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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS
1
Application of an STFT-Based Seismic Even and
Odd Decomposition Method for Thin-Layer
Property Estimation
Jian Zhou , Jing Ba , John P. Castagna, Qiang Guo , Cun Yu, and Ren Jiang
Abstract— For seismically thin-reservoir layers, variations in
rock properties may not be directly linked to seismic amplitude
due to the wave interference of layer top and base reflections.
In addition, thin-layer reflection signal locally has a different
phase from that of the signal wavelet. Signal even and odd
components can be considered as amplitudes at different signal
phases, which may have a different sensitivity to the variations in
thin layer and surrounding layer properties. A novel extension of
the spectral decomposition concept is proposed that decomposes
seismic signal into its even and odd components via the short-time
Fourier transform. Amplitude attributes for the original signal
and even and odd part components are compared for their ability
to restore the correct “amplitude-layer property” correlation
without resolving the thin layer. Numerical modeling analysis
shows that amplitude at peak frequency (APF) of the seismic
data odd component APF (OAPF) is more sensitive to thinreservoir property change compared to the conventional APF
and even component APF attributes. When applied in analyzing
real seismic data in a tight-dolomite reservoir, conventional APF
and conventional acoustic impedance inversion did not provide a
correct relationship to porosity variations. Meanwhile, the OAPF
attribute responds well to porosity measured in boreholes. This
suggests that the interpretability of amplitude attributes in thin
layers can be improved by signal even and odd decomposition.
Index Terms— Amplitude, even and odd decomposition,
porosity, spectral decomposition, thin layer.
I. I NTRODUCTION
S
PECTRAL decomposition has been proven to be an
effective method for seismic signal analysis in petroleum
exploration. It decomposes seismic signal into a discrete
number of signal subsets, each corresponding to amplitude
and phase for a narrow frequency band. These narrowband
amplitudes indicate seismic reflection strength at the frequency
that is characteristic to different geologic bodies [1], [2].
Spectral decomposition has been applied in direct hydrocarbon
detection, seismic attenuation analysis, channel identification,
and thin-layer thickness estimation [3]–[6]. Plain-view
amplitude images of the narrowband subsets show different
interfering patterns within the analysis window [2]. However,
this also suggests that spectral amplitude of a thin reservoir
is not directly linked to layer properties, but is instead,
a complex combination of variations in reservoir properties,
overburden velocity, and layer thickness.
Reflection signal from a seismically thin layer is a composite waveform that is the superposition of the reflections
from the top and base of the layer. Directly relating seismic
amplitude to thin-layer properties often requires resolving
the top and base reflections first. Although there has been
progress made recently in improving resolution in spectral
decomposition algorithms [4], [7]–[9], however, as indicated
by the Heisenberg uncertainty principle, there is always a
limit beyond which it is not possible to improve time and
frequency resolution at the same time [10]. Second, methods that aim to restore the correct “seismic amplitude—thin
reservoir property” relationship often involve removing the
thickness factor, by adapting an amplitude–thickness curve
based on a model. However, the popular Widess [11] and
Kallweit and Wood [12] models assume that the top and
base reflections have equal magnitude and opposite or equal
sign, respectively, which does not always fit real geology
and is not generally applicable [13]. Thus, conventional spectral decomposition amplitude attributes, e.g., peak frequency
amplitude or amplitude at peak frequency (APF), usually are
less sensitive to true reflection strength and thin-reservoir
properties.
Phase decomposition is a recent extension of the spectral
decomposition concept proposed by Castagna et al. [14].
The concept of phase decomposition assumes that amplitude
variation induced by thin-reservoir property variation has a
different phase from the phase of the original seismic wavelet.
By separating seismic amplitudes of different phases, e.g.,
the 90°/270° and 0°/180° phase components for a zero-phase
seismic data set, thin-layer property change is more clearly
shown in the 90°/270° phase component amplitude. As a
quick and simple test of the phase decomposition concept
and its applicability, the seismic even and odd component
decomposition can be achieved using forward and inverse
Fourier transform [15].
The objective of this letter is to introduce two attributes, odd
component APF (OAPF) and even component APF (EAPF),
based on seismic odd and even component decomposition, and
Manuscript received December 2, 2018; revised January 3, 2019; accepted
February 20, 2019. This work was supported in part by the SpeciallyAppointed Professor Program of Jiangsu Province, China, in part by the
Cultivation Program of 111 Plan of China under Grant BC2018019, and in
part by the Fundamental Research Funds for the Central Universities, China,
under Grant 2016B13114. (Corresponding author: Jing Ba.)
J. Zhou is with the School of Earth Sciences and Engineering, Hohai
University, Nanjing 211100 China, and also with the Department of Earth
and Atmospheric Sciences, University of Houston, Houston, TX 77021 USA.
J. Ba, Q. Guo, and C. Yu are with the School of Earth Sciences and Engineering, Hohai University, Nanjing 211100, China (e-mail: jba@hhu.edu.cn).
J. P. Castagna is with the Department of Earth and Atmospheric Sciences,
University of Houston, Houston, TX 77021 USA.
R. Jiang is with PetroChina Company Ltd., Research Institute of Petroleum
Exploration and Development, Beijing 100083, China.
Color versions of one or more of the figures in this letter are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LGRS.2019.2901261
1545-598X © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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2
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS
time thickness, and r1 and r2 are the reflection coefficients
at the top and base of the layer, respectively.
As any time series can be uniquely divided into even and
odd parts [15], we now define even, ge (t), and odd, go (t),
reflectivity series components by
T
T
ge (t) = re δ t −
+ re δ t +
(2)
2
2
and
T
T
− ro δ t +
go (t) = ro δ t −
2
2
(3)
g(t) = ge (t) + go (t).
(4)
where
re and ro are the even and odd reflection coefficients given by
re =
Fig. 1. Reflection pair from a 10-ms-thick thin layer, and amplitude and
phase spectra of its even and odd parts. (a) Reflection pair. (b) Even part.
(c) Odd part. (d)–(f) Amplitude spectrum of original reflection, even part,
and odd part, respectively. (g)–(i) Phase spectrum of original reflections, even
part, and odd part, respectively.
then demonstrate their unique sensitivities to thin-layer and
overburden-layer property changes, respectively. These new
attributes serve as indirect proxies that can be used to create an
accurate “seismic amplitude—reservoir property” relationship
for thin reservoirs, without having to resolve the thin layers.
We first present a method of decomposing seismic signal into
its even and odd components, and the calculation of EAPF
and OAPF attributes. Then, we provide numerical examples
to assess the effectiveness of the OAPF and EAPF attributes
in estimating porosity of thin tight-dolomite reservoir and
overburden rock property changes. Finally, we show a realdata example using a 3-D seismic data set from the Sichuan
Basin, China, and compare with the conventional amplitude
attribute and seismic impedance inversion results.
II. M ETHODS
A. Generalized Thin Layer
Restoring the correct “seismic amplitude—thin reservoir
property” relationship requires understanding complex factors
that affect seismic reflection amplitude from a generalized thin
layer. We define a simple layer as one with only two reflection
coefficients: one at the top and one at the base (Fig. 1).
This is a good starting point to understand and approximate
most complex real layers, particularly in an earth with a
blocky impedance profile. As compared to the treatments by
Widess [11] and Kallweit and Wood [12], we investigate
the case of a generalized simple layer where the reflection
coefficients need not be equal in magnitude and can have
arbitrary signs.
Following [16], we represent the generalized simple layer as
T
T
g(t) = r1 δ t −
+ r2 δ t +
(1)
2
2
where g(t) is the reflectivity series as a function of time,
δ is the Dirac-delta function, T is the layer two-way
r1 + r2
2
(5)
and
r1 − r2
.
(6)
2
Fig. 1 shows the comparison of different properties of a
thin-layer reflection and its even and odd parts, in the example
of a 10-ms-thick generalized simple layer. First, by using a
time window that is centered at the middle of the layer, the
amplitude spectrum of the even and odd parts of reflectivity is
a rectified cosine and sine function, respectively. The conventional APF attribute is defined as the maximum amplitude at
the frequency peaks, which equals |r1 | + |r2 |. The APF for the
amplitude spectrum of the even part reflectivity (EAPF) and
the odd part reflectivity (OAPF) is equal to |2re | and |2ro |,
respectively. This is the definition of the two proposed new
APF attributes. Second, in this case, since r1 and r2 have the
same sign, the conventional APF equals to |r1 + r2 |, which
equals to the EAPF. In the case where the thin layer top and
base reflections have opposite signs, the conventional APF
would equal |r1 − r2 |, which equals the OAPF. Beside this
proof in the time domain, the property can also be simply
proved in the frequency domain. Finally, the comparison of
phase spectra [Fig. 1(g)–(i)] shows that the original thin-layer
reflection pair has a variable phase over frequencies, while
its even and odd parts have constant phases at 0° and 90°,
respectively.
ro =
B. Even and Odd Component Decomposition and APFs
From previous analysis, it is obvious that the even and
odd component of reflection from a generalized thin layer
can be solved in the time domain if the top and base
reflection coefficients are known. Unfortunately, this is usually
not possible in real seismic data. However, reflection signal
from a seismically thin layer can be decomposed into its
even and odd components in the frequency domain using the
Fourier transform without having to know r1 and r2 . Convert
the reflection signal g(t) from time to frequency domain,
the frequency spectrum of a thin layer, G( f ), is then
G( f ) = 2re cos(π f T ) + i 2ro sin(π f T ).
(7)
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ZHOU et al.: APPLICATION OF AN STFT-BASED SEISMIC EVEN AND ODD DECOMPOSITION METHOD
Fig. 2.
3
Schematic of a simple one-layer-two-half-space model.
In (7), the real part of the complex spectrum, Re[G( f )], is the
spectrum of the even part of the reflectivity series
Re[G( f )] = 2re cos(π f T )
(8)
while the imaginary part of the complex spectrum, Im[G( f )],
is the spectrum of the odd part of the reflectivity series
Im[G( f )] = 2ro sin(π f T ).
(9)
This property enables a unique and simple solution to
extract even and odd components of the input seismic signal,
by using forward and inverse Fourier transform. For a
broadband wavelet with a flat spectrum, as pointed out by
Partyka et al. [2], amplitude at any peak frequency is independent of thickness, which makes it a fundamentally different
quantity than peak amplitude (PA) in the time domain.
III. M ODEL T EST
In this section, we compare the two new APF attributes
(EAPF and OAPF) with the conventional APF attribute as
a response to perturbation in both reservoir and overburden
layer property, to compare different sensitivities of the two new
APF attributes to the conventional APF. We create a simple
one-layer-two-half-space model, as shown in Fig. 2, where the
middle layer is the thin reservoir. Thickness of the thin layer
can vary between zero to the tuning thickness of the wavelet
used in this model. Physical parameters of the model, including P-wave velocity (v p ), density (ρ), and porosity (ς), are
obtained from blocked well logs in a tight-dolomite reservoir
in the Sichuan Basin. P-wave velocity of overburden rock and
reservoir porosity are set as variables, where underlying rock
properties are set to remain unchanged. Reservoir layer v p is
predicted using an empirical trend between porosity (%) and
v p (m/s) from log data from 14 wells in this reservoir, which is
v p = 6824.6 − 122.1ς.
(10)
Rock density used in the model (g/cc) is estimated using the
Gardner equation
ρ = 0.31v p0.25 .
(11)
Four sets of parameters are chosen to represent endmember scenarios. The transitions from scenario I to II and
scenario III to IV indicate the increase of reservoir rock
Fig. 3. (a) Schematic of model parameters. (b)–(d) Model APF attributes:
conventional APF, EAPF, and OAPF, respectively.
porosity, which in turn leads to the decrease of its acoustic
impedance while overburden rock properties remain the same.
The changes from scenario I to III and scenario II to IV
represent processes where overburden v p decreases while
the reservoir layer properties remain the same. This process
simulates the effect of change in overburden lithology.
Results of the two new APF attributes and the conventional
APF attributes (Fig. 3) show that the EAPF attribute is relatively unaffected by reservoir property variation, i.e., porosity,
but decreases as overburden v p decreases [Fig. 3(c)]. This
makes it a good indicator for lateral lithological variation
of overburden rocks. On the other hand, the OAPF attribute
varies almost linearly as a function of both reservoir layer
porosity and overburden v p [Fig. 3(d)]. This suggests that the
OAPF attribute is independent of the EAPF and conventional
APF attribute, and thus could potentially provide more insights
about changes of reservoir porosity. Compared to EAPF and
OAPF, the APF trend is more complex [Fig. 3(b)].
The APF attribute could be equal to either the
OAPF or EAPF attribute, depending on the combination of
model parameters. This has been proven by the example
in Fig. 1. In this case, since the original reflection coefficients
form primarily an even pair, the conventional APF attribute
is mostly controlled by EAPF. Because variation in thinlayer properties mainly results in change in the signal odd
part, while variation in overburden properties mainly causes
change in the signal even part [14], the conventional APF,
although commonly used to indicate thin-layer properties, can
be dominated by either the even or odd component of the
reflection pair, i.e., the changes in either overburden- or thinlayer property, respectively. This suggests that the conventional
APF for thin-layer property interpretation is less reliable in the
case of thin layers compared to EAPF or OAPF.
IV. R EAL -DATA E XAMPLE
A. Study Area, Log Data, and Conventional Attributes
We test the two proposed APF and conventional APF
attributes in a field seismic data set from the Sichuan Basin.
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4
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS
Fig. 4. Log curves, well correlation, and inverted Z p results around the
LWM group and dolomite reservoir at well “A.”
Fig. 6. (a) 2-D seismic and APF attribute profiles of the LWM tight-dolomite
reservoirs in Sichuan Basin. (b) Comparison between OAPF, Z p, and porosity
log curves at the two wells from the profile (log scale of OAPF does not match
color profile).
Fig. 5. Dolomite Z p—porosity relationships. (a) Log Z p and porosity in
log scale. (b) Log Z p and porosity in seismic scale. (c) Seismic inverted Z p
and porosity in seismic scale.
The 3-D poststack seismic data have a trace interval of 20 m
and a sampling rate of 2 ms. The main tight-dolomite reservoir
of the early Cambrian period Longwangmiao (LWM) geologic
group is usually over 4500 m deep in the study area. The
LWM group is characterized with high-impedance tight
dolomite encased by a suite of complex dolomite on the top
and mudrock at the bottom. Porosity of dolomite reservoir
rocks is usually less than 8%.
Log data from one representative well in the study area, well
“A,” are shown in Fig. 4. Lithological analysis indicates that
the reservoir is made up predominantly of dolomite rock in the
LWM formation, where increasing acoustic impedance (Z p) is
strongly correlated with decreasing dolomite porosity. A synthetic seismogram is created as part of the log correlation
process, with the intention of extracting an acceptable seismic
wavelet for further analysis. The wavelet is calculated by
deconvolving the true seismic trace with reflectivity calculated from a blocked acoustic impedance profile, which was
obtained from well logs. However, due to the limitation in
seismic data resolution, it seems difficult to achieve a near
perfect match between the synthetic and true seismic signals
and to directly interpret thin-layer properties of the reservoirs
from seismic data alone.
The inverted acoustic impedance data, inverted from seismic
data using a conventional sparse-spike algorithm, define the
upper and lower boundaries of the reservoir. However, detailed
analysis suggests that the inverted Z p data alone are not necessarily suitable for quantitative prediction of porosity within the
reservoirs. Fig. 5 shows cross-plots between Z p from log and
inverted seismic to porosity in both log and seismic scales.
In log scale (0.125 m per sample), acoustic impedance of
dolomite rock in reservoirs shows strong negative correlation
to porosity. After converting to seismic scale (2 ms per
sample), the blocked log data show the same trend. However,
the seismically inverted acoustic impedance data show a weak
positive correlation to the measured reservoir porosity. This
false correlation might be related to the low resolution of
original seismic data and limitation in the inversion algorithm
to resolve such a thin layer.
B. APF Attribute Results
The proposed APF attributes are compared with the seismic
data and conventional APF attribute on a 2-D seismic profile
that passes through four gas wells [Fig. 6(a)]. The wavelet
extracted during the log correlation process of well “A” (Fig. 4)
is used to remove the wavelet effect on amplitudes in the
attributes [5]. The amplitude and attribute data are plotted
using a relative scale, where original values are normalized by
subtracting mean values and dividing by standard deviations.
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ZHOU et al.: APPLICATION OF AN STFT-BASED SEISMIC EVEN AND ODD DECOMPOSITION METHOD
5
the OAPF and the measured porosities at wells provides a
support for the proposition that the OAPF attribute is directly
related to, and can be used to estimate, reservoir porosity in
this case.
Fig. 7. Dolomite reservoir APF attributes and true porosity relationship from
wells. (a) Conventional APF. (b) EAPF. (c) OAPF.
As predicted in the previous modeling analysis, the highporosity dolomite exhibits strong negative OAPF values in
real seismic data, due to lower Z p value for high-porosity
rock and hence smaller reflection strength on the reflection
odd part. Compared to the seismic amplitude, EAPF, and APF
attribute profile, the OAPF profile correlates better to well log
porosity in four wells and characterizes the main reservoir
layers with better lateral continuity. Reservoirs in well “A”
are composed of two thin dolomite layers, with the shallow
layer being the main reservoir, while the other three wells
only contain one main reservoir layer. This suggests that the
OAPF attribute generally provides a more robust estimation
of reservoir porosity in thin-reservoir layers than conventional
seismic amplitude and related attributes.
The OAPF attribute results at the two wells are compared
in more detail with log-measured Z p, seismically inverted Z p,
and true porosity log curves at the seismic scale [Fig. 6(b)].
Compared to the log-measured Z p curves, Z p curves inverted
from seismic data depict the general trend in the true
impedance log. However, within the dolomite layer (highlighted area), Z p values of inverted seismic trace do not fit the
Z p trend in the log data. On the other hand, the OAPF curve
shows good negative correlation to the log porosity curve.
Furthermore, Fig. 7 shows the comparison of the three
APF attributes with the true porosity by using the data points
only within the reservoir. The APF attribute is relatively
unchanged with the dolomite porosity variation, similar to that
in the modeling results. The EAPF and OAPF attributes show
stronger correlation to porosity. Especially, the OAPF attribute
shows a clear negative correlation which is also evident in
both modeling results and log data. This suggests that the
OAPF is a relatively better proxy for the high-resolution rock
porosity prediction, compared to the conventional APF and
seismic impedance inversion analysis. The agreement between
V. C ONCLUSION
We propose that the signal even and odd decomposition method and EAPF and OAPF attributes are useful for
improving thin-layer reservoir property prediction. The OAPF
attribute is more sensitive to perturbations in thin-reservoir
properties compared to the conventional APF, while the
EAPF is preferentially indicative of background velocity-depth
trends. Numerical analysis and real seismic data show the
effectiveness of the workflow for delineating reservoir property
variation. During the analysis of field data, the conventional
seismic impedance inversion results show poor resolution of
porosity change within the thin reservoir. However, the OAPF
attribute shows good correlation to the true reservoir porosity
measured in wells. These results suggest that in the future,
incorporating EAPF and OAPF attributes may potentially
improve the robustness of seismic multiattribute analysis and
rock property prediction results.
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