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TUTORIAL in This Session, We Invite Authors to Submit Articles Which Illustrate Important Concepts Relevant to Exploration Geophysics TUTORIAL In this session, we invite authors to submit articles which illustrate important concepts relevant to exploration geophysics. WHY ZERO-PHASE FILTER SHOULD NOT BE APPLIED PRIOR TO DECONVOLUTION? C. H. Mehta* and Deepak Sareen ** * Consultant, 109/12, Mohit Nagar, GMS Rd, Dehradun, 248006, India. E-mail : [email protected] ** Geodata Processing & Interpretation Center, ONGC, DehraDun 248195, India When the seismic wavelet is not known, the seismic data are deconvolved using one-sided deconvolution which is based on an assumption that the seismic wavelet is of minimum phase. What harm may come to deconvolution if the data are acquired with a non-minimum phase wavelet or rendered non-minimum phase during pre processing as, for instance, using a zero-phase band pass filter? We answer this question in this tutorial and illustrate the requirement for the input data to be of minimum phase for effective deconvolution using a numerical and field example. Let us first review different definitions of a minimum w t + 1 = {0,…………, , 0, a, 1, b, 0,……………} (7) phase time series, also called a minimum-delay time series. As applied to seismic data, it could be a time series of a seismic trace, a source wavelet, or a response t=0 of a linear filter. A reader already knowledgeable about these definitions may like to proceed directly to Note that the discrete Fourier transform, X(ω), of the time series examples. (1), is obtained simply by substituting (3) into (2). ∞ Let, ω ∑ ω ∆ X( ) = xk exp(i k t) (8) ∞ x = { x , x , x ,…….. x ,……… } (1) - t -1 0 1 n for - Ny ≤ ω ≤ Ny, be a discrete time series obtained by sampling a function x(t) π ∆ at a time sampling interval ∆ t. For simplicity, we shall restrict where Ny = / t discussion to only real functions x(t). We define a Z- transform is the Nyquist frequency (angular). of x t as In this sense, the definition (2) for Z-transform X(Z) has a ∞ useful characteristic that one can compute / examine the X(z) = ∑ x Zk (2) k discrete F.T. of x in frequency domain by substituting the - ∞ t complex value of Z as in (8) and also view the underlying where time series x as the coefficients of different powers of Z. Z = exp(i ω ∆t), i= √-1 (3) t Z-transform has some other useful features: As an example, if we have a time series wt, Convolution in the time domain corresponds to w = { 0,……….a, 1, b, 0,0……………..} (4) t multiplication in the z domain, that is, if X(z), W(z), and R(z) are the Z-transforms of the sequences xt, wt, and rt, respectively, then the convolution in time domain, t=0 its Z-transform is given by xt = wt * rt (9) W(Z) = a / Z + 1 + b Z (5) becomes a multiplication in the Z-domain, Z is called as a unit-delay operator since multiplying the X(z) = W(z) R(z) (10) expression (2) by Z generates another time series which is delayed by one sample with respect to (1). For instance, A sequence for which multiplying (5) by Z, gives xt = 0 for t < 0 Z W(Z) = a + Z + b Z2 (6) is called a casual sequence. Another term for such a sequence is a "realizable" sequence. The word realizable comes from which defines a time-shifted series w t + 1 as, GEOHORIZONS July 2006 /24 the fact that for any real time system (as different from a The inverse of a finite discrete sequence {x0, x1, …………xn} system designed numerically on a computer), the response is defined by the equation of the system cannot come before the input; that is, the -1 δ impulse response has to vanish for t<0. Conversely, one xt * (x )t = t, 0 (15) cannot design in real life a system for which xt # 0 for t<0, i.e. you cannot change the past (Philosophical, isn't it?). An where, example of a non-realizable filter is the zero-phase band pass -1 -1 -1 -1 -1 filter. If it is of finite length, then one can make it "realizable", (x )t = {….., ,x-1 , x0 , x1 , x2 ………..} by delaying the response by half the length of the filter which introduces a similar delay in the output. t = 0 Further, if is , in general, a two-sided time series , and 2 ∑ ∞ δ = { 0,…………,0, 1, 0, ………………} xn < (11) t, 0 is a spike at time t = 0 i.e., if a causal sequence xn has a finite "energy", the sequence is called a realizable stable sequence. In Z-domain, equation (15) becomes, Consider a realizable sequence of a finite length, X(Z) X-1(Z) = 1 (16) x ={x, x , x ,…………… ,x } (12) t 0 1 2 N Or, with a Z-transform, 1 X-1 (Z) = 1 / X(Z) = (17) X (Z-z ) ………….(Z-z ) 2 N N 1 N X(Z) = x0+ x1Z + x2Z +…………… xNZ If all the roots of X(Z) are outside the unit circle, = xN (Z-z1) (Z-z2)…………………(Z-zN) (13) i.e. if zi > 1, for i =1, N, where z1, z2, …………zN are the roots of the polynomial X(Z). then From the property (10), the multiple products (13) in Z-domain represent a cascaded convolution in time domain. That is, X(Z) # 0 for Z ≤ 1 . the sequence xt can be written as i.e. X-1 (Z) is analytic for Z≤ 1. It is this analyticity property x = x ( -z , 1) * (-z , 1) * ……………* (-z , 1) (14) t N 1 2 N of X-1(Z) which guarantees us that it can be expanded in a convergent power series around Z=0: ≥ A couplet (c0, c1) is said to be minimum-delay if c0 c1 , i.e., if the energy is loaded in the front. A couplet (c0, c1) is ∞ ≥ -1 -1 k said to be maximum-delay if c1 c0 . In case c1=c0, the ∑ ≤ X (Z) = (x ) k Z , for Z 1, (18) couplet can be regarded as either minimum-delay or as a k=0 maximum-delay. with the property that Definition of a Minimum Phase Sequence: ∑ -1 2 ∞ (x )k < (19) A sequence x is called a Minimum Phase sequence or -1 t This proves the existence of a one-sided sequence (x ) k of minimum-delay sequence if all the zeroes of polynomial X(Z) finite energy, which when convolved with xk would give a lie outside the unit circle in the complex z-plane. spike. This existence criterion is not just of academic interest; i.e. we shall shortly see the relevance of this property of a minimum phase sequence for seismic deconvolution. zi >1, for i = 1, 2,………N. Let us note a few more important definitions and properties associated with minimum phase sequences. Conversely, if all the roots zi in (14) lie inside the unit circle, the sequence is called a maximum- delay sequence. If some For the sequence xt as in (1), let us define a time reversed roots are outside the unit circle and some inside, the sequence sequence is called a mixed-phase sequence. An important significance x = { , x , x , x , x ,…………..} (20) of the minimum phase property of a sequence xn is that such t 1 0 -1 -2 a sequence has a one-sided inverse of finite energy. This is simple to see as follows. t=0 GEOHORIZONS July 2006 /25 ω ω Φ ω For real valued xt , the Z-transform of (20) is given by, X( ) = X( ) exp {i ( )} (29) X* (Z) = X (1/Z) (21) where, X(ω) = P(ω) 1/2 which is just a complex conjugate function of X(Z). P(ω) is the power spectrum which is also the Fourier Autocorrelation of x defined by t Transform of the Auto-correlation A(t) defined in (23).. ∞ Although we shall not prove it here, we state that if x is of ∑ t A(t) = xk x t+k (22) minimum phase, then log X(ω) and phase Φ (ω) form a - ∞ Hilbert Transform pair. i.e, can also be expressed as convolution of xt with x t (23) Φ(ω) = (1/ πω) * Log X(ω) (30) A(t) = x * x t t where * indicates convolution. Thus, given the power-spectrum P(ω) we can construct the In Z-domain, therefore, we have minimum phase Φ (ω). Note that the property of minimum phase is not just a property of the phase function Φ(ω), but a property A(z) = X(z) X*(z) = X(Z) X (1/Z) (24) associated with a pair of functions - the power spectrum and the phase function. In other words, it is a property of the entire Now, if X(Z) is a polynomial of minimum phase, then its roots time series, in which the phase and the amplitude are intimately z1, z2, …….zN defined in (13) all lie outside the unit circle. coupled and derivable from each other. As a consequence, if Accordingly, X(1/Z) which is a power series in (1/Z) we alter only the amplitude spectrum of a minimum phase time representing an anti-casual sequence is given by series without affecting its phase spectrum, say, for example, by applying a band-pass zero phase filter, then the resulting time X(1/Z) = XN (1/Z - z1 )(1/Z - z2) …(1/Z -zN) (25) series is no longer of minimum phase. This, as we shall see, has tremendous significance for seismic deconvolution. N = ( XN / Z ) (1 - z1 Z) (1-z2 Z)…….
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