This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 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 , . Spectral decomposition has been applied in direct hydrocarbon detection, seismic attenuation analysis, channel identification, and thin-layer thickness estimation –. Plain-view amplitude images of the narrowband subsets show different interfering patterns within the analysis window . 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 , –, 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 . 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  and Kallweit and Wood  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 . 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. . 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 . 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: [email protected]). 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. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 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 , 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  and Kallweit and Wood , 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 , 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) This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 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. , 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 , 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. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 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 . 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. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 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. 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