M.Sc Student Mirel Merabi Multichannel Semi-Blind Sparse Deconvolution of Seismic Signals Department of Electrical Engineering Professor Israel Cohen Abstract

Seismic waves are waves that can propagate in the subsurface and are reflected from underground layers of the earth. Reflection seismology involves emission of seismic waves by a seismic source on the earth's surface, and obtaining the reflected waves by an array of geophones. The reflected signals are measured on the earths' surface and the measurements combine the seismic data. Important features are extracted from this data in order to better understand the subsurface structure. The kernel defining the channel is called wavelet and the signal describing the reflected impulses is called reflectivity. Convolution is a well-defined operator that is used to mathematically model the connection between the input and output signals of an LTI system given a third signal, called the kernel, that describes the system. Deconvolution is a process where we hold the output signal and the kernel and try to find the input signal, this is called the non-blind deconvolution. Sometimes the kernel is also unknown and then the problem is called blind deconvolution. Seismic deconvolution is a general problem associated with recovering the reflectivity series from the seismic data when the wavelet is known. Many methods and algorithms were presented for solving this kind of problem, using different kind of models and assuming different a-priori knowledge on the signal and on the system. Many deconvolution methods were developed to solve the seismic deconvolution problem. When partial information is given on the wavelet, the blind deconvolution methods can ignore it and solve it as a blind problem. Alternatively, non-blind deconvolution methods can assume the partial information is the full one if possible or make an educated guess to fill the missing information and solve it as a non-blind problem. In this research, we solve the problem of semi-blind seismic deconvolution, where the wavelet is known up to some error. The multichannel semi-blind deconvolution (MSBD) model was developed for cases where there is some uncertainty in the assumed wavelet.

We model the wavelet uncertainty as an additive noise to the wavelet and analyze that

noise with our developed method. We present a novel two-stage iterative algorithm that recovers both the reflectivity and the wavelet. While the reflectivity series is recovered using sparse modeling of the signal, the wavelet is recovered using L2 minimization, exploiting the fact that all channels share the same wavelet. The L2 minimization solution is revised to suit the multichannel case.

As mentioned before, our method assumes an additive noise to the wavelet. We test 2 use cases. In the first one we test the straightforward case where an additive white Gaussian noise is added to the wavelet, and in the second we test the case where we are uncertain with a parameter defining the wavelet. In the second case the noise is not additive to the true wavelet but we show how we model it so we can apply our general method on this case also. We show that our algorithm outperforms both the non-blind and the blind methods.