fracspy.modelling.kirchhoff.Kirchhoff

class fracspy.modelling.kirchhoff.Kirchhoff(z, x, t, recs, vel, wav, wavcenter, y=None, mode='eikonal', wavfilter=False, trav=None, engine='numpy', dtype='float64', name='K')

Kirchhoff single-sided, demigration operator.

Kirchhoff-based demigration/migration operator for single-sided propagation (from subsurface to surface). Uses a high-frequency approximation of Green’s function propagators based on trav.

Parameters:

znumpy.ndarray

Depth axis

xnumpy.ndarray

Spatial axis

tnumpy.ndarray

Time axis for data

recsnumpy.ndarray

Receivers in array of size \(\lbrack 2 (3) \times n_r \rbrack\) The first axis should be ordered as (y,) x, z.

velnumpy.ndarray or float

Velocity model of size \(\lbrack (n_y\,\times)\; n_x \times n_z \rbrack\) (or constant)

wavnumpy.ndarray

Wavelet.

wavcenterint

Index of wavelet center

ynumpy.ndarray

Additional spatial axis (for 3-dimensional problems)

modestr, optional

Computation mode (analytic, eikonal or byot, see Notes for more details)

wavfilterbool, optional

Apply wavelet filter (True) or not (False)

travnumpy.ndarray or tuple, optional

Traveltime table of size \(\lbrack (n_y) n_x n_z \times n_r \rbrack\) (to be provided if mode='byot').

enginestr, optional

Engine used for computations (numpy or numba).

dtypestr, optional

Type of elements in input array.

namestr, optional

Name of operator (to be used by pylops.utils.describe.describe)

Attributes:

shapetuple

Operator shape

explicitbool

Operator contains a matrix that can be solved explicitly (True) or not (False)

Raises:

NotImplementedError

If mode is neither analytic, eikonal, or byot

Notes

The Kirchhoff single-sided demigration operator synthesizes seismic data given a propagation velocity model \(v\) and a source distribution \(m\). In forward mode:

\[d(\mathbf{x_r}, \mathbf{x_s}, t) = \widetilde{w}(t) * \int_V G(\mathbf{x_r}, \mathbf{x_s}, t) m(\mathbf{x_s})\,\mathrm{d}\mathbf{x_s}\]

where \(m(\mathbf{x_s})\) represents the source distribution at every location in the subsurface, \(G(\mathbf{x_r}, \mathbf{x_s}, t)\) is the Green’s function from source-to-receiver, and finally \(\widetilde{w}(t)\) is a filtered version of the wavelet \(w(t)\) [1] (or the wavelet itself when wavfilter=False). In our implementation, the following high-frequency approximation of the Green’s functions is adopted:

\[G(\mathbf{x_r}, \mathbf{x_s}, \omega) = e^{j \omega t(\mathbf{x_r}, \mathbf{x_s})}\]

where \(t(\mathbf{x_r}, \mathbf{x})\) is the traveltime and no amplitude term is applied

Depending on the choice of mode the traveltime and amplitude of the Green’s function will be also computed differently:

• mode=analytic or mode=eikonal: traveltimes are computed for every source-receiver pair and the Green’s functions are implemented from traveltime look-up tables, placing the source distribution values at corresponding source-to-receiver time in the data.

• byot: bring your own tables. The traveltime table is provided directly by user using trav input parameter.

Finally, the adjoint of the demigration operator is a migration operator which projects the data in the model domain creating an image of the source distribution.

[1]

Safron, L. “Multicomponent least-squares Kirchhoff depth migration”, MSc Thesis, 2018.

Methods

__init__(z, x, t, recs, vel, wav, wavcenter)
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()