Using equation (1), the corresponding mixed-phase inverse filter is. F(Z) = W(Z)-1 = M(Z)-1 P(Z)-1 = FM(Z) FP(Z) ,. (2) representing the product of a stable causal ...
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73rd EAGE Conference & Exhibition incorporating SPE EUROPEC 2011 Vienna, Austria, 23-26 May 2011
Introduction Estimating the source wavelet from reflection data and removing it by means of inverse filtering to retrieve the earth’s reflectivity series is a key step to enhance the resolution and accuracy of reflection images. Deconvolution involves determining both the source wavelet and the reflectivity series from noisy data. In order to solve this ambiguous problem, additional assumptions about the wavelet, reflectivity, and noise have to be made. Standard stochastic deconvolution techniques (e.g., spiking deconvolution; Robinson and Treitel, 2000) are based on the assumption that the source wavelet is minimum phase, whereas blind deconvolution approaches omit a priori assumptions about the source wavelet phase but rather include constraints on the reflectivity series. A commonly made assumption is that the reflectivity series is sparse. Statistical measures such as the varimax norm (Wiggins, 1978) may be employed to evaluate the spikiness of the deconvolution output. Although the varimax norm is a robust sparseness measure in the presence of noise, it is insensitive to polarity and time shifts, and associated with a multi-modal objective function (e.g., Wiggins, 1985). We present a blind deconvolution technique that includes prior information about the wavelet in a Bayesian framework to allow stable mixed-phase wavelet estimation based on varimax maximization of the deconvolution output. Taking into account that a mixed-phase wavelet can be decomposed into its minimum-phase equivalent and a dispersive all-pass filter, we split the mixed-phase wavelet estimation into two steps. While it is straightforward to compute the minimum-phase equivalent from the data in the first stage, a global optimization technique is used to invert for the all-pass filter in the frequency domain in the second stage. Theoretical background We assume that the Z-transform of a recorded trace X(Z) = R(Z) W(Z) is the product of a reflectivity series R(Z) with a wavelet W(Z), where Z indicates the unit delay operator. A mixed-phase wavelet W(Z) can be expressed as the product of a minimum-phase wavelet M(Z) and a dispersive all-pass filter P(Z) (e.g., Robinson and Treitel, 2000): W(Z) = M(Z) P(Z) .
(1)
Solving the deconvolution problem involves finding an inverse filter F(Z) that optimally removes W(Z) from X(Z), and results in an estimate of the reflectivity series R’(Z), where ‘ marks estimates. Using equation (1), the corresponding mixed-phase inverse filter is F(Z) = W(Z)-1 = M(Z)-1 P(Z)-1 = FM(Z) FP(Z) ,
(2)
representing the product of a stable causal minimum-phase inverse filter FM(Z) and a purely noncausal inverse all-pass filter FP(Z). In the frequency domain, the amplitude spectra of P(Z) and FP(Z) are unity and the action of the all-pass filter is exclusively contained in its phase spectrum Ip(f), where f denotes frequency. Methodology We propose a two-stage procedure to estimate F’(Z) from X(Z) in order to obtain R’(Z). First, a truncated F’M(Z) is computed from the autocorrelation of X(Z) by solving the corresponding normal equations. When applied to X(Z), F’M(Z) is the optimum filter to cancel M(Z): XM = X(Z) F’M(Z) § R(Z) P(Z) ,
(3)
resulting in a trace XM with whitened amplitude spectrum but distorted phase because of P(Z) remaining in the right-hand side of equation 3. M’(Z) is obtained by inverting F’M(Z). Once F’M(Z) is found, F’P(Z) needs to be determined to completely describe F’(Z) and to compute an estimate of the reflectivity: R’(Z) = XM F’P(Z) .
73rd EAGE Conference & Exhibition incorporating SPE EUROPEC 2011 Vienna, Austria, 23-26 May 2011
(4)
Reflecting F’P(Z) about time index 0 yields P’(Z) and multiplication by M’(Z) provides W’(Z). However, the splitting of the phase spectrum of XM into the all-pass filter phase spectrum Ip(f) and the phase spectrum of R’(Z) is non-unique. We therefore constrain the phase of R’(Z) by assuming that R(Z) is as sparse as possible and measure the spikiness of R’(Z) using the varimax norm v (Wiggins, 1978). Because the varimax norm is insensitive to polarity and time shifts, we include prior information about the wavelet in a Bayesian framework to additionally constrain the solution of equation 4 and to obtain the most probable polarity and onset time for W’(Z). It is reasonable to assume that the wavelet polarity, onset time, and duration can be guessed from, for example, first arrivals or prominent reflections. A “master waveform” Wm(Z) defining the shape of the onset and the wavelet duration then allows incorporating this prior knowledge in the inversion. A prior probability mass function U(Z) summarizes our believe in how much W’(Z) should correspond to Wm(Z). Once a W’(Z) is available through the course of the inversion, Bayes’ rule states that the posterior probability Eq that W’(Z) corresponds to W(Z) given Wm(Z) is obtained by multiplying a likelihood function, which is a function of the sample-by-sample difference between W’(Z) and Wm(Z), with U(Z). For numerical stability, we define W’(Z) as two-sided time series that is symmetric about time zero, and, for computational efficiency, we split W’(Z) into a signal and a noise window. Then, we require the root-mean-square of the amplitudes within the noise window EN to be below a noise level. Finally, we formulate an objective function to minimize
) = ln((1 - ¥v(R’(Z))) - O1(Eq(W’(Z))) + O2EN(W’(Z))) ,
(5)
where O1 and O2 are appropriate weights. Equation 5 is multi-modal and non-linear with respect to Ip(f). We employ the differential evolution (DE) algorithm due to Storn and Price (1997), which is a genetic algorithm for global optimization, to find the optimum phase values. We generate I’p(f) in the frequency domain using three model-parameterization approaches: (1) a phase value is estimated for every signal frequency (technique 1), (2) the phase gradients for logarithmically spaced frequency ranges are estimated (technique 2), and (3) a linear phase spectrum of the form I = a + bf is estimated (technique 3). Depending on the wavelet length and signal frequency bandwidth, the number of parameters to estimate is 10-40 phase values for technique 1, 5-10 gradient values for technique 2, and 2 parameters for technique 3. Computing time scales with the number of parameters and fewer parameters generally also need fewer DE iterations until convergence. 1-D synthetic example We illustrate the performance of our algorithm using a 1-D reflectivity model representing the mixture of two Laplace distributions (Walden and Hosken, 1986) shown in Figure 1a-1 convolved with the first derivative of a truncated Gaussian pulse with a dominant frequency of ~40 Hz (Figures 1a-2 and 1b). Standard spiking deconvolution (i.e., equation 3) only removes M’(Z) (Figure 1d) and leaves traces with considerable phase distortions relative to the reflectivity series (Figure 1a-3). In order to remove the distortions from the minimum-phase deconvolved trace, we begin by gathering the prior information about the wavelet. We extract the shape of the prominent reflection at ~0.43 s as Wm(Z) and define U(Z) that reflects a high probability that the wavelet to estimate should match Wm(Z) for the first ~0.01 s, and assign a high probability that the wavelet has a duration of
