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Sparsity enhanced wavelet deconvolution Disclosure Number: IPCOM000245255D
Publication Date: 2016-Feb-22
Document File: 7 page(s) / 394K

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Sparsity enhanced wavelet deconvolution


I describe an enhancement of Wiener wavelet deconvolution, i.e. the deconvolution of wavelets that are known in amplitude and phase, employing gradually information from sparse-spike wavelet deconvolution at frequencies at which the wavelet amplitude is weak. The basic assumption is that the wavelet is known, for example after wavelet estimation based on well-log data. Usually, such wavelets will be band-limited, i.e. have very low amplitudes at low and high frequencies, and eventually also at certain frequencies in between. The conventional inverse filter will hence amplify the noise in the data at such frequencies. To avoid this effect, I suggest blending in at those frequencies the sparse-spike deconvolution result, such that these contributions are stronger if the wavelet amplitude gets weaker. I demonstrate this new technique with absorption simulation wavelets, i.e. in the context of Q-inverse filtering. This new technique may be used on many types of collected data, including without limitation, seismic data.


The invention combines two approaches to wavelet deconvolution, i.e., combines the inverse Wiener filtering approach with sparse spike wavelet deconvolution. The novelty is in the frequency dependent weighted summation of the sparse-spike deconvolution result and the inverse Wiener filtering result; i.e., the sparse-spike result is used gradually more where the wavelet amplitude spectrum is small.

Let 𝑊(𝑓) denote the wavelet transfer function in the temporal frequency domain. Let's 𝐷(𝑓) denote the data, and 𝑅(𝑓) the reflectivity, of which we assume that the corresponding time-series is sparse, i.e., consists of a limited number of spikes. Further, let's 𝐿1(𝑓) denote the Fourier representation of the result of sparse-spike deconvolution applied to the data,
i.e. 𝑙1(𝑡𝑛) = arg 𝑚𝑖𝑛𝑦(𝑡𝑛)‖𝑦‖1 𝑠. 𝑡. 𝑑(𝑡𝑛) = (𝑤 ∗ 𝑦)(𝑡𝑛), and 𝐿2(𝑓) =

|𝑊(𝑓)|2+⁵ 𝐷(𝑓)the

The combination of the inverse Wiener filtering deconvolution result and the sparse-spike deconvolution result is given by the following equation in the temporal frequency domain

𝑋(𝑓) = 𝐿2(𝑓) + 𝑀(𝑓)𝐿1(𝑓) eq.1

In which 𝑀(𝑓) denotes the frequency dependent weight function being used for adding the sparse-spike deconvolution result to the inverse Weiner filtering result. In the general case this weight function could be given by,

𝑀(𝑓) =


result of inverse Wiener filtering deconvolution.

   ⁵ |𝑊(𝑓)|2+⁵, eq.2

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which depends only on the wavelet power spectrum and the pre-whitening parameter. However, other weight function to blend in the sparse-spike result could be envisaged.

If the wavelet powers spectrum is small, the weight of the sparse-spike de...