pylops.waveeqprocessing.Kirchhoff¶
- class pylops.waveeqprocessing.Kirchhoff(*args, **kwargs)[source]¶
Kirchhoff Demigration operator.
Kirchhoff-based demigration/migration operator. Uses a high-frequency approximation of Green’s function propagators based on
trav
.- Parameters
- z
numpy.ndarray
Depth axis
- x
numpy.ndarray
Spatial axis
- t
numpy.ndarray
Time axis for data
- srcs
numpy.ndarray
Sources in array of size \(\lbrack 2 (3) \times n_s \rbrack\) The first axis should be ordered as (
y
,)x
,z
.- recs
numpy.ndarray
Receivers in array of size \(\lbrack 2 (3) \times n_r \rbrack\) The first axis should be ordered as (
y
,)x
,z
.- vel
numpy.ndarray
orfloat
Velocity model of size \(\lbrack (n_y\,\times)\; n_x \times n_z \rbrack\) (or constant)
- wav
numpy.ndarray
Wavelet.
- wavcenter
int
Index of wavelet center
- y
numpy.ndarray
Additional spatial axis (for 3-dimensional problems)
- mode
str
, optional Computation mode (
analytic
,eikonal
orbyot
, see Notes for more details)- wavfilter
bool
, optional New in version 2.0.0.
Apply wavelet filter (
True
) or not (False
)- dynamic
bool
, optional New in version 2.0.0.
Include dynamic weights in computations (
True
) or not (False
). Note that whenmode=byot
, the user is required to provide such weights inamp
.- trav
numpy.ndarray
, optional Traveltime table of size \(\lbrack (n_y) n_x n_z \times n_s n_r \rbrack\) (to be provided if
mode='byot'
)- amp
numpy.ndarray
, optional New in version 2.0.0.
Amplitude table of size \(\lbrack (n_y) n_x n_z \times n_s n_r \rbrack\) (to be provided if
mode='byot'
)- aperture
float
ortuple
, optional New in version 2.0.0.
Maximum allowed aperture expressed as the ratio of offset over depth. If
None
, no aperture limitations are introduced. If scalar, a taper from 80% to 100% of aperture is applied. If tuple, apertures below the first value are accepted and those after the second value are rejected. A tapering is implemented for those between such values.- angleaperture
float
ortuple
, optional New in version 2.0.0.
Maximum allowed angle (either source or receiver side) in degrees. If
None
, angle aperture limitations are introduced. Seeaperture
for implementation details regarding scalar and tuple cases.- anglerefl
np.ndarray
, optional New in version 2.0.0.
Angle between the normal of the reflectors and the vertical axis in degrees
- snell
float
ortuple
, optional New in version 2.0.0.
Threshold on Snell’s law evaluation. If larger, the source-receiver-image point is discarded. If
None
, no check on the validity of the Snell’s law is performed. Seeaperture
for implementation details regarding scalar and tuple cases.- engine
str
, optional Engine used for computations (
numpy
ornumba
).- dtype
str
, optional Type of elements in input array.
- name
str
, optional New in version 2.0.0.
Name of operator (to be used by
pylops.utils.describe.describe
)
- z
- Raises
- NotImplementedError
If
mode
is neitheranalytic
,eikonal
, orbyot
Notes
The Kirchhoff demigration operator synthesizes seismic data given a propagation velocity model \(v\) and a reflectivity model \(m\). In forward mode [1], [2]:
\[d(\mathbf{x_r}, \mathbf{x_s}, t) = \widetilde{w}(t) * \int_V G(\mathbf{x_r}, \mathbf{x}, t) m(\mathbf{x}) G(\mathbf{x}, \mathbf{x_s}, t)\,\mathrm{d}\mathbf{x}\]where \(m(\mathbf{x})\) represents the reflectivity at every location in the subsurface, \(G(\mathbf{x}, \mathbf{x_s}, t)\) and \(G(\mathbf{x_r}, \mathbf{x}, t)\) are the Green’s functions from source-to-subsurface-to-receiver and finally \(\widetilde{w}(t)\) is a filtered version of the wavelet \(w(t)\) [3] (or the wavelet itself when
wavfilter=False
). In our implementation, the following high-frequency approximation of the Green’s functions is adopted:\[G(\mathbf{x_r}, \mathbf{x}, \omega) = a(\mathbf{x_r}, \mathbf{x}) e^{j \omega t(\mathbf{x_r}, \mathbf{x})}\]where \(a(\mathbf{x_r}, \mathbf{x})\) is the amplitude and \(t(\mathbf{x_r}, \mathbf{x})\) is the traveltime. When
dynamic=False
the amplitude is disregarded leading to a kinematic-only Kirchhoff operator.\[d(\mathbf{x_r}, \mathbf{x_s}, t) = \tilde{w}(t) * \int_V e^{j \omega (t(\mathbf{x_r}, \mathbf{x}) + t(\mathbf{x}, \mathbf{x_s}))} m(\mathbf{x}) \,\mathrm{d}\mathbf{x}\]On the other hand, when
dynamic=True
, the amplitude scaling is defined as \(a(\mathbf{x}, \mathbf{y})=\frac{1}{\|\mathbf{x} - \mathbf{y}\|}\), that is, the reciprocal of the distance between the two points, approximating the geometrical spreading of the wavefront. Moreover an angle scaling is included in the modelling operator added as follows:\[d(\mathbf{x_r}, \mathbf{x_s}, t) = \tilde{w}(t) * \int_V a(\mathbf{x}, \mathbf{x_s}) a(\mathbf{x}, \mathbf{x_r}) \frac{|cos \theta_s + cos \theta_r|} {v(\mathbf{x})} e^{j \omega (t(\mathbf{x_r}, \mathbf{x}) + t(\mathbf{x}, \mathbf{x_s}))} m(\mathbf{x}) \,\mathrm{d}\mathbf{x}\]where \(\theta_s\) and \(\theta_r\) are the angles between the source-side and receiver-side rays and the normal to the reflector at the image point (or the vertical axis at the image point when
reflslope=None
), respectively.Depending on the choice of
mode
the traveltime and amplitude of the Green’s function will be also computed differently:mode=analytic
ormode=eikonal
: traveltimes, geometrical spreading, and angles are computed for every source-image point-receiver triplets and the Green’s functions are implemented from traveltime look-up tables, placing scaled reflectivity values at corresponding source-to-receiver time in the data.byot
: bring your own tables. Traveltime table are provided directly by user usingtrav
input parameter. Similarly, in this case one can provide their own amplitude scalingamp
(which should include the angle scaling too).
Three aperture limitations have been also implemented as defined by:
aperture
: the maximum allowed aperture is expressed as the ratio of offset over depth. This aperture limitation avoid including grazing angles whose contributions can introduce aliasing effects. A taper is added at the edges of the aperture;angleaperture
: the maximum allowed angle aperture is expressed as the difference between the incident or emerging angle at every image point and the vertical axis (or the normal to the reflector ifanglerefl
is provided. This aperture limitation also avoid including grazing angles whose contributions can introduce aliasing effects. Note that for a homogenous medium and slowly varying heterogenous medium the offset and angle aperture limits may work in the same way;snell
: the maximum allowed snell’s angle is expressed as the absolute value of the sum between incident and emerging angles defined as in theangleaperture
case. This aperture limitation is introduced to turn a scattering-based Kirchhoff engine into a reflection-based Kirchhoff engine where each image point is not considered as scatter but as a local horizontal (or straight) reflector.
Finally, the adjoint of the demigration operator is a migration operator which projects data in the model domain creating an image of the subsurface reflectivity.
- 1
Bleistein, N., Cohen, J.K., and Stockwell, J.W.. “Mathematics of Multidimensional Seismic Imaging, Migration and Inversion”, 2000.
- 2
Santos, L.T., Schleicher, J., Tygel, M., and Hubral, P. “Seismic modeling by demigration”, Geophysics, 65(4), pp. 1281-1289, 2000.
- 3
Safron, L. “Multicomponent least-squares Kirchhoff depth migration”, MSc Thesis, 2018.
- Attributes
Methods
__init__
(z, x, t, srcs, recs, vel, wav, ...)Initialize this LinearOperator.
adjoint
()Hermitian adjoint.
apply_columns
(cols)Apply subset of columns of operator
cond
([uselobpcg])Condition number of linear operator.
conj
()Complex conjugate operator
div
(y[, niter, densesolver])Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
dot
(x)Matrix-matrix or matrix-vector multiplication.
eigs
([neigs, symmetric, niter, uselobpcg])Most significant eigenvalues of linear operator.
matmat
(X)Matrix-matrix multiplication.
matvec
(x)Matrix-vector multiplication.
reset_count
()Reset counters
rmatmat
(X)Matrix-matrix multiplication.
rmatvec
(x)Adjoint matrix-vector multiplication.
todense
([backend])Return dense matrix.
toimag
([forw, adj])Imag operator
toreal
([forw, adj])Real operator
tosparse
()Return sparse matrix.
trace
([neval, method, backend])Trace of linear operator.
transpose
()Transpose this linear operator.