pylops.waveeqprocessing.MDD

pylops.waveeqprocessing.MDD(G, d, dt=0.004, dr=1.0, nfmax=None, wav=None, twosided=True, causality_precond=False, adjoint=False, psf=False, dtype='float64', dottest=False, saveGt=True, add_negative=True, **kwargs_lsqr)[source]

Multi-dimensional deconvolution.

Solve multi-dimensional deconvolution problem using scipy.sparse.linalg.lsqr iterative solver.

Parameters:
G : numpy.ndarray

Multi-dimensional convolution kernel in time domain of size \([n_s \times n_r \times n_t]\) for twosided=False or twosided=True and add_negative=True (with only positive times) or size \([n_s \times n_r \times 2*n_t-1]\) for twosided=True and add_negative=False (with both positive and negative times)

d : numpy.ndarray

Data in time domain \([n_s (\times n_vs) \times n_t]\)

dt : float, optional

Sampling of time integration axis

dr : float, optional

Sampling of receiver integration axis

nfmax : int, optional

Index of max frequency to include in deconvolution process

wav : numpy.ndarray, optional

Wavelet to convolve to the inverted model and psf. If None, the outputs of the inversion are returned directly

twosided : bool, optional

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

add_negative : bool, optional

Add negative side to MDC operator and data (True) or not (False)- operator and data are already provided with both positive and negative sides. To be used only with twosided=True.

causality_precond : bool, optional

Apply causality mask (True) or not (False)

adjoint : bool, optional

Compute and return adjoint(s)

psf : bool, optional

Compute and return Point Spread Function (PSF) and its inverse

dtype : bool, optional

Type of elements in input array.

dottest : bool, optional

Apply dot-test

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

**kwargs_lsqr

Arbitrary keyword arguments for scipy.sparse.linalg.lsqr solver
Returns:
minv : numpy.ndarray

Inverted model of size \([n_r (\times n_{vs}) \times n_t]\) for twosided=False or \([n_r (\times n_vs) \times 2*n_t-1]\) for twosided=True

madj : numpy.ndarray

Adjoint model of size \([n_r (\times n_{vs}) \times n_t]\) for twosided=False or \([n_r (\times n_r) \times 2*n_t-1]\) for twosided=True

psfinv : numpy.ndarray

Inverted psf of size \([n_r \times n_r \times n_t]\) for twosided=False or \([n_r \times n_r \times 2*n_t-1]\) for twosided=True

psfadj : numpy.ndarray

Adjoint psf of size \([n_r \times n_r \times n_t]\) for twosided=False or \([n_r \times n_r \times 2*n_t-1]\) for twosided=True

See also

MDC
Multi-dimensional convolution

Notes

Multi-dimensional deconvolution (MDD) is a mathematical ill-solved problem, well-known in the image processing and geophysical community [1].

MDD aims at removing the effects of a Multi-dimensional Convolution (MDC) kernel or the so-called blurring operator or point-spread function (PSF) from a given data. It can be written as

\[\mathbf{d}= \mathbf{D} \mathbf{m}\]

or, equivalently, by means of its normal equation

\[\mathbf{m}= (\mathbf{D}^H\mathbf{D})^{-1} \mathbf{D}^H\mathbf{d}\]

where \(\mathbf{D}^H\mathbf{D}\) is the PSF.

[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”, Geophyscial Journal International, vol. 185, pp. 1335-1364. 2011.

Examples using pylops.waveeqprocessing.MDD