pylops.waveeqprocessing.WavefieldDecomposition¶
-
pylops.waveeqprocessing.
WavefieldDecomposition
(p, vz, nt, nr, dt, dr, rho, vel, nffts=(None, None, None), critical=100.0, ntaper=10, scaling=1.0, kind='inverse', restriction=None, sptransf=None, solver=<function lsqr at 0x7fbb90e8c950>, dottest=False, dtype='complex128', **kwargs_solver)[source]¶ Up-down wavefield decomposition.
Apply seismic wavefield decomposition from multi-component (pressure and vertical particle velocity) data. This process is also generally referred to as data-based deghosting.
Parameters: - p :
np.ndarray
Pressure data of size \(\lbrack n_{r_x} (\times n_{r_y}) \times n_t \rbrack\) (or \(\lbrack n_{r_{x,sub}} (\times n_{r_{y,sub}}) \times n_t \rbrack\) in case a
restriction
operator is provided. Note that \(n_{r_{x,sub}}\) (and \(n_{r_{y,sub}}\)) must agree with the size of the output of this operator)- vz :
np.ndarray
Vertical particle velocity data of same size as pressure data
- nt :
int
Number of samples along the time axis
- nr :
int
ortuple
Number of samples along the receiver axis (or axes)
- dt :
float
Sampling along the time axis
- dr :
float
ortuple
Sampling along the receiver array (or axes)
- rho :
float
Density along the receiver array (must be constant)
- vel :
float
Velocity along the receiver array (must be constant)
- nffts :
tuple
, optional Number of samples along the wavenumber and frequency axes
- critical :
float
, optional Percentage of angles to retain in obliquity factor. For example, if
critical=100
only angles below the critical angle \(\frac{f(k_x)}{v}\) will be retained- ntaper :
float
, optional Number of samples of taper applied to obliquity factor around critical angle
- kind :
str
, optional Type of separation:
inverse
(default) oranalytical
- scaling :
float
, optional Scaling to apply to the operator (see Notes of
pylops.waveeqprocessing.wavedecomposition.UpDownComposition2D
for more details)- restriction :
pylops.LinearOperator
, optional Restriction operator
- sptransf :
pylops.LinearOperator
, optional Sparsifying operator
- solver :
float
, optional Function handle of solver to be used if
kind='inverse'
- dottest :
bool
, optional Apply dot-test
- dtype :
str
, optional Type of elements in input array.
- **kwargs_solver
Arbitrary keyword arguments for chosen
solver
Returns: - pup :
np.ndarray
Up-going wavefield
- pdown :
np.ndarray
Down-going wavefield
Raises: - KeyError
If
kind
is neitheranalytical
norinverse
Notes
Up- and down-going components of seismic data (\(p^-(x, t)\) and \(p^+(x, t)\)) can be estimated from multi-component data (\(p(x, t)\) and \(v_z(x, t)\)) by computing the following expression [1]:
\[\begin{split}\begin{bmatrix} \mathbf{p^+}(k_x, \omega) \\ \mathbf{p^-}(k_x, \omega) \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & \frac{\omega \rho}{k_z} \\ 1 & - \frac{\omega \rho}{k_z} \\ \end{bmatrix} \begin{bmatrix} \mathbf{p}(k_x, \omega) \\ \mathbf{v_z}(k_x, \omega) \end{bmatrix}\end{split}\]if
kind='analytical'
or alternatively by solving the equation inptcpy.waveeqprocessing.UpDownComposition2D
as an inverse problem, ifkind='inverse'
.The latter approach has several advantages as data regularization can be included as part of the separation process allowing the input data to be aliased. This is obtained by solving the following problem:
\[\begin{split}\begin{bmatrix} \mathbf{p} \\ s*\mathbf{v_z} \end{bmatrix} = \begin{bmatrix} \mathbf{R}\mathbf{F} & 0 \\ 0 & s*\mathbf{R}\mathbf{F} \end{bmatrix} \mathbf{W} \begin{bmatrix} \mathbf{F}^H \mathbf{S} & 0 \\ 0 & \mathbf{F}^H \mathbf{S} \end{bmatrix} \mathbf{p^{\pm}}\end{split}\]where \(\mathbf{R}\) is a
ptcpy.basicoperators.Restriction
operator and \(\mathbf{S}\) is sparsyfing transform operator (e.g.,ptcpy.signalprocessing.Radon2D
).[1] Wapenaar, K. “Reciprocity properties of one-way propagators”, Geophysics, vol. 63, pp. 1795-1798. 1998. - p :