Tackling mixed-phase wavelets in blind deconvolution using hybrid solver

By assuming the reflectivity series to be sparse rather than ”white”, we introduced the hybrid norm spiking deconvolution (Zhang, 2010). However one theoretical drawback of spiking deconvolution is that it assumes the source wavelet to be minimum-phase, which might not be true in practice. We propose a new formulation of spiking deconvolution with the hybrid solver that does not require such assumption. We show the effectiveness of this method using various examples. INTRODUCTION In the previous report (Zhang, 2010), we introduced the spiking deconvolution problem using the hybrid norm solver (Claerbout, 2009b). Synthetic examples (Zhang, 2010) showed that given a minimum-phased wavelet, it retrieved the sparse reflectivity model almost perfectly even with a reflection series that is far from white, while conventional L2 deconvolution did a poor job. However, if the assumption of a minimum-phase wavelet was broken, the hybrid norm spiking deconvolution failed quickly and gave a poor result similar to conventional L2 deconvolution. In this paper, we still rely on the hybrid norm solver to retrieve the sparse model, but we use a slightly more complex formulation that avoids the mininmum-phase wavelet constraint. We start by realizing that any (mixed-phase) wavelet C(Z) can be decomposed into a minimum-phase part A(Z) and a maximum-phase part B(1/Z) plus a certain time shift: C(Z) = A(Z)B(1/Z)Z, (1) where B(Z) is also a minimum-phase wavelet (therefore B(1/Z) is a maximum-phase wavelet) and k is the order of B(Z). This Z term makes the wavelet C(Z) causal. In time domain, (1) can be written as c = a ∗ b ∗ δ(n− k), (2) where b stands for the time reverse of series b. Our original spiking deconvolution can find only the inverse of the minimum-phase counterpart of c. It can be defined as an inverse problem as follows: [d]fc = r, (3) SEP–142 Zhang and Claerbout 2 the hybrid solver deconvolution where [d] is the data convolution operator, and fc is the unknown filter. In this formulation, the filter is the only unknown, the hybrid norm is applied on the residue term r to enforce the sparseness constraint. In theory, the residual r itself is the reflectivity model. Such a method requires the wavelet in the data to be minimumphase because only a minimum-phase wavelet has a causal stable inverse. The following new formulation utilizes a pair of conventional deconvolutions, trying to invert components a and b separately: [(d ∗ f r b )] fa = ra, [(d ∗ fa)] fb = rb, (4) in which fa and fb is the corresponding filters that are the inverses of a and b denoted above, the superscript r means time-reverse. The operator in each equation is the convolution operator. Again the hybrid norm is applied to ra and rb, and the reflectivity model is simply ra plus a time shift. Notice that this is a non-linear inversion, since the operator itself depends on the unknown fa and fb. In practice we have to solve these two inversions alternately and iteratively. To understand the meaning of (4), let d = m ∗ c = m ∗ a ∗ b ∗ δ(n− k), (5) where m is the reflectivity model and the δ term is just a time shift. Assume fa and fb is perfectly known in the operators (which is not true in reality), i.e. fa ∗ a = δ(n), fb ∗ b = δ(n) Substituting (5) into (4), since d ∗ f r b = m ∗ δ(n− k) ∗ a, (6) (d ∗ fa) = (m ∗ b ∗ δ(n− k)) = m ∗ δ(n+ k) ∗ b, (7) we have [(m ∗ δ(n− k)) ∗ a] fa = ra, [(m ∗ δ(n+ k)) ∗ b] fb = rb. (8) From (8) it is easier to see what is behind the new formulation (4): It tries to separate the two parts of the wavelet, turning each one into a traditional deconvolution problem in which the wavelet (a, b) is always minimum-phase. Because of the non-linear nature of this method, it is not guaranteed that we find the exact inverse of a, b. The following section shows several examples (complexity starting from low to high) illustrating the effectiveness and limitations of the method. DATA EXAMPLES Inverting a single wavelet To verify the new method’s ability to handle mixed-phase wavelets, we first set the input data to be a single wavelet, to see whether the data can be compressed to a SEP–142 Zhang and Claerbout 3 the hybrid solver deconvolution single spike. We choose three types of wavelets as inputs 1. a minimum-phase wavelet used in the previous report (Zhang, 2010), referred to as wavelet 1. 2. a wavelet proposed by Claerbout that deviates slightly from minimum-phase, it is created by integrating a triangular function twice and applying a leaky integration on the resulting output; this wavelet is referred to as wavelet 2. 3. a zero-phase wavelet created by convolving the minimum-phase with its own time-reverse wavelet. Such wavelet has identical a and b component, referred to as wavelet 3. (a) (b) Figure 1: (a) Input wavelet 1 and (b) its deconvolution result. [ER] (a) (b) Figure 2: (a) Input wavelet 2 and (b) its deconvolution result. [ER] (a) (b) Figure 3: (a) Input wavelet 3 and (b) its deconvolution result. [ER] Figure 1 show wavelet 1 (minimum-phase wavelet) and the result of reflectivity model. Figure 2 show wavelet 2 (Jon’s wavelet) and the result of reflectivity model. Figure 3 show wavelet 3 (zero-phase wavelet) and the result of reflectivity model. In all 3 cases, our new method is able to compress the wavelet into a spike.