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Multiplicative random seismic noise caused by small-scale near-surface
scattering and its transformation during stacking
Andrey Bakulin1, Dmitry Neklyudov2, and Ilya Silvestrov1
ABSTRACT
Even after sophisticated processing, land seismic data in com-
plex areas exhibit weak and distorted prestack reflections with
low coherency. Usually, the local stacking methods reveal clear
reflections. However, the absolute level of amplitude spectra after
such stacking experiences a substantial decline across the entire
frequency band, reaching −10 to −25 dB. In addition, stacking
leads to a significant and progressive loss of higher frequencies.
We describe mathematical and intuitive physical models for
multiplicative random noise that could consistently explain these
field observations at least semiquantitatively. Multiplicative noise
is represented by random timeshifts (residual statics) and random
phase perturbations different for each frequency. Residual statics
explain the progressive loss of higher frequencies. On the other
hand, phase perturbations lead to a severe loss of coherency on
prestack gathers and produce a strong downward bias or loss of
broadband amplitudes after stacking. We find that both types of
multiplicative noise can be physically generated by near-surface
scattering layers with small-to-medium-scale geologic hetero-
geneities. We speculate that such multiplicative distortions can
be referred to as seismic speckle noise well established in optics
and ultrasonics. Furthermore, we derive the fundamental proper-
ties of how random multiplicative noise transforms while stacking.
The first essential finding reveals that stacking produces an un-
biased estimate of the clean signal phase. The second finding finds
the mathematical relationship between the frequency-dependent
loss of stacked amplitude and the standard deviation of residual
statics and phase perturbations. These findings serve as a theoreti-
cal justification for the previously proposed methods of phase sub-
stitution and phase corrections and open the way to efficiently
address random multiplicative noise in seismic processing.
INTRODUCTION
In exploration geophysics, it is commonly assumed that the
biggest challenge of land seismic data is associated with strong
near-surface arrivals (surface waves, refractions, guided modes,
etc.) superimposed on top of weak but coherent reflections. It is fur-
ther reasoned that, when slow near-surface arrivals are not adequately
sampled (aliased), the resulting energy can appear as incoherent
noise, causing ambiguity between signal and noise (Ait-Messaoud
et al., 2005). In the past, large source/receiver arrays were used to
suppress near-surface noise, particularly with lower apparent veloc-
ities (Meunier, 2011). However, although intraarray dense sampling
was practical, 3D sampling remained more sparse for economic rea-
sons. With time, more dense acquisition geometries (at least in two
out of four spatial directions) are gradually becoming more prevalent
(Regone et al., 2015), although with smaller arrays or single sensors
attempting to reduce aliasing and eliminate undesirable array effects.
Nevertheless, raw and processed data from desert environments often
look incredibly challenging, suggesting that something else may be
happening in our data. A simple illustration of this fact is that a rou-
tine first-break picking of the strongest first-arrivals events often be-
comes extremely challenging on single-sensor data from such regions
(Khalil and Gulunay, 2011;Bakulin et al., 2019a,2019b).
Broadly speaking, noise can be organized (coherent) or unorgan-
ized (random). If we focus on primary reflections as a seismic sig-
nal, all other coherent seismic arrivals represent organized coherent
noise. Such noise can be considered deterministic. If a shot is re-
peated, this type of noise would perfectly reproduce. Random ad-
ditive noise is another category caused by natural or manmade
activities that have irregular appearances on the seismic records.
If a shot is repeated, the signal remains the same but random noise
changes. Because activities causing the noise occur independently
Manuscript received by the Editor 28 December 2021; revised manuscript received 23 March 2022; published ahead of production 24 May 2022; published
online 20 July 2022.
1Saudi Aramco, EXPEC Advanced Research Center, Dhahran, Saudi Arabia. E-mail: a_bakulin@yahoo.com; ilya.silvestrov@aramco.com.
2Institute of Petroleum Geology and Geophysics SB RAS, Novosibirsk, Russia. E-mail: dmitn@mail.ru (corresponding author).
© 2022 Society of Exploration Geophysicists. All rights reserved.
V419
GEOPHYSICS, VOL. 87, NO. 5 (SEPTEMBER-OCTOBER 2022); P. V419–V435, 18 FIGS.
10.1190/GEO2021-0830.1
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DOI:10.1190/geo2021-0830.1
from a seismic operation, random noise is typically not correlated
with signal justifying its assumed additive nature.
A different kind of “random”noise is caused by scattering in the
highly heterogeneous near-surface or deeper layers (Figure 1a). If
small and medium-scale heterogeneities cause scattering, the result-
ing wavefield is known to change rapidly even with a slight change
in an observation geometry such as source/receiver, angle, azimuth,
etc. Several studies considered wavefield perturbations caused by
such heterogeneities assuming that the media can be represented
as a random velocity field with some stochastic properties (Ikelle
et al., 1993;Shapiro et al., 1996;Sivaji et al., 2002;Borcea et al.,
2006). They show that traces experience complex distortions, and
the noise appears random on multichannel seismic records. Never-
theless, such noise remains perfectly reproducible, meaning that if
the experiment is repeated with the same acquisition geometry, the
noise will be identical because seismic scattering is a deterministic
physical phenomenon.
The preceding description closely resembles what is known as
speckle noise or just speckle in optics and acoustics (Goodman,
1976,2020). In contrast to more conventional additive noise, such
noise is multiplicative, meaning that the signal is subject to filtering
or multiplication in the frequency domain. In optics and acoustics,
speckle is generated when light interacts with rough surfaces or
volumetric scatterers. In both cases, surface feature or scatterer size
is smaller than the wavelength, whereas a significant number of
features are present inside the Fresnel zone or the elementary scat-
tering volume (Figure 1a, magnified). The resulting reflected light
signal represents a superposition of multiple interfering arrivals
with different phases. Such interference causes strong variations
of amplitude/intensity that appear random and obscure optical im-
ages. Forward scattering of seismic arrivals propagating through a
near surface with small-scale scatterers would lead to a very similar
phenomenon manifested as random variations of amplitudes and
phases of every arrival traversing through such a scattering layer.
An example of a synthetic aperture radar (SAR) image with speckle
noise characterized by granular amplitude/intensity distortions is
shown in Figure 2a. For comparison, Figure 2b and 2c displays pre-
stack gather and final stack images from a land single-sensor seis-
mic data set. All images possess characteristic random granular
noise that is abundantly present and does not stack out on the final
image (Figure 2c). Therefore, we can refer to such distortions as
“seismic speckle”—a natural extension of an established defi-
nition used in optics and acoustics.
The multiplicative noise model itself is not new
in geophysics. For example, a widely used surface-
consistent deconvolution relies on a multiplicative
noise model. It assumes that the frequency-depen-
dent phase variations are caused by a combination
of source/receiver coupling and near-surface ef-
fects (Taner and Koehler, 1981;Cary and Lorentz,
1993). However, multipliers in this model are pre-
sumed to be deterministic operators independent
of time and satisfy the surface consistency hypoth-
esis. Surface consistency assumes that operators
depend only on source and receiver positions
and offset. They are applied to the entire trace, as-
suming that all arrivals require identical correc-
tions (no time dependency). Although surface-
consistent deconvolution has proven useful in cor-
recting for some phase variations in seismic data, it
fails to address strong random phase variations
caused by small- and medium-scale near-surface
scattering. Physical scattering does not satisfy as-
sumptions of surface consistency because interfer-
ence patterns vary quickly and unpredictably in
time and space. Therefore, scattering distortions
are unique for each wavepath (Figure 1a)and
time-dependent. In summary, geophysics thus
far has only used deterministic multiplicative mod-
els with additional simplifying assumptions of sur-
face consistency that make the problem tractable
and overdetermined (the number of equations is
larger than the number of unknowns). A more gen-
eral multiplicative model is required to address
scattering distortions. However, such a problem re-
quires a stochastic approach (random noise) as
done in speckle to become tractable.
Wave propagation theory in random media sug-
gests that scattering noise can be represented as
space- and frequency-dependent multipliers with
Figure 1. (a) Origin of seismic multiplicative random noise caused by seismic speckle
born in the near-surface scattering layer and (b) conceptual depiction of local ensemble
used for prestack data enhancement.
Figure 2. Images with speckle noise from SAR and seismic data: (a) SAR image from
Valsesia (2020) has granular artifacts on the 2D areal image; prestack gather (b) and
(c) poststack image from a land single-sensor seismic data after processing (from
Bakulin et al., 2020a) show time-offset section with a “speckled”appearance due to
random phase and amplitude variation. A similar granular texture of the amplitude
is observed on every time slice. Neither surface objects nor subsurface geologic layers
of interest under imaging are expected to contain observed high-frequency physical
variation.
V420 Bakulin et al.
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DOI:10.1190/geo2021-0830.1
specific distribution functions depending on the media’s stochastic
properties (e.g., Müller and Shapiro, 2001). However, random-media
wave propagation theory developments appear far removed from
seismic processing practice. Finally, seismic studies of random-media
wave propagation did not relate to multiplicative speckle noise,
unequivocally recognized and analyzed in acoustics and ultrasound.
This study highlights the abundance and significance of a highly var-
iable multiplicative noise in seismic data, at least in a desert environ-
ment. We believe that a stronger appreciation of multiplicative
speckle noise may lead to substantial practical implications similar
to what happened in optics/laser/SAR domains since the 1960s
(Beckmann, 1962,1964;Goodman, 1976,2000,2007,2020;Mor-
eira et al., 2013) and ultrasonic/acoustics fields since the 1980s (Ab-
bot and Thurstone, 1979;Wells and Halliwell, 1981;Fink and
Derode, 1998;Damerjian et al., 2014). Mitigating the multiplicative
speckle noise is a complex problem that is still not fully solved
(Goodman, 2020). However, the growing arsenal of methods based
on powerful speckle-based statistical models led to significant
advances in laser, SAR, and ultrasound imaging (Goodman, 2007;
Moreira et al., 2013;Damerjian et al., 2014) as well as cross polli-
nation between different domains (Goodman, 2020).
The optical and ultrasound domains often focus on monochromatic
waves and intensity images. As a result, phase and especially the fre-
quency dependence of the phase are less studied. In contrast, seismic
data rely on broadband data where accurate reconstruction of phase
information is a key to successful seismic processing and imaging. It
is well acknowledged that if the amplitude spectrum is perturbed, but
the phase spectrum remains intact, the main signal events can still
be accurately detected in terms of the time and spatial positions
(Oppenheim and Lim, 1981;Blackledget, 2006;Ulrych et al.,
2007;Bakulin et al., 2020b). Although this fact is known, it often
is described qualitatively, and its implications are not fully appreciated
in seismic imaging. To correct this, we put a mathematical derivation
in Appendix Ainspired by Lichman (1999) to demonstrate that arrival
time information of seismic events is encoded in the phase spectrum.
More specifically, it underscores that the amplitude spectrum (equa-
tion A-3) depends only on traveltime differences. The phase spectrum
is a function of actual traveltimes (equation A-4). This critical fact is a
cornerstone for understanding the effects of multiplicative noise and
designing methods to mitigate it.
Bakulin et al. (2018a,2020a) show that reflections can be
extremely weak and distorted in land seismic data when acquired
with small arrays or single sensors. Weak signals must be collected
and averaged using the multichannel seismic data redundancy to
become processable and imageable. Furthermore, phase variations
appear to be a critical culprit behind severe distortions. Enhance-
ment techniques use information about local traveltimes of desired
arrivals to provide output data with an improved signal-to-noise ra-
tio (S/N). Bakulin et al. (2020b,2020c,2021) put forward phase
substitution and phase correction methods. They speculate that
the phase spectra of enhanced data should be a relatively close es-
timate of the local phases of “actual”undistorted signals. In other
words, they assume that the positions and phases of reflected events
in a trace become more or less correct after enhancement. Khalil and
Gulunay (2011) also make a similar assumption for the first arrivals
when deriving intraarray statics for single-sensor data from the des-
ert environment. This study puts forward a physical and mathemati-
cal model that can theoretically justify this statement using realistic
assumptions about random multiplicative noise.
We start the paper by examining typical seismic records from the
desert environment exhibiting the effects of complex surface scat-
tering and summarizing the main observations on how such data
transform during the local stacking process. Next, we show the
numerical modeling results with a near-surface clutter layer that rep-
licates essential features of the real data. Modeling confirms that
multiple scattering causes strong frequency-dependent phase distor-
tions that appear random and cause severe loss of coherency. We
then build a mathematical model for seismic speckle approximating
the noise as random multiplicative noise. Finally, we examine the
effects of random multiplicative noise on local stacking. Two re-
markable fundamental properties are discovered. First, summation
leads to an estimate of the true undistorted phase spectrum. Second,
the amplitude of the stacked response shows a characteristic fre-
quency-dependent loss that closely replicates observations from
the field data. These findings cement our understanding of the seis-
mic speckle as a random multiplicative noise abundantly present in
challenging seismic data. At the same time, discovered fundamental
properties pave the way to the elimination of seismic speckle by
novel processing algorithms. This new recognition also opens
the door to transfer learning from a plethora of existing approaches
to suppress multiplicative speckle noise developed in optics, acous-
tics, and ultrasonic.
WHAT DO WE OBSERVE IN REAL DATA?
We first examine two different data sets from the desert environ-
ment paying attention to subtle details such as the variation of phase
and amplitudes and their transformation during the stacking process.
A common-midpoint (CMP) gather from the first 3D data set is
shown in Figure 3. This is the legacy data with 72-geophone groups
from a very challenging geophysical area. Although ordinarily, such
legacy data are of a high S/N, this case represents an exception from
an area with the most complex near surface. The input data have al-
ready been passed through a standard time processing flow for land
data, including linear and random noise removal, surface-consistent
scaling, deconvolution, etc. (see e.g., Taner and Koehler, 1981;Cary
and Lorentz, 1993;Chan and Stewart, 1994;Meunier, 1999;Liu et al.,
2006). Nevertheless, the prestack signal remains very weak, and there
are no visible reflections in the gather. In addition, we observe that the
entire gather shown in Figure 3a remains similarly speckled from top
to bottom and from small offset to large. Although it is not uncom-
mon that unsuppressed remnants of intense direct and scattered
groundroll often can create a residual “noise cone”restricted by direct
surface-wave arrivals, even events outside such a noise cone remain
obscured in this case, suggesting a different mechanism for distor-
tions. Figure 3b shows the same CMP gather after data enhancement
based on local stacking using nonlinear beamforming (NLBF) (Ba-
kulin et al., 2020a). The detail of the actual enhancement technique
itself is of secondary importance here. Other local-stacking tech-
niques could achieve a similar result (Baykulov and Gajewski,
2009;Berkovitch et al., 2011;Buzlukov and Landa, 2013;Bakulin
et al., 2018a,2018b). We note that approximately 200 neighboring
traces are used in the local summation to enhance each original trace
in this case. After the enhancement, the reflections are easily recog-
nizable in the entire offset range. However, the high-frequency con-
tent of the signal becomes strongly suppressed (Figure 3c)whereas
reflection events become overly smoothed. In other words, beam-
forming replaces the jittery amplitude and phase along the event with
relatively smoothly varying quantities.
Multiplicative scattering noise V421
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DOI:10.1190/geo2021-0830.1
We jump to a second data set from a more recent and dense ac-
quisition using smaller nine-geophone arrays in a different area also
from the desert environment. Again, we observe very similar behav-
ior. Figure 4a shows a CMP gather after processing with a little
visible signal. Local stacking with NLBF reveals underlying reflec-
tions (Figure 4b). Likewise, higher frequencies greater than 20 Hz
are greatly diminished (Figure 4c).
In addition to challenging reflection data quality, such data also
typically exhibit very complex first arrivals. Khalil and Gulunay
(2011) and Bakulin et al. (2018b,2019b) present multiple examples
of jumbled early arrivals with low coherency from nine-geophone
arrays and single-sensor data. To enable first-break picking and full-
waveform inversion (FWI), early arrivals must be significantly pre-
processed using local stacking. Finally, there were cases of single-
sensor data deemed not usable for near-surface model building us-
ing traveltime tomography and FWI. Suppose we distill common
traits shared by these and many other challenging data sets from
the desert environment. In that case, we arrive at three critical ob-
servations:
Observation 1: Even after sophisticated processing, reflections
on prestack data remain distorted with low coherency or even
untrackable. Such distortions are not localized in time-space but
instead spread over the entire gather. In addition, first arrivals also
are cluttered, making it difficult to pick first breaks or use early
arrivals for FWI.
Observation 2: After applying local stacking, reflections crop up
very clearly, but the absolute level of amplitude spectra experiences
significant bias downward (−15 to −20 dB in the shown example)
across all frequencies.
Observation 3: There is a significant and progressive loss of
higher frequencies after local stacking.
Although it is expected that stacking in the presence of residual
statics acts as a low-pass filter (Berni and Roever, 1989;Marsdsen,
1993), the lowest frequencies below 10–20 Hz should not be signifi-
cantly affected. However, we observe −15 to −20 dB amplitude bias
at those same low frequencies, even though a standard fold normali-
zation is used in stacking. Although seismic waves do attenuate with
depth, note that we operate with prestack events already propagated
to the same depth in local stacking. We merely examine the trans-
formation of the frequency content of the same event before and after
stacking. Therefore, the loss of higher frequencies at hand (observa-
tion 3) has little to do with any actual attenuation during propagation.
Finally, we stress that data are examined after preprocessing that in-
clude conventional linear and random noise attenuation steps in both
examples. Something does not add up in a traditional point of view of
coherent reflections superimposed by “background near-surface
noise,”even if we allow some residual statics to be present. We shall
seek and offer an alternative explanation to explain observations 1–3
without contradictions. This alternative explanation is a distortion
of the signal itself caused by medium- and small-scale scattering
(Figure 1a).
Figure 3. NMO-corrected CMP gather from an area with complex
near surface: (a) after standard processing, (b) the same gather, but
after additional enhancement with NLBF based on local stacking
(aperture of 300 m), and (c) amplitude spectra of original (blue) and
enhanced (red) data. Observe low coherency in (a) and high coher-
ency in (b), but associated with significant amplitude bias and
progressive loss of higher frequencies. Figure 4. Same as Figure 3, but for a different data set from the
desert environment. The beamforming aperture is 300 m.
V422 Bakulin et al.
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DOI:10.1190/geo2021-0830.1
NEAR-SURFACE SCATTERING LAYER AND ITS
IMPACT ON SIGNAL PHASE
To illustrate the impact of scattering on the seismic signal, we
resort to the simple acoustic model shown in Figure 5a with a
near-surface scattering layer. We deliberately select acoustic mod-
eling to exclude the effects of groundroll, converted waves, and
other elastic arrivals. Thus, we take a large part of the organized
coherent near-surface noise out of the equation from the start to
emphasize that the proposed mechanism for signal distortion does
not require elastic noise. Instead, only sufficient small- and
medium-scale heterogeneities in the near-surface layer are neces-
sary to create volumetric random scattering noise referred to as
speckle in optics and ultrasonics.
To interpret somewhat jumbled seismic gathers, let us remind
ourselves of the meaning of coherency that we usually expect on
multichannel seismic records in terms of the signal phase. A visible
coherent event in multichannel data corresponds to a time window
with mild phase angles variations. A constant phase surface pro-
vides a coherent event in multichannel data in a noise-free case.
Let us illustrate this statement using synthetic and real data.
We contrast two common-shot gathers calculated using the finite-
difference modeling with the same subsurface model except for the
differing near-surface layer. The first case has a homogeneous near-
surface layer (Figure 5b). In contrast, the second has a near-surface
scattering layer (Figure 5c) modeled as a random clutter.
For analysis, we choose the third reflection event marked by the
arrows. A time window with a width of 200 ms was extracted along
the chosen reflector. Corresponding windowed data are shown in
Figure 6after normal moveout corrections. As one can see, the tar-
get reflected arrivals are perfectly aligned for “clean”data (homo-
geneous near surface). In contrast, the same arrival is severely
broken up in the presence of a near-surface scattering layer (Fig-
ure 6b). A Fourier transform is applied to the extracted windowed
data. In Figure 7, we present phase angles as a function of trace
index at a fixed frequency of 10 and 20 Hz (dominant frequency
used in acoustic finite-difference modeling). Phases of clean data
are nearly constant, whereas phases of perturbed data vary abruptly.
A similar behavior holds for any other frequency with randomly
looking phase jumps. In addition, there is no apparent connection
between phase jumps observed at different frequencies. Such ran-
dom phase fluctuations are the primary cause of why one cannot
see coherent arrivals in the “perturbed”data (Figure 6b). After
beamforming, phase angles acquire a relatively smooth behavior
(Figure 7). As a result, one can see the coherent event in the cor-
responding gather (Figure 6c).
From this numerical experiment, we can say that the main reason
why even after elaborate processing, one can barely see coherent
events in land seismic data is random phase variations of “distorted
reflections.”We do call them distorted reflections for two main
reasons. First, they reside in the same time window as undistorted
arrival in the clean model, called ballistic arrival. Second, they are
not contaminated by the superposition of “background”noise but
rather consistently experience complex interference. Random-like
phase variations make reflections poorly visible. Displaying ampli-
tude at a fixed frequency (Figure 8), we observe random-like fluc-
tuations, characteristic for optical or ultrasound speckle noise
(Goodman, 2007). Local stacking of data with such phase perturba-
tions leads to comparable amplitude bias and progressive loss of
higher frequencies, as noted in the field data (observations 2 and 3).
Similar phase behavior is seen in real seismic data. Figure 9
presents the time-windowed data taken from NMO-corrected
CMP gather shown in Figure 3a. For analysis, we time-gated a sin-
gle visible reflected event. Phase angles taken at the frequencies 10
and 30 Hz for the original and enhanced data are presented in Fig-
ure 10. The phase angles of the original data vary chaotically in the
allowable interval ½−π;π. In contrast, phases of enhanced data have
very smooth variations.
In the synthetic example with the acoustic model, observed noise
allows a more straightforward physical interpretation. We have
eliminated surface waves and shear waves from the equation. There-
fore, we are left with a scattering of the P-wave energy only. It may
be tempting to assume that what we see in Figure 5c is a superpo-
sition of many diffractions from the near-surface scatterers that
overlay and complicate undistorted primary reflections. However,
simple numerical experiments disprove such an interpretation.
Figure 5. The synthetic model with near-surface clutter and associated data: (a) five-layer acoustic model with near-surface layer modeled as a
random clutter (velocity variation of 200 m/s and correlation length of 30 m), (b) common-shot gather when a homogeneous layer replaces
near-surface clutter, (c) gather in the model (a) with near-surface clutter, and (d) enhanced gather from (c) after NLBF. The arrow marks the
target reflector. Ricker wavelet with 20 Hz central frequency is used as a source signature.
Multiplicative scattering noise V423
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DOI:10.1190/geo2021-0830.1
Figure 11a–11c shows wavefields in the full five-layer model versus
reduced two-layer model (no deep reflectors) and their difference.
Figure 11b shows the entire scattered wavefield induced by hetero-
geneities of the near-surface layer. As expected, such noise is the
strongest near early arrivals (single scattering). It decays away from
the first breaks or toward later times, as is expected for multiple
scattering. In particular, Figure 11c shows the difference containing
four primary reflectors and all noise generated squarely by them-
selves. First, we can see that reflection events on the full gather
(Figure 11a) and difference field (Figure 11c) remain similarly dam-
aged, suggesting that the superimposed background noise removed
by differencing is not a significant explanation to observed distor-
tions. In addition, the reflected events themselves induce the halos
around them. Finally, stronger reflectors create more intense halos
(compare Figures 5b and 11c).
Figure 11d further emphasizes the point that a single direct arrival
propagating through the near-surface layer is indeed strongly dis-
torted in the vicinity of the ballistic arrival itself. All these observa-
tions could be readily explained by the mechanism from Figure 1a,
typical for (volumetric) speckle noise. Comparing reflector R3 in
Figure 11c with direct arrival in Figure 11d, it is evident that distor-
tions due to two-way propagation are more pronounced than those
caused by a one-way transmission through a near-surface scattering
layer. Because reflections and multiples themselves induce scattering
noise, it is plausible to relate Figures 3a,4a,and11a. In the synthetic
case, only a handful of reflectors induced overlapping halos of scat-
tering noise. The desert environment is characterized by an almost
continuous sequence of subsurface reflectors with strong contrasts
(Alexandrov et al., 2015). Therefore, the proposed mechanism can
plausibly explain the almost continuous carpet of scattering noise
on real data, supporting the previous observation 1.
We conclude that the noise mechanism is likely related to the
speckle concept, as depicted in Figure 1a. Many forward-scattered
waves arrive near-ballistic traveltime, creating a complex interfer-
ence that jumbles the total waveform (Figure 11c and 11d). Prunty
and Snieder (2017) highlight the critical role of such near-ballistic
forward scattering to explain the memory effect associated with
acoustic speckle. The multiplicative nature of speckle noise ex-
plains why a stronger signal generates stronger
noise. The sequence of primary reflections and
multiples seen in Figure 11a induces correspond-
ing “halos”or “ornaments”around each of them,
with stronger events creating more energetic
halos. Seismic processing and imaging rely on
tracking and eventually summing redundant seis-
mic traces providing illumination from multitu-
dinal offsets and azimuth. In particular, time
processing and imaging rely on consistent wave-
forms within prestack data that imply smoothly
varying phases. We observe that local stacking
performs phase “ordering.”As a result, events
become coherent and visible (Figures 6c and
7), thus conditioning the data for processing
and imaging. However, significant side effects
occur while stacking, such as amplitude bias
and loss of higher frequencies that limit the ver-
tical resolution of the seismic images. Currently
used seismic processing algorithms are not de-
signed to tackle random multiplicative noise. If
we know the true velocity model, migration
may correct such phase variations and lead to
correct images. However, it is almost impossible
to precisely recover small-scale velocity varia-
tions in practice. If we only estimate the
smoothed background velocity model, migration
would similarly suffer from phase perturbations,
producing poor focusing (He et al., 2017).
Armed with the preceding intuitive understand-
ing of the nature of seismic speckle (Figure 1a)
and numerical experiments, we can mathemati-
cally describe it as multiplicative random noise.
As outlined in the “Introduction”section, such a
description is well established in optics and ultra-
sound. In the next section, we put forward a
mathematical model of multiplicative random
seismic noise and analyze how it affects the am-
plitude and phase of the data during the local
stacking process.
Figure 6. Time-windowed (200 ms) synthetic data around the target reflector from
Figure 5: (a) clean data with the homogeneous near surface, (b) perturbed data with
near-surface scattering layer, and (c) data from (b) enhanced using beamforming.
Figure 7. Signatures of the phase spectrum for data from Figure 6: (a) phase extracted at a
fixed frequency as a function of trace index at 10 Hz, (b) same as (a) but for 20 Hz, and
(c) phase spectrum of a single trace at an offset of 500 m. The blue lines denote clean data,
green marks “clutter”data, whereas red corresponds to enhanced data after beamforming.
Observe substantial phase variations of data with the clutter (green). In contrast, beam-
formed data (red) have a smoothly varying phase closer to the clean phase (blue). Note that
the phase of the beamformed data approaches the clean signal phase below 40 Hz (c).
V424 Bakulin et al.
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DOI:10.1190/geo2021-0830.1
MATHEMATICAL MODEL OF SEISMIC TRACE
WITH MULTIPLICATIVE RANDOM NOISE AND
ITS TRANSFORMATION WHILE STACKING
Random model of speckle and interpretation of
ensembles in local stacking
In the typical speckle treatment (Goodman, 1976,2000), it is
reasoned that we lack an exact knowledge of the detailed micro-
scopic structure that causes multiple forward scattering shown in
Figure 1a. Therefore, it is necessary to discuss the properties of
speckle patterns in statistical terms. The statistics of interest are
defined over an ensemble of different experiments. If we fix
the source-receiver pair, a different experiment will correspond
to placing another fine-scale realization of near-surface hetero-
geneity (with the same statistical properties), generating another
version of the distorted or noisy outcome. For a fixed pair, such
changes need to be made only in the vicinity of the wavepath
because properties outside those areas would not affect the out-
come. Because the outcome of the interference changes quickly
(Goodman, 2000,2020;Fink and Derode, 1998), another way
to produce a different experiment is to change the source and/
or receivers’positions slightly. The wavepath in the near-surface
layer would shift, leading to another interference outcome. Let us
restrict our initial investigation to the local stacking of prestack
data instead of global stacking. Although generating stacked or
migrated images may involve huge ensembles such as CMP gath-
ers with wavepaths traversing vast amounts of the subsurface, this
is not the case for local data enhancement that only stacks a limited
amount of data inside a local ensemble (Buzlukov and Landa,
2013;Bakulin et al., 2018a,2020a). By design, a typical ensemble
of traces is drawn from a certain spatial vicinity limited by a local
aperture (Figure 1b). Bakulin et al. (2020a) discuss ensemble se-
lection and typical examples of various ensembles represented by
local subsets of various gathers such as CMP, cross spread, etc.
Therefore, all traces inside such an ensemble share two key com-
monalities.
1) All traces are expected to contain similar
reflected signals produced by a limited
segment of a geologic interface of interest.
Hence, we assume that, in the absence of
multiplicative noise, the reflected signals
detected by each trace are nearly identical
inside the ensemble (after local moveout
correction).
2) Each trace carries a different realization
of multiplicative noise. By design with
sources and receivers selected within a lim-
ited vicinity, all wavepaths are similarly
clustered around a certain vicinity of the
near-surface scattering layer (Figure 1b).
As a result, exact microscopic properties
(e.g., the positions of small scatterers) in-
side each wavepath differ and lead todiffer-
ent interference patterns on each trace.
However, assuming that statistical proper-
ties (e.g., the number of scatterers per unit
cell and their contrast) are fixed or slowly
varying between neighboring wavepaths
—we can consider traces from the ensemble providing
distinct realizations of the same seismic speckle.
One point to mention here is that reflections from different depths
would lead to a different scattering distortion because their wave-
paths in the highly scattering near surface could be sufficiently
different (Figure 1a), even though both events belong to the same
seismic trace. Let us suppose that near-surface properties vary
laterally and with depth. In that case, distinctive distortions are ex-
pected for different reflectors, suggesting that simplified assump-
tions such as surface consistency do not hold for the case of
complex 3D wave propagation when small-scale heterogeneities
are present in the near surface.
Multiplicative random noise model
Because the signal itself induces speckle noise, it is universally
accepted from optics to ultrasound that it may be accurately repre-
sented as random multiplicative noise (Goodman, 1976,2020;Jain,
1989;Moreira et al., 2013;Damerijan et al., 2014). Based on the
considerations previously, we represent the trace recorded by the kth
channel of the ensemble as
xkðtÞ¼rkðtÞsðtÞþnkðtÞ;(1)
where sðtÞis the desired signal, rkðtÞis random multiplicative noise
with the same distribution within all channels, nkðtÞis additive ran-
dom noise, “*”denotes convolution, and k=1, :::,K(Kis the
number of the channels in the local ensemble).
In the Fourier domain (equation 1), it can be written as
XkðωÞ¼RkðωÞSðωÞþNkðωÞ;(2)
where Xk,Rk,S, and Nkare Fourier transforms of the corresponding
time-domain functions in equation 1.
Figure 8. Signatures of the amplitude spectrum for data from Figure 6: (a) amplitude
extracted at a fixed frequency as a function of trace index at 10 Hz, (b) same as
(a) but for 20 Hz, and (c) averaged amplitude spectra of all traces inside the window.
The blue lines denote clean data, green marks perturbed data, whereas red corresponds
to enhanced data after beamforming. Observe random variation of the data with near-sur-
face scattering layer (green) around a certain trend. In addition, pay attention to the am-
plitude bias marked by arrows with the average amplitude of beamformed data (red) being
lower than distorted data before stacking (green). Finally, spectrum (c) exhibits character-
istic amplitude reduction and roll-off at high frequencies, similar to what is seen in real
data from Figures 3c and 4c.
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It is assumed that additive noise NkðωÞis uncorrelated between
different channels and has a zero mean, i.e., E½NkðωÞ ¼ 0, where
an Estands for the mathematical expectation. In addition, signal and
noise are assumed to be uncorrelated.
Local stacking as an approximation of mathematical
expectation of random trace
The effect of stacking on additive random noise is well under-
stood. Hence, it serves as a cornerstone of increasing S/N during
acquisition, processing, and imaging (Meunier, 2011). However,
since multiplicative noise is not well recognized in seismic data,
a similar understanding needs to be built for seismic multiplicative
noise because stacking is the essential element of processing and
imaging. Therefore, we shall establish some basic foundation using
the simplified mathematical model (equation 1).
The local stack over an ensemble of traces is calculated as
¯
XðωÞ¼1
KX
K
k¼1
XkðωÞ:(3)
A large ensemble of several hundreds of traces for local stacking
provides a sufficient data set to approximate the mathematical expect-
ation of the described randomprocess. Let us examine its information
content and properties. Mathematically, this is expressed as
¯
XðωÞ≈E½XkðωÞ ¼ E½RkðωÞSðωÞ þ E½NkðωÞ
¼SðωÞE½RkðωÞ:(4)
Here, we assume that the signal remains constant and assume that
the additive noise’s mathematical expectation vanishes. Equation 4
provides the theoretical prediction of how summation transforms
the amplitude and phase of the stacked result. Most importantly,
it allows abstracting from specific fine-scale geologic heterogeneity
and replaces it with just knowledge of statistical distribution. Be-
cause we assume fixed flat signals, then the function
ΦðωÞ¼E½RkðωÞ (5)
can be thought of as a distorting filter ΦðωÞapplied to the original
signal as shown in equation 4.
Let us now consider two types of multiplicative noise. The first
type is inspired by numerical experiments and data observations
shown in Figures 5–11, summarized as random frequency-depen-
dent phase fluctuations. The second is the well-known residual stat-
ics or random timeshifts between channels. Initially, we analyze the
effects of each type of multiplicative noise on stacking separately.
Finally, we consider the combined effects of both types of noises on
stacked amplitude and phase and relate the results to real data ob-
servations.
The first type of multiplicative random noise: Random
frequency-dependent phase fluctuations
Let us consider multiplicative seismic speckle noise as random
frequency-dependent phase fluctuations acting on each time win-
dow:
RkðωÞ¼eiφkðωÞ:(6)
Figure 9. Time-windowed land data around the target reflector (100 ms) taken from Figure 3a (blue window): (a) original data, (b) data after
beamforming based on local stacking, and (c) associated absolute (nonnormalized) amplitude spectra.
Figure 10. Phase angles for land data from Figure 9: (a) phase an-
gles (in radian) at 20 Hz as a function of trace index and (b) phase
angles at 30 Hz as a function of trace index. Observe large random
phase variations of original data (blue) with a much smoother phase
of beamformed data (red). Notice no apparent correlation between
phase perturbations at (a) 20 Hz and (b) 30 Hz.
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Following the footsteps of an established model of speckle noise
as a random-walk phenomenon (Goodman, 1976,2000,2020), we
assume that, at a given frequency ω, random variables x¼φkðωÞ
are independent for all channels and have the same probability den-
sity function Pðω;xÞat a given frequency ω.
Some general properties of stacked sums with multiplicative
noise can be established by defining the distributions’symmetry.
For example, using a known formula for the mathematical expect-
ation of a random variable function, we can write (equation 5)as
ΦðωÞ¼Z
∞
−∞
Pðω;xÞeixdx: (7)
Equation 7leads to an important property that if the probability
density function of the phase fluctuations of the multiplicative noise
is even, i.e., Pðω;xÞ¼Pðω;−xÞ(implying that the expected value
is zero), then ΦðωÞis real-valued, and equation 4transforms to
E½XkðωÞ ¼ jSðωÞjeiφSðωÞΦðωÞ;(8)
where jSðωÞj is the amplitude spectrum of the signal and φSis its
phase. We can make essential conclusions under these rather general
assumptions without specifying the actual symmetric distributions of
the phase perturbations. First, we observe the phase “cleanup”proc-
ess in which random symmetric phase perturbations average out dur-
ing stacking and lead either to signal phase (if ΦðωÞ>0), or flipped
signal phase rotated by π(if ΦðωÞ<0). Second, real-valued ΦðωÞ
describes the transformation or filtering of the signal amplitude spec-
tra during stacking. To arrive at quantitative numerical results, let us
derive a specific form of ΦðωÞfor the case of normal and uniform
distribution of phase perturbations.
Normal (Gaussian) distribution of phase fluctuations
If random variables x¼φkðωÞdescribing phase perturbations all
have the same normal (Gaussian) distribution with zero mean and
standard deviation σφðωÞ, then
Pðω;xÞ¼ 1
ffiffiffiffiffi
2π
pσφðωÞe−x2
2σ2
φðωÞ:(9)
After substitution of expression 9in expres-
sion 7, it follows that
ΦðωÞ¼e−σ2
φðωÞ
2:(9a)
As a result, mathematical expectation can be
expressed as
E½XkðωÞ ¼ jSðωÞjeiφSðωÞe−σ2
φðωÞ
2:(9b)
Although it was predicted from symmetry
considerations, it is nevertheless remarkable to
note that the resulting phase spectrum after stack-
ing in expression 9b is the same as the phase of the clean signal,
i.e.,
argfE½XkðωÞg ¼ φSðωÞ;(10)
whereas a real-valued filter reduces amplitude compared with a clean
signal. If standard deviations remain constant for all frequencies, am-
plitude loss is identical across the entire band. If we assume that the
standard deviation increases with frequency (i.e., larger phase pertur-
bations at higher frequencies), then the loss of signal amplitude after
stacking would progressively increase with frequency.
Symmetric uniform distribution of phase fluctuations
If we assume that those phase perturbations φkðωÞare uniformly
distributed within the interval ½−φ0ðωÞ;φ0ðωÞ (in radians), then
probability density function has the form:
Pðω;xÞ¼1
2φ0ðωÞ;if jxj<φ0ðωÞ
0;if jxj>φ0ðωÞ;(11)
ΦðωÞ¼sin½φ0ðωÞ
φ0ðωÞ;(11a)
with mathematical expectation
E½XkðωÞ ¼ jSðωÞjeiφSðωÞsin φ0ðωÞ
φ0ðωÞ:(11b)
Again, it is remarkable to note that, after taking the mathematical
expectation, phase distortions are eliminated, and the resulting phase
spectrum of the broadband signal after stacking tracks the phase of
the clean signal provided φ0<π.Notethatifφ0ðωÞ¼π,then
E½XkðωÞ ¼ 0that could be interpreted as perfect destructive inter-
ference. Finally, if φ0>π, then the stacked phase still tracks the
clean signal phase but with the opposite sign (polarity flip). As in
the previous example, the stacked amplitude is reduced compared
with the clean signal, although the real-valued factor has a more com-
plex frequency dependency.
Figure 11. Visualizing multiplicative scattering noise: (a) surface shot gather in five-layer
model with near-surface scattering layer from Figure 5a, (b) same shot in the presence of
near-surface clutter layer, but with deep reflections eliminated (layers replaced by homo-
geneous half-space), (c) difference between (a and b) highlighting distorted reflection
events with signal-induced halos, and (d) buried shot in the same model as (b) showing
the impact of near-surface scattering layer on one-way propagation. Reflector R3 in
(c) with two-way propagation appears more broken when compared with direct arrival
in (d) experiencing single transmission through near-surface scattering layer.
Multiplicative scattering noise V427
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The second type of multiplicative noise: Random
timeshifts between channels (residual statics)
Although residual statics is a well-known type of seismic distor-
tion, it also can be mathematically described as a specialized type of
multiplicative noise caused by random time shifts τkbetween chan-
nels. In this case, multiplicative noise has a specific form:
RkðωÞ¼e−iωτk(12)
and
ΦðωÞ¼E½e−iωτk:(13)
Please note that while it may appear as if the residual statics may be
represented as a particular case of the first type of multiplicative noise,
this is generally not the case. Indeed, once a specific value of static
shift is drawn for a selected channel, then phase perturbations versus
frequency for multiplicative noise (equation 12) would always follow
a linear dependence. Therefore, phase perturbations are synchronized.
In contrast, the first type of multiplicative noise is entirely random
fluctuations occurring within a single trace at each frequency. In other
words, if phase perturbations are visualized for each trace realization,
then the first type would exhibit random jumps in phase with fre-
quency (as in Figure 10). In contrast, phase variations for the second
type would follow a straight line as a function of frequency.
Normal (Gaussian) distribution of timeshifts (residual statics)
Let the residual statics τkhave a normal distribution with zero
mean and the same standard deviation στ:
PðxÞ¼ 1
ffiffiffiffiffi
2π
pστ
e−x2
2σ2
τ:(14)
Then, associated quantities can be expressed as
ΦðωÞ¼e−ω2σ2
τ
2;(14a)
E½XkðωÞ ¼ jSðωÞjeiφsðωÞe−ω2σ2
τ
2:(14b)
We arrive at the same conclusion that the phase spectrum of the
stack (equation 14b) is the same as that of the clean signal, i.e., expres-
sion 10 is valid again. Amplitude experiences exponential loss with
frequency. For field arrays, so-called intraarray residual statics were
always blamed as a significant analog grouping limitation that reduces
high-frequency content. Exponential decay was noted by Berni and
Roever (1989) in an alternative derivation that did not use a multipli-
cative noise model and did not analyze the phase behavior.
Symmetric uniform distribution of timeshifts (residual statics)
Let residual statics τknow be uniformly distributed within the
interval ½−τ0;τ0. Then, probability density function has the form:
PðxÞ¼1
2τ0;if jxj<τ0
0;if jxj>τ0
;(15)
ΦðωÞ¼sinðωτ0Þ
ωτ0
;(15a)
E½XðωÞ ¼ jSðωÞjeiφsðωÞsinðωτ0Þ
ωτ0
:(15b)
The mathematical expectation has a similar structure with the
phase spectrum of the signal multiplied by the real-valued ampli-
tude loss factor that could change the sign with frequency. We fur-
ther observe that the amplitude loss factor sinðωτ0Þ∕ωτ0oscillates
and has periodic notches at ωτ0¼π;2π;3π;:::, where the change
of sign occurs. Before the first notch ω¼π∕τ0, the stacked phase
would be equal to the signal phase, whereas it will become rotated
by πafter that. A similar pattern repeats for subsequent notches.
For small residual statics, the first notch also may happen at
a frequency exceeding the maximum frequency fmax of the broad-
band signal. In this case, the stacked trace will equal the clean sig-
nal. Such a case occurs when the following condition is satisfied
fmax <1
2τ0
:(16)
The joint effect of residual statics and
frequency-dependent random phase fluctuations
Let us consider the last case when both types of multiplicative
noise (equations 6and 12) are present. Assuming that τkand
φkðωÞare independent of each other, we obtain
ΦðωÞ¼ZZ
∞
−∞
Pð1ÞðxÞPð2ÞðyÞe−iðωxþyÞdxdy; (17)
where Pð1Þand Pð2Þare the probability density functions of varia-
bles τkand φk, respectively.
If phase fluctuations and random timeshifts are normally distrib-
uted with standard deviations σφand στ, respectively, then the math-
ematical expectation is given by
E½XkðωÞ ¼ jSðωÞjeiφsðωÞe−ω2σ2
τ
2e−σ2
φ
2:(18)
Likewise, if phase perturbations and random timeshifts are uni-
formly distributed with intervals [−φ0ðωÞ;φ0ðωÞ and ½−τ0;τ0,
respectively, we arrive at the following mathematical expectation:
E½XkðωÞ ¼ jSðωÞjeiφsðωÞsinðωτ0Þ
ωτ0
sinðφ0ðωÞÞ
φ0ðωÞ:(19)
Similar conclusions apply to both cases. For the normal distribu-
tion, the phase of mathematical expectation is equal to the clean
signal phase, whereas the amplitude loss factor is a product of
two terms —one caused by phase perturbations and another
by residual statics.
SYNTHETIC SIMULATIONS OF LOCAL
STACKING WITH MULTIPLICATIVE
RANDOM NOISE
For numerical simulations, let us consider a simple case when the
signal is fixed and represented by the Klauder wavelet (Figure 12),
typical of land vibroseis acquisition after correlation. First, an ensem-
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DOI:10.1190/geo2021-0830.1
ble of traces is generated using different realizations of multiplicative
noise (Figure 13) following a mathematical model (equation 1).
Then, we numerically stack traces from the ensemble (with a finite
and realistic number of traces) and compare with the amplitude and
phase predicted by theoretical equations for the preceding mathemati-
cal expectation. Our goal is to validate the usefulness of the theoreti-
cal predictions for realistic local stacking scenarios with a limited
number of traces.
Example 1. Random timeshifts between channels
(stack of 1000 traces)
We first perform a numerical simulation of the effect of residual
statics on stacked amplitude and phase and compare them with the
theoretical formulas. We consider normal and uniform distributions
to seek an initial indication of which type may provide a better match
to the preceding experimental observations. We use an ensemble of
1000 traces that could be a typical number for local stacking enhance-
ment of high-density 3D seismic land data with medium and large
apertures of 200–400 m (Bakulin et al., 2020a).
Normal distribution of residual statics
We start with the normal distribution of residual
statics and consider two realistic cases of
στ¼4ms and στ¼8ms.Figure14 visualizes
the distribution of residual statics for one
ensemble in the case of στ¼8ms.Althoughap-
proximately 2/3 of the channels have statics less
than the standard deviation of 8 ms, there are oc-
currences of higher statics reaching up to 20 ms.
Figure 15 shows the effect of residual statics on
the phase and amplitude after local stacking.
The phase of stacked data in red overlays the clean
phase (Figure 15b) as predicted by equation 14b.
This is a remarkable fact that holds the key
to mitigating the effects of residual statics in
processing. Figure 15a shows a dramatic attenu-
ation or roll-off of the amplitude at high frequen-
cies that agrees with equation 14b. The steepness
of the roll-off increases with increasing standard
deviation. Residual statics may be one possible
explanation of observation 3 about the significant
and progressive loss of higher frequencies after
stacking real data. However, it does not explain
observations 1 and 2. If data suffer only from
residual statics, waveforms remain identical (Fig-
ure 13a), whereas low frequencies are virtually
untouched; therefore, additional noise types also
must be present.
Uniform distribution of residual statics
Let us consider in a similar manner the uniform
distribution of residual statics with several ranges
τ0¼4ms,τ0¼8ms,andτ0¼12 ms. Figure 16
shows the effect of residual statics on the phase
and amplitude after local stacking. Whereas in
the time domain (Figure 16c), we can see a similar
broadening of the wavelet, the details of amplitude
and phase spectra are quite different. Similar to the previous case of
normal distribution, there also is a roll-off of amplitudes at higher
frequencies proportional to the statics ranges. However, amplitude
spectra are populated with characteristic notches (Figure 16a)pre-
dicted by equation 15b. Before the first notch, the phase of the stacked
response agrees with the clean phase (Figure 16b) consistent with
equation 15b. However, the phase changes by πafter the notch,
and this pattern repeats for the subsequent notches. Such
characteristic notches are not observed after stacking real data from
Figures 3and 4, suggesting that uniform distribution is not a plausible
description for geologic heterogeneity.
Example 2. Random phase perturbations (stack
of 1000 traces)
Similarly, let us evaluate the effect of the first type of multipli-
cative noise represented by random phase perturbations. For brevity,
we present only results for the case of normal distribution. Uniform
distribution leads to a similar effect. We selected normal distribution
as the primary case because the actual phase distribution of the
phase for synthetic or real data indicates such a type of distribution
as more common (not shown).
Figure 12. Klauder wavelet used as a clean signal in the synthetic experiments: (a) time-
domain representation and (b) amplitude spectrum. Klauder wavelet corresponds to an
autocorrelation of the linear sweep 5–80 Hz with appropriate tapers on each side.
Figure 13. A subset of traces from the ensembles used for numerical examples with
different types of multiplicative noise: (a) random timeshifts with a normal distribution
(στ¼4ms) and (b) combination of normally distributed random phase perturbations
(σφ¼π∕2) and timeshifts (στ¼4ms). Synthetic ensemble generated using fixed signal
(equation 1) and random realizations of multiplicative noise on each channel. Although
(a) exhibits only slight arrival time variation, panel (b) shows severe waveform changes
from trace to trace, leading to a reduced coherency.
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Normal distribution of phase perturbations
We consider the standard deviations of σφ¼π∕2and
σφ¼π∕1.2 rad. Figure 17 demonstrates the effect of perturbations
on the phase and amplitude of the stacked signal. Similar to pre-
vious observations, the stacked phase tracks the phase of the clean
signal (Figure 17b). However, amplitude spectra experience a se-
vere constant loss at all frequencies (Figure 17a) because we assume
a fixed value of the standard deviation for all frequencies. The mag-
nitude of amplitude bias or loss increases with increasing standard
deviation. Phase distortions may explain observations 1 and 2. In-
deed, phase distortions change the wavelet shapes and thus reduce
coherency significantly (Figure 13b). They also lead to severe am-
plitude bias or loss after stacking. However, they do not predict am-
plitude roll-off if we assume a constant standard deviation with
frequency. However, suppose that the standard deviation of phase
perturbations will increase with a frequency that sounds plausible
because higher frequencies become more and more affected by
small-scale heterogeneities. In that case, we also could reproduce
a similar amplitude roll-off as observed in the case of residual statics
(Figure 15a). Therefore, all three observations could be explained.
The concept of statics remains a useful simplification. However, we
should never forget its limitation, characterized by Meunier (2011),
as an “impossibility for such a simplistic propagation model (infi-
nitely low velocity) to account fully for the perturbation of the
wavefield associated with the weathered-layer variations.”We be-
lieve that we can move to a more plausible way to describe near-
surface distortions by using phase perturbations. Although still
being statistical (as residual statics), it provides a more realistic
(wave propagation-based) representation of wavefield perturbations
by the near-surface scattering layer. Again, such an upgrade would
be consistent with speckle studies in other domains with significant
progress achieved.
Example 3. The joint effect of residual statics and
random phase perturbations (stack of 1000 traces)
We have established the basic building blocks to understand and
explain the real data by analyzing the individual effects of two types
of multiplicative noise. We have shown that phase perturbations can
explain observations 1 and 2, whereas a residual statics can mimic
observation 3. Although we have seen that a more complex form of
phase perturbations also could explain observation 3, let us consider
one simpler additional case that also consistently explains all ob-
servations 1–3 from real data at once. In this example, two types
of multiplicative noise act together or superim-
posed. We stress that the normal distribution
of residual statics and phase corrections appears
to be the best matching effects seen on real data.
Normally distributed phase fluctuations and
residual statics
Let us consider the normal distribution of
phase corrections with σφ¼π∕2and residual
statics with στ¼4ms. Figure 18 demonstrates
the combined action of two types of multiplica-
tive noise on the stacked signal’s phase and am-
plitude. Figure 18c shows how phase fluctuations
can distort the input trace shown in green. In the
time domain, stacked waveform experiences
broadening (Figure 18c). The phase of the
stacked response agrees with the phase of the
true signal (Figure 18b). Amplitude spectrum ex-
periences substantial loss caused by a combina-
tion of downward shift induced by phase
perturbations and roll-off of the higher frequen-
cies caused by residual statics (Figure 18a).
Comparison with real data
Figure 18a can be directly compared with
Figures 3c and 4c from real data. Although
we are not aiming for a quantitative match,
we have fully replicated all observations 1–3,
including distorted waveforms, amplitude bias,
Figure 15. Effect of residual statics on amplitude and phase spectra: (a) amplitude spec-
tra, (b) phase spectra, and (c) time-domain representation. The clean signal is shown in
blue; numerically stacked quantities are in red, whereas theoretical predictions are in
black: black, red, and blue curves all overlay each other in (b). Panel (a) shows a good
match between numerically stacked amplitude spectra in red with theoretical predictions
in black. The time-domain representation (c) shows the broadening of the wavelet
caused by mis-stacking due to multiplicative noise.
Figure 14. Normally distributed random timeshifts (residual
static) as a function of trace index in the ensemble (στ¼8ms).
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and roll-off of amplitudes at higher frequencies. Note that the
amplitude bias of phase perturbations (Figures 17a and 18a) rep-
licates the approximate magnitude of the massive 15–30 dB reduc-
tion seen in real data (Figures 3c and 4c). We have ignored the
effects of random additive noise because stacking perfectly han-
dles such noise without affecting the magnitude
of the signal. Data were used in the preceding
field examples after a typical processing se-
quence that was expected to reduce additive
noise to manageable levels. Even if we allow
the presence of −10 dB additive noise in Fig-
ures 3and 4, local stacking in the presence of
additive noise is expected to reduce energy lev-
els by only −5 dB uniformly. In contrast, we see
a more massive amplitude reduction of −15 to
−20 dB. Although we do not dispute the pres-
ence of additive noise, we argue that it is not the
main culprit in explaining field observations
1–3. Such conclusions completely agree with
studies of acoustic and ultrasonic speckle noise
(Damerjian et al., 2014;Goodman, 2020).
The normal distribution of residual statics, as
well as phase fluctuations, appears more likely.
Numerical experiments with cluttered near-sur-
face layers and real data support the hypothesis
of the normal distribution of phase perturbations
because histograms of phase deviations are sim-
ilar to Gaussian-like shapes instead of uniform
rectangles (not shown). Likewise, experimental
and numerical studies of ultrasonic speckle from
a rough water-sand boundary also suggest the
Gaussian distribution of phase variations (Hare
and Hay, 2020). We have made the simplest
assumption of frequency independence of stan-
dard deviation for phase perturbations based
on initial observations on synthetic and field
data. We also have noted that if the standard
deviation of phase perturbations would increase
with frequency, it also could explain amplitude
roll-off at higher frequencies, similar to residual
statics. Future studies should examine these as-
sumptions in more detail to fine-tune this under-
standing.
DISCUSSION AND IMPLICATIONS
FOR SEISMIC PROCESSING
Recognizing the essential role of multiplica-
tive random noise in seismic data has important
implications. Armed with the new understand-
ing, it is insightful to have a retrospective look
at previous seismic practices. Nonsurface-consis-
tent trim statics correction is one known process-
ing step that tries to address small residual
timeshifts in prestack data that vary with time.
Reilly et al. (2010) describe a 3D dynamic trace
alignment procedure similar to trim statics. Re-
markably, this procedure also was inspired by
challenging offshore seismic data from the
Middle East with complex near surface and over-
burden. They speculate that time-dependent traveltime jitter might
be caused by small-to-medium-scale heterogeneities in the velocity
model not captured by typical velocity model building. According
to their work, addressing such small-scale velocity variations may
be equally crucial for time and depth imaging because they remain
Figure 16. Same as Figure 15 but for uniform distribution of residual statics with differ-
ent maximum absolute timeshift values (τ0). Phase spectra for τ0¼12 ms only are
shown in (b), with a black arrow marking the location of the phase change by πthat
occurred after the first notch.
Figure 17. Same as Figure 15 but for normal distribution of phase perturbations with
σφ¼π∕2and σφ¼π∕1.2 rad.
Multiplicative scattering noise V431
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DOI:10.1190/geo2021-0830.1
missed or improperly characterized in the velocity model building
process even with advanced techniques such as FWI. However,
the statics-based approaches mentioned previously simplistically
assume that distortions consist of only small timeshifts but not
waveform variations. Also, previous studies did not provide a math-
ematical model and only described the impact of residual statics on
the amplitude (Berni and Roever, 1989), but not the phase. Neklyu-
dov et al. (2017) and Stork (2020) emphasize the role of waveform
distortions in land seismic data. They point to the outsized impact of
the near-surface layers. Xie et al. (2016) present numerical exam-
ples demonstrating that seismic migration suffers a severe loss of
focusing when small-scale near-surface heterogeneities are not
captured in the velocity model. Khalil and Gulunay (2011) show
the robustness and usefulness of local stacking of single-sensor land
data as pilots for deriving intraarray statics. Bakulin et al. (2020b,
2020c) suggest phase substitution and phase correction methods
that derive an estimate of the clean phase from the beamformed re-
flected data and show significant improvement in eliminating asso-
ciated distortions for processing and imaging. Bakulin et al. (2021)
further propose seismic time-frequency masking that could use raw
amplitude and beamformed amplitude to arrive at an improved es-
timate of the signal amplitude reducing the loss of higher frequen-
cies. Still, these results lacked a mathematical model and a proper
basis for these methods.
The results of this study serve as a theoretical justification explain-
ing that the phase derived from local stacking indeed provides an ac-
curate estimation of the signal phase in the presence of random
multiplicative noise caused by near-surface scattering. Remarkably,
both types of multiplicative noise in the form of random static shifts
and random phase perturbations can be treated using the same
approach. In particular, phase perturbations cause waveform distortion
and variation that are challenging to fix. Surface-consistent deconvo-
lution is the only process attempting to address the prestack data’s
variable waveforms, assuming that they can be modeled as determin-
istic multiplicative distortions. It further relies on the overly simplify-
ing assumptions of surface consistency and time independence that do
not capture the essential complexities of 3D wave propagation with
small-scale near-surface scattering shown in Figure 1a. The presented
mathematical model with random multiplicative noise provides a uni-
fied description of residual statics and phase distortions that more ac-
curately capture small-scale scattering effects.
The obtained understanding serves as a foundation for success-
fully addressing multiplicative speckle noise in seismic processing.
Using a random statistical model inspired by studies of optical and
ultrasonic speckle (Goodman, 2020), we theoretically justify that
the phase derived from local stacking provides an accurate estima-
tion of the signal phase. We believe that such a model has the po-
tential to mitigate many damaging effects of multiplicative seismic
noise if we can adequately guess or characterize the distribution
of small-scale near-surface heterogeneity. This belief is based on
the remarkable progress achieved in addressing the optical and
ultrasonic speckle using similar statistical models (Goodman,
2020). Thus, we can still make significant improvements in seismic
processing and imaging without ever knowing the exact micro-
scopic details of the near surface.
Why may recognition of speckle noise be so late? As Goodman
(2000) points out: “Speckle suppression remains one of the most
important unsolved problems of coherent imaging.”Speckle noise
in optics has been studied since the 1960s, with an initial focus
on reflections from rough surfaces having details less than a wave-
length (Beckmann, 1962,1964;Goodman, 1976).
The acoustic and ultrasonic speckle came later,
starting in the 1980s, recognizing the small-scale
volumetric scattering (Abbot and Thurstone,
1979;Fink and Derode, 1998). However, these
fields primarily focused on studying and mitigat-
ing speckle noise on intensity images at a fixed
frequency. In contrast, seismic imaging is always
done with broadband signals making speckle
noise much harder to recognize among many
other effects present in the data. Also, in medical
ultrasound, variability between different parts of
the population is relatively small compared with
vast variations between geologic near-surface
heterogeneity present in different parts of the
world. In particular, challenging land seismic data,
especially from the desert environment, may pro-
vide the most substantial evidence of random
multiplicative distortions. However, some of that
evidence was out of sight because exploration in
the desert environment was conducted with large
geophone arrays mitigating the effects of multipli-
cative noise in the field. For example, 72-geo-
phone arrays were used for decades in a desert
environment. Currently, acquired data with
nine-geophone arrays are significantly more chal-
lenging. Single-sensor data in areas with a com-
plex near surface exhibit an ultimate complexity
and bring severe processing challenges. The
Figure 18. Same as Figure 15 but for normal distribution of phase corrections with
σφ¼π∕2and residual statics with στ¼4ms. For illustrative purposes, we also post
on (c) a trace from a single channel in green. Note that each ensemble channel is dis-
torted by a different realization of multiplicative noise, as visualized in Figure 13b,
showing 25 additional traces from such an ensemble.
V432 Bakulin et al.
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DOI:10.1190/geo2021-0830.1
analysis of the latest data with nine-geophone arrays and single sen-
sors leads us to present findings on multiplicative noise.
CONCLUSION
We have carefully analyzed the amplitude and phase spectra’s
transformation during local stacking of the challenging prestack
land seismic data from the desert environment. To explain the
key observations from the data, we put forward a seismic trace
model with random multiplicative noise describing the effects of
small-scale near-surface scattering. We further specified two types
of such noise: random phase perturbations and random timeshifts
(residual statics). Although not usually described by the multipli-
cative model, the second type of distortion is well recognized. In
contrast, the first type is new and not currently acknowledged in
seismic processing. Synthetic modeling confirms that small- and
medium-scale near-surface heterogeneities generate random-look-
ing nonsurface-consistent phase perturbations that are different at
each frequency. Invoking both types of multiplicative noise, we
can semiquantitatively reproduce all three critical observations here
from the real data that could not be consistently explained by a con-
ventional model with additive background noise:
1) Even after sophisticated processing, reflections on prestack
data remain distorted with low coherency or even untrack-
able. Such reflection cluttering is not localized in time-space
but instead spread over the entire gather in which reflections
may be potentially present. In addition, first arrivals also are
distorted, making it difficult to pick first breaks or use early
arrivals for FWI.
2) After applying local stacking, reflections crop up very
clearly; however, the absolute level of amplitude spectra ex-
periences a strong downward bias or loss (−15 to −20 dB or
more) across the entire band of frequencies.
3) There is a significant and progressive loss of higher frequen-
cies after local stacking.
Multiplicative noise with random phase perturbations might ap-
pear new to the seismic community. However, after closer exami-
nation, we suggest that it could be thought of as a seismic version of
speckle noise well established in optical, acoustic, and ultrasonic
imaging. In a volumetric case, speckle noise occurs when multiple
scatterers are present in an elementary volume. Instead of a single
ballistic arrival, a wavefield is composed of a superposition of
multiple forward-scattered arrivals with adjacent near-ballistic trav-
eltimes. Optical and ultrasonic images with speckle noise exhibit a
strong granular imprint that obscures objects under examination.
Likewise, seismic prestack data often have substantial amplitude
and phase variations even when the underlying reflected signals
are expected to be constant or smoothly varying. Specifically, a
near-surface layer with small- and medium-scale heterogeneities
could be the most likely source of speckle noise, as illustrated
by a synthetic example.
Based on this model, we demonstrate that similar to the case of
additive noise, stacking remains a useful primary instrument to
combat multiplicative noise. Local stacking is particularly useful
to mitigate the random nature of speckle noise because the wave-
paths involved traverse only a small vicinity of the near surface.
Under these conditions, underlying reflection signals are expected
to be similar, whereas a fixed local statistical distribution can
approximate random multiplicative noise. With this in mind, we
have proven that the stacked phase gives an unbiased estimate of
the clean signal. This phase estimate’s accuracy or standard
deviation depends on the number of traces. It requires further stud-
ies to find a practical trade-off between accuracy and potential loss
of resolution. The recovery of the true signal phase via stacking is
the cornerstone to providing a path to mitigate random multiplica-
tive noise in seismic processing and imaging. Such phase recovery
was already used in the phase substitution method but without
proper justification. With additive noise, stacking estimates the
phase and amplitude of the clean signal. With multiplicative noise,
stacking provides only an estimate of the clean phase, whereas the
recovered amplitude represents a severely distorted version of the
signal amplitude. However, providing a simple statistical model de-
scribing the anatomy of these distortions could pave the way to mit-
igating them. Random statistical models proved very productive in
addressing the adverse effects of optical and ultrasonic speckle
noise. By recognizing seismic speckle, we expect similar progress
to mitigate the impact of small-scale scattering on seismic data.
We demonstrate that the proposed model of random multiplica-
tive noise can fully explain all three preceding field observations.
Phase perturbations lead to the loss of coherency and amplitude bias
or a downward reduction across the entire broadband spectrum,
supporting the first and second preceding field observations. In con-
trast, residual statics only slightly affect amplitudes at low frequen-
cies and instead lead to characteristic amplitude roll-off at medium
and higher frequencies explaining the third observation from the
field data. When both types of random multiplicative noise act to-
gether, we fully replicate the effects seen while stacking complex
field data from the desert environment with single sensors or small
geophone arrays.
DATA AND MATERIALS AVAILABILITY
Data associated with this research are confidential and cannot be
released.
APPENDIX A
MATHEMATICAL EXPRESSIONS FOR AMPLI-
TUDE AND PHASE SPECTRA OF PRESTACK DATA
CONSISTING OF MULTIPLE EVENTS
In the frequency domain, each seismic trace XðωÞcan be char-
acterized by its amplitude jXðωÞj and phase φXðωÞspectra uniquely
defining the frequency signature of the signal XðωÞ¼jXðωÞjeiφXðωÞ
and its time-domain xðtÞrepresentation.
Let the seismic trace xðtÞconsist of Marrivals. Each arrival has a
waveform fjðtÞand corresponding arrival time τj:
xðtÞ¼X
M
j¼1
fjðt−τjÞ:(A-1)
In the frequency domain, equation A-1 is rewritten as
XðωÞ¼X
M
j¼1
FjðωÞeiωτj¼X
M
j¼1jFjðωÞjei½ωτjþφjðωÞ;(A-2)
Multiplicative scattering noise V433
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DOI:10.1190/geo2021-0830.1
where jFjðωÞj and φjðωÞare amplitude and phase spectra of jth
arrival, respectively.
The following expression gives the power spectrum of the trace
(equation A-1):
jXðωÞj2¼X
M
n¼1jFnðωÞjei½ωτnþφnðωÞ ·X
M
m¼1jFmðωÞje−i½ωτmþφmðωÞ
¼X
M
j¼1jFjðωÞj2þ2X
M−1
n¼1jFnðωÞj·X
M
m¼nþ1jFmðωÞj·
cosfωðτn−τmÞþφnðωÞ−φmðωÞg:(A-3)
A phase spectrum can be derived from the expression:
cosfφXðωÞg ¼ RefXðωÞg
jXðωÞj
¼PM
j¼1jFjðωÞj ·cosfωτjþφjðωÞg
jXðωÞj :(A-4)
As one can see, the amplitude spectrum depends only on trav-
eltime differences. The phase spectrum is a function of actual trav-
eltimes (Lichman, 1999). In the simplest case, if there is only one
arrival (M¼1), the amplitude spectrum jXðωÞj ¼ jFðωÞj, i.e., it
does not depend on traveltimes at all. Hence, we conclude that
arrival time information of seismic events is encoded in the phase
spectrum.
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