Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approachVikram Jayaram, Dylan Copeland, Carola Ellinger, Charles Sicking, Stu Nelan, Josh Gilberg and Chris CarterGlobal Geophysical Services, Dallas, TX

SUMMARY

In this study, we implemented and tested a new processing-based broadband solution for mitigating F-K transform arti-facts for receiver deghosting in a marine environment. The F-K transform has traditionally been used for flat cable (constantdepth) deghosting and often times tailored to meet the slanted(variable depth) cable criteria. Recently, the usage of τ− p do-main deterministic deghost operator has been more prominentwith slant cable deghosting. Irrespective of the type of trans-form or deghost operator used, a windowed process is essentialdue to the time and offset varying character of the ghost. Thisuse of a windowed process usually results in poor reconstruc-tion of deghosted signals and artifacts beyond the control of thetransform(s) itself. The windowing in time and offset producesedgy effects which can be clearly seen in the difference plots.Our method, using a non-windowing approach, demonstrates abetter representation of the deghosted signals without the arti-facts caused by the boundary of the windows. This method hasalso been well-tested for both the flat and slant cable receiverdeghosting workflows in synthetic and field data examples.

INTRODUCTION

In recent years, the industry has seen tremendous growth andattention to high quality broadband marine seismic processingand acquisition. The receiver ghost is a well known problemin marine seismic acquisition where the seismic resolution iscorrupted by the presence of sea-surface reflections. Remov-ing the receiver ghost before migration provides better low andhigh frequency response as well as a higher signal-to-noise ra-tio for preprocessing steps such as multiple suppression andvelocity analysis.Since Posthumus (1993) seminal study of utilizing simultane-ously towed shallow and deep cables, different configurationsof receiver arrangements have been put to test trying to dealwith the receiver deghosting problem. Traditionally, the con-stant depth cable deghosting is performed in F-K (frequency-wave number) space. The major limitation of deghosting in F-K space is the fact that the receiver depths need to be constant(Fokkema and van den Berg, 1993). Other deconvolution tech-niques in pre- and post-migrations were more recently studiedby Soubaras (2010). They presented a deghosting method thatuses a multichannel deconvolution in the stack or the commonimage gathers after migration and mirror migration. In anotherstudy, Wang and Li (2013) proposed to use the recorded dataand the mirror data which is created from the recorded datato remove both shot and receiver ghosts in the pre-migrationstage. It uses a bootstrap iteration in τ − p space to determinethe ghost-delay time for a local t-x window.Nevertheless, the cost (computational) of the transform whetherit is τ − p or F-K and its artifacts predicates the quality of the

deghosted output. In this paper we present a processing-basedsolution that mitigates such artifacts in F-K deghosting.

THEORY

In typical marine acquisition the upward going seismic wave-field reflected from subsea bottom layers are first recorded bythe receivers of the towed cables. The waves continue to prop-agate to the free surface boundary of air and water and thenreflect back down. This downward traveling reflected wave isagain recorded by the receivers causing a destructive interfer-ence resulting in formation of receiver ghosts.Since the reflectivity r at the free surface is theoretically closeto −1, the downward going wavefield has similar amplitudebut reversed polarity of the upward going wavefield, as illus-trated in Figure 1. As a result, some frequencies in the acquiredsignal are attenuated near the ghost notches. It is a knownfact that removing the receiver ghost can potentially infill theghost notches and thus help obtain images with higher qualityin terms of frequency channels and much improved signal-to-noise ratio (S/N). A standard operator (in complex form) forremoving the receiver ghost in a F-K space for a particular re-ceiver depth z, is given as:

D( f ,kx) =1

1+ r ei 4 π z√

( fc )

2 − k2x

. (1)

Here f is the frequency steps and kx is the wave-number. Toperform deghosting, the above filter should be developed suchthat the phase and the amplitude effect due to ghosting canbe corrected. The sea surface ghost reflections modulate thespectrum of conventional seismic data, reducing energy at theso-called notch frequencies (Amundsen and Zhou, 2013) givenby

fn =nc2z

, n = 0,1,2, · · · · . (2)

In Equation 2 it becomes clear that the first notch is always atzero frequency. The second and following notches are steeredby depth z. As a result, there is a strong loss of useful low-frequency energy in seismic data, in addition to similar lossesat the second and higher notch frequencies. The usable seismicpressure bandwidth is normally between the first and secondnotch.

Methodology

The effective source signal not only includes the direct sig-nal, but also the ghost as well. A similar kind of effect ariseson the receiver side too; therefore, in marine seismic data ac-quisition, a receiver measures the whole wavefield that is dis-torted due to both the source and receiver side ghosts. In most

Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approach

Figure 1: Illustration of a signal wave field vs. the downwardreflected ghost wavefield. In this paper for the sake of sim-plicity we consider the case of a constant c. ∆T (dt) is thedifferential ghost delay time.

field data examples a source deghosting precedes the receiverside deghosting in a processing workflow. Due to the non-stationarity character of the ghost, window-based processingbecomes essential to correctly deconvolve the receiver ghost.In most conventional processing workflows, the windows arepartitioned in the offset direction to take into account the vari-able depth of the cable. This variable depth is usually between6m to 30m, with a receiver spacing of 6.25m. Our syntheticexperiments were designed keeping in mind such geometries.A full waveform acoustic modeling was performed to generatethe synthetics shown in Figure 2 consisting of primary-ghostwaveforms gathered from modeled sub-seafloor layers. It isobserved that the ghost waveform is approximately 50ms be-low the primary waveforms in the near offsets. Figure 3 is the

Figure 2: (a)-(b) shows before and after deghosting of the syn-thetic cable experiments.

result of applying the deterministic F-K operator mentioned inEquation 1. In order to do so, the input data needs to trans-formed in the F-K space. A very similar operator is also usedin the τ − p space when the input data is in slant stack Radonspace. As a reminder, the F-K transform is defined as

F(kx,ω) =

∫ +∞

−∞

∫ +∞

−∞f (x, t)ei(kxx−ωt)dxdt, (3)

with the inverse F-K transform as

f (x, t) =∫ +∞

−∞

∫ +∞

−∞F(kx,ω)e−i(kxx−ωt)dkxdω. (4)

In the proposed approach we do not set a length and width of a

Figure 3: (a)-(b) F-K (using discrete version of Eqn. 3) trans-form of synthetic example before and after application of thedeghosting operator. The arrow marks within the transformplots indicate boosting of notch frequencies.

localized windowed process corresponding to time and offsetrespectively. Instead we operate within the limitation set bythe F-K transform. We perform a forward F-K transform forevery depth in the cable, apply the operator, and perform theinverse transform at the corresponding receiver depths. Thisway we do not create any window related boundaries. Thisimplementation is highly optimized and is extremely fast. Inseveral situation when there are undulations in the cable geom-etry leading to erroneous receiving depths our implementationalso utilizes a smoothing depth function. A window opera-tor in both overlapping or non-overlapping mode in offset andtime does have vertical strumming noise artifact as shown inFigure 6 d. This difference plot in our field data examplesshows that the proposed methodology does not produce suchartifacts and outputs a better estimate of the deghosted signal.The top and bottom plots in Figure 4 shows the average am-plitude spectra of the input data vs. the deghosted data. Theplot on the right clearly indicates a 12dB+ boost at the notchfrequencies.An interesting point to be noted here is when we plot the av-erage power spectrum of the deghosted signal (primary only)and the estimated ghost only spectrum, they are identical. Inother words the primary signal and the ghost have the same

Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approach

average spectra but shifted in time. Therefore, subtracting theghost in the waveform space fills in the notch by removing theghost waveform from the trace data. Figure 5 shows the differ-ences in input stack vs. deghosted stack. It can be seen in thethe stack plots that the diffractions appear much clearer afterdeghosting.

Figure 4: Top plot shows the ghost notch occurring betweenfrequencies 45Hz and 70Hz. Bottom plot shows the ghostnotch being filled up after deghosting.

Figure 5: The before and after figure panels shows the stacksections of our field data example before and after deghosting.The circles depict the improvements in the diffractions and thearrows point to one of the locations of the removed ghost be-sides the locations along the water bottom.

Utilizing conventional windowed processing schemes cannotcircumvent the effect of such artifacts. These results are demon-strated in Figure 6. The difference plots show how the bound-ary edge between depths can be removed by utilizing the pro-posed approach. Even the autocorrelations shown in Figure 7

confirm that the vertical-line artifacts are not present using theproposed deghosting approach.

FUTURE DIRECTIONS

The future improvements to this implementation include a moretargeted approach to deal with the varying character of theghost versus offset and 2-way time (depth z). The ghost is non-stationary and changes with offset and 2-way time. i.e. the ∆T(illustrated in Figure 1) of the ghost delay changes with depthof the cable, the offset, and the reflection depth of the signal.We suggest computing the ∆T of the ghost using the velocityfunction at that shot and the offset of each receiver to ray traceto get the angle of the signal propagation for the reflected sig-nal for each receiver versus 2-way time.Using the angle of the signal, a ∆T for the ghost can be com-puted and converted to a equivalent receiver depth. The for-ward F-K, the operator for the ghost and the inverse F-K willprovide the deghosted signal for that receiver for every 2-waytime. Therefore, the deghost operator can be made contin-uously variable and will provide a much superior deghostedsignal.

CONCLUSION

We present a new processing-based broadband solution formitigating F-K transform artifacts for receiver deghosting. Ourdeghosting approach does not use any windowing to effec-tively deconvolve the non-stationary character of the ghost sig-nals. As a result of which we do not produce artifacts that areusually seen in case of the windowing approach. Both our syn-thetic and real field data examples confirm not producing theseartifacts and outputs a cleaner estimate of the deghosted sig-nals. Interesting studies of the spectral plots indicated the pri-mary (deghosted) signal and estimated ghost signal have iden-tical spectra without the notch.

ACKNOWLEDGMENTS

The authors thank Global Geophysical Services for the permis-sion to publish this work. The authors would also like to thanktheir colleagues Dan Nietupski, Steve Svatek, Bill Mclain, EricKylberg for their contributions in these studies and preparationof this expanded abstract.

Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approach

Figure 6: (a) is the input example of the field data before deghosting, (b) shows a conventional windowed based deghosting, (c) isthe proposed deghosting approach, (d) is the difference between (b) & (c) showing edge artifacts showing up as residuals.

Figure 7: (a) is the autocorrelation of the traces in the field data example using conventional deghosting, the arrows indicatevertical-line artifact (b) shows the autocorrelation output using proposed approach without the vertical-line artifact.