PPop.A.T,
datar.reshape(nt0 * ntheta, nspatprod))
if not simultaneous:
# solve regularized normal eqs. trace-by-trace
minv = get_lstsq(data)(PP, datarn, **kwargs_solver)[0]
else:
# solve regularized normal equations simultaneously
PPop_reg = MatrixMult(PP, dims=nspatprod)
if ncp == np:
minv = lsqr(PPop_reg, datarn.ravel(),
**kwargs_solver)[0]
else:
minv = cgls(PPop_reg,
datarn.ravel(),
x0=ncp.zeros(int(PPop_reg.shape[1]),
PPop_reg.dtype),
**kwargs_solver)[0]
#else:
# # create regularized normal eqs. and solve them simultaneously
# PP = np.dot(PPop.A.T, PPop.A) + epsI * np.eye(nt0*nm)
# datarn = PPop.A.T * datar.reshape(nt0*ntheta, nspatprod)
# PPop_reg = MatrixMult(PP, dims=ntheta*nspatprod)
# minv = lstsq(PPop_reg, datarn.flatten(), **kwargs_solver)[0]
else:
# solve unregularized normal equations simultaneously with lop
if ncp == np:
minv = lsqr(PPop, datar, **kwargs_solver)[0]
else:
minv = cgls(PPop,
datar,
x0=ncp.zeros(int(PPop.shape[1]), PPop.dtype),
**kwargs_solver)[0]
else:
# Create Thicknov regularization
if epsI is not None:
if isinstance(epsI, (list, tuple)):
if len(epsI) != nm:
raise ValueError('epsI must be a scalar or a list of'
'size nm')
RegI = Diagonal(np.array(epsI),
dims=(nt0, nm, nspatprod),
dir=1)
else:
RegI = epsI * Identity(nt0 * nm * nspatprod)
if epsRL1 is None:
# L2 inversion with spatial regularization
if dims == 1:
Regop = SecondDerivative(nt0 * nm,
dtype=PPop.dtype,
dims=(nt0, nm))
elif dims == 2:
Regop = Laplacian((nt0, nm, nx), dirs=(0, 2), dtype=PPop.dtype)
else:
Regop = Laplacian((nt0, nm, nx, ny),
dirs=(2, 3),
dtype=PPop.dtype)
if epsI is None:
Regop = (Regop, )
epsR = (epsR, )
else:
Regop = (Regop, RegI)
epsR = (epsR, 1)
minv = \
RegularizedInversion(PPop, Regop, data.ravel(),
x0=m0.flatten() if m0 is not None
else None, epsRs=epsR,
returninfo=False, **kwargs_solver)
else:
# Blockiness-promoting inversion with spatial regularization
if dims == 1:
RegL1op = FirstDerivative(nt0 * nm, dtype=PPop.dtype)
RegL2op = None
elif dims == 2:
RegL1op = FirstDerivative(nt0 * nx * nm,
dims=(nt0, nm, nx),
dir=0,
dtype=PPop.dtype)
RegL2op = SecondDerivative(nt0 * nx * nm,
dims=(nt0, nm, nx),
dir=2,
dtype=PPop.dtype)
else:
RegL1op = FirstDerivative(nt0 * nx * ny * nm,
dims=(nt0, nm, nx, ny),
dir=0,
dtype=PPop.dtype)
RegL2op = Laplacian((nt0, nm, nx, ny),
dirs=(2, 3),
dtype=PPop.dtype)
if dims == 1:
if epsI is not None:
RegL2op = (RegI, )
epsR = (1, )
else:
if epsI is None:
RegL2op = (RegL2op, )
epsR = (epsR, )
else:
RegL2op = (RegL2op, RegI)
epsR = (epsR, 1)
epsRL1 = (epsRL1, )
if 'mu' in kwargs_solver.keys():
mu = kwargs_solver['mu']
kwargs_solver.pop('mu')
else:
mu = 1.
if 'niter_outer' in kwargs_solver.keys():
niter_outer = kwargs_solver['niter_outer']
kwargs_solver.pop('niter_outer')
else:
niter_outer = 3
if 'niter_inner' in kwargs_solver.keys():
niter_inner = kwargs_solver['niter_inner']
kwargs_solver.pop('niter_inner')
else:
niter_inner = 5
minv = SplitBregman(PPop, (RegL1op, ),
data.ravel(),
RegsL2=RegL2op,
epsRL1s=epsRL1,
epsRL2s=epsR,
mu=mu,
niter_outer=niter_outer,
niter_inner=niter_inner,
x0=None if m0 is None else m0.flatten(),
**kwargs_solver)[0]
# compute residual
if returnres:
if epsR is None:
datar -= PPop * minv.ravel()
else:
datar = data.ravel() - PPop * minv.ravel()
# re-swap axes for explicit operator
if explicit:
if m0 is not None:
m0 = m0.swapaxes(0, 1)
# reshape inverted model and residual data
if dims == 1:
if explicit:
minv = minv.reshape(nm, nt0).swapaxes(0, 1)
if returnres:
datar = datar.reshape(n_lins, ntheta, nt0).squeeze().swapaxes(
0 + multi, 1 + multi)
else:
minv = minv.reshape(nt0, nm)
if returnres:
datar = datar.reshape(n_lins, nt0, ntheta).squeeze()
elif dims == 2:
if explicit:
minv = minv.reshape(nm, nt0, nx).swapaxes(0, 1)
if returnres:
datar = datar.reshape(n_lins, ntheta, nt0,
nx).squeeze().swapaxes(
0 + multi, 1 + multi)
else:
minv = minv.reshape(nt0, nm, nx)
if returnres:
datar = datar.reshape(n_lins, nt0, ntheta, nx).squeeze()
else:
if explicit:
minv = minv.reshape(nm, nt0, nx, ny).swapaxes(0, 1)
if returnres:
datar = datar.reshape(n_lins, ntheta, nt0, nx,
ny).squeeze().swapaxes(
0 + multi, 1 + multi)
else:
minv = minv.reshape(nt0, nm, nx, ny)
if returnres:
datar = datar.reshape(n_lins, nt0, ntheta, nx, ny).squeeze()
if m0 is not None and epsR is None:
minv = minv + m0
if returnres:
return minv, datar
else:
return minv
def PoststackInversion(data,
wav,
m0=None,
explicit=False,
simultaneous=False,
epsI=None,
epsR=None,
dottest=False,
**kwargs_solver):
r"""Post-stack linearized seismic inversion.
Invert post-stack seismic operator to retrieve an acoustic
impedance profile from band-limited seismic post-stack data.
Depending on the choice of input parameters, inversion can be
trace-by-trace with explicit operator or global with either
explicit or linear operator.
Parameters
----------
data : :obj:`np.ndarray`
Band-limited seismic post-stack data of size :math:`[n_{t0} \times n_x]`
wav : :obj:`np.ndarray`
Wavelet in time domain (must had odd number of elements and centered to zero)
m0 : :obj:`np.ndarray`, optional
Background model of size :math:`[n_{t0} \times n_x]`
explicit : :obj:`bool`, optional
Create a chained linear operator (``False``, preffered for large data)
or a ``MatrixMult`` linear operator with dense matrix
(``True``, preffered for small data)
simultaneous : :obj:`bool`, optional
Simultaneously invert entire data (``True``) or invert
trace-by-trace (``False``) when using ``explicit`` operator
(note that the entire data is always inverted when working with linear operator)
epsI : :obj:`float`, optional
Damping factor for Tikhonov regularization term
epsR : :obj:`float`, optional
Damping factor for additional Laplacian regularization term
dottest : :obj:`bool`, optional
Apply dot-test
**kwargs_solver
Arbitrary keyword arguments for :py:func:`scipy.linalg.lstsq`
solver (if ``explicit=True`` and ``epsR=None``)
or :py:func:`scipy.sparse.linalg.lsqr` solver (if ``explicit=False``
and/or ``epsR`` is not ``None``))
Returns
-------
minv : :obj:`np.ndarray`
Inverted model of size :math:`[n_{t0} \times n_x]`
dr : :obj:`np.ndarray`
Residual data (i.e., data - background data) of
size :math:`[n_{t0} \times n_x]`
Notes
-----
The cost function and solver used in the seismic post-stack inversion module depends on the choice of
``explicit``, ``simultaneous``, ``epsI``, and ``epsR`` parameters:
* ``explicit=True``, ``epsI=None`` and ``epsR=None``: the explicit solver :py:func:`scipy.linalg.lstsq`
is used if ``simultaneous=False`` (or the iterative solver :py:func:`scipy.sparse.linalg.lsqr`
is used if ``simultaneous=True``)
* ``explicit=True`` with ``epsI`` and ``epsR=None``: the regularized normal equations
:math:`\mathbf{W}^T\mathbf{d} = (\mathbf{W}^T \mathbf{W} + \epsilon_I^2 \mathbf{I}) \mathbf{AI}`
are instead fed into the :py:func:`scipy.linalg.lstsq` solver if ``simultaneous=False``
(or the iterative solver :py:func:`scipy.sparse.linalg.lsqr` if ``simultaneous=True``)
* ``explicit=False`` and ``epsR=None``: the iterative solver :py:func:`scipy.sparse.linalg.lsqr` is used
* ``explicit=False`` with ``epsR``: the iterative solver
:py:func:`pylops.optimization.leastsquares.RegularizedInversion` is used
Note that the convergence of iterative solvers such as :py:func:`scipy.sparse.linalg.lsqr`
can be very slow for this type of operator. It is suggested to take a two steps
approach with first a trace-by-trace inversion using the explicit operator,
followed by a regularized global inversion using the outcome of the previous
inversion as initial guess.
"""
if data.ndim == 1:
dims = 1
nt0 = data.size
nspat = None
nother = nx = 1
elif data.ndim == 2:
dims = 2
nt0, nx = data.shape
nspat = (nx, )
nother = nx
else:
dims = 3
nt0, nx, ny = data.shape
nspat = (nx, ny)
nother = nx * ny
data = data.reshape(nt0, nother)
# check if background model and data have same shape
if m0 is not None and data.shape != m0.shape:
raise ValueError('data and m0 must have same shape')
# create operator
PPop = PoststackLinearModelling(wav,
nt0=nt0,
ndims=nspat,
explicit=explicit)
if dottest:
assert Dottest(PPop, nt0 * nother, nt0 * nother, verb=True)
# create and remove background data from original data
datar = data.flatten(
) if m0 is None else data.flatten() - PPop * m0.flatten()
# invert model
if epsR is None:
# inversion without spatial regularization
if explicit:
if epsI is None and not simultaneous:
# solve unregularized equations indipendently trace-by-trace
minv = lstsq(PPop.A,
datar.reshape(nt0, nother).squeeze(),
**kwargs_solver)[0]
elif epsI is None and simultaneous:
# solve unregularized equations simultaneously
minv = lsqr(PPop, datar, **kwargs_solver)[0]
elif epsI is not None:
# create regularized normal equations
PP = np.dot(PPop.A.T, PPop.A) + epsI * np.eye(nt0)
datar = np.dot(PPop.A.T, datar.reshape(nt0, nother))
if not simultaneous:
# solve regularized normal equations indipendently trace-by-trace
minv = lstsq(PP, datar.reshape(nt0, nother),
**kwargs_solver)[0]
else:
# solve regularized normal equations simultaneously
PPop_reg = MatrixMult(PP, dims=nother)
minv = lsqr(PPop_reg, datar.flatten(), **kwargs_solver)[0]
else:
# create regularized normal equations and solve them simultaneously
PP = np.dot(PPop.A, PPop.A) + epsI * np.eye(nt0)
datar = PPop.A.T * datar.reshape(nt0, nother)
PPop_reg = MatrixMult(PP, dims=nother)
minv = lstsq(PPop_reg, datar.flatten(), **kwargs_solver)[0]
else:
# solve unregularized normal equations simultaneously with lop
minv = lsqr(PPop, datar, **kwargs_solver)[0]
else:
# inversion with spatial regularization
if dims == 1:
Regop = SecondDerivative(nt0, dtype=PPop.dtype)
elif dims == 2:
Regop = Laplacian((nt0, nx), dtype=PPop.dtype)
else:
Regop = Laplacian((nt0, nx, ny), dirs=(1, 2), dtype=PPop.dtype)
minv = RegularizedInversion(PPop, [Regop],
data.flatten(),
x0=m0.flatten(),
epsRs=[epsR],
returninfo=False,
**kwargs_solver)
if dims == 1:
minv = minv.squeeze()
datar = datar.squeeze()
elif dims == 2:
minv = minv.reshape(nt0, nx)
datar = datar.reshape(nt0, nx)
else:
minv = minv.reshape(nt0, nx, ny)
datar = datar.reshape(nt0, nx, ny)
if m0 is not None and epsR is None:
minv = minv + m0
return minv, datar
def SeismicInterpolation(data,
nrec,
iava,
iava1=None,
kind='fk',
nffts=None,
sampling=None,
spataxis=None,
spat1axis=None,
taxis=None,
paxis=None,
p1axis=None,
centeredh=True,
nwins=None,
nwin=None,
nover=None,
design=False,
engine='numba',
dottest=False,
**kwargs_solver):
r"""Seismic interpolation (or regularization).
Interpolate seismic data from irregular to regular spatial grid.
Depending on the size of the input ``data``, interpolation is either
2- or 3-dimensional. In case of 3-dimensional interpolation,
data can be irregularly sampled in either one or both spatial directions.
Parameters
----------
data : :obj:`np.ndarray`
Irregularly sampled seismic data of size
:math:`[n_{r_y} (\times n_{r_x} \times n_t)]`
nrec : :obj:`int` or :obj:`tuple`
Number of elements in the regularly sampled (reconstructed) spatial
array, :math:`n_{R_y}` for 2-dimensional data and
:math:`(n_{R_y}, n_{R_x})` for 3-dimensional data
iava : :obj:`list` or :obj:`numpy.ndarray`
Integer (or floating) indices of locations of available samples in
first dimension of regularly sampled spatial grid of interpolated
signal. The :class:`pylops.basicoperators.Restriction` operator is
used in case of integer indices, while the
:class:`pylops.signalprocessing.Iterp` operator is used in
case of floating indices.
iava1 : :obj:`list` or :obj:`numpy.ndarray`, optional
Integer (or floating) indices of locations of available samples in
second dimension of regularly sampled spatial grid of interpolated
signal. Can be used only in case of 3-dimensional data.
kind : :obj:`str`, optional
Type of inversion: ``fk`` (default), ``spatial``, ``radon-linear``,
``chirpradon-linear``, ``radon-parabolic`` or , ``radon-hyperbolic``
and ``sliding``
nffts : :obj:`int` or :obj:`tuple`, optional
nffts : :obj:`tuple`, optional
Number of samples in Fourier Transform for each direction.
Required if ``kind='fk'``
sampling : :obj:`tuple`, optional
Sampling steps ``dy`` (, ``dx``) and ``dt``. Required if ``kind='fk'``
or ``kind='radon-linear'``
spataxis : :obj:`np.ndarray`, optional
First spatial axis. Required for ``kind='radon-linear'``,
``kind='chirpradon-linear'``, ``kind='radon-parabolic'``,
``kind='radon-hyperbolic'``, can also be provided instead of
``sampling`` for ``kind='fk'``
spat1axis : :obj:`np.ndarray`, optional
Second spatial axis. Required for ``kind='radon-linear'``,
``kind='chirpradon-linear'``, ``kind='radon-parabolic'``,
``kind='radon-hyperbolic'``, can also be provided instead of
``sampling`` for ``kind='fk'``
taxis : :obj:`np.ndarray`, optional
Time axis. Required for ``kind='radon-linear'``,
``kind='chirpradon-linear'``, ``kind='radon-parabolic'``,
``kind='radon-hyperbolic'``, can also be provided instead of
``sampling`` for ``kind='fk'``
paxis : :obj:`np.ndarray`, optional
First Radon axis. Required for ``kind='radon-linear'``,
``kind='chirpradon-linear'``, ``kind='radon-parabolic'``,
``kind='radon-hyperbolic'`` and ``kind='sliding'``
p1axis : :obj:`np.ndarray`, optional
Second Radon axis. Required for ``kind='radon-linear'``,
``kind='chirpradon-linear'``, ``kind='radon-parabolic'``,
``kind='radon-hyperbolic'`` and ``kind='sliding'``
centeredh : :obj:`bool`, optional
Assume centered spatial axis (``True``) or not (``False``).
Required for ``kind='radon-linear'``, ``kind='radon-parabolic'``
and ``kind='radon-hyperbolic'``
nwins : :obj:`int` or :obj:`tuple`, optional
Number of windows. Required for ``kind='sliding'``
nwin : :obj:`int` or :obj:`tuple`, optional
Number of samples of window. Required for ``kind='sliding'``
nover : :obj:`int` or :obj:`tuple`, optional
Number of samples of overlapping part of window. Required for
``kind='sliding'``
design : :obj:`bool`, optional
Print number of sliding window (``True``) or not (``False``) when
using ``kind='sliding'``
engine : :obj:`str`, optional
Engine used for Radon computations (``numpy/numba``
for ``Radon2D`` and ``Radon3D`` or ``numpy/fftw``
for ``ChirpRadon2D`` and ``ChirpRadon3D`` or )
dottest : :obj:`bool`, optional
Apply dot-test
**kwargs_solver
Arbitrary keyword arguments for
:py:func:`pylops.optimization.leastsquares.RegularizedInversion` solver
if ``kind='spatial'`` or
:py:func:`pylops.optimization.sparsity.FISTA` solver otherwise
Returns
-------
recdata : :obj:`np.ndarray`
Reconstructed data of size :math:`[n_{R_y} (\times n_{R_x} \times n_t)]`
recprec : :obj:`np.ndarray`
Reconstructed data in the sparse or preconditioned domain in case of
``kind='fk'``, ``kind='radon-linear'``, ``kind='radon-parabolic'``,
``kind='radon-hyperbolic'`` and ``kind='sliding'``
cost : :obj:`np.ndarray`
Cost function norm
Raises
------
KeyError
If ``kind`` is neither ``spatial``, ``fl``, ``radon-linear``,
``radon-parabolic``, ``radon-hyperbolic`` nor ``sliding``
Notes
-----
The problem of seismic data interpolation (or regularization) can be
formally written as
.. math::
\mathbf{y} = \mathbf{R} \mathbf{x}
where a restriction or interpolation operator is applied along the spatial
direction(s). Here :math:`\mathbf{y} = [\mathbf{y}_{R1}^T, \mathbf{y}_{R2}^T,...,
\mathbf{y}_{RN^T}]^T` where each vector :math:`\mathbf{y}_{Ri}`
contains all time samples recorded in the seismic data at the specific
receiver :math:`R_i`. Similarly, :math:`\mathbf{x} = [\mathbf{x}_{r1}^T,
\mathbf{x}_{r2}^T,..., \mathbf{x}_{rM}^T]`, contains all traces at the
regularly and finely sampled receiver locations :math:`r_i`.
Several alternative approaches can be taken to solve such a problem. They
mostly differ in the choice of the regularization (or preconditining) used
to mitigate the ill-posedness of the problem:
* ``spatial``: least-squares inversion in the original time-space domain
with an additional spatial smoothing regularization term,
corresponding to the cost function
:math:`J = ||\mathbf{y} - \mathbf{R} \mathbf{x}||_2 +
\epsilon_\nabla \nabla ||\mathbf{x}||_2` where :math:`\nabla` is
a second order space derivative implemented via
:class:`pylops.basicoperators.SecondDerivative` in 2-dimensional case
and :class:`pylops.basicoperators.Laplacian` in 3-dimensional case
* ``fk``: L1 inversion in frequency-wavenumber preconditioned domain
corresponding to the cost function
:math:`J = ||\mathbf{y} - \mathbf{R} \mathbf{F} \mathbf{x}||_2` where
:math:`\mathbf{F}` is frequency-wavenumber transform implemented via
:class:`pylops.signalprocessing.FFT2D` in 2-dimensional case
and :class:`pylops.signalprocessing.FFTND` in 3-dimensional case
* ``radon-linear``: L1 inversion in linear Radon preconditioned domain
using the same cost function as ``fk`` but with :math:`\mathbf{F}`
being a Radon transform implemented via
:class:`pylops.signalprocessing.Radon2D` in 2-dimensional case
and :class:`pylops.signalprocessing.Radon3D` in 3-dimensional case
* ``radon-parabolic``: L1 inversion in parabolic Radon
preconditioned domain
* ``radon-hyperbolic``: L1 inversion in hyperbolic Radon
preconditioned domain
* ``sliding``: L1 inversion in sliding-linear Radon
preconditioned domain using the same cost function as ``fk``
but with :math:`\mathbf{F}` being a sliding Radon transform
implemented via :class:`pylops.signalprocessing.Sliding2D` in
2-dimensional case and :class:`pylops.signalprocessing.Sliding3D`
in 3-dimensional case
"""
ncp = get_array_module(data)
dtype = data.dtype
ndims = data.ndim
if ndims == 1 or ndims > 3:
raise ValueError('data must have 2 or 3 dimensions')
if ndims == 2:
dimsd = data.shape
dims = (nrec, dimsd[1])
else:
dimsd = data.shape
dims = (nrec[0], nrec[1], dimsd[2])
# sampling
if taxis is not None:
dt = taxis[1] - taxis[0]
if spataxis is not None:
dspat = np.abs(spataxis[1] - spataxis[0])
if spat1axis is not None:
dspat1 = np.abs(spat1axis[1] - spat1axis[0])
# create restriction/interpolation operator
if iava.dtype == float:
Rop = Interp(np.prod(dims),
iava,
dims=dims,
dir=0,
kind='linear',
dtype=dtype)
if ndims == 3 and iava1 is not None:
dims1 = (len(iava), nrec[1], dimsd[2])
Rop1 = Interp(np.prod(dims1),
iava1,
dims=dims1,
dir=1,
kind='linear',
dtype=dtype)
Rop = Rop1 * Rop
else:
Rop = Restriction(np.prod(dims), iava, dims=dims, dir=0, dtype=dtype)
if ndims == 3 and iava1 is not None:
dims1 = (len(iava), nrec[1], dimsd[2])
Rop1 = Restriction(np.prod(dims1),
iava1,
dims=dims1,
dir=1,
dtype=dtype)
Rop = Rop1 * Rop
# create other operators for inversion
if kind == 'spatial':
prec = False
dotcflag = 0
if ndims == 3 and iava1 is not None:
Regop = Laplacian(dims=dims, dirs=(0, 1), dtype=dtype)
else:
Regop = SecondDerivative(np.prod(dims),
dims=(dims),
dir=0,
dtype=dtype)
SIop = Rop
elif kind == 'fk':
prec = True
dimsp = nffts
dotcflag = 1
if ndims == 3:
if sampling is None:
if spataxis is None or spat1axis is None or taxis is None:
raise ValueError('Provide either sampling or spataxis, '
'spat1axis and taxis for kind=%s' % kind)
else:
sampling = (np.abs(spataxis[1] - spataxis[1]),
np.abs(spat1axis[1] - spat1axis[1]),
np.abs(taxis[1] - taxis[1]))
Pop = FFTND(dims=dims, nffts=nffts, sampling=sampling)
Pop = Pop.H
else:
if sampling is None:
if spataxis is None or taxis is None:
raise ValueError('Provide either sampling or spataxis, '
'and taxis for kind=%s' % kind)
else:
sampling = (np.abs(spataxis[1] - spataxis[1]),
np.abs(taxis[1] - taxis[1]))
Pop = FFT2D(dims=dims, nffts=nffts, sampling=sampling)
Pop = Pop.H
SIop = Rop * Pop
elif 'chirpradon' in kind:
prec = True
dotcflag = 0
if ndims == 3:
Pop = ChirpRadon3D(
taxis, spataxis, spat1axis,
(np.max(paxis) * dspat / dt, np.max(p1axis) * dspat1 / dt)).H
dimsp = (spataxis.size, spat1axis.size, taxis.size)
else:
Pop = ChirpRadon2D(taxis, spataxis, np.max(paxis) * dspat / dt).H
dimsp = (spataxis.size, taxis.size)
SIop = Rop * Pop
elif 'radon' in kind:
prec = True
dotcflag = 0
kindradon = kind.split('-')[-1]
if ndims == 3:
Pop = Radon3D(taxis,
spataxis,
spat1axis,
paxis,
p1axis,
centeredh=centeredh,
kind=kindradon,
engine=engine)
dimsp = (paxis.size, p1axis.size, taxis.size)
else:
Pop = Radon2D(taxis,
spataxis,
paxis,
centeredh=centeredh,
kind=kindradon,
engine=engine)
dimsp = (paxis.size, taxis.size)
SIop = Rop * Pop
elif kind == 'sliding':
prec = True
dotcflag = 0
if ndims == 3:
nspat, nspat1 = spataxis.size, spat1axis.size
spataxis_local = np.linspace(-dspat * nwin[0] // 2,
dspat * nwin[0] // 2, nwin[0])
spat1axis_local = np.linspace(-dspat1 * nwin[1] // 2,
dspat1 * nwin[1] // 2, nwin[1])
dimsslid = (nspat, nspat1, taxis.size)
if ncp == np:
npaxis, np1axis = paxis.size, p1axis.size
Op = Radon3D(taxis,
spataxis_local,
spat1axis_local,
paxis,
p1axis,
centeredh=True,
kind='linear',
engine=engine)
else:
npaxis, np1axis = nwin[0], nwin[1]
Op = ChirpRadon3D(taxis, spataxis_local, spat1axis_local,
(np.max(paxis) * dspat / dt,
np.max(p1axis) * dspat1 / dt)).H
dimsp = (nwins[0] * npaxis, nwins[1] * np1axis, dimsslid[2])
Pop = Sliding3D(Op,
dimsp,
dimsslid,
nwin,
nover, (npaxis, np1axis),
tapertype='cosine')
# to be able to reshape correctly the preconditioned model
dimsp = (nwins[0], nwins[1], npaxis, np1axis, dimsslid[2])
else:
nspat = spataxis.size
spataxis_local = np.linspace(-dspat * nwin // 2, dspat * nwin // 2,
nwin)
dimsslid = (nspat, taxis.size)
if ncp == np:
npaxis = paxis.size
Op = Radon2D(taxis,
spataxis_local,
paxis,
centeredh=True,
kind='linear',
engine=engine)
else:
npaxis = nwin
Op = ChirpRadon2D(taxis, spataxis_local,
np.max(paxis) * dspat / dt).H
dimsp = (nwins * npaxis, dimsslid[1])
Pop = Sliding2D(Op,
dimsp,
dimsslid,
nwin,
nover,
tapertype='cosine',
design=design)
SIop = Rop * Pop
else:
raise KeyError('kind must be spatial, fk, radon-linear, '
'radon-parabolic, radon-hyperbolic or sliding')
# dot-test
if dottest:
Dottest(SIop,
np.prod(dimsd),
np.prod(dimsp) if prec else np.prod(dims),
complexflag=dotcflag,
raiseerror=True,
verb=True)
# inversion
if kind == 'spatial':
recdata = \
RegularizedInversion(SIop, [Regop], data.flatten(),
**kwargs_solver)
if isinstance(recdata, tuple):
recdata = recdata[0]
recdata = recdata.reshape(dims)
recprec = None
cost = None
else:
recprec = FISTA(SIop, data.flatten(), **kwargs_solver)
if len(recprec) == 3:
cost = recprec[2]
else:
cost = None
recprec = recprec[0]
recdata = np.real(Pop * recprec)
recprec = recprec.reshape(dimsp)
recdata = recdata.reshape(dims)
return recdata, recprec, cost
def PoststackInversion(data,
wav,
m0=None,
explicit=False,
simultaneous=False,
epsI=None,
epsR=None,
dottest=False,
epsRL1=None,
**kwargs_solver):
r"""Post-stack linearized seismic inversion.
Invert post-stack seismic operator to retrieve an elastic parameter of
choice from band-limited seismic post-stack data.
Depending on the choice of input parameters, inversion can be
trace-by-trace with explicit operator or global with either
explicit or linear operator.
Parameters
----------
data : :obj:`np.ndarray`
Band-limited seismic post-stack data of size
:math:`[n_{t0} (\times n_x \times n_y)]`
wav : :obj:`np.ndarray`
Wavelet in time domain (must have odd number of elements
and centered to zero). If 1d, assume stationary wavelet for the entire
time axis. If 2d of size :math:`[n_{t0} \times n_h]` use as
non-stationary wavelet
m0 : :obj:`np.ndarray`, optional
Background model of size :math:`[n_{t0} (\times n_x \times n_y)]`
explicit : :obj:`bool`, optional
Create a chained linear operator (``False``, preferred for large data)
or a ``MatrixMult`` linear operator with dense matrix
(``True``, preferred for small data)
simultaneous : :obj:`bool`, optional
Simultaneously invert entire data (``True``) or invert
trace-by-trace (``False``) when using ``explicit`` operator
(note that the entire data is always inverted when working
with linear operator)
epsI : :obj:`float`, optional
Damping factor for Tikhonov regularization term
epsR : :obj:`float`, optional
Damping factor for additional Laplacian regularization term
dottest : :obj:`bool`, optional
Apply dot-test
epsRL1 : :obj:`float`, optional
Damping factor for additional blockiness regularization term
**kwargs_solver
Arbitrary keyword arguments for :py:func:`scipy.linalg.lstsq`
solver (if ``explicit=True`` and ``epsR=None``)
or :py:func:`scipy.sparse.linalg.lsqr` solver (if ``explicit=False``
and/or ``epsR`` is not ``None``)
Returns
-------
minv : :obj:`np.ndarray`
Inverted model of size :math:`[n_{t0} (\times n_x \times n_y)]`
datar : :obj:`np.ndarray`
Residual data (i.e., data - background data) of
size :math:`[n_{t0} (\times n_x \times n_y)]`
Notes
-----
The cost function and solver used in the seismic post-stack inversion
module depends on the choice of ``explicit``, ``simultaneous``, ``epsI``,
and ``epsR`` parameters:
* ``explicit=True``, ``epsI=None`` and ``epsR=None``: the explicit
solver :py:func:`scipy.linalg.lstsq` is used if ``simultaneous=False``
(or the iterative solver :py:func:`scipy.sparse.linalg.lsqr` is used
if ``simultaneous=True``)
* ``explicit=True`` with ``epsI`` and ``epsR=None``: the regularized
normal equations :math:`\mathbf{W}^T\mathbf{d} = (\mathbf{W}^T
\mathbf{W} + \epsilon_I^2 \mathbf{I}) \mathbf{AI}` are instead fed
into the :py:func:`scipy.linalg.lstsq` solver if ``simultaneous=False``
(or the iterative solver :py:func:`scipy.sparse.linalg.lsqr`
if ``simultaneous=True``)
* ``explicit=False`` and ``epsR=None``: the iterative solver
:py:func:`scipy.sparse.linalg.lsqr` is used
* ``explicit=False`` with ``epsR`` and ``epsRL1=None``: the iterative
solver :py:func:`pylops.optimization.leastsquares.RegularizedInversion`
is used to solve the spatially regularized problem.
* ``explicit=False`` with ``epsR`` and ``epsRL1``: the iterative
solver :py:func:`pylops.optimization.sparsity.SplitBregman`
is used to solve the blockiness-promoting (in vertical direction)
and spatially regularized (in additional horizontal directions) problem.
Note that the convergence of iterative solvers such as
:py:func:`scipy.sparse.linalg.lsqr` can be very slow for this type of
operator. It is suggested to take a two steps approach with first a
trace-by-trace inversion using the explicit operator, followed by a
regularized global inversion using the outcome of the previous
inversion as initial guess.
"""
ncp = get_array_module(wav)
# check if background model and data have same shape
if m0 is not None and data.shape != m0.shape:
raise ValueError('data and m0 must have same shape')
# find out dimensions
if data.ndim == 1:
dims = 1
nt0 = data.size
nspat = None
nspatprod = nx = 1
elif data.ndim == 2:
dims = 2
nt0, nx = data.shape
nspat = (nx, )
nspatprod = nx
else:
dims = 3
nt0, nx, ny = data.shape
nspat = (nx, ny)
nspatprod = nx * ny
data = data.reshape(nt0, nspatprod)
# create operator
PPop = PoststackLinearModelling(wav,
nt0=nt0,
spatdims=nspat,
explicit=explicit)
if dottest:
Dottest(PPop,
nt0 * nspatprod,
nt0 * nspatprod,
raiseerror=True,
backend=get_module_name(ncp),
verb=True)
# create and remove background data from original data
datar = data.flatten() if m0 is None else \
data.flatten() - PPop * m0.flatten()
# invert model
if epsR is None:
# inversion without spatial regularization
if explicit:
if epsI is None and not simultaneous:
# solve unregularized equations indipendently trace-by-trace
minv = \
get_lstsq(data)(PPop.A, datar.reshape(nt0, nspatprod).squeeze(),
**kwargs_solver)[0]
elif epsI is None and simultaneous:
# solve unregularized equations simultaneously
if ncp == np:
minv = lsqr(PPop, datar, **kwargs_solver)[0]
else:
minv = \
cgls(PPop, datar,
x0=ncp.zeros(int(PPop.shape[1]), PPop.dtype),
**kwargs_solver)[0]
elif epsI is not None:
# create regularized normal equations
PP = ncp.dot(PPop.A.T, PPop.A) + \
epsI * ncp.eye(nt0, dtype=PPop.A.dtype)
datarn = ncp.dot(PPop.A.T, datar.reshape(nt0, nspatprod))
if not simultaneous:
# solve regularized normal eqs. trace-by-trace
minv = get_lstsq(data)(PP, datarn, **kwargs_solver)[0]
else:
# solve regularized normal equations simultaneously
PPop_reg = MatrixMult(PP, dims=nspatprod)
if ncp == np:
minv = lsqr(PPop_reg, datar.ravel(),
**kwargs_solver)[0]
else:
minv = cgls(PPop_reg,
datar.ravel(),
x0=ncp.zeros(int(PPop_reg.shape[1]),
PPop_reg.dtype),
**kwargs_solver)[0]
else:
# create regularized normal eqs. and solve them simultaneously
PP = ncp.dot(PPop.A.T, PPop.A) + \
epsI * ncp.eye(nt0, dtype=PPop.A.dtype)
datarn = PPop.A.T * datar.reshape(nt0, nspatprod)
PPop_reg = MatrixMult(PP, dims=nspatprod)
minv = \
get_lstsq(data)(PPop_reg.A, datarn.flatten(),
**kwargs_solver)[0]
else:
# solve unregularized normal equations simultaneously with lop
if ncp == np:
minv = lsqr(PPop, datar, **kwargs_solver)[0]
else:
minv = \
cgls(PPop, datar,
x0=ncp.zeros(int(PPop.shape[1]), PPop.dtype),
**kwargs_solver)[0]
else:
if epsRL1 is None:
# L2 inversion with spatial regularization
if dims == 1:
Regop = SecondDerivative(nt0, dtype=PPop.dtype)
elif dims == 2:
Regop = Laplacian((nt0, nx), dtype=PPop.dtype)
else:
Regop = Laplacian((nt0, nx, ny), dirs=(1, 2), dtype=PPop.dtype)
minv = RegularizedInversion(
PPop, [Regop],
data.flatten(),
x0=None if m0 is None else m0.flatten(),
epsRs=[epsR],
returninfo=False,
**kwargs_solver)
else:
# Blockiness-promoting inversion with spatial regularization
if dims == 1:
RegL1op = FirstDerivative(nt0,
kind='forward',
dtype=PPop.dtype)
RegL2op = None
elif dims == 2:
RegL1op = FirstDerivative(nt0 * nx,
dims=(nt0, nx),
dir=0,
kind='forward',
dtype=PPop.dtype)
RegL2op = SecondDerivative(nt0 * nx,
dims=(nt0, nx),
dir=1,
dtype=PPop.dtype)
else:
RegL1op = FirstDerivative(nt0 * nx * ny,
dims=(nt0, nx, ny),
dir=0,
kind='forward',
dtype=PPop.dtype)
RegL2op = Laplacian((nt0, nx, ny),
dirs=(1, 2),
dtype=PPop.dtype)
if 'mu' in kwargs_solver.keys():
mu = kwargs_solver['mu']
kwargs_solver.pop('mu')
else:
mu = 1.
if 'niter_outer' in kwargs_solver.keys():
niter_outer = kwargs_solver['niter_outer']
kwargs_solver.pop('niter_outer')
else:
niter_outer = 3
if 'niter_inner' in kwargs_solver.keys():
niter_inner = kwargs_solver['niter_inner']
kwargs_solver.pop('niter_inner')
else:
niter_inner = 5
if not isinstance(epsRL1, (list, tuple)):
epsRL1 = list([epsRL1])
if not isinstance(epsR, (list, tuple)):
epsR = list([epsR])
minv = SplitBregman(PPop, [RegL1op],
data.ravel(),
RegsL2=[RegL2op],
epsRL1s=epsRL1,
epsRL2s=epsR,
mu=mu,
niter_outer=niter_outer,
niter_inner=niter_inner,
x0=None if m0 is None else m0.flatten(),
**kwargs_solver)[0]
# compute residual
if epsR is None:
datar -= PPop * minv.ravel()
else:
datar = data.ravel() - PPop * minv.ravel()
# reshape inverted model and residual data
if dims == 1:
minv = minv.squeeze()
datar = datar.squeeze()
elif dims == 2:
minv = minv.reshape(nt0, nx)
datar = datar.reshape(nt0, nx)
else:
minv = minv.reshape(nt0, nx, ny)
datar = datar.reshape(nt0, nx, ny)
if m0 is not None and epsR is None:
minv = minv + m0
return minv, datar
