Reflection-response retrieval with seismic interferometry by multidimensional deconvolution from surface reflection data

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We N112 06

Reflection-response Retrieval with Seismic

Interferometry by Multidimensional

Deconvolution from Surface Reflection Data

B. Boullenger* (Delft University of Technology), J. Hunziker (Delft

University of Technology) & D. Draganov (Delft University of Technology)

SUMMARY

Seismic interferometry (SI) allows retrieval of virtual-source responses at positions of receivers, where no actual source is shot, by cross-correlating (CC) the seismic responses between receivers. The theory requires a boundary of subsurface sources to retrieve the surface reflection response. With reflection data acquired with both sources and receivers at the Earth’s surface, the retrieved virtual-source reflection responses suffer from non-physical arrivals and amplitude errors that may be significant. Instead of using the common CC method, we propose an approximate method to apply SI by multidimensional

deconvolution (MDD). The method is data-driven and does not require a priori information about the subsurface. Numerical results show that, although its effect on the non-physical arrivals is limited, the MDD method clearly improves the retrieved amplitudes by flattening the spectrum and balancing the illumination of the virtual-source responses. Therefore, the virtual-source response retrieved by MDD is a better estimate of the reflection response than the response retrieved by CC. The additional reflection data retrieved by MDD have higher potential of filling in possibly missing source data in the original dataset.

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Introduction

Seismic reflection surveys, with deployment of seismic receivers and sources at the Earth’s surface, aim to produce a reliable image of the subsurface structures. Successful imaging of, possibly complex, geological targets requires a good source coverage, in particular in the presence of strong scattering bodies obscuring the target. In practice, high-density coverage with sources is not always possible in the field because of, for example, the difficult terrain or environment preservation. Therefore the targets may not be imaged with conventional processing of the seismic data due to the missing source illumination. By turning receivers into virtual sources while exploiting multiple reflections, seismic interferometry (SI) allows retrieving the reflection responses from additional source locations (Schuster, 2009) and may provide the missing responses to fill in the illumination gap observed in the surface reflection data. The most commonly used method of SI involves simple cross-correlation (CC) operations. With reflection data acquired only at the surface, the retrieved virtual responses may contain significant non-physical arrivals and amplitude errors precisely because sources are only available at the surface (one-sided illumination) (Boullenger et al, 2014). It is then difficult to merge the virtual responses with the existing data set with confidence. Alternatively, we propose a multidimensional deconvolution (MDD) approach to retrieve the virtual responses while suppressing limitations encountered with CC.

Method

Virtual-source responses can be retrieved using SI by CC. To retrieve the reﬂection response at receivers at the surface, the theory requires sources in the subsurface. Using stationary-phase arguments it can be show that with sources distributed at the surface, the reﬂection response can still be retrieved (e.g., Draganov et al (2012)). The virtual-source response ˆCf ull, but not the Green’s function, between two

surface receivers at xAand xBis obtained by using, in the space-frequency domain,

ˆ

Cf ull(xB,xA,ω) =

∑

xS

ˆ

V(xB,xS,ω) ˆV∗(xS,xA,ω), (1)

where ˆV is the vertical particle velocity and xSis the source position; the hat denotes frequency-domain

parameter and the asterisk∗means complex conjugation. For practical situations, SI by CC is robust but the retrieved ˆCf ull for the surface acquisition geometry suffers from retrieved non-physical arrivals and

often from inaccurate amplitudes and radiation patterns of the virtual sources.

For sources in the subsurface and receivers at the surface, SI by MDD can be written as (Wapenaar et al, 2011) ˆ Vscat(xB,xS,ω) = ˆV(xB,xS,ω) − 2 ˆV0(xB,xS,ω) = Srec ˆ Gscat(xB,x,ω) ˆP0(x,xS,ω)dx, (2)

where Srec is the receiver boundary (at the surface), ˆV0 and ˆP0 are the vertical particle velocity and

pressure ﬁelds, respectively, in the absence of a free-surface, and ˆGscat is the reﬂection response at the

surface. Correlating both sides of equation 2 with ˆV0 and summing over multiple source positions xs

yields ˆ C(xB,xA,ω) = Srec ˆ Gscat(xB,x,ω)ˆΓ(xS,xA,ω), (3) where ˆC(xB,xA,ω) = ∑xSVˆscat(xB,xS,ω) ˆV ∗ 0(xA,xS,ω) and ˆΓ(xS,xA,ω) = ∑xSPˆ0(x,xS,ω) ˆV ∗ 0(xA,xS,ω).

Equation 3 shows that what we obtain from SI by CC is not the desired impulse (reﬂection) response, but a correlation function which is smeared version of the desired scattered Green’s function due to a convolution with a point-spread function ˆΓ. The point-spread function characterizes the inhomogeneity of the illumination of the receivers by the sources. In matrix notation (monochromatic matrices with receivers and sources in rows and columns), equation 3 can be written as

ˆ

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In several situations, such as for OBC data or passive seismic data, MDD can overcome the intrinsic limitations of the CC approach (Van der Neut et al, 2011). With surface reflection data, using stationary-phase arguments like for CC, MDD can also be applied, but will also suffer from retrieval of non-physical arrivals. A further difficulty is to obtain the point-spread function, which is the deconvolution operator. ˆΓ consists not only of the focus point at around t=0 s in ˆC, but also of other terms that are more difficult to separate and extract from the correlation function. To apply MDD to purely surface reflection data, we propose to approximate ˆC by ˆCf ull = ˆV ˆVH (sign H denotes complex adjoint) and

the point-spread function ˆΓ by the retrieved arrivals in ˆCf ull that focus around t=0 s. We label this

approximated deconvolution operator ˆΓest. The proposed MDD is then

{ ˆGscat}est= { ˆCf ull− ˆΓest}{ˆΓest+εI}−1, (5)

where ε is a stabilization factor. The method is data-driven. It does not require the knowledge of the primary responses. The main processing during the method’s application consist of extracting the estimated point-spread function ˆΓest from the correlation function ˆCf ull to then perform the stabilized

inversion. In practice, ˆΓest is obtained by selecting in ˆCf ullthe energy focusing at around t=0 s, which is

the result from mainly primary-primary correlations. Numerical Example

The subsurface model for the numerical example is shown in Figure 1. Sources are placed from 0 m to 6000 m every 30 m. Receivers are placed from 0 m to 6000 m every 15 m. An example common-source gather from the modelled surface reﬂection data for a source at x=3000 m is shown in Figure 1(b). We apply equation 1 to all pairs of receivers to retrieve ˆCf ull. The correlation function ˆCf ull is illustrated in

Figure 2(a) for a virtual-source position at x=1485 m. No actual source is present at that position. The next step is to select the signal focusing at around t=0 s and separate it from ˆCf ull (for all virtual-source

positions). The result is an estimate of the point-spread function ˆΓest. ˆΓest for the virtual-source position

x=1485 m is shown in Figure 2(b).

We apply MDD as per equation 5 to obtain virtual-source responses. Figure 3(b) shows the result of

0 1 2 3 4 Time (s) 0 2000 4000 6000 Receiver position (m) a) b) 0 1000 2000 3000 Depth (m) 0 2000 4000 6000 Horizontal position (m) 1500 2000 2500 3000 P-wave velocity (m/s)

Figure 1 a) Velocity model and b) common-source gather at x=3000 m.

the inversion for a virtual-source position at x=1485 m. For comparison, the CC-based virtual-source and the directly modelled reﬂection responses are shown in Figures 3(a) and 3(c), respectively. The MDD result provides a better estimate of the reﬂection response than the simple cross-correlation. Later arrivals are boosted and the wavelet is sharpened. The most dramatic improvement with respect to ˆCf ull

is the better accuracy and balance of the amplitudes along the receivers. This is particularly visible by comparing the right side of the reﬂection hyperbolas in Figures 3(a) and 3(b). The corresponding frequency-wavenumber spectra of the virtual-source gathers are shown in Figures 4(a) and 4(b) for

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-4 -3 -2 -1 0 1 2 3 4 Time (s) 0 2000 4000 6000 Receiver position (m) -4 -3 -2 -1 0 1 2 3 4 Time (s) 0 2000 4000 6000 Receiver position (m)

a)

b)

Figure 2 a) Cross-correlation function ˆCf ull for a virtual-source position at x=1485 m, and b) the

corresponding estimated point-spread function ˆΓest.

ˆ

Cf ull and MDD, respectively. The frequency-wavenumber spectrum of the directly modelled

common-source gather is also shown in Figure 4(c) for comparison. We clearly see that, although the proposed MDD method does not permit to suppress a number of non-physical arrivals completely, it ﬂattens the frequency spectrum and, as a signiﬁcant improvement, balances the illumination of from the virtual-source. 0 1 2 3 4 Time (s) 0 2000 4000 6000 Receiver position (m) a) b) c) 0 1 2 3 4 Time (s) 0 2000 4000 6000 Receiver position (m) 0 1 2 3 4 Time (s) 0 2000 4000 6000 Receiver position (m)

Figure 3 Retrieved common-source gather for a virtual source at x=1485 m using a) cross-correlation and b) multidimensional deconvolution. c) Directly modelled common-source gather for an active source at x=1485 m.

Conclusions

We propose seismic interferometry (SI) by multidimensional deconvolution (MDD) to retrieve responses from additional source positions using a surface acquisition geometry. The method is fully data-driven:

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a)

b)

c)

Figure 4 Frequency-wavenumber spectrum of the common-source gathers for a source at x=1485 m obtained a) using cross-correlation, b) multidimensional deconvolution, and c) from direct modelling. it does not require any a priori information about the subsurface neither estimate of primary reflections. The results from the numerical experiments show that the results retrieved by MDD are a clear improve-ment compared to the more common approach of using SI by cross-correlation. Although the MDD method employs several approximations such as the estimate of the correlation function as well as the point-spread function, it successfully corrects the radiation patterns of the virtual sources by flattening the frequency spectrum and balancing the illumination. This results in a better estimate of the reflection response and that may be used to fill in possibly missing data in the original dataset.

Acknowledgements

This research is funded by the Division for Earth and Life Sciences (ALW) with ﬁnancial aid from the Netherlands Organization for Scientiﬁc Research (NWO) with grant VIDI 864.11.009.

References

Boullenger, B., Wapenaar, K. and Draganov, D. [2014] A method to suppress spurious multiples in virtual-source gathers retrieved using seismic interferometry with reﬂection data. 84th annual international meeting, SEG, Extended Abstracts.

Draganov, D., Heller, K., and Ghose, R. [2012] Monitoring CO2 storage using ghost reﬂections retrieved from seismic interferometry. International Journal of Greenhouse Gas Control, 11, S35–S46.

Schuster, G. [2009] Seimsic interferometry. Cambridge Press.

Van der Neut, J., Thorbecke, J., Metha, K., Slob, E. and Wapenaar, K. 2011, Controlled-source interferomtric redatuming by cross-correlation and multi-dimensional deconvolution in elastic media. Geophysics, 76, no. 4, SA63–SA76.

Wapenaar, K., and Fokkema, J. [2006] Green’s function representations for seismic interferometry. Geophysics, 71, no. 4, SI33–SI46.

Wapenaar, K., Van der Neut, J., Ruigrok, E., Draganov, D., Hunziker, J., Slob, E., Thorbecke, J. and Snieder, R. [2011] Seismic interferometry by cross-correlation and by multidimensional deconvolution: a systematic comparison. Geophysical Journal International, 185, 1335–1364.