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 (constant depth) 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 prominent with slant cable deghosting. Irrespective of the type of trans- form or deghost operator used, a windowed process is essential due to the time and offset varying character of the ghost. This use of a windowed process usually results in poor reconstruc- tion of deghosted signals and artifacts beyond the control of the transform(s) itself. The windowing in time and offset produces edgy effects which can be clearly seen in the difference plots. Our method, using a non-windowing approach, demonstrates a better representation of the deghosted signals without the arti- facts caused by the boundary of the windows. This method has also been well-tested for both the flat and slant cable receiver deghosting workflows in synthetic and field data examples.

1. Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approach
Vikram Jayaram, Dylan Copeland, Carola Ellinger, Charles Sicking, Stu Nelan, Josh Gilberg and Chris Carter
Global 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 (constant
depth) 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 prominent
with slant cable deghosting. Irrespective of the type of trans-
form or deghost operator used, a windowed process is essential
due to the time and offset varying character of the ghost. This
use of a windowed process usually results in poor reconstruc-
tion of deghosted signals and artifacts beyond the control of the
transform(s) itself. The windowing in time and offset produces
edgy effects which can be clearly seen in the difference plots.
Our method, using a non-windowing approach, demonstrates a
better representation of the deghosted signals without the arti-
facts caused by the boundary of the windows. This method has
also been well-tested for both the flat and slant cable receiver
deghosting workflows in synthetic and field data examples.
INTRODUCTION
In recent years, the industry has seen tremendous growth and
attention to high quality broadband marine seismic processing
and acquisition. The receiver ghost is a well known problem
in marine seismic acquisition where the seismic resolution is
corrupted by the presence of sea-surface reflections. Remov-
ing the receiver ghost before migration provides better low and
high frequency response as well as a higher signal-to-noise ra-
tio for preprocessing steps such as multiple suppression and
velocity analysis.
Since Posthumus (1993) seminal study of utilizing simultane-
ously towed shallow and deep cables, different configurations
of receiver arrangements have been put to test trying to deal
with 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 studied
by Soubaras (2010). They presented a deghosting method that
uses a multichannel deconvolution in the stack or the common
image gathers after migration and mirror migration. In another
study, Wang and Li (2013) proposed to use the recorded data
and the mirror data which is created from the recorded data
to remove both shot and receiver ghosts in the pre-migration
stage. It uses a bootstrap iteration in τ − p space to determine
the ghost-delay time for a local t-x window.
Nevertheless, the cost (computational) of the transform whether
it is τ − p or F-K and its artifacts predicates the quality of the
deghosted output. In this paper we present a processing-based
solution 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 by
the receivers of the towed cables. The waves continue to prop-
agate to the free surface boundary of air and water and then
reflect back down. This downward traveling reflected wave is
again 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 close
to −1, the downward going wavefield has similar amplitude
but reversed polarity of the upward going wavefield, as illus-
trated in Figure 1. As a result, some frequencies in the acquired
signal are attenuated near the ghost notches. It is a known
fact that removing the receiver ghost can potentially infill the
ghost notches and thus help obtain images with higher quality
in terms of frequency channels and much improved signal-to-
noise ratio (S/N). A standard operator (in complex form) for
removing 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 ( f
c )2 − k2
x
. (1)
Here f is the frequency steps and kx is the wave-number. To
perform deghosting, the above filter should be developed such
that the phase and the amplitude effect due to ghosting can
be corrected. The sea surface ghost reflections modulate the
spectrum of conventional seismic data, reducing energy at the
so-called notch frequencies (Amundsen and Zhou, 2013) given
by
fn =
nc
2z
, n = 0,1,2, ···· . (2)
In Equation 2 it becomes clear that the first notch is always at
zero frequency. The second and following notches are steered
by depth z. As a result, there is a strong loss of useful low-
frequency energy in seismic data, in addition to similar losses
at the second and higher notch frequencies. The usable seismic
pressure bandwidth is normally between the first and second
notch.
Methodology
The effective source signal not only includes the direct sig-
nal, but also the ghost as well. A similar kind of effect arises
on 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

2. Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approach
Figure 1: Illustration of a signal wave field vs. the downward
reflected ghost wavefield. In this paper for the sake of sim-
plicity we consider the case of a constant c. ∆T (dt) is the
differential ghost delay time.
field data examples a source deghosting precedes the receiver
side deghosting in a processing workflow. Due to the non-
stationarity character of the ghost, window-based processing
becomes essential to correctly deconvolve the receiver ghost.
In most conventional processing workflows, the windows are
partitioned in the offset direction to take into account the vari-
able depth of the cable. This variable depth is usually between
6m to 30m, with a receiver spacing of 6.25m. Our synthetic
experiments were designed keeping in mind such geometries.
A full waveform acoustic modeling was performed to generate
the synthetics shown in Figure 2 consisting of primary-ghost
waveforms gathered from modeled sub-seafloor layers. It is
observed 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 in
Equation 1. In order to do so, the input data needs to trans-
formed in the F-K space. A very similar operator is also used
in the τ − p space when the input data is in slant stack Radon
space. 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 the
deghosting operator. The arrow marks within the transform
plots indicate boosting of notch frequencies.
localized windowed process corresponding to time and offset
respectively. Instead we operate within the limitation set by
the F-K transform. We perform a forward F-K transform for
every depth in the cable, apply the operator, and perform the
inverse transform at the corresponding receiver depths. This
way we do not create any window related boundaries. This
implementation is highly optimized and is extremely fast. In
several situation when there are undulations in the cable geom-
etry leading to erroneous receiving depths our implementation
also utilizes a smoothing depth function. A window opera-
tor in both overlapping or non-overlapping mode in offset and
time does have vertical strumming noise artifact as shown in
Figure 6 d. This difference plot in our field data examples
shows that the proposed methodology does not produce such
artifacts 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. The
plot on the right clearly indicates a 12dB+ boost at the notch
frequencies.
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. In
other words the primary signal and the ghost have the same

3. Receiver deghosting method to mitigate F-K transform artifacts: A non-windowing approach
average spectra but shifted in time. Therefore, subtracting the
ghost in the waveform space fills in the notch by removing the
ghost waveform from the trace data. Figure 5 shows the differ-
ences in input stack vs. deghosted stack. It can be seen in the
the stack plots that the diffractions appear much clearer after
deghosting.
Figure 4: Top plot shows the ghost notch occurring between
frequencies 45Hz and 70Hz. Bottom plot shows the ghost
notch being filled up after deghosting.
Figure 5: The before and after figure panels shows the stack
sections of our field data example before and after deghosting.
The circles depict the improvements in the diffractions and the
arrows point to one of the locations of the removed ghost be-
sides the locations along the water bottom.
Utilizing conventional windowed processing schemes cannot
circumvent 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 the
proposed deghosting approach.
FUTURE DIRECTIONS
The future improvements to this implementation include a more
targeted approach to deal with the varying character of the
ghost 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 depth
of the cable, the offset, and the reflection depth of the signal.
We suggest computing the ∆T of the ghost using the velocity
function at that shot and the offset of each receiver to ray trace
to 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 will
provide the deghosted signal for that receiver for every 2-way
time. Therefore, the deghost operator can be made contin-
uously variable and will provide a much superior deghosted
signal.
CONCLUSION
We present a new processing-based broadband solution for
mitigating F-K transform artifacts for receiver deghosting. Our
deghosting 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 are
usually seen in case of the windowing approach. Both our syn-
thetic and real field data examples confirm not producing these
artifacts 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 thank
their colleagues Dan Nietupski, Steve Svatek, Bill Mclain, Eric
Kylberg for their contributions in these studies and preparation
of this expanded abstract.

4. 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) is
the 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 indicate
vertical-line artifact (b) shows the autocorrelation output using proposed approach without the vertical-line artifact.