pymarchenko.neumarchenko.NeumannMarchenko

class pymarchenko.neumarchenko.NeumannMarchenko(R, dt=0.004, nt=None, dr=1.0, nfmax=None, wav=None, toff=0.0, nsmooth=10, dtype='float64', saveRt=True, prescaled=False)

Iterative Marchenko redatuming

Solve multi-dimensional Marchenko redatuming problem using Neumann iterative substitution.

Parameters:

Rnumpy.ndarray

Multi-dimensional reflection response in time or frequency domain of size \([n_s \times n_r \times n_t/n_{fmax}]\). If provided in time, R should not be of complex type. If provided in frequency, R should contain the positive time axis followed by the negative one. Note that the reflection response should have already been multiplied by 2.

dtfloat, optional

Sampling of time integration axis

ntfloat, optional

Number of samples in time (not required if R is in time)

drfloat, optional

Sampling of receiver integration axis

nfmaxint, optional

Index of max frequency to include in deconvolution process

wavnumpy.ndarray, optional

Wavelet to apply to direct arrival when created using trav

tofffloat, optional

Time-offset to apply to traveltime

nsmoothint, optional

Number of samples of smoothing operator to apply to window

dtypebool, optional

Type of elements in input array.

saveRtbool, optional

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

prescaledbool, optional

Apply scaling to R (False) or not (False) when performing spatial and temporal summations within the pylops.waveeqprocessing.MDC operator. In case prescaled=True, the R is assumed to have been pre-scaled by the user.

Raises:

TypeError

If t is not numpy.ndarray.

Notes

Marchenko redatuming is a method that allows to produce correct subsurface-to-surface responses given the availability of a reflection data and a macro-velocity model [1].

The Marchenko equations can be solved via Neumann iterative substitution:

\[\mathbf{f_m^+} = \Theta \mathbf{R^*} (\Theta \mathbf{R} \mathbf{f_d^+} + \Theta \mathbf{R} \mathbf{f_m^+})\]

and isolating \(\mathbf{f_m^+}\):

\[(\mathbf{I} - \Theta \mathbf{R^*}\Theta \mathbf{R}) \mathbf{f_m^+} = \Theta \mathbf{R^*} \Theta \mathbf{R} \mathbf{f_d^+}\]

We can then expand the term within parenthesis as a Neumann series and write:

\[\mathbf{f^+} = \sum_{k=0}^\inf (\Theta \mathbf{R^*}\Theta \mathbf{R})^k \mathbf{f_d^+}\]

Finally the subsurface Green’s functions can be obtained applying the following operator to the retrieved focusing functions

\[\begin{split}\begin{bmatrix} -\mathbf{g^-} \\ \mathbf{g^{+ *}} \end{bmatrix} = \mathbf{I} - \begin{bmatrix} \mathbf{0} & \mathbf{R} \\ \mathbf{R^*} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{f^-} \\ \mathbf{f^+} \end{bmatrix}\end{split}\]

Here \(\mathbf{R}\) is the monopole-to-particle velocity seismic response (already multiplied by 2).

[1]

Wapenaar, K., Thorbecke, J., Van der Neut, J., Broggini, F., Slob, E., and Snieder, R., “Marchenko imaging”, Geophysics, vol. 79, pp. WA39-WA57. 2014.

Attributes:

nsint

Number of samples along source axis

nrint

Number of samples along receiver axis

shapetuple

Operator shape

explicitbool

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

Methods

__init__(R[, dt, nt, dr, nfmax, wav, toff, ...])
apply_onepoint(trav[, G0, nfft, rtm, ...])	Marchenko redatuming for one point

Examples using pymarchenko.neumarchenko.NeumannMarchenko

1. Marchenko redatuming by iterative substitution

1. Marchenko redatuming by iterative substitution