pylops.waveeqprocessing.MDC

pylops.waveeqprocessing.MDC(G, nt, nv, dt=1.0, dr=1.0, twosided=True, fftengine='numpy', saveGt=True, conj=False, usematmul=False, prescaled=False, name='M')[source]

Multi-dimensional convolution.

Apply multi-dimensional convolution between two datasets. Model and data should be provided after flattening 2- or 3-dimensional arrays of size \([n_t \times n_r \;(\times n_{vs})]\) and \([n_t \times n_s \;(\times n_{vs})]\) (or \(2n_t-1\) for twosided=True), respectively.

Parameters:
G : numpy.ndarray

Multi-dimensional convolution kernel in frequency domain of size \([n_{f_\text{max}} \times n_s \times n_r]\)

nt : int

Number of samples along time axis for model and data (note that this must be equal to \(2n_t-1\) when working with twosided=True.

nv : int

Number of samples along virtual source axis

dt : float, optional

Sampling of time integration axis \(\Delta t\)

dr : float, optional

Sampling of receiver integration axis \(\Delta r\)

twosided : bool, optional

MDC operator has both negative and positive time (True) or only positive (False)

fftengine : str, optional

Engine used for fft computation (numpy, scipy or fftw)

saveGt : bool, optional

Save G and G.H to speed up the computation of adjoint of pylops.signalprocessing.Fredholm1 (True) or create G.H on-the-fly (False) Note that saveGt=True will be faster but double the amount of required memory

conj : str, optional

Perform Fredholm integral computation with complex conjugate of G

usematmul : bool, optional

Use numpy.matmul (True) or for-loop with numpy.dot (False) in pylops.signalprocessing.Fredholm1 operator. Refer to Fredholm1 documentation for details.

prescaled : bool, optional

Apply scaling to kernel (False) or not (False) when performing spatial and temporal summations. In case prescaled=True, the kernel is assumed to have been pre-scaled when passed to the MDC routine.

name : str, optional

New in version 2.0.0.

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

Raises:
ValueError

If nt is even and twosided=True

See also

MDD
Multi-dimensional deconvolution

Notes

The so-called multi-dimensional convolution (MDC) is a chained operator [1]. It is composed of a forward Fourier transform, a multi-dimensional integration, and an inverse Fourier transform:

\[y(t, s, v) = \mathscr{F}^{-1} \Big( \int_S G(f, s, r) \mathscr{F}(x(t, r, v))\,\mathrm{d}r \Big)\]

which is discretized as follows:

\[y(t, s, v) = \sqrt{n_t} \Delta t \Delta r\mathscr{F}^{-1} \Big( \sum_{i_r=0}^{n_r} G(f, s, i_r) \mathscr{F}(x(t, i_r, v)) \Big)\]

where \(\sqrt{n_t} \Delta t \Delta r\) is not applied if prescaled=True.

This operation can be discretized and performed by means of a linear operator

\[\mathbf{D}= \mathbf{F}^H \mathbf{G} \mathbf{F}\]

where \(\mathbf{F}\) is the Fourier transform applied along the time axis and \(\mathbf{G}\) is the multi-dimensional convolution kernel.

[1]Wapenaar, K., van der Neut, J., Ruigrok, E., Draganov, D., Hunziker, J., Slob, E., Thorbecke, J., and Snieder, R., “Seismic interferometry by crosscorrelation and by multi-dimensional deconvolution: a systematic comparison”, Geophysical Journal International, vol. 185, pp. 1335-1364. 2011.

Examples using pylops.waveeqprocessing.MDC