IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 2, MARCH 1999
867
Instantaneous Parameters Extraction via Wavelet Transform Jinghuai Gao, Xiaolong Dong, Wen-Bing Wang, Youming Li, and Cunhuan Pan
Abstract—In this paper, a novel theorem on the wavelet transform and Hilbert transform (HT) is proposed and applied to extract the instantaneous parameters of energy-limited, real signals. Numerical simulations shows advantages of the presented method in both precision and antinoise performance. Index Terms—Analytic signal, instantaneous attributes, wavelet transform.
I. INTRODUCTION The instantaneous parameters (IP’s) of seismic signal can be used to analyze the strata characteristics, such as rock properties and subsurface structure, and can be used for parameters inversion, such as a quality factor [1]–[3]. Although the concept of the IP’s has been introduced for decades, improving the estimation precision, investigating the physical interpretation, and extending their applications are still the active areas for researching [4]–[5]. The commonly used method for IP estimation is as follows. First, the analytic counterpart corresponding to the real-valued signal is found via the Hilbert transform (HT); then, the IP’s are calculated with the analytic signal. Due to the infinite length and the slow decaying rate of the impulse response (IR) of HT, the Gibbs effect in digital implementation will introduce reduction of estimation precision. Moreover, this method is sensitive to noise. In this paper, a wavelet-based method is proposed for the calculation of the analytic counterpart of a real-valued signal in L2 (R); (which implies finite energy); comparisons between the new method and HT method are made, the numerical simulations showing higher precision and antinoise performance of the new method. II. DEFINITIONS
AND
NOTATIONS
Considering an analytic wavelet function transform g^(! ) satisfying
g(t) 2 L1 (R; dt) g^(!) 2 L1 (Rnf0g; d!=j!j)
g(t)
and its Fourier
\ L (R; dt)
(1)
\ L (Rnf0g; d!=j!j)
(2)
2
2
Fig. 1. Simulated signal.
where t; b 2 R; R is the real number set and conjugate of g (t): III. METHODS A. Theorem
Theorem 1: If s(t) is a signal with finite energy and g (t) is an analytic wavelet function, then S (b; a); the WT of s(t), with respect to g (t); is a complex function with respect to real-valued variable b and the scale factor a(a > 0): For a fixed value of a; the imaginary part of this complex function is the HT of the real part; i.e., S (b; a) is an analytic function with respect to b: The proof is given in [1] and [9]. Theorem 2: If g (t) is an analytic wavelet function with its real part gR (t) being even and Cg = s01 (^gR(!) =!) d!; with 0 < Cg < 1: Then for an arbitrary real s(t) 2 L2 (R; dt); we have
Cg
1
a 01
s(t)g t 0 b dt a
= s(t) + jH [s(t)]
1 S (t; a) = 1 s(b)g b 0 t db a 01 a 1 1 = s(b) gR b 0 t 0 jgI b 0 t a 01 a a 1 1 = s(b)gR b 0 t db a 01 a 1 1 0ja s(b)gI b 0 t db a 01 = SR (t; a) + jSI (t; a)
(3)
Manuscript received August 13, 1996; revised March 23, 1998. J. Gao and Y. Li are with the Institute of Geophysics, the Chinese Academy of Sciences, Beijing, 100101 China (e-mail:
[email protected]). X. Dong is with the Center of Space, the Chinese Academy of Sciences, Beijing, 100101 China. W.-B. Wang is with the Institute of Microwave Engineering and Optical Communication, Xi’an Jiaotong University, Xi’an, 710049 China. C. Pan is with the Institute of ChangQing Oil Exploring Company, QingYang, GanShu 745113, China. Publisher Item Identifier S 0196-2892(99)00123-0.
0
S (t; a) da a
(4)
where S (t; a) is defined as (3), H [s(t)] is the HT of s(t): Proof: From (3), we have
for a given signal s(t) 2 L (R; dt); the WT of s(t) with respect to wavelet g (t) is defined as 1
1
1
2
S (b; a) =
a > 0; g (t) is complex
db
(5)
where
gR (t) = Re(g(t)); gI (t) = Im(g(t)) 1 s(b)gR b 0 t db SR (t; a) = 1 a 01 a 1 0 1 b SI (t; a) = s(b)gI 0 t db: a 01 a
0196–2892/99$10.00 1999 IEEE
(6) (7)
868
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 2, MARCH 1999
(a)
(b)
(c)
(d)
Fig. 2. IF’s calculated with (a) HT and (b) WT. Errors of the numerical values with respect to the theoretical values. (c) With HT and (d) WT, respectively. The MSE is 0.16 for HT and 0.0114 for WT.
2 R; a > 0; we can
According to Theorem 1, for an arbitrary a obtain
The modified Morlet wavelet [11] (Morlet wavelet modified for seismic data processing) is
p2m=(2 )t] 0(1=2)[p2m=(2 )t] = cos(mt)e 0(1=2)[p2m=(2 )t]
g (t) = eimt e0(1=2)[
S (t; a) = SR (t; a) + jSI (t; a) = SR (t; a) + jH [SR (t; a)] =
1
1
a
01
+ jH
b
s(b)gR
0t a
1
1
a
01
s(b)gR
db b
0t a
db
(8)
both sides of (6) multiplied by 1=(Cg a); and then integrate it with respect to a from zero to infinity
1
1
Cg 0 =
S (t; a) 1
da a
1
Cg 0 = s(t)
1
1
a2
01
s(b)gR
b
0t a
db da
(9)
both sides of (8) multiplied by 1=(Cg a); and then integrate it with respect to a from zero to infinity, accounting for the linearity of HT and the result of (9), the (4) can be obtained.
+ j sin(mt)e
p
:
(11)
Let C = 2m=(2 ): In (10) and (11), m is the angular frequency, is the number of cycles of the carrier wave in an envelope, and is a real number related to precision (when jg (t)j e0 g (t) can be approximated as zero). In a view point of numerical computation, when m2 =(4C 2 ) is large enough, the wavelet defined in (11) satisfies Theorem 2. All examples in this paper are with the wavelet in (11). When g (t) is a compactly supported wavelet, the analytic counterpart of a real-valued signal can be numerically calculated without cutoff error. The wavelets given in (10) and (11) are not compactly supported, but their amplitudes decay at the rate of e0(1=2)t or e0(1=2)(ct) (C > 1) when away from their centers, so (4) will lead to more accurate results than that by HT (the filter factor of HT decays at the rate of 1=t): C. Definition of IP’s Define the IP’s of a real-valued signal s(t) as follows:
B. Some Analytic Wavelets Satisfying Theorem 2 The Morlet wavelet is g (t) = ejmt e0t =2 (m > 6):
e(t) =
(10)
It is easy to verify that g (t) satisfies Theorem 2 in a numerically approximate sense [9].
s2 (t) + H 2 [s(t)] H [s(t)] (t) = arctan s(t) 1 d H [s(t)] f (t) = arctan 2 dt s(t)
(12)
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 2, MARCH 1999
869
(a)
(b)
(c)
(d)
(e)
Fig. 3. Simulated signal with Gaussian-distributed white noise of a peak–peak SNR of 3%. (a) IF’s calculated with HT and (b) (filter factor length is 801 points) and WT. (c) Errors of the numerical values with respect to the theoretical value. (d) With HT and (e) WT, respectively. The MSE is 2.09 0:03: (HT) and 0.56 (WT), respectively, where "2
=
where e(t); (t) and f (t) is the instantaneous amplitude (IA), instantaneous phase (IPh), and instantaneous frequency (IF) of s(t); respectively. In this paper, the formula for IF estimation is
f (t) =
1
s(t)
2
dH[s(t)]
0
ds(t) H[s(t)] dt dt e2 (t) + 2 e2max
(13)
where 2
2
emax = max(e (t))
(14)
and 0 < < 1: IV. NUMERICAL EXAMPLES In this section, comparisons of IF estimation with the method based on (4) and HT are presented. Let
0(1=2)(c
s(t) = e
t)
0(1=2)(c
cos(m1 t) + e
t)
cos(m2 t)
where c1 = 2 = 4; m1
p
p
2m1 =(2 1 ); c2 = 2m2 =(2 2 ); = 5; 1 = = 28:28; m2 = 42:43; we may have
0(1=2)(c
H[s(t)] = e
t)
0(1=2)(c
sin(m1 t) + e
t)
sin(m2 t):
Fig. 1 is s(t). Fig. 2 gives the calculated IF’s and the estimation errors of HT and WT method, where 2 = 0:001: Figs. 3 and 4 give the results of signals with additive white noise. Comparing (a) and (b) in Fig. 2, the numerical result of IF with the wavelet-based method is very close to the theoretical value; while the one with HT method has notable deviations in the vicinity of A and B. The MSE of the former is 0.0114, and the latter is 0.16. Fig. 3 is an example with additive random noise of 0.03 peak–peak SNR. Fig. 3(e) shows that the result of wavelet-based method is closer to the theoretical one (with a MSE of 0.56), while the result of HT method has a greater estimation error (with an MSE of 2.09). Fig. 4 is with additive noise 0.13 peak–peak SNR. It is obvious that the wavelet-based method is with a much more high antinoise performance. The length of the filter factor of HT is 801 points. From Figs. 2–4, all results in the vicinities of the starting and ending points of the signal, the IF calculated with HT method, have
870
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 2, MARCH 1999
(a)
(b)
(c)
Fig. 4. Simulated signal with Gaussian-distributed white noise of a peak–peak SNR of 13% (a). (b) IF’s calculated with HT (the filter factor length : : is 801 points). (c) WT, where "2
= 0 08
obvious deviations; the ones from the wavelet-based method are with higher precision due to its time-frequency localization properties. V. CONCLUSIONS In this paper, the wavelet-based method of IP’s estimation is discussed. Theorem 2 reveals the relationship between the WT and HT of a signal. Based on the results, the analytic counterpart of a real-valued signal can be calculated with its WT and the IP’s can be estimated. Numerical simulations show advantages of the waveletbased method to the HT method, both in precision and antinoise performances. Due to the localization property in time-scale space, the wavelet-based method can lead to a good method for the analysis of seismic data from a complex-structured area by choosing a suitable wavelet function. REFERENCES [1] M. Taner, F. Koehler, and R. E. Sheriff, “Complex seismic trace analysis,” Geophysics, vol. 44, pp. 1041–1063, 1979. [2] M. Taner and R. Sheriff, “Application of amplitude, frequency and other attributes to strati-graphic and hydrocarbon determination,” in Applications to Hydrocarbon Exploration: Memoir 26, C. Payton. Ed. Tulsa, OK: Amer. Assoc. Petroleum Geologists, 1977, pp. 389–416.
[3] M. P. Matheney and R. L. Nowack, “Seismic attenuation values obtained from instantaneous-frequency matching and spectral ratios,” Geophys. J. Int., vol. 123, pp. 1–15, 1995. [4] M. Sun and R. J. Sclabassi, “Discrete-time instantaneous frequency and its computation,” IEEE Trans. Signal Processing, vol. 41, pp. 1867–1900, Nov. 1993. [5] L. Mandel, “Interpretation of instantaneous frequencies,” Amer. J. Phys., vol. 42, pp. 840–846, 1974. [6] A. Grossmann and J. Morlet, “Decomposition of function into wavelets Physics, of constant shape, and related transforms,” in Mathematics Lecture on Recent Results, L. Streit, Ed. Singapore: World Scientific, 1985, pp. 135–165. [7] P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transform in seismic signal analysis,” Geoexploration, vol. 23, pp. 85–102, 1984. [8] A. Grossmann, R. Kroland-Martinet, and J. Morlet, “Reading and understanding continues wavelet transforms,” in Proc. Wavelets, TimeFrequency Methods Phase Space, 1st Int. Wavelets Conf., Marseille, France, 1989, pp. 2–20. [9] G. Jinghuai, W. Wenbing, and Z. Guangming, “Wavelet transform and instantaneous attributes analysis of a signal,” Acta Geophys. Sinica, vol. 40, pp. 821–832, 1997. [10] M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech., vol. 24, pp. 395–457, 1992. [11] G. Jinghuai, W. Wenbing, Z. Guangming, P. Yuhua, and W. Yugui,” On the choice of wavelet functions in seismic data processing,” Acta Geophys. Sinica, vol. 39, pp. 392–400, 1996.
+
