Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation

Journal of Earth Science, Vol. 4, No. 6, p. 168 178, December 13 ISSN 1674-487X Printed in China DOI: 1.17/s1583-13-387-1 Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation Luping Sun* ( 孙鲁平 ) School of Geophysics and Information Technology, China University of Geosciences, Beijing 183, China; Key Laboratory of Geo-Detection, Ministry of Education, China University of Geosciences, Beijing 183, China Xiaodong Zheng ( 郑晓东 ), Hao Shou ( 首皓 ) Institude of Geophysical Exploration Technology, RIPED, Beijing 183, China ABSTRACT: Quantitative thickness estimation of thin-layer is a great challenge in seismic exploration, especially for thin-layer below tuning thickness. In this article, we analyzed the seismic response characteristics of rhythm and gradual type of thin-layer wedge models and presented a new method for thin-layer thickness estimation which uses relative peak frequency increment. This method can describe the peak frequency to thickness relationship of rhythm and gradual thin-layers in unified equation while the traditional methods using amplitude information cannot. What s more, it won t be influenced by the absolute value of thin-layer reflection coefficient and peak frequency of wavelet. The unified equations were presented which can be used for rhythm and gradual thin-layer thickness calculation. Model tests showed that the method we introduced has a high precision and it doesn't need to determine the value of top or bottom reflection coefficient, so it has a more wide application in practice. The application of real data demonstrated that the relative peak frequency increment attribute can character the plane distribution feature and thickness characteristic of channel sand bodies very well. KEY WORDS: thin-layer, quantitative estimation, relative peak frequency increment, Ricker wavelet. INTRODUCTION Quantitative thin-layer thickness estimation is a frequently touched topic in seismic exploration. This study was supported by the Fundamental Research Funds for the Central Universities, Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 1114), and the China National Key S & T Project on Marine Carbonate Reservoir Characterization (No. 11ZX543). *Corresponding author: sunluping@cugb.edu.cn China University of Geosciences and Springer-Verlag Berlin Heidelberg 13 Manuscript received January 3, 13. Manuscript accepted May 9, 13. Widess (1973) concluded that the limit of detectable thickness for a single bed is λ/8, where λ is the dominant wavelength. Neidell and Poggiagliolmi (1977) defined tuning-thickness as λ/4. Koefoed and Voogd (198) gave the definition of thin-layer by the linear relation between the amplitude of composite reflection and thickness. According to Kiefoed and Voogd s study, thin-layer was defined as the layer of which the error between true amplitude line and linear approximation line is under 1% (Koefoed and Voogd, 198). In practice, from the Rayleigh s criteria, λ/4 is usually considered as the limit of seismic resolution considering the influence of noise and wavelet (Neidell and Poggiagliolmi, 1977). In this article, we also considered thin-layer as thickness less than λ/4. For quantitative thin-layer thickness estimation,

Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation 169 there are three major methods. The first type is methods in time domain, which calculates the thickness of thin-layer using the characteristics of composite waveform. The second type is methods in frequency domain, which estimates thin-layer thickness with frequency spectrum. The third type can be called integrated attributes inversion method, which predicts thin-layer thickness by integrated non-linear inversion algorithms. The most widely used information in time domain is amplitude of composite waveform. This idea was firstly indicated by Widess (1973), he pointed out that the thickness less than λ/8 can be estimated by exploiting the linear relation between thickness and reflection amplitude. But he only studied the rhythm type of thin-layer of which the upper and lower reflection coefficient are equal magnitude but opposite polarity. Chung and Lawton (1995) did a deeper research and extended the method into rhythm and gradual thin-layers. Spectral decomposition (Gridley and Partyka, 1997) and spectral inversion (Portniaguine and Castagna, 4) are two common techniques in frequency domain for thin-layer thickness estimation. The theoretical basement of spectral decomposition to calculate thin-layer thickness is the reciprocal relation between the period of spectral notches and thickness. This method is widely used in the imaging and mapping of bed thickness and geologic discontinuities (Partyka, 5; Huang, ; Marfurt and Kirlin, 1; Partyka et al., 1999). However, the spectral notches phenomenon is very complex in real. If seismic bandwidth is insufficient to identify the period of spectral notches, this method will be ineffective. Spectral inversion is another way to determine thin-layer thickness (Puryear and Castagna, 8; Chopra and Castagna, 7, 6; Portniaguine and Castagna, 4). Although this method has a high precision, it is very sensitive to noise. Besides, Yao (1991) estimated thin-layer thickness from low frequency component and Liu and Marfurt (5) used peak instantaneous frequency instead. However, both of them only discussed the case of rhythm thin-layer (Yao, 1991) or almost rhythm thin-layer (Liu and Marfurt, 5). The method presented by Dou (1995) also fell into this type. Sun et al. (9) presented the theoretical expression and approximate formulas for thin-layer thickness calculation using peak frequency. The real data application showed that it is very useful in channel sand bodies prediction (Sun et al., 1). For the integrated attributes inversion method, the integrated non-linear inversion algorithms are used to predict thickness of thin-layer, which assume there are non-linear relationship between seismic attributes and thin-layer thickness (Zhang et al., 7; Bai et al, 6; Huang et al., 1997). However, this method needs well logging information to train samples and only can be used in areas with a certain number of wells. From the above discussion, we can see some of these methods have additional requirements and some are only suitable for special thin-layer type. All of these limit their practical applications. In this article, a method for quantitative thin-layer thickness estimation using relative peak frequency increment was proposed, which belongs to the frequency domain method. Compared to traditional amplitude method in time domain, this new method can unify the rhythm and gradual type of thin-layers. The thickness calculation equations for these two types of thin-layer were presented with the identical form. The equations have high precision and will not be influenced by the thin-layer reflection coefficients. THEORY AND METHOD Wavelet and Thin-Layer Wedge Models In this article, peak frequency is defined as the frequency corresponding to the maximum spectral amplitude in the frequency domain. Suppose the seismic source wavelet is Ricker wavelet (Ricker, 1953). The main wavelength and period of wavelet are denoted by λ and T m, respectively. The Ricker wavelet and its amplitude spectrum are showed in Figs. 1a and 1b. The peak frequency of wavelet is f p =4 Hz. Then the main period of wavelet is Tm = 6/( πf p) ms (Kallweit and Wood, 198), with the corresponding tuning thickness equal to 1.5 m. Figures a and b show the reflectivity wedge models of the rhythm and gradual thin-layer respectively. The top and bottom reflection coefficients are denoted by R 1 and R. For convenience, we did not consider the density variation here. The X-axis of Figs. a and b is trace number and the Y-axis is two-way travel time thickness. Since we are only interested in

17 Luping Sun, Xiaodong Zheng and Hao Shou thin-layer, the maximum thickness of all the wedge models designed in this article is below the tuning thickness. The thickness of the wedge models in Figs. a and b varies from ms on the left to 1 ms on the right. The maximum thickness is 1.5 m. We set two pulses at the top and bottom interfaces position. With the Fourier transform, the amplitude spectrum of two wedge models was obtained (Figs. c and d). The X-axis is trace number and the Y-axis is frequency. We can see the amplitude spectrum of both rhythm and gradual thin-layers show periodical notches (Gridley and Partyka, 1997). The period of notches is Tf =1/ τ, whereτ is the two-way travel time thickness of thin-layer. This is the theoretical basement of spectral decomposition to calculate thin-layer thickness (Marfurt and Kirlin, 1; Gridley and Partyka, 1997; Okaya, 1995). Figure 3 are seismograms and amplitude spectrum of seismograms of the wedge models. The seismograms are generated by convolving wavelet with reflectivity series. Here we showed the amplitude spectrum of all the 5 traces together. From the 1. 6 fp=4 Hz Amplitude.5. Amplitude 4 -.5 -.4 -....4 t (s) 4 8 1 16 f (Hz) Figure 1. Ricker wavelet and its amplitude spectrum. Two way time (ms) 4 6 8 1 V1= m/s V3= m/s V= 5 m/s Two way time (ms) 4 6 8 1 V1= m/s V3=3 15 m/s V= 5 m/s 1 1 1 3 4 5 1 3 4 5 ( c ) (d) 5 5 f (Hz) 1 f (Hz) 1 15 15 1 3 4 5 1 3 4 5 Figure. Thin-layer wedge model and its amplitude spectrum. Rhythm (R 1 = -R ) type wedge model; gradual (R 1 =R ) type wedge model; (c) amplitude spectrum of rhythm type; (d) amplitude spectrum of gradual type wedge model. Dark is positive amplitude.

Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation 171 1 1 Spectrum peak Spectrum Two way time (ms) Two way time (ms) 4 6 8 1 1 1 3 4 5 4 6 8 1 Amplitude Amplitude 1 ( c ) (d) 8 6 4 4 6 8 f (Hz) 1 8 6 4 Spectrum peak Spectrum 1 1 3 4 5 4 6 8 f (Hz) Figure 3. and are synthetic seismograms and amplitude spectrum of rhythm type wedge model (R 1 = -R ); (c) and (d) are synthetic seismograms and amplitude spectrum of gradual type wedge model (R 1 =R ). The blue real line and red dotted line show the location of peak frequency of source wavelet and peak frequency of seismograms respectively. The red arrowheads instruct thickness diminution direction. seismograms (Figs. 3a and 3c), we can see the reflection waveform of top and bottom interface superposed together and cannot be identified. The blue solid line and red dotted line mark location of peak frequency of source wavelet (f=4 Hz) and peak frequency of seismograms respectively. The arrowheads instruct thickness decreasing direction. From Figs. 3b and 3d, we found the amplitude spectrum for rhythm (R 1 = -R ) and gradual (R 1 =R ) thin-layer have differences and similarities. From Fig. 3b, we can see that for rhythm case (R 1 = -R ) the maximum amplitude of spectrum decreases gradually with thickness decreasing (the red arrowheads are downward). What s more, peak frequency of seismogram is always higher than that of the source wavelet (the red dotted line is on the right-hand of the blue line in Fig. 3b). However, for gradual case (R 1 =R ) showed in Fig. 3d, the situation is opposite. The maximum amplitude of spectrum increases gradually with thickness decreasing (the red arrowheads are upward) and the peak frequency of seismogram is lower than that of the source wavelet (the red dotted line is on the left-hand of the blue line in Fig. 3d). The similarity is the peak frequency of seismograms for both case R 1 = -R and R 1 =R increases gradually with thin-layer thickness decreasing (both of Figs. 3b and 3d the red arrowheads are rightward). This phenomenon indicates that the information of thin-layer thickness must be contained in the peak frequency of seismograms, which constitutes the theoretical basis of this paper. Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Calculation Widess (1973) showed the thickness less than λ/8 can be estimated by exploiting the relation between thickness and reflection amplitude. In his formula the maximum relative amplitude of seismograms in time domain was used for thickness calculation. Figure 4 illustrates the maximum relative amplitude of seismograms A max. The X-axis is the thin-layer thickness d

17 Luping Sun, Xiaodong Zheng and Hao Shou expressed by the ratio of real thickness to the dominant wavelength. With thin-layer thickness increasing, A max curve for rhythm case firstly increases linearly then increases slowly, finally reaches maximum at tuning thickness (Fig. 4a). But for gradual case showed in Fig. 4b, with thin-layer thickness increasing, A max curve firstly decreases slowly then a little faster. Near the tuning thickness, the decrease rate of A max slows down suddenly. The minimum of A max shows at tuning thickness position. The A max curve for case of rhythm and gradual are quite different and it is difficult to express these two curves in a unified equation. As frequency is one of the most closely related attributes to thin-layer thickness, we presented a new method for quantitative thin-layer thickness estimation fp_s - fp -.331 d +.4 4 R1 = - R = 1/9 = fp -5.116 d -.5 17 R1 = R = 1/9 using relative peak frequency increment. In this paper, peak frequency increment f p_s f p is defined as the difference between peak frequency of seismogram f p_s and that of wavelet f p. Then relative peak frequency increment (f p_s f p )/f p is defined as f p_s f p divided by peak frequency of wavelet f p. The curves of (f p_s f p )/f p are showed in Fig. 5. It can be seen that (f p_s f p )/f p curves are similar for the cases R 1 = -R and R 1 =R. and can be easily expressed in an unified form. Using a quadratic curve f(x)=ax +c to fit the relationship of relative peak frequency increment (f p_s -f p )/f p with thin-layer thickness d, the results are shown in Fig. 6. Black dotted line is the real data and red real line is fitting result. The fitting equation is (1).3.3 Amax..1 Maximum relative amplitude...5.1.15..5 Amax.5..15 Maximum relative amplitude.1..5.1.15..5 Figure 4. Relationship between maximum relative amplitude and thin-layer thickness. Rhythm type wedge model (R 1 = -R ); gradual type wedge model (R 1 =R )... -.5.18.14 -.1 -.15 -..1 Relative peak frequency increment..5.1.15..5 -.5 Relative peak frequency increment -.3..5.1.15..5 Figure 5. Relationship between relative peak frequency increment and thin-layer thickness. Rhythm type wedge model (R 1 =-R ); gradual type wedge model (R 1 =R ).

Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation 173 From Equation (1), the thin-layer thickness d is obtained by (- f +.4 4)/.331 R = - R = 1/9 f - f d = f = (- f -.5 17)/5.116 R1 = R = 1/9 fp 1 p_s p () DISCUSSION The calculation Equation () is obtained by a wavelet with peak frequency f p =4 Hz and wedge models with reflection coefficients R 1 = R =1/9. However, in actual application the value of f p and R 1 or R maybe not as that we used. Moreover, both f p and R 1, R are not so easy to determine accurately. So in order to test the universality and applicability of Equation (), we did some model tests as follows to discuss the influence of f p and R 1, R to Equation (). Wavelet Peak Frequency f p With the gradual type (R 1 =R ) of thin-layer wedge model showed in Fig. b, we generated five seismograms by wavelets with different peak frequency f p equal to, 4, 6, 8, and 1 Hz. The curves of peak frequency increment f p_s -f p and relative peak frequency increment (f p_s f p )/f p are showed in Figs. 7a and 7b, respectively. In Fig. 7a, five curves with different f p are separated. In other words, peak frequency increment f p_s f p is influenced by peak frequency of wavelet. But as shown in Fig. 7b, five curves are superposed together which indicates relative peak frequency increment (f p_s f p )/f p is not influenced by f p. The new method we proposed in this paper is using (f p_s f p )/f p to calculate thin-layer thickness, so it can remove the influence of f p.. Real data Fit line. Real data Fit line.16 General model: fx ( ) = a x +c Coefficients:.1 a=-.331; c=.4 4 Goodness of fit: R :.999 7 RMSE:. 89 5.8..5.1.15..5 Thickness ( ) λ -.1 -. -.3 General model: fx ( ) = a x +c Coefficients: a=-5.116; c=-.5 17 Goodness of fit: R :.995 9 RMSE:.5 463..5.1.15..5 Thickness ( ) Figure 6. Fitting results of relative peak frequency increment of seismograms (f p_s f p )/f p and relative thickness d. Rhythm type wedge model (R 1 = -R ); gradual type wedge model (R 1 =R ). Black dotted line is the real data and red real line is fitting result. λ. -5 -.5 p_s p f - f -1-15 fp= Hz - fp=4 Hz fp=6 Hz -5 fp=8 Hz -3 fp=1 Hz..5.1.15..5 -.1 -.15 -. -.5 -.3 fp= Hz fp=4 Hz fp=6 Hz fp=8 Hz fp=1 Hz..5.1.15..5 Figure 7. Relationship between f p_s f p and d; relationship between (f p_s f p )/f p and d.

174 Luping Sun, Xiaodong Zheng and Hao Shou Reflection Coefficients of Thin-Layer To systematically study the influence of reflection coefficients, we generated a series of wedge models (Table 1, 1 models) with different R 1 and R. Fig. 8a and Fig. 8b show the maximum relative amplitude of seismograms A max in time domain and relative peak frequency increment (f p_s f p )/f p in frequency domain, respectively. The number on the right side is the value of R 1 /R. Table 1 Reflection coefficients of different wedge models R1: -..4 -.3.4 -.1.. -.4 -.3.4.1 -. R: 1. -..6 -.8.1 -..1 -. -.6.8.1 -. R/R 1 : - - -.5 -.5-1 -1.5.5 1 1 Diagram 1.6 -.5. Amax -.5.1-1 1. -.5. -.8.5 -.1-1.4 - -. -1 1 1 -.3 1...5.1.15..5..5.1.15..5 Figure 8. Relationship between A max and d; relationship between (f p_s f p )/f p and d. In Fig. 8a, all curves of the 1 models are different. It shows the values of R 1 and R influence the maximum relative amplitude of seismogram A max. If use A max to estimate thickness, we need to firstly determine the value of R 1 and R, otherwise it is unable to make sure the corresponding relationship. However, there are only four curves in Fig. 8b. The four models of R 1 /R = - and R 1 /R = -.5 fall on the same curve (blue curve). There are also the same results for the four models of R 1 /R = and R 1 /R =.5 (green curve), two models of R 1 /R = -1 (red curve) and two models of R 1 /R =1 (pink curve). This indicates the relative peak frequency increment (f p_s f p )/f p is not dependent on the absolute values of R 1 or R, but is only determined by R 1 /R. This is quite different from A max. What s more, models with reciprocal ratio of top to bottom reflection coefficients have the same (f p_s f p )/f p curve, as indicated by the cases of R 1 /R = and R 1 /R =.5. According to the discussion above, we concluded that the absolute values of R 1 or R wouldn t influence the relative peak frequency increment. So the Equation () can be written as follows. (- f +.4 4)/.331 R = -R f -f d = f = (- f -.5 17)/5.116 R1 = R f p 1 p_s p (3) Equation (3) suits for all the thin-layers with equal magnitude reflection coefficients, both rhythm and gradual type. So it has universal significance. Compared with traditional amplitude method in time domain, the method proposed in this paper can express the rhythm and gradual type of thin-layers in unified equation. And more convenience, we need not to determine the value of R 1 or R in actual ap-

Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation 175 plication. Here, we gave a quantitative explanation for Equation (3). Sand-shale formation is very common in oil and gas exploration especially in China. Considering a thin sand formation surrounded by mudstone and the thickness of sand layer is less than λ/4, λ is the main wavelength. The purpose of our work is to make clear the thickness of this sandstone reservoir. In this case, the top and bottom reflection coefficients of sand are of equal magnitude and opposite polarity (R 1 = -R ), which is a typical rhythm type of thin bed. Suppose the peak frequency of seismic wavelet f p equal to 4 Hz. At the position of our target formation, due to the tuning effect of thin sandstone, the peak frequency of seismic record f p_s will be higher than f p. Suppose the increment of seismic peak frequency is equal to 6 Hz, which means f p_s =46 Hz. Then according to Equation (3), the relative peak frequency increment f=(f p_s f p )/f p =.15 and the relative thickness of sand formation d= (- f +.4 4)/.331 =.178 6. The results indicate that the thickness of sand at this position is equal to.178 6 λ. If the velocity of the target sandstone can be determined by other method, the real thickness will be calculated. For example, if the velocity of the target sandstone V s =3 m/s, then the real thickness of the sand reservoir at this position can be estimated by h=.176 8 λ.176 8 V s /f p =13.6 m. PRECISION ANALYSIS The precision of Equation (3) was discussed. Figure 9 show the curves of real thickness (red line) and calculated thickness (black line) with Equation (3). The corresponding error curves of rhythm type (blue line) and gradual type (red line) of thin-layer are showed in Fig. 1. The error was estimated as calculated thickness minus real thickness then divided by real thickness. From Figs. 9 and 1, we can see the precision of Equation (3) is pretty good, especially when the thickness of thin-layer is more than.5λ. While the thickness is more than λ/16, the error of Equation (3) for gradual type thin-layer is under 5%. For rhythm type, the precision is much higher; the error is less than 1%. When the thickness is very small, less than.5λ, the peak frequency will be difficult to pick accurately, so the error becomes bigger. ACTUAL APPLICATION In order to verify the applicability of this method to real data, we used it in area A. The tectonic setting of area A is a Mesozoic rift basin. The data we used in this article is from the south depression of the basin. The lithology of our interested formation is clastic. The exploration degree of this area is in early stage. There are some channels deposits in Upper Jurassic. The plane distribution features and thickness determination of these channel sand bodies are very important for this area s reservoir prospecting. We calculated the relative peak frequency increment attribute (Fig. 11a) along the target horizon and converted it to two-way time thickness (Fig. 11b) using.5.5.. Thickness ( λ ).15.1 Thickness ( λ ).15.1.5 Calculated thickness Real thickness.5 Calculated thickness Real thickness..5.1.15..5..5.1.15..5 Figure 9. The curves of real and calculated thickness. Rhythm type wedge model (R 1 = -R ); gradual type wedge model (R 1 =R ). Black line is the real thickness and red line is calculated result.

176 Luping Sun, Xiaodong Zheng and Hao Shou Relative error.4.. -. -.4 -.6 -.8 Error curve for rhythm thin-layer Error curve for gradual thin-layer -1..5.1.15..5 Figure 1. Error curves. Blue and red lines are the error curves of rhythm and gradual type, respectively. Equation (3). The relative peak frequency increment attribute varies in different position on the plane map. At the boundary of the channels, the sand thickness is relative thin and peak frequency value is bigger than in the center of the channel. With the inverted P-wave velocity (Fig. 11c) we finally arrived at sand thickness in depth domain (Fig. 11d). From these results we found that the relative peak frequency increment attributes can character the plane distribution feature and thickness characteristic of channel sand bodies very well. The thickness of channel sand bodies is about 4 to 3 meters. The results of this article provide a good basis for deeper exploration in this area. Figure 11. Relative peak frequency increment attribute; time thickness of sand bodies; (c) P-wave velocity; (d) calculated thickness of sand bodies. CONCLUSIONS A new method for quantitative thin-layer thickness estimation using relative peak frequency incre ment was presented. According to the theoretical study and model simulation, we arrived at the following conclusions. (1) Under the range of tuning thickness, the maximum relative amplitude curves in time domain are quite different for rhythm and gradual thin-layer. While the relative peak frequency increment curves are similar and can be easily expressed in an unified form. And the certain unified equations were presented with a high precision. () The relative peak frequency increment is not influenced by peak frequency of wavelet and it is determined by the ratio of top to bottom reflection coef-

Relative Peak Frequency Increment Method for Quantitative Thin-Layer Thickness Estimation 177 ficients instead of their absolute values. So in actual application, we just make sure the thin-layer type, and need not to pay much attention to the absolute value of reflection coefficients. (3) The real data application demonstrated the relative peak frequency increment attribute is suitable to describe the plane distribution of channel sand bodies and indicate the favorable reservoir position. The method for quantitative thin-layer thickness estimation using relative peak frequency increment is feasible. (4) The spectrum for real data maybe complex in some situation. Sometimes, the seismic spectrum may have more than one peak and the peak frequency with the definition in this paper may be not appropriate. In this case, we advise to use peak frequency of the spectral envelope to estimate thin-layer thickness. ACKNOWLEDGMENTS This work was supported by the Fundamental Research Funds for the Central Universities, Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 1114), and the China National Key S & T Project on Marine Carbonate Reservoir Characterization (No. 11ZX543). Thanks go to Prof. Yan Zhang for his great support to this research. REFERENCES CITED Bai, G. J., Wu, H. N., Zhao, X. G., et al., 6. Research on Prediction of Thin Bed Thickness Using Seismic Data and Its Application. Progress in Geophysics, 1(): 554 558 (in Chinese with English Abstract) Chopra, S., Castagna, J. P., 6. Thin-Bed Reflectivity Inversion. SEG 76th Annual International Meeting Expanded Abstracts, New Orleans. 57 61 Chopra, S., Castagna, J. P., 7. Thin-Bed Reflectivity Inversion and Seismic Interpretation. SEG 77th Annual International Meeting Expanded Abstracts, San Antonio. 193 197 Chung, H. M., Lawton, D. C., 1995. Amplitude Responses of Thin Beds: Sinusoidal Approximation versus Ricker approximation. Geophysics, 6(3): 3 3 Dou, Y. S., 1995. Thin Bed Interpretation with Amplitude Spectrum Square Ratio Method. Oil Geophysical Prospecting, 3(): 57 65 (in Chinese with English Abstract) Gridley, J. A., Partyka, G. A., 1997. Processing and Interpretational Aspects of Spectral Decomposition. SEG Technical Program Expanded Abstracts, Dallas. 155 158 Huang, Z. P., Wang, X. H., Wang, Y. Z., 1997. Parameter Analysis of Seismic Attributes and Thickness Prediction for Thin Bed. Geophysical Prospecting for Petroleum, 36(3): 8 38 (in Chinese with English Abstract) Huang, X. D.,. Discussion on Notches in Thin Bed. Progress in Exploration Geophysics, 5(5): 1 6 (in Chinese with English Abstract) Kallweit, R. S., Wood, L. C., 198. The Limits of Resolution of Zero-Phase Wavelets. Geophysics, 47(7): 135 146 Koefoed, O., Voogd, N. D., 198. The Linear Properties of Thin Layers: With an Application to Synthetic Seismograms over Coal Seams. Geophysics, 45(8): 154 168 Liu, J. L., Marfurt, K. J., 6. Thin Bed Thickness Prediction Using Peak Instantaneous Frequency. SEG 76th Annual International Meeting Expanded Abstracts, New Orleans. 968 97 Marfurt, K. J., Kirlin, R. L., 1. Narrow-Band Spectral Analysis and Thin-Bed Tuning. Geophysics, 66(4): 174 183 Neidell, N. S., Poggiagliolmi, E., 1977. Stratigraphic Modeling and Interpretation-Geophysical Principles and Techniques. American Association of Petroleum Geologists, Special Memoir, 6: 389 416 Okaya, D., 1995. Spectral Properties of the Earth s Contribution to Seismic Resolution. Geophysics, 6(1): 41 51 Partyka, G. A., 5. Spectral Decomposition. SEG Distinguished Lecture, Houston Partyka, G. A., Gridley, J. A., Lopez, J. A., 1999. Interpretational Aspects of Spectral Decomposition in Reservoir Characterization. The Leading Edge, 18(3): 353 36 Portniaguine, O., Castagna, J. P., 4. Inverse Spectral Decomposition. SEG 74th Annual International Meeting Expanded Abstracts, Denver. 1786 1789 Puryear, C. I., Castagna, J. P., 8. Layer-Thickness Determination and Stratigraphic Interpretation Using Spectral Inversion: Theory and Application. Geophysics, 73(): 37 48 Ricker, N., 1953. Wavelet Contraction, Wavelet Expansion, and the Control of Seismic Resolution. Geophysics, 18(4): 769 79 Sun, L. P., Zheng, X. D., Shou, H., et al., 1. Quantitative Prediction of Channel Sand Bodies Based on Seismic

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