8/2/2019 Optimizing Reconstruction for Sparse Spatial Sampling

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Optimizing reconstruction for sparse spatial sampling

Paul M. Zwartjes (PhD student)

Laboratory of Seismics and Acoustics, Department of Applied Physics

A succesfull algorithm for the reconstruction of irregularly and sparsely sampled

seismic data has recently been developed. Various assumptions are necessary forsuccesfull application of this method, such as spatially bandlimited data that has no

or small azimuthal dependence. In this research project I investigate how to improve

on these assumptions to make the reconstruction mthod more widely applicable.

In seismic acquisition a regular coverage of the survey area is desired. In practice, however the data

are irregularly sampled. The overcome this (and to reduce the data volume) the irregularly distribut-

ed traces are collected into regularly distributed bins. This binning procedure is one of the largest

sources of error in seismic acquisition. A more sophisticated algorithm for regularizing sparsely

sampled seismic data has been developed in recent years in order to reduce this error source. By

estimating the frequency content of the data, we can generate the wavefield that would be recordedat the regularly spaced locations. This procedure seems daunting, because it the requires the so-

lution of a large inverse problem. However, by clever use of mathematics the calculations can be

done with a number of FFTs, which drastically reduces the computation time

The regularisation of the midpoint/offset distribution in 3D seismic data is done by combining the

information of several CMP gathers. Sampling irregularities occur in both midpoint and offset

directions. Since the total number of data is limited we can only estimate a limited number of

events in the frequency domain. In the current approach it is assumed that the data is band-limited

and concentrated around the midpoint and offset wavenumber axis (by applying NMO) and that

there is no (strong) azimuthal variation in the subsurface. These assumptions mean the coverage

in the 2D wavenumber domain can be restricted to several samples on both sides of each axis.

In this research I investigate algorithms that allow a greater coverage in the Fourier domain, prefer-

ably the entire domain. Using parametric inversion this can be done in two ways. The first approach

simply uses a larger parameter set for the inversion. To solve such an underdetermined problem, we

need to add more information than is present in the seismic data alone. This is done by choosing a

form of regularisation suitable for the reconstruction problem. The second approach uses statistical

methods to determine a limited set of the most significantly contributing wavenumbers.

So far I have tested the first approach on 2D synthetic dataset. The regularisation of the inverse

problem is given by the a priori knowledge that we seek a sparse distribution of frequencies to

explain the recorded wavefield. This new approach yields better results than the commonly used

Gaussian distribution and yields a stable solution even for overdetermined problems. This is an

advantageous property for increasing the frequency domain coverage in the reconstruction of 3-Dseismic data. Also I have shown the feasibility of using non-adjacent wavenumber regions in the

inversion, which is a necessary requirement for the statistical subset selection methods that can be

used to determine the most significant parameters in the Fourier domain.

8/2/2019 Optimizing Reconstruction for Sparse Spatial Sampling

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Reconstructed data, using Gaussian norm

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Reconstructed data, using Cauchy norm

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Fig. 1 (a) Original sparsely sampled synthetic data. (b) Reconstruction using the current method. (c) Reconstruction

using new algorithm.

Research Plan 2000 After succesfull tests on 2D data the new algorithm will be applied to 3D seismic data in order to

increase the coverage in the Fourier domain. Subset selection methods will be investigated for their succesfullness in selecting the most sig-

nificantly contributing spatial frequencies present the recorded seismic wavefield.

References

Duijndam, A. J. W., Volker, A. W. F., and Zwartjes, P. M., 2000, Reducing the acquisition footprint

in imaging: least squares migration vs. reconstruction: , DOLPHIN, chapter 7.

Zwartjes, P. M., and Duijndam, A. J. W., 2000, Optimizing reconstruction for sparse spatial sam-

pling: , DOLPHIN, chapter 10.

Completed CTG PhD courses An overview of standard seismic processing applications (TG001) Wave field imaging and inversion in electromagnetics and acoustics (TG101)

Two-way and one-way representations of seismic wavefields (TG112)