amplitudes
A bootstrap deghosting approach for horizontal and non-horizontal streamers
Thomas Andre Larsen Greiner
Master Thesis, Spring 2016
effects on amplitudes
A bootstrap deghosting approach for horizontal and non-horizontal streamers
Thomas Andre Larsen Greiner
Thesis submitted for the degree of
Master in Petroleum Geology and Petroleum Geophysics (30 credits)
Department of Geosciences
Faculty of Mathematics and Natural Sciences University of Oslo
Spring 2016
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Preface
This thesis project is the final work of a 2 year M.Sc degree in Applied Geophysics, and a part of the study program Petroleum Geology and Petroleum Geophysics at the Department of Geosciences at the University of Oslo. The thesis work was carried out during the spring semester of 2016 (4/1/2016 to 1/6/2016). The thesis subject was proposed and supervised by Lundin Norway, with an internal co-supervisor at the University of Oslo for ensuring that the work was suitable for a M.Sc degree. The work was carried out at Lundin Norway’s office at Lysaker. The supervisors for this project were Jan Erik Lie (Lundin), Adjunct Associate Professor Isabelle Lecomte (Norsar/UiO) and Andreas Kjelsrud Evensen (Lundin).
Oslo, 2016-01-16 Thomas Andre Larsen Greiner
Acknowledgements
I would like to express my gratitude to my supervisors Jan Erik Lie, Isabelle Lecomte and Andreas Kjelsrud Evensen. Thank you for assigning me to this project and for your guidance and valuable discussions throughout the thesis work. I would also like to express my gratitude to Odd Kolbjørnsen for the highly educative, inspirational and valuable discussions. I am also grateful for the discussions and help from Espen Harris Nilsen, Piotr Lewczuk, and Emilie Davenne related to the Promax software, and Per Eivind Dhelie for reading through this thesis and providing with helpful advice. A special thanks to Kjetil Eik Haavik for the help and discussions related to the Madagascar software.
I would also like to thank my family and friends, who have supported me throughout my academic years, and a special thanks to my brother Christopher who inspired my interest in science. Finally, I would like to thank Maria Eidsaa Larsen for her massive support and encouragement throughout these years, and for taking care of Larsen almost 24/7 throughout this thesis work. Your are the best. I am also truly grateful for the encouragement provided by the Eidsaa Larsen family.
T.L.G.
Abstract
The large impedance contrast between the sea-water and the air above causes an inevitable reflection event in marine seismic data known as the ghost reflection, which causes an alteration of the primary recorded signal. This results in inferior bandwidth compared to the bandwidth of the primary reflection. In order to achieve the broadest bandwidth possible, the ghost has to be removed from the seismic record. The ghost removal method is known as deghosting.
The purpose of this thesis was to develop algorithms for pressure-only receiver deghosting for horizontal and non-horizontal streamers and investigate possible amplitude distortions due to deghosting and streamer configuration. A finite difference program was used to develop the synthetic pressure-only models for testing the deghosting algorithms. Three streamer configurations were used for creating the different pressure wavefields, a shallow streamer (10 m), an ultra deep streamer (60 m) and a linear slanted streamer (10-60 m). A bootstrap deghosting algorithm which self-determines the time-delay parameter of the ghost was written, and implemented on synthetic and real data.
The bootstrap algorithm performes well on estimating the correct time-delay parameter of the ghost. Deghosting in the frequency-slowness (f −p) domain with bootstrap for flat streamers works well on the synthetic data. The frequency-offset (f −x) domain deghosting with partial windowing for the slanted streamer works well if only the events of interest are included. The partial window approach is not optimal due to problems at window edges, causing possible events of interest to be removed. The tests on synthetic data has shown that varying the streamer depth with offset gives a better response in the post-stack domain than the flat streamer configurations, if the deghosting results are similar.
The AVO study have shown that the amplitudes after deghosting are slightly dampened, but show similar relative magnitude to the ghost-free models, which implies a scalable effect.
It is also shown that deghosting could be crucial for achieving a more reliable AVO response due to the overestimation in AVO after phase rotation. In addition, no significant deviations in relative amplitudes due to streamer configurations were observed.
Deghosting on real data showed insignificant results for the shallow streamer, and poor results for the ultra deep streamer. One of the possible reasons for the poor results with the ultra deep streamer was most likely due to the streamer depth not being constant, leading to non-correct deghosting in theτ−pdomain.
Preface . . . i
Acknowledgements . . . ii
Abstract . . . iii
List of Figures xx 1 Introduction 1 1.1 Aim and objectives . . . 1
1.2 Purpose and methods . . . 1
1.3 Background . . . 3
1.4 Structure of the Report . . . 9
2 Theory 11 2.1 Pressure signals in marine seismic data . . . 11
2.2 The Ghost reflection . . . 14
2.3 Deghosting . . . 21
2.4 Elastic wave propagation and amplitude analysis . . . 26
3 Deghosting examples and results 33 3.1 Synthetic example 1: 1D deghosting . . . 35
3.1.1 1D deghosting results . . . 39
3.2 Synthetic example 2: 2D deghosting and finite difference modeling . . . 52
3.2.1 Flat streamer deghosting in the tau-p domain . . . 58
3.2.2 Slanted streamer deghosting in the time-offset domain . . . 71
3.2.3 Post-stack wavelet comparison . . . 78
3.3 Deghosting of real data . . . 82 v
4 Deghosting effects on amplitudes 93 4.1 1D deghosting and AVO . . . 93 4.2 2D deghosting effects on AVO . . . 103
5 Discussion 113
5.1 Deghosting of synthetic data . . . 113 5.2 Deghosting of real data . . . 117 5.3 Amplitude effects . . . 118
6 Conclusions 123
6.1 Conclusions . . . 123 6.2 Recommendations for Further Work . . . 124
Bibliography 126
A Generating synthetic seismogram 131
B Matlab and Madagascar scripts and deghosting algorithms 135
C Physics of elastic wave propagation 145
D Estimating the time-delay with reflection depth and offset 151
E Additional figures 155
1.1 Conceptual sketch of a source (air gun) which gives rise to a pressure wave propagating downward and reflects at interfaces (boundary conditions). The reflected waves are recorded at the receiver in the time domain. . . 3 1.2 The concept of temporal resolution illustrated by use of reflection coefficients
and a Ricker wavelet converging as the reflection coefficients move closer to- gether. The acoustic impedance (AI) (from Amundsen and Landrø (2013)). . . . 4 1.3 The concept of temporal resolution illustrated by altering the frequency re-
sponse of the pulse (modified from Amundsen and Landrø (2013)). . . 5 1.4 Conceptual sketch of the source and receiver ghost, which reflects at the sea-
surface and interferes with the primary reflection in the time domain. As the upcoming wavefields contains a source ghost, a ghost of the receiver ghost will arise. . . 6 1.5 Basic concept of acquisition with conventional streamer compared to a variable
depth streamer. From this sketch it is rather implicit that the ghost arrival depends on the depth of the streamer, and by slanting the streamer we will achieve a diversity of ghost arrivals. . . 7 2.1 Conceptual sketch of a wave front (plane wave) propagating upwards and re-
flecting at the free surface. At the sea-surface the pressure vanishes, and is therefore considered to act as an free surface. . . 12 2.2 Basic concept of the ghost reflection recorded at different depths, and how
the combined pressure is recorded in the time- and frequency domain. The characteristic shape of the frequency domain arising from the ghost reflection is dependent on both receiver depth and angle of propagation. . . 15
vii
2.3 Basic illustration of the ghost filters effect on a primary signal in the frequency domain. . . 16 2.4 Time-delay model, which illustrates in a general matter how the time-delay
(t-d) could vary dependent on the streamer configuration, by using a constant- angle approximation. The figure clearly shows that a shallow streamer at 10m depth has a limited diversity of ghost time-delays and the ultra deep and slanted streamer have large diversities. . . 17 2.5 Conceptual sketch of a plane wave propagating in the negative z-direction. Here
we illustrate that the ghost signalP(ω,kz)e−j kz2hcan be regarded as a spatially shifted versionP(ω,kz), which is recorded at a streamer, mirrored at the sea surface, instead of following the reflection path(green line) . . . 19 2.6 Ghost filters amplitude spectrum for receiver depth of 10m in the f −k do-
main. This filter takes the angle of propagation into account for a more correct estimation of the ghost time-delay . . . 20 2.7 Left: Illustration of the ghost filters amplitude spectrum for receiver depths at
10m and 60m, given by Equation 2.20. Right: The inversion of this equation, including a dampening factor as shown in Equation 2.34. . . 22 2.8 Bootstrap algorithm illustrated schematically (pseudo code). A time-delayT(i)
within a certain time range is picked and used for initial deghosting. The sub- traction between the extracted ghost (shifted withT(i)) and primary gives the residual energy (α(i)). The time-delay (T(i)) which gives the minimum error energy, is the best estimated time-delay. . . 23 2.9 Conceptual sketch of NMO-correcting and windowing before deghosting. . . . 24 2.10 Conceptual sketch of slant stacking in thet−xdomain. The dark green, light
green , brown and light blue line corresponds to the linear summation in thet−x domain, which corresponds to points in theτ-p domain. After slant stacking, every individual slowness value should ideally consists of one time-delay, which stabilize the deghosting procedure. . . 25 2.11 Conceptual sketch of a P-wave partitioning at an interface. In this illustration it
is assumed that both layers can support shear stress. This is not the case at the sea floor, where in that case, there will only be a reflected P-wave. . . 29
2.12 Different AVO models created from the impedance contrast between two layers.
A) show a negative amplitude at the intercept (zero offset) and is increasing in negative value with offset. This signature could probably produce a class 2 anomaly from a linear regression (see Figure 2.13). B) show a signature which is negative at zero offset, but decreasing in amplitude with offset. This signature could probably produce a class 4 anomaly (modified from Chopra and Castagna (2014)). . . 30 2.13 The AVO crossplot for intercept and gradient analysis. If the intercept and
gradient range falls outside the reservoir values, it is plotted in the background trend. The possible linear regression line from Figure 2.12 could be plotted as a class 2 trap (modified from Chopra and Castagna (2014)). . . 31
3.1 Simple conceptual sketch of 1D convolution with the reflection-series from the earth’s and sea-water reflectivity. The sea surface reflection coefficient is, in this illustration, set to -1, which gives the ghost reflection inverted replicas of the earth’s reflection coefficients. The time-delay here is simplified by assuming the wave propagation is vertical. . . 35 3.2 Convolution of the reflection coefficients with the primary response (Gaussian
pulse) with three different receiver depths, giving the primary and ghost re- sponse (Primary + ghost) results. Observe that the shallow receiver (10m) only have the zero-frequency notch within the frequency band. . . 36 3.3 A comparison of each amplitude spectrum from the convolution result. For
deeper receivers (higher time-delays) the low side of the spectrum is less damp- ened. However, we loose more frequencies due to notches. . . 37 3.4 Deghosted results from Algorithm 1 with ε=0.1. The upper model shows
deghosting results of the shallow receiver (10m), the center model for deep receiver (30m) and the lower model for ultra deep receiver (60m). . . 40 3.5 Deghosting results from Algorithm 1 withε=0.01. The upper model shows
deghosting results of the shallow receiver (10m), the center model for deep receiver (30m) and the lower model for ultra deep receiver (60m) . . . 41
3.6 Deghosting results using Equation 2.34 with aε= 0.1. In this model random white noise is included. The upper model shows deghosting results of the shallow receiver (10 m), the center model for deep receiver (30 m) and the lower model for ultra deep recevier (60 m) . . . 43 3.7 Deghosting results using Equation 2.34 with aε= 0.01. In this model random
white noise is included. The upper model shows deghosting results of the shallow receiver (10 m), the center model for deep receiver (30 m) and the lower model for ultra deep recevier (60 m) . . . 44 3.8 Stacking of the traces from different receiver depths. The notches corresponding
to the 30 and 60mreceivers are now filled inn by complimentary amplitudes . 45 3.9 Synthetic model illustration without ghost (left) and with ghost (right). Top
show rays, middle is the time signals and bottom is the frequency spectrum.
The three events are recorded with three different angle of incidence, which will result in a time-delay variation on one single trace. . . 47 3.10 The deghosted result from Algorithm 1 and its corresponding amplitude spec-
trum (bottom). The ghost-free response (primary) is included to compare the deghosted response. We see that the algorithm have amplified or suppressed at wrong frequencies. . . 48 3.11 The deghosted result from Algorithm 2 with partial window applied and its
corresponding amplitude spectrum (bottom). The algorithm works well on producing the ghost-free response (primary). The amount of artifact increases as a result of introducing more notches into the frequency band. . . 49 3.12 The modeled 1D trace (with white noise added) after deghosting with Algorithm
2 (top) and its corresponding amplitude spectrum (bottom). The algorithm performs well on random white noise. . . 50 3.13 A time-delay error sensitivity comparison between the shallow and ultra deep
receiver. The inversion process at large time-delay is more sensitive to errors, and even a small error will give unacceptable results. . . 51 3.14 Velocity and density models for finite difference modelling. The explosive stress
source was positioned at 5kmand 98mbelow top model. . . 52
3.15 Conceptual sketch of shot-gather design. The second simulation, which has the streamer mirrored at the sea surface, is subtracted from the first simulation, to give an output with ghost wavefield included. . . 54 3.16 The modeled scalar pressure wavefield from the flat streamers at 10m (left)
and 60m(middle), and a linear slant from 10−60m(right). The arrivals have been static shifted up to sea-level. The gathers are displayed with a time power constant of 1.75 (in Promax). We see that the ghost wavefield characteristics are different for every streamer. . . 56 3.17 The modeled scalar pressure wavefield after NMO-correction without stretch
limit applied. We see that the shallow streamer ghost wavefield is parallel to sub- parallel to the primary. The ultra deep ghost wavefield is more or less parallel close to near-offset, and starts to diverge at larger offset. . . 57 3.18 The modeled wavefield from the shallow streamer in the time-offset domain
(left) and its corresponding wavefield in theτ−pdomain (right). We see that the hyperbolic reflection-events (in the time domain) are now transformed into elliptic events (τ−p domain). Ideally, each individual slowness value is now corresponding to a single ghost time-delay. . . 59 3.19 The modeled wavefield from the ultra deep streamer in the time-offset domain
(left) and its corresponding wavefield in theτ−pdomain (right). We see that the hyperbolic reflection-events (in the time domain) are now transformed into elliptic events (τ−pdomain). Ideally, a single ghost time-delay is now sorted into each individual slowness value. . . 60 3.20 Comparison of the two frequency domains, f −x(left) and f −p(right), for the
shallow streamer. The amount of notch-control in the f −xdomain compares well to the notch-control in thef −pdomain, as the time-delays variations for shallow streamers are limited. . . 61 3.21 Comparison of the two frequency domains, f −x(left) and f −p(right), for the
ultra deep streamer. The amount of notch-control in the f −xdomain (left) is somewhat limited. The chaotic frequencies at far offset is caused by interfering events with large time-delay variations. In the f −p domain we see a good sorting of notches for each slowness value. . . 62
3.22 Wavefield modeled with the shallow streamer before (left) and afterτ−pdeghost- ing (right). The wavefield before deghosting show energy corresponding to the downgoing wavefield (with opposite polarity) at the base of the reflection, which is removed in the deghosted result, and with no visual artifacts created. . . 64 3.23 Modeled wavefield within a local time-offset window of the shallow streamer
before deghosting (left) and after deghosting (right). We see that the wavelet signature corresponding to a combined primary and ghost when the streamer is located at this depth. . . 65 3.24 Modeled wavefield of the shallow streamer afterτ−pdeghosting (left), without
the ghost (middle) and the residual energy between the two (right). The residual energy is apparently constant both temporally and spatially for the deepest reflection, which implies a scaleable difference between the deghosted wavefield and the primary wavefield. However, the shallower reflections show relative magnitude difference pointed out by the green and red arrows. . . 66 3.25 Local time-offset window of the shallow streamer modeling after deghosting
(left), modeling without ghost (middle) and the residual energy after subtraction between the two (right). . . 67 3.26 Modeled wavefield of the ultra deep streamer before (left) and afterτ−pdeghost-
ing (right). The wavefield before deghosting shows energy corresponding to the upgoing wavefield with a mirrored wavefield (with opposite polarity) below. . . 68 3.27 Local time-offset window of the ultra deep streamer-modeling before (left) and
after deghosting (right). At this streamer depth we see that the primary- and ghost wavefield are largely separated. The residual energy after deghosting is considerably removed, leaving a hint of artifact corresponding to the ghost wavefield. . . 69 3.28 Modeled wavefield of the ultra deep streamer afterτ−pdeghosting (left), with-
out the ghost (middle) and the residual energy (difference) between the two (right). The residual energy is apparently constant both temporally and spatially, which implies a scalable difference between the deghosted wavefield and the primary wavefield. . . 70
3.29 Local time-offset window of the shallow streamer modeling after deghosting (left), modeling without ghost (middle) and the residual energy after subtraction between the two (right). . . 71 3.30 The pressure wavefield modeled with the slanted streamer (top left) and its cor-
responding frequency-offset domain (top right). The zoomed in event (bottom left) shows a good notch diversity in the frequency-offset domain (bottom right). 72 3.31 Modeled wavefield of the slanted streamer before deghosting (left) and after
deghosting (right). The pre-deghosting wavefield shows separate down going wavefields corresponding to the ghost reflection, which is noticeable reduced in the post-deghsoted wavefield. . . 74 3.32 Local time-offset window of the slanted streamer-modeling before (left) and af-
ter deghosting (right). From the varying streamer depth we see that the primary- and ghost wavefield are diverging with offset. The residual energy after deghost- ing is considerably removed close to near-offset. . . 75 3.33 Modeled wavefield of the slanted streamer after deghosting (left), without the
ghost (middle) and the residual energy (difference) between the two (right). The deghosted result compares well to the ghost free pressure at small offsets, but does not compare well at interfering events. . . 76 3.34 Local time-offset window of the slanted streamer-modeling after deghosting
(left), modeling without ghost (middle) and the residual energy after subtraction between the two (right). . . 77 3.35 Local time-offset window of the linear slanted streamer wavefield. The artifacts
are noticeably increasing with offset. . . 77 3.36 Stacked sections from the deghosted and ghost-free data, from all streamer
configurations. The events that are used for comparison are marked in the figure as Event 2 and Event 3. . . 78 3.37 A comparison of the wavelets extracted from the second reflection-event. The
wavelet extracted from the shallow streamer shows larger side-lobes, but a slightly narrower main-lobe than the ultra deep and slanted streamer wavelets.
In the frequency domain we see that the shallow streamer wavelet has a less optimal reconstruction of the amplitudes below 15 Hz compared to the ultra deep and slanted ones. . . 79
3.38 A comparison of the wavelets extracted from the third reflection-event. The wavelet extracted from the shallow streamer show a greater side-lobe magnitude than the other two wavelets. The wavelet from the ultra deep and slanted streamer show similar shape, only with a slight lesser side-lobe effect on the slanted streamer wavelet. . . 80 3.39 A comparison of the wavelets extracted from the third reflection-event, only
with a mute applied to exclude possible sub-optimal deghosting dampening of amplitudes. . . 81 3.40 The shot gather from the shallow streamer (left) and ultra deep streamer (right)
displayed the first 3seconds. The shot gathers have been static shifted up to sea-level, bandpass filtered (2.5 Hz lowcut and 120 Hz highcut), and a power constant of 1.75 and with a non-correct NMO-correction applied. The red boxes indicate the local time-offset window for deghosting. . . 83 3.41 A local time-offset window of the shallow (left) and the ultra deep receiver (right).
The ghosts are significant for the ultra deep streamer, indicated by the green arrows. . . 84 3.42 Mean amplitude spectrum extracted from the two first seconds (0 s<t<2 s)
and from near- to 1 km offset. The green arrows points out where we should expect the notches to occur (considering below 40 Hz) from the ultra deep receiver (blue), and where the amplitudes should be suppressed relative to the shallow streamer (red). There is no clear notch occurrences at these values, no significant relative value difference. However, the low side of the spectrum below 8 Hz is clearly amplified. . . 85 3.43 The target reflections before deghosting (left) and after deghosting (right). The
post-deghosting result show a noticeable difference in the near-offset reange (indicated by the green arrows) compared to the pre-deghosted reflections. At larger offset the difference becomes less observable. . . 86 3.44 Wiggle plot of the target reflections pre-deghosting (top left) and post-deghosting
(top right) and their corresponding autocorrelograms (bottom). There is no noticeable difference from the autocorrelograms, which implies a poor result from deghosting. . . 87
3.45 The mean autocorrelation value across the entire offset for the shallow streamer.
The insignificant difference between the two implies a poor deghosting result across the offfset. A deghosting result of significant value should show smaller sidelobes post-deghosting than pre-deghosting. . . 87 3.46 The target reflections after deghosting (right) are less than satisfying, and does
not come close to the ghost-free response. The image is smeared out and leaves a result which is unacceptable. . . 88 3.47 Wiggle plot of the target reflections pre-deghosting (top left) and post-deghosting
(top right) and their corresponding autocorrelograms (bottom). . . 88 3.48 The mean autocorrelation value across the entire offset. The insignificant dif-
ference between the two implies a poor deghosting result across the offfset. A good deghosting result should show smaller sidelobes post-deghosting than pre-deghosting. . . 89 3.49 Extracted sea-bottom event within a limited time-window. Frequencies around
25 Hz are over-amplified, giving a poor result in the time domain. . . 90 3.50 Extracted sea-bottom events from the synthetic ultra deep model (top) and
the real data (bottom). The notch occurrence in the real data imply a much shallower receiveer than 60 m. . . 90 3.51 Extracted sea-bottom events from the synthetic ultra deep model (top) and the
real data (bottom) after deghosting with Algorithm 2. We see that the result has improved compared to the first deghosting attempt. . . 91 3.52 Extracted sea-bottom events from the synthetic ultra deep model (top) and the
real data (bottom) with a 90ophase rotation applied before deghosting with Algorithm 2. . . 92 3.53 A local time-offset window of target reflections before deghosting (left) and
after deghosting (right). The target reflections after deghosting are still at an unacceptable level, and does not come close to the ghost-free response. In the near-trace range for the sea bottom reflection we see a minor improvement (green arrow). . . 92
4.1 Conceptual sketch of a constant-angle model, used for AVO modeling. The ray path is assumed to have a straight linear path down to the interface, and reflects back to the receiver. It is assumed that the ray obeys Snell’s law at the interface and refracts with an angle dependent on the angle of incidence and velocity contrast of each layer. . . 94 4.2 Angles extracted from the constant angle model (Figure D.2) at 2kmdepth. The
angles at this depth is near equality. In this case we can expect the AVO response from each streamer configuration to be similar. . . 95 4.3 Shallow streamer AVO modeling results. The amplitudes are picked at the
troughs (red) min value. The amplitudes before deghosting (primary + ghost) show a tuning effect which completely cancels out the AVO response above 4km.
The deghosting result show additional dampening of the responses, but appears to be constant an scalable. . . 98 4.4 Ultra deep streamer AVO results from the 1D convolution. The amplitudes are
picked at the troughs (red) min value. Both pre- and post-deghosting result show a good relative amplitude match with the reference (primary) AVO response. 99 4.5 Slanted streamer AVO results from the 1D convolution. The amplitudes are
picked at the troughs (red) min value. Both pre- and post-deghosting result show a good relative amplitude match with the reference (primary) AVO response.100 4.6 Comparison of AVO responses. The AVO response from the ultra deep and
slanted streamer configurations show a near perfect match (primary + ghost) or only a scaleable effect (deghosted) to the response without the ghost (primary).
The shallow streamer shows a tuning effect from the ghost, which increases with offset, giving a false relative AVO response. . . 101 4.7 A comparison of ghost wavelet time-delays. The bootstrap time-delay estima-
tion for the shallow streamer shows estimation errors due to ghost tuning. When the wavelets are well separated or no tuning occurs, the bootstrap algorithm works well. . . 102 4.8 Deghosted gather (left) from the slanted streamer, and its corresponding f −k
filtered result (right). The result show that a significant amount of the coherent noise are removed from the shot gather. . . 103
4.9 A comparison of the pre- (top) and post f −k filtered data (bottom) of the first reflection-event. Notice the relative dampening from near- to far-offset.
However, the dampening effect is not as prominent from near- to mid-offset. . 104 4.10 Wiggle-plot of the modeled wavefield with the slanted streamer after deghosting,
phase-rotation and NMO-correction (left). The deghosted wavefield was then f −k-filtered to show and the reflectors for AVO analysis (right). The amplitudes was picked on the peak (black) of the wavelet. . . 105 4.11 Comparison of the AVO responses after deghosting. Event 2 shows no major
difference due to optimal deghosting on all streamer configurations. However, a slightly higher relative dampening effect is observed from the slanted streamer at offsets larger than 1.2 km (Event 2). For Event 3 the sub-optimal deghosting result from the slanted streamer give a large relative dampening effect at offsets larger than 1.75 km. . . 107 4.12 Comparison of AVO responses modeled without the ghost. The different streamer
configurations show no significant difference in AVO. . . 108 4.13 Comparison of AVO responses with the ghost included. The phase rotation lifts
the shallow streamer AVO response because of interference between the primary and ghost. The ultra deep streamer primary and ghost are well separated, which gives no amplification after the phase rotation. . . 109 4.14 2D AVO comparison of the different AVO responses for the shallow streamer.
The over-estimated AVO response is most likely due to the phase rotation. . . . 110 4.15 2D AVO comparison of the different AVO responses for the ultra deep streamer. 111 4.16 2D AVO comparison of the different AVO responses for the slanted streamer. . . 112 5.1 Basic illustration of partial windows missing the event of interest and will there-
fore lead to removal of the wrong reflections. Inside Time window n, the ghost from another event would act as a primary, hence removing the primary. Time window n+1 would remove the primary from another event. . . 115 5.2 Ghost filter for source and receiver separately, and if both are taken into consider-
ation. At first one could believe that the source and receiver are complimentary at these depth, but since the source and receiever ghosts defines a product (green plot), they are not complimentary. . . 117
5.3 Conceptual sketch of the tuning effect when the primary and ghost interferes.
The primary and ghost are separated with a time-delay equal tot0. Because of ghost tuning the time-delay is estimated to be closer tot1. . . 119 5.4 Tuning model for the ghost reflection modeled with the receiver ghost only. . . 120 5.5 Model showing the difference between reflection-angle using the concept of ray
paths with constant angles. We see that the difference between the slanted- and ultra deep- versus the shallow streamer is more or less insignificant. . . 122 D.1 Geometrical interpretation of a ray path from shot to receiver where the streamer
is slanted with a angleα . . . 152 D.2 Constant angle model as function of offset and reflection depth, computed with
Equation D.5 . . . 153 E.1 The Gaussian pulse used in the single wavelet deghosting examples.σ= 0.01. . 155 E.2 The Gaussian pulse used in the multiple pulse deghosting examples, where the
σvalue was changed from 0.01 to 0.005, to give the pulse a narrower main lobe and broader frequency band. . . 156 E.3 The 0-40 Hz bandpass wavelet used in the 2D experiment. This wavelet has a
more satisfying frequency band than f.ex. a Ricker wavelet. . . 157 E.4 Raw shot gathers used for deghosting. . . 158 E.5 Header information from the near-channel and channel at 1 km. The receiver
elevations are highlighted in the red boxes. . . 159 E.6 Header from ultra deep streamer (60 m). . . 160 E.7 Header from shallow streamer (8 m). . . 161 E.8 NMO-corrected shot gather. . . 162 E.9 The stacked results from the shallow streamer modeling. Stacking the data
pre-deghosting show wavelets with the characteristic shape of a Ricker wavelet (inverted). The post-deghosted wavelets show the characteristic shape of the input wavelet, and compares well to the ghost-free response. . . 163 E.10 Amplitude spectrum after stack. . . 164 E.11 NMO-corrected shot gather . . . 165
E.12 The stacked results from the ultra deep streamer modeling. Stacking the data pre-deghosting show wavelets with the characteristic shape of a distorted mirror wavelet with opposite polarity. The post-deghosted wavelets show the charac- teristic shape of the input wavelet, and compares well to the ghost-free response.
Artifacts show up in the post-stack response which implies a sub-optimal de- structive interference of the artifacts. . . 166 E.13 Post-stack amplitude spectrum of the ultra deep streamer. Good recovery on
the low frequencies below 8Hz. However, . . . 167 E.14 NMO-corrected shot gather . . . 168 E.15 Post-stack amplitude spectrum of the slanted streamer. The recovery is good for
frequencies below 8Hz. However, the frequencies above 10Hz are dampened, most likely due to the sub-optimal deghosting. . . 169 E.16 Brute stack of the slanted streamer. The first and second event compares well
to the ghost-free stack (primary). The third event does not show a good match.
The lack in amplitude is most likely due to the sub-optimal deghosting. . . 170 E.17 Primary wavefield modeled with the shallow streamer after NMO-correction
andf −kfilter applied. . . 171 E.18 Wavefield modeled with the shallow streamer after deghosting, NMO-correction
andf −kfilter applied. . . 172 E.19 Wavefield modeled with the shallow streamer after NMO-correction and f −k
filter applied. . . 173 E.20 Primary wavefield modeled with the ultra deep streamer after NMO-correction
andf −kfilter applied. . . 174 E.21 Wavefield modeled with the ultra deep streamer after deghosting, NMO-correction
andf −kfilter applied. . . 175 E.22 Wavefield modeled with the shallow streamer after NMO-correction and f −k
filter applied. . . 176 E.23 Primary wavefield modeled with the slanted streamer after NMO-correction
andf −kfilter applied. . . 177 E.24 Wavefield modeled with the slanted streamer after deghosting, NMO-correction
andf −kfilter applied. . . 178
E.25 Wavefield modeled with the slanted streamer after NMO-correction and f −k filter applied. . . 179 E.26 1D AVO comparison between primaries, data with ghost (primary + ghost) and
deghosted data. . . 180 E.27 Model build for the relative tuning effect from the ghost reflection as function of
offset and reflection depth. It is based on a constant angle model, and therefore represents the extreme case scenario. For non-constant angles (ray bending at interfaces), the angle of propagation will be more vertical with depth, and therefore decrease the relative tuning. . . 181 E.28 Velocity analysis for the shallow streamer. . . 182 E.29 Velocity analysis for the ultra deep streamer, . . . 183 E.30 Velocity analysis for the slanted streamer. . . 183 E.31 Forward/inverse quality control of the shallow streamer. Reference shot gather
A) and the forward/inverse (τ−p) shot gather B). . . 184 E.32 Forward/inverse quality control of the ultra deep streamer. Reference gather A)
and the forward/inverse gahter B). . . 185
Introduction
1.1 Aim and objectives
This thesis presents the results from a study proposed and supervised by Lundin Norway AS. The aim of this thesis is to increase the understanding of broadband seismic methods and what possible effects (positive or negative) it could introduce to pre-stack attribute interpretations. The evolution of broadband seismic method is derived from both technology and creative acquisition techniques, as well as in the processing stage. This thesis will focus on the acquisition- and processing based broadband seismic methods.
The common, and more or less established truth, is that shallow streamers yields a non- optimal response on low frequencies. By increasing the streamer depth the response on the low frequency content improves, but could damage the higher part of the frequency band, and create mirror images. A proposed solution is that by varying the streamer depth with offset, this would improve the response on both the higher and the lower part of the frequency band, hence increasing the bandwidth of the signal. Achieving a broader frequency band includes the removal of an inevitable sea-surface reflection known as the ghost reflection. The goal of increasing the bandwidth of the seismic wavelets is to increase resolution of seismic data and improve amplitude versus offset (AVO) interpretations and inversion.
1.2 Purpose and methods
The main purpose of this thesis was to develop synthetic studies on various pressure only streamer configurations and apply ghost removal algorithms in order to increase the frequency
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content. The inevitable sea-surface reflection gives rise to ghosts, which occur on the shot- and receiver side. The source and receiver ghost can be separately removed, and in this thesis, the focus will be on the receiver ghost reflection. In addition, we would like to see if the removal of the ghost reflection has any positive or negative effects on amplitude-related analysis. If for instance the algorithms used for removing the ghost, distorts the relative amplitude effect, it would lead to erroneous AVO interpretations, which could lead to increased exploration risk, if not accounted for.
This study comprise forward modeling by using a finite difference simulation to create the seismic wavefields, and processing of the generated synthetics in both Promax and Matlab.
The seismic modeling was carried out in the Madagascar software (www.ahay.org). Both conventional deghosting algorithms which have been derived by many others in the past, and more advanced algorithms are presented in this thesis. This thesis introduce the novel methods presented by Wang and Peng (2012) and Wang et al. (2013), which describes a bootstrap approach for self-determination of the time-delay parameter of the ghost. The bootstrap algorithm implemented in this thesis is an iterative method where one first defines a time vector, which is assumed to contain the correct time-delay. The algorithm then computes the error from each time-delay within the time-range and outputs the time-delay which gives the lowest error.
Because of the limited access to deghosting algorithms, they were written as direct imple- mentations of the equations presented by Wang and Peng (2012) and Wang et al. (2013) by using Matlab. The mirror migration step described by Wang and Peng (2012) before deghost- ing is not part of the algorithms presented. The presented results were analyzed, discussed and combined in to an overall discussion trying to link important observation with each other.
Modifications of the algorithms presented in this paper have also been discussed.
1.3 Background
In recent year’s hydrocarbon exploration in more difficult and complex geological areas have led the industry to develop new technology and new ideas for improving the resolution of seis- mic data. In reflection seismic acquisition we are dealing with band limited waves, originating from an air canon situated close to the sea-surface. These waves propagate as wavefields through the subsurface and reflect at boundaries (interfaces) which give rise to impedance contrast. Figure 1.1 illustrates a simplified seismic acquisition system comprising the pressure source and receivers, and illustrated by zero-phase wavelets propagating downwards and reflects at the interfaces. The closer the interfaces are from each other, the closer the wavelets get on our recorded seismic trace, which implies that vertical resolution is dependent on the wavelength and the separation between the two interfaces.
Figure 1.1:Conceptual sketch of a source (air gun) which gives rise to a pressure wave propagating downward and reflects at interfaces (boundary conditions). The reflected waves are recorded at the receiver in the time domain.
Figure 1.2:The concept of temporal resolution illustrated by use of reflection coefficients and a Ricker wavelet converging as the reflection coefficients move closer together. The acoustic impedance (AI) (from Amundsen and Landrø (2013)).
Sheriff (1989) defined resolution as the ability to distinguish two features from one another.
In order to separate the two interfaces from each other we have to be within what is commonly known as the ’resolvable limit’, which is the Rayleigh limit of resolution (Amundsen and Landrø, 2013). This limit is commonly accepted to be at one quarter of the wavelength (λ/4) of the seismic wave. Figure 1.2 illustrates the resolution problem using a Ricker wavelet in the time domain (temporal resolution). This stepwise illustration starts with the acoustic impedance (AI), which is the primary wave (P-wave) velocity multiplied with the density
AI =Vρ (1.1)
whereV is the velocity in the medium andρis the density. From the contrast in AI at a given interface we can compute the zero-incident reflection coefficient (considering pressure)
R=Vi+1ρi+1−Viρi
Vi+1ρi+1+Viρi
(1.2) where i denotes the ith layer. From (d) to (f ) the separation between interfaces, illustrated by the reflection coefficients (R), is decreasing, and we move from a resolved- to an unresolved separation. The wavelength is related both to the velocity and the frequency. There is not much to do about the velocity of the subsurface; however, the frequency response of the wavelet is up for alteration.
Figure 1.3:The concept of temporal resolution illustrated by altering the frequency response of the pulse (modified from Amundsen and Landrø (2013)).
Seismic resolution is closely connected to bandwidth (frequency range) of the wavelet.
In Figure 1.3 we can see that by increasing the high frequency part of the spectrum gives the wavelet a sharper mainlobe. In addition, by increasing the low frequency content of the wavelet gives smaller sidelobes. If the high frequency content is increased without taking into account the lower side of the spectrum we get large sidelobe-amplitudes, which could cause small-amplitude events to be lost within the sidelobes of adjacent high-amplitude reflections (Amundsen and Landrø, 2013). By taking into account the low frequencies, we get better resolution into deeper targets, and better stability for inversion (Soubaras and Dowle, 2010).
The ability of acquiring a broad frequency band is closely related to the removal of a special kind of multiple known as the ghost reflection, which is an inevitable reflection occurring at the sea surface. The ghost reflection has been described as far back as the 1960s (Lindsay, 1960; Hammond, 1962; Schneider et al., 1964), and is therefore a well-known phenomenon in marine seismic acquisition. Lindsay (1960) describes this reflection event as energy which reflects at an overlying discontinuity, and follows the directly downgoing wavefront (primary) from the shotpoint.
What Lindsay (1960) described is commonly known as the source ghost. Commonly, there are two kinds of ghosts described in the marine system, the source ghost and the receiver ghost, which are illustrated in Figure 1.4. From a physical point of view, the ghost reflection is a polarity reversed replica which interferes with the primary reflection, either destructively or constructively, and contaminates the original frequency band.
Figure 1.4:Conceptual sketch of the source and receiver ghost, which reflects at the sea-surface and interferes with the primary reflection in the time domain. As the upcoming wavefields contains a source ghost, a ghost of the receiver ghost will arise.
Dependent on the source and/or receiver depth it will either contaminate the low frequen- cies or high frequencies. In order to reconstruct the original seismic bandwidth we need to remove the effect from the ghost on both the source and receiver side. The main objective in broadband seismic is to remove the ghost from the seismic record, which gives a wider band than acquired in conventional seismic acquisition (Amundsen and Landrø, 2013). Different solutions have been proposed for both the source side and the receiver side. The common technique of removing the ghost signature from the seismic record is known as deghosting. As discussed by Sablon et al. (2013), Siliqi et al. (2013) and Fu et al. (2015) recent tests of marine synchronized multi-level source has shown great results to the source ghost problem. In order to handle the receiver ghost, creative acquisition based solutions such as the variable- depth streamer (VDS) (Soubaras and Lafet, 2011) and technological solutions such as the dual-sensor streamer (Tenghamn et al., 2007) has been commercialized. A simplified sketch of a conventional- compared to a VDS acquisition based system is illustrated in Figure 1.5.
In VDS acquisition, the main object is to create a diversity of the ghost arrival times, in which case the destructive and constructive interference is desynchronised with offset. This would lead to an enhanced ghost-free response in the final stacked section. Soubaras (2010) proposed a solution where the ghosts were removed at the imaging stage by performing a migration and mirror migration and apply a joint deconvolution algorithm of the two images after stack. This means that one has to preserve the receiver ghost throughout the processing sequence, which could cause problems using conventional processing programs (Sablon
Figure 1.5:Basic concept of acquisition with conventional streamer compared to a variable depth streamer. From this sketch it is rather implicit that the ghost arrival depends on the depth of the streamer, and by slanting the streamer we will achieve a diversity of ghost arrivals.
et al., 2012). In addition, the ghost can affect important pre-stack attribute interpretations, and is therefore beneficial to remove before imaging, as discussed by Zhang et al. (2012) and Soubaras and Lafet (2013).
Problems related to broadband seismic using hydrophone-only data will be investigated in this thesis. The thesis will be constrained to the ghost problem related to the receiver side.
Horizontal and VDS streamer configurations will be studied. The VDS acquisition methods are directly related to improved ghost removal, and problems related to the streamer con- figurations and deghosting will therefore be investigated. In conventional seismic the ghost time-delay is small, and therefore the primary and ghost interferes, which AVO commonly can performed directly on. If we want to enhance the low frequency part of the spectrum we can lower the streamer depth, but eventually the primary and ghost becomes separated, giving mirror images and possible interference from close features. In this case, deghosting is unavoidable. The same applies if the streamer is varying with depth, as the ghost eventually separates from the primary or causes variations in amplitudes with offset because of vari-
able ghost interference. In addition, one can question whether non-conventional streamer configurations could give a false AVO response compared to the conventional, because of the angle of incidence of non-vertical propagating waves could differ from configuration to configuration.
1.4 Structure of the Report
The rest of the report is structured as follows. Chapter 2 gives a short introduction to theory of pressure signals in marine seismic data, a more theoretical introduction to the ghost reflection, and how we can remove it from the seismic record. Also a short introduction to elastic wave propagation and AVO analysis will be given.
In chapter 3, results from the synthetic modeling and deghosting will be presented, with short explanations on observations. This chapter starts with a 1D example, to give a detailed analysis on to which extent we were able to recreate the pressure signal using the presented algorithms. The next example comprises a 2D study by introducing a finite difference method for simulating elastic wave propagation. This data was later used for AVO interpretations. At the end of the chapter the deghosting algorithms is applied to a real seismic data set, to see how the algorithms works on realistic data sets.
In chapter 4, results from the deghosting effect on AVO will be presented. The chapter starts with a simplified 1D approach, by using a constant angle model concept and uses seismic reflection theory to produce the AVO responses. The second example introduces the AVO responses created by the 2D synthetic finite difference simulation.
The main results from chapter 3 and 4 will be discussed in chapter 5. Modifications to the deghosting algorithms will also be discussed, along with an overall discussion which relates the observations to what others have presented in the past.
Theory
This chapter contains basic ghost reflection theory and how we can remove it from the seismic record, by using both conventional and non-conventional approaches. As mentioned in the previous chapter, this thesis will be constrained to the ghost problems arising at the receiver side, and therefore, the term ghost will now be referred to as the receiver ghost. The chapter will also contain some basic introduction to elastic wave propagation and its relation to AVO interpretation.
2.1 Pressure signals in marine seismic data
In conventional marine acquisition the streamer is kept at a certain level below the sea surface, recording difference in hydrostatic pressure. A pressure wavefield propagating in four dimensions can be written mathematically as the scalar wave equation (Yilmaz, 2001)
∇2P(t,x,y,z)= 1 c2
∂2P(t,x,y,z)
∂t2 (2.1)
wherex,yandzare the spatial coordinates,tis the time coordinate andc is the velocity in the acoustic medium. The operator∇2is the spatial Laplacian operator which is defined as
∇2= ∇ · ∇ = ∂2
∂x2+ ∂2
∂y2+ ∂2
∂z2 (2.2)
Any function which describes a wave propagating in time and space is useful, however, these derivations will deal with complex exponentials. By following the example of Stein and Wysession (2015) for harmonic plane waves, a wave propagating in time and 2D space (xand
11
z) can be written as
P(t,x,z)=A0e−i(ωt−kxx−kzz) (2.3) whereA0is the amplitude of the plane wave,i is the complex number (i.e. p
−1),ωis the angular frequency,kx andkz are the wavenumbers in the x-direction and z-direction respectively. By inserting Equation 2.3 into Equation 2.1 and taking the partial derivative with respect tot,xandzwe get
i2kx2e−i(ωt−kxx−kzz)+i2kz2e−i(ωt−kxx−kzz)= 1
c2i2ω2e−i(ωt−kxx−kzz) (2.4) which gives the solution to the wavenumberk, or dispersion relation.kcan be defined as
k=ω c =
q
k2x+k2z (2.5)
To see what happens at the sea surface, consider a pressure-fieldP−(t,z) propagating vertical in the negative z-direction (upwards), as shown in Figure 2.1. When this pressure field reflects at the sea surface, it gives rise to a pressure fieldP+(t,z) which propagates in the opposite direction, i.e. positive z-direction.
Figure 2.1:Conceptual sketch of a wave front (plane wave) propagating upwards and reflecting at the free surface. At the sea-surface the pressure vanishes, and is therefore considered to act as an free surface.
The combined pressure field is then at a superposition of these
P(t,z)=P−(t,z)+P+(t,z) (2.6) At the sea-surface (free surface) the pressure field vanishes (Amundsen, 1993) and we can write the up- and downwards propagating pressure as
P−(t,z=0)+P+(t,z=0)=0 (2.7) which gives
P−(t,z=0)= −P+(t,z=0) (2.8) Writing the vertical propagating pressure field as simple harmonics gives
P−(t,z=0)=A0e−i(ωt) (2.9) and
P+(t,z=0)=A1e−i(ωt) (2.10) Inserting Equation 2.9 and Equation 2.10 into Equation 2.8 gives
A0
A1= −e−iωt
e−iωt = −1 (2.11)
whereA0/A1is the sea-surface reflection coefficient.
2.2 The Ghost reflection
1D approximation
The sea-surface reflection, described in the previous section, is known as the ghost reflection, which occurs at the source and receiver side. This reflection contaminates the data, which is best illustrated in the frequency domain, where we observe amplification or suppression at certain frequencies. Figure 2.2 illustrates the effect from the receiver ghost on the primary reflection in the frequency domain, using a simplified non-negative pulse as the initial source pulse. As described in the former section the combined pressure-field can be written as a sum of the upwards and the downwards propagating pressure-fields. It can also be written as a convolution in the time domain (Gelius and Johansen, 2010), as the upwards propagating pressure is convolved with the filter known as the ghost filter
(P−∗G)(t)= Z +∞
−∞
P−(τ)G(t−τ)dτ (2.12)
or more simply as
P(t)=P−(t)∗G(t) (2.13)
where∗ denotes the convolution,G(t) is the ghost filter andP−(t) is the primary re- flection. Equation 2.13 can be expressed in the frequency domain, using the Fourier trans- form/convolution relation
P(t)=P−(t)∗G(t)↔P(ω)=P−(ω)G(ω) (2.14) This implies that convolution in the time domain is equal to multiplication in the fre- quency domain. One way to derive the 1D ghost filterG(ω) is by using the linear relationship and time shift of the Fourier transform. We have for the combined pressure
P(ω)= Z +∞
−∞
P(t)e−iωtdt (2.15)
and for the upwards propagating pressure
P−(ω)= Z +∞
−∞
P−(t)e−iωtdt (2.16)
and for the down going pressure with a time-delayt0
P+(ω)= Z +∞
−∞
P+(t−t0)e−iωtdt=P−(ω)e−iωt0 (2.17) wheret0is the time-delay between the primary and ghost. Using the linear relationship of the Fourier transform we can write the recorded pressure in the frequency domain as
P(ω)=P−(ω)−P−(ω)e−iωt0 (2.18) Inserting Equation 2.18 into Equation 2.14 (frequency domain) gives the ghost filter
G(ω)=P−(ω)−P−(ω)e−iωt0
P−(ω) =1−e−iωt0 (2.19)
Figure 2.2:Basic concept of the ghost reflection recorded at different depths, and how the combined pressure is recorded in the time- and frequency domain. The characteristic shape of the frequency domain arising from the ghost reflection is dependent on both receiver depth and angle of propagation.
Figure 2.3:Basic illustration of the ghost filters effect on a primary signal in the frequency domain.
The amplitude spectrum of the ghost filter can be expressed further by using trigonometric identities, which gives
|G(ω)| =2|sin
³ωt0 2
´
| (2.20)
The characteristic shape of the ghost filter and its effect on the primary reflection in the frequency domain are illustrated in Figure 2.3. The time-delay of the ghost is dependent on receiver depth and the incident angle. By using simple geometrical relation the time-delay can be defined as
t0(θ)=2hcos(θ)
c (2.21)
whereh is the receiver depth (or streamer depth) andθ is the reflection-angle at the sea-surface. In Figure 2.3 we can see that the ghost filter repeat zero points in the spectrum.
These are known as notches, and from Equation 2.20 we see that the notches repeats at ωnt0
2 =nπ (2.22)
wherenis any integer. Using the relationω=2πf, the notch frequencies in Hz are written as
fn=n/t0(θ) (2.23)
or for zero incident angle
fn=n c
2h (2.24)
The sinusoidal shape of the ghost filter gives the first notch always at 0 frequency.
In addition, deeper streamer depth lead to more notches within our desired frequency band. The time-delay parameter can be generalized if we consider constant-angle raypaths (not obeying Snell’s law), and the ghost reflection-angle is the same as the primary waves angle of incidence (i.e. plane waves), we can further express the time-delay dependency on reflection depth and offset
ti(xi,zi)=
2hcos³
tan−12z xi
i−xitan(α)
´
c (2.25)
wherexi is the offset andzi is the depth down to the reflector, andαis streamer-angle.
See Appendix C for the derivation of this equation. Figure 2.4 show the time-delay variation with reflection depth and offset for a constant-angle model.
Figure 2.4:Time-delay model, which illustrates in a general matter how the time-delay (t-d) could vary dependent on the streamer configuration, by using a constant-angle approximation. The figure clearly shows that a shallow streamer at 10m depth has a limited diversity of ghost time-delays and the ultra deep and slanted streamer have large diversities.
The combination of Equation 2.20 and the time-delay models in Figure 2.4 illustrates basically the idea behind varying the streamer-depth with offset. A diversity of time-delays with offset gives a diversity of notches, which will ideally fill in lost frequencies after stack.
2D approximation
If we want to take the angle of propagation into account, the ghost filter can be further expressed in the frequency-wavenumber (f −k) domain by taking the temporal and spatial Fourier transform (2D) of the wavefields. The 2D Fourier transform ofP(t,z) is defined in the Fourier domain as (Yilmaz, 2001)
P(ω,kz)= Z +∞
−∞
Z +∞
−∞
P(t,z)e−i(ωt+kzz)dtdz (2.26) In the 2D case we could consider the ghost wavefield to be a spatially shifted wavefield dependent on the streamer depth. We write the downgoing ghost wavefield as
P+(ω,kz)= Z +∞
−∞
Z +∞
−∞
P(t,z−2h)e−i(ωt+kzz)dtdz=P−(ω,kz)e−i kz2h (2.27) The concept of a spatially shifted plane wave is illustrated in Figure 2.5, using the concept of a mirrored streamer at the sea-surface.
Figure 2.5:Conceptual sketch of a plane wave propagating in the negative z-direction. Here we illus- trate that the ghost signalP(ω,kz)e−j kz2hcan be regarded as a spatially shifted versionP(ω,kz), which is recorded at a streamer, mirrored at the sea surface, instead of following the reflection path(green line)
By using the same Fourier relation as with the 1D time domain, the 2D ghost filter can now be written as
G(ω,kz)=1−e−i kz2h (2.28)
We can write the wavenumber in the z-direction as
kz= q
k2−kx2 (2.29)
and the wavenumber as
k=ω c =2πf
c (2.30)
Which gives the ghost filter
G(f,kx)=1−e−i4πh
q
(cf)2−kx2
(2.31) wheref is frequency inH zandkx is the wavenumber in the x-direction with unit 1/m. A 2D ghost filter in theF K domain with streamer depthh=10mis shown in Figure 2.6.
Figure 2.6:Ghost filters amplitude spectrum for receiver depth of 10min thef−kdomain. This filter takes the angle of propagation into account for a more correct estimation of the ghost time-delay
2.3 Deghosting
Conventional approach
In order to remove the ghost reflection we can in theory multiply the combined signal with the inverse of the ghost filter, which is defined in 1D as
P−(ω)=P(ω)
G(ω) (2.32)
and in 2D as
P−(ω,kz)=P(ω,kz)
G(ω,kz) (2.33)
A ghost filter inversion in the frequency domain for receiver depths at 10mand 60mare displayed in Figure 2.7. The problem is that the inverse ghost filter approaches∞towards notches (|G(ω)| →0). In an ideal world this simple inversion would reconstruct the primary signal, but is unstable towards notch frequencies, and undefined at notches because of the division by zero.
One solution is to place the streamer at a depth where our notches occur outside our desired frequency band, and then do a low cut close to the zero notch and high cut before the second. According to Soubaras (2013) it is possible to bypass the lowcut/highcut by stabilizing the zero division at notches with a dampening factorε, which gives the 1D ghost filter
P−(ω)=P(ω) G∗(ω)
G(ω)G∗(ω)+ε2 (2.34)
and for 2D
P−(ω,kz)=P(ω,kz) G∗(ω,kz)
G(ω,kz)G∗(ω,kz)+ε2 (2.35) WhereG∗(ω) denotes the complex conjugate ofG(ω). This algorithm dampens the in- version close to notches, stabilizing the inverse filter. However, since theεis controlling the amount of amplification near the notches, which can lead to over-amplification of noise, it has to be picked with caution.