TIME frequency (TF) transform maps a 1-D nonstationary

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1 142 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 15, NO. 1, JANUARY 2018 Time Frequency Analysis of Seismic Data Using a Three Parameters S Transform Naihao Liu, Jinghuai Gao, Bo Zhang, Fangyu Li, and Qian Wang Abstract The S transform (ST) is one of the most commonly used time frequency (TF) analysis algorithms and is commonly used in assisting reservoir characterization and hydrocarbon detection. Unfortunately, the TF spectrum obtained by the ST has a low temporal resolution at low frequencies, which lowers its ability in thin beds and channels detection. In this letter, we propose a three parameters ST (TPST) to optimize the TF resolution flexibly. To demonstrate the validity and effectiveness of the TPST, we first apply it to a synthetic data and a synthetic seismic trace and then to a filed data. Synthetic data examples show that this TPST achieves an optimized TF resolution, compared with the standard ST and modified ST with two parameters. Field data experiments illustrate that the TPST is superior to the ST in highlighting the channel edges. The lateral continuity of the frequency slice produced by the TPST is more continuous than that of the ST. Index Terms S transform (ST), seismic time frequency (TF) analysis, three parameters S transform (TPST). I. INTRODUCTION TIME frequency (TF) transform maps a 1-D nonstationary signal, such as seismic signal in time domain into a 2-D plane in the TF domain [1], [2]. The TF analysis methods can be categorized into three main classes [3]. The first class is the linear TF transform, such as the short-time Fourier transform (STFT), continuous wavelet transform (CWT), and S transform (ST). The linear TF methods are intuitive, but suffer from the Heisenberg uncertain principle [4]. The second class is based on the quadratic transform, such as Wigner Ville distribution [5]. The quadratic methods achieve a high TF resolution, and the disadvantage is that the cross Fig. 1. Proposed window function in time domain. (a) Proposed window in time domain with = (1.5, 0.5, 3) (blue line), = (1.5, 1, 3) (green line), and = (1.5, 1.5, 3) (red line), f = 10 Hz. (b) Proposed window in time domain with f = 10 Hz (blue line), f = 20 Hz (green line), and f = 30 Hz (red line), = (1.2, 0.9, 5). Manuscript received August 22, 2017; revised October 9, 2017 and November 4, 2017; accepted November 20, Date of publication December 13, 2017; date of current version December 27, This work was supported in part by the National Natural Science Foundation of China under Grant and Grant and in part by the Major National Science and Technology Projects under Grant 2016ZX and Grant 2017ZX (Corresponding author: Jinghuai Gao.) N. Liu is with the National Engineering Laboratory for Offshore Oil Exploration, School of Electronic and Information Engineering, Xi an Jiaotong University, Xi an , China, and also with the Department of Geological Sciences, The University of Alabama, Tuscaloosa, AL USA ( lnhfly@163.com). J. Gao and Q. Wang are with the National Engineering Laboratory for Offshore Oil Exploration, School of Electronic and Information Engineering, Xi an Jiaotong University, Xi an , China ( jhgao@mail.xjtu.edu.cn; wqlq668930@126.com). B. Zhang is with the Department of Geological Sciences, The University of Alabama, Tuscaloosa, AL USA ( bzhang33@ua.edu). F. Li was with the ConocoPhillips School of Geology and Geophysics, The University of Oklahoma, Norman, OK USA. He is now with the College of Engineering, University of Georgia, Athens, GA USA ( fangyu.li@uga.edu). Color versions of one or more of the figures in this letter are available online at Digital Object Identifier /LGRS Fig. 2. Changing rule of the proposed window width about frequency. The width of the proposed window is narrower than that of ST at low frequencies, and has the same size with that of the ST at high frequencies when p is greater than 1. The width of the proposed window is wider than that of the ST at high frequencies when p is smaller than 1. terms interfere the TF result. The third class is based on the principle of sparse spike inversion, such as the least-squares approaches and matching pursuit algorithms [6], [7]. Those sparse spike inversion approaches are designed to overcome the window effects of projection methods [8], and the disadvantage is that the computation cost is very high. The ST is proposed by Stockwell et al. [9], which is regarded as a hybrid of the STFT and CWT. By employing X 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 LIU et al.: TF ANALYSIS OF SEISMIC DATA USING A TPST 143 Fig. 3. Synthetic data example. (a) Noise-free (red line) and noisy (green line) synthetic signals s(t), consisting of three components. Noise-free TF spectra produced by (b) ST, (c) MST, and (d) TPST. Noisy TF spectra calculated by (e) ST, (f) MST, and (g) TPST. the Fourier transformation kernel, the ST retains the absolute phase information. By using a frequency-dependent window function, it can produce high time resolution at high frequencies and high frequency resolution at low frequencies. The standard deviation of the Gaussian function scales inversely with the frequency, which produces a frequency-dependent resolution for the ST [10]. Also, the ST maintains a direct relationship with the Fourier transform, which is helpful to reconstruct the analyzed signal. However, there is no parameter to adjust the window width of the ST resulting in a poor time resolution at low frequencies and degrade the frequency resolution at high frequencies. Various modified window functions were proposed to optimize the TF resolution. Mansinha et al. [11] proposed the 2-D ST and applied it to the pattern analysis. McFadden et al. [12] proposed a generalized ST (GST) by using an asymmetrical window function. Pinnegar and Mansinha [13] developed another form GST and applied it to determine the P-wave arrival time in a noisy seismogram. Pinnegar and Mansinha [13] used two prescribed frequency functions to control the scale and the shape of the analyzing window. Li and Castagna [14] proposed a modified ST (MST) with two parameters and employ it to direct hydrocarbon detection. Sejdić et al. [15] improved energy concentration in the TF domain by introducing a parameter to the window in the ST. In this letter, we propose a three parameters ST (TPST) and apply it to detect the channels and characterize the channel features. We first briefly review the theory of the standard ST. We then propose to optimize the TF resolution of the ST using a TPST. The proposed method is first tested by a synthetic signal and a synthetic seismic trace and then to real seismic data. II. S TRANSFORM The ST of a signal s(t) is defined as [9] ST(τ, f ) = s(t) f 2π e (t τ)2 f 2 2 e i2π ft dt (1)

3 144 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 15, NO. 1, JANUARY 2018 Fig. 4. Seismic synthetic trace example. (a) Reflectivity. (b) Seismic synthetic trace. TF spectra calculated by (c) standard ST, (d) MST, and (e) TPST. There are three reflection sets in this model. The first set only contains one reflectivity with a positive magnitude of 0.3. The magnitude of the second reflection set is 0.3 with the opposite sign. The third reflection set contains two reflectivities with a positive magnitude of 0.3. The time thicknesses of the second and third pairs are 30 and 40 ms. where t and f are time and frequency variables. τ denotes the localization of the frequency-dependent Gaussian window. The advantage of the ST over the STFT is that the ST performs a multiresolution of the analyzed signal s(t). According to Parseval s theorem, (1) can be rewritten in the frequency domain as ST(τ, f ) = S(α + f )e 2π2 α 2 f 2 e i2πατ dα (2) where S(α) is the Fourier transform of s(t). Obviously, taking advantage of the fast Fourier transform, the ST has a fast implementation. III. THREE PARAMETERS STRANSFORM The window used in the ST has standard deviation varied over frequency. However, the changing rule of the standard deviation remains invariable, and this is the main limitation of the ST. To remedy this drawback, a simple improvement can be obtained by modifying the standard deviation of the window in the ST as 1 δ( f ) = kf p (3) + m where = (k, p, m) are adjustable parameters. k is a parameter defining the mode of the window width. p adjusts the changing rate of the window width and m controls the tradeoff between the ST and STFT. Based on (3), the TPST can be expressed as TPST(τ, f ) = s(t) kf p + m e (t τ)2 (kf p +m) 2 2 e i2π ft dt. 2π The width of the proposed window is independent of frequency, and the TPST is degraded into the STFT when considering k = 0andm = 1. The TPST approaches the standard ST when = (1, 1, 0). According to Parseval s theorem, (4) can be rewritten in the Fourier domain as TPST(τ, f ) = S(α + f )e 2π2 α 2 (kf p +m) 2 e i2πατ dα. (5) Note that the three parameters = (k, p, m) control the width and shape of the window function in the time and frequency domains, by squeezing or stretching it. As a result, this distinctly achieves an adaptive and optimized TF resolution. Fig. 1 illustrates the proposed window in time domain for various parameters and various frequencies, respectively. Fig. 1(a) shows that the time resolution is increasing with increasing p. Fig. 1(b) illustrates that the time resolution is increasing with increasing frequency. Thus, our proposed method has the capability to improve the time resolution at the low frequency and compared with the standard ST. (4)

4 LIU et al.: TF ANALYSIS OF SEISMIC DATA USING A TPST 145 Fig. 2 shows the varying of the window width with varying frequencies for a different control parameters set = (k, p, m). The window width of the TPST (red line) is more compact than that of the ST (blue line) at low frequencies when p is greater than 1.This phenomenon indicates that the TPST has a higher time resolution than the ST when p is greater than 1. The window width of the TPST has the same size with that of the ST at high frequencies. This phenomenon indicates that the TPST has the same time resolution as the ST. The window width of the TPST is wider than that of the standard ST at high frequencies when p is smaller than 1. This phenomenon indicates that the TPST has an improved frequency resolution when p is smaller than 1. The above features indicate that we can obtain an adaptive TF representation with optimized TF resolution by choosing suitable parameters in the TPST. Note that we optimize the three parameters in the TPST using the concentration measure (CM) approach [17] in this letter. By optimizing the three parameters in the TPST using the CM method, we achieve an optimized TF representation. IV. INVERSION OF THREE PARAMETERS STRANSFORM The window function in the TPST satisfies the admissibility condition of unit area as kf p + m e (t τ)2 (kf p +m) 2 2 dτ = 1. (6) 2π This normalization condition guarantees the energy inversion of the TPST. By integrating the TF spectrum over time, the relationship between the Fourier spectrum and TF spectrum of the TPST is defined as TPST(τ, f )dτ = S( f ) (7) where S( f ) is the Fourier transform of s(t). Therefore, the input signal can be simply reconstructed as s(t) = TPST(τ, f )e i2π ft dτdf. (8) V. SYNTHETIC AND FIELD DATA EXAMPLES In this section, we apply the TPST to synthetic signals and field data to compare its performance with the standard ST and MST with two parameters [17]. Fig. 3(a) shows a composited signal s(t) in (9) (red line) containing three slowly varying frequency components [14], [15], where the green line denotes the noisy signal added with Gaussian white noise. The SNR equals 5 db. We optimize and choose the parameters in the TPST as (0.5, 0.8, 2) in this example. Fig. 3(b) (d) shows the noise-free TF spectra calculated by using the ST, MST, and TPST, respectively. Due to the invariable changing rule of the standard deviation, the TF spectrum produced by the ST illustrates a smearing TF result. Although the TF spectrum calculated by the MST distinguishes three components, the TF resolution needs to be improved. Compared with the ST and MST, the TF spectrum produced by the TPST separates the three slowly varying frequency components and achieves a higher TF resolution. Fig. 3(e) (g) shows the noisy TF spectra calculated by the ST, MST, and TPST, respectively. Although the TF spectrum calculated by the TPST is affected Fig. 5. Field data example. (a) Time slice of seismic amplitude at 1.81 s. (b) RGB blending image of the 20-, 50-, and 70-Hz slices calculated by the ST. (c) RGB blending image of the 20-, 40-, and 70-Hz slices calculated by the TPST. by heavy noise, it achieves a more accurate and stable result than those produced by the ST and MST s(t) = cos(60πt + 8πt 2 ) + cos(40πt + 4πt 2 ) + cos(20πt 2πt 2 ). (9) Fig. 4(b) shows a seismic synthetic trace, through the convolution between a-three pair reflectivity in Fig. 4(a) and a Ricker wavelet. The dominant frequency of the Ricker wavelets is 50 Hz. The first reflectivity pair only has one positive reflectivity with a magnitude of 0.3. The magnitude of the second reflection pair is 0.3 with the opposite sign. The third reflection set contains two reflectivities with a positive magnitude of 0.3. The time thicknesses of the second and third reflectivity pairs are 30 and 40 ms, respectively. Fig. 4(c) and (d) show the TF spectra generated by the ST, MST, and TPST, respectively. Note that (0.9, 1.1, 8) is chosen in the TPST in this example. Unfortunately, the TF spectrum computing by using the ST is smeared to certain degree to the second and third reflection pairs. The TF spectrum calculated by the MST almost distinguishes the second and third pairs. The TF spectrum achieved by the TPST distinguishes the second and third pairs clearly. The synthetic seismic test illustrates that the TPST is superior to the ST and MST in detecting thin beds. At last, we apply the TPST to field data to characterize channels. The studied seismic survey, Watonga, is located at the west central Oklahoma and in the eastern part of the Anadarko Basin, Oklahoma. The target formation is the Middle Pennsylvanian age Red Fork Formation and is composed of clastic facies deposited in a deep marine

5 146 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 15, NO. 1, JANUARY 2018 by using the TPST, respectively. The black ellipses shown in Fig. 6(a) (c) shows the location of the main channels. The channel in Fig. 6(c) is well separated from about the above formations, while the channel in Fig. 6(b) smears along the vertical time axis. Please also note the lateral continuity of the TF features in Fig. 6(c) is superior to that in Fig. 6(b). VI. CONCLUSION In this letter, we propose a TPST to obtain an optimized TF representation. The proposed method employs three parameters = (k, p, m) to flexibly control the window width to achieve an optimized TF resolution. Both synthetic signals and real seismic data illustrate that the TPST has an improved time resolution at the low-frequency band compared with those of the ST and MST. At the same time, the TPST has the same TF resolution at the high-frequency band. Thus, our proposed TPST is superior to detect the channel features than the ST and MST. Fig. 6. Two-dimensional field data example. (a) 2-D seismic data from Fig. 5(a) with Inline number Hz spectral components calculated by (b) standard ST and (c) TPST. The black ellipses indicate the channels, which are indicated by white arrows in Fig. 5(b) and (c). (shale/silt) to shallow-water fluvial-dominated environment. The Red Fork Sandstone is sandwiched between two limestone layers with the Pink Limestone on the top and the Inola Limestone on the bottom [16]. The Upper Red Fork incised-valley system consists of multiple stages of incision and fill, resulting in a stratigraphically complex internal architecture [8], [16]. The sample rate of the seismic data is 4 ms. Fig. 5(a) shows the amplitude time slice through the incised Red Fork channels at 1.81 s (all the spectral component time slices in this letter have the same 1.81 s). The red and white arrows indicate wide branch channels, whereas the yellow arrows denote narrow branch channels. Improving the time resolution of the TF spectrum is the essential to characterize channels at different scales. Using the CM method, we optimize the parameters in the TPST as (0.9, 1.2, 7) in this field data. Fig. 5(b) shows the RGB blending image of the 20-, 50-, and 70-Hz spectral slices calculated by using the ST. Fig. 5(c) indicates the RGB blending image of the 20-, 40-, and 70-Hz spectral slices calculated by using the TPST. Both RGB images correctly highlight the wide channels. However, the RGB image produced by the TPST is superior to that of the ST in highlight the narrow channels indicated by blue arrows. The RGB blending image indicates channels at different scales clearly, especially the edges and thickness of channels. We next exam the channel features vertical sections along the red line shown in Fig. 5(a). 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