A comparison of 2D Seismic data regularization with MWNI and MP

Regularization and interpolation of prestack seismic data have become an essential part of modern data preconditioning flows before imaging and inversion. In this paper, we present a comparative study of two techniques for seismic data regularization that operate in the Fourier domain: Matching Pursuit (MP) and Minimum Weighted Norm Interpolation (MWNI). We illustrate the performance of both methods on 1D harmonic functions and afterwards we apply them to the 2D prestack regularization of the Parnaíba basin dataset. Our numerical tests show that MWNI is faster than MP, but the MP produce slightly better results. Introduction Migration and inversion methods assume that prestack data are densely and regularly sampled in space. However, acquisition often fails in providing dense and regularly sampled data. The latter is due to operational constraints and the presence of obstacles. In marine acquisition, another type of irregularity occurs due to ocean currents that do not allow hydrophone cables to remain equally spaced. All these difficulties make the seismic data irregularly sampled or to have large empty gaps. Therefore, regularization is needed to represent prestack data in a regular complete grid. Currently, there are many seismic regularization techniques. They can be grouped into three categories: 1) techniques for interpolating regularly sampled data, such as f-x interpolation (Spitz, 1991); 2) techniques that require the data to be on a regular grid, with optimal missing traces (Cabrera and Parks, 1991; Liu and Sacchi, 2004); and 3) techniques that use the exact input locations of the data (Xu et al., 2005; Mallat and Zhang, 1993, Schonewille et al., 2013). Nguyen and Winnet (2011) compare two methods of the third group (Xu et all., 2005, and Mallat and Zhang, 1993). This paper concentrate on the categories two and three of the methods and pays particular attention to two method of them (Liu and Sacchi, 2004; and a similar method to that proposed by Schonewille et al., 2013). As it was already mentioned, we work with two interpolation methods that operate in the Fourier domain, Matching Pursuit (Mallat and Zhang, 1993) and MWNI (Liu and Sacchi, 2004). First, each trace is transformed into the frequency domain (from t-x domain to f-x domain). Then, for each temporal frequency f, an MP or MWNI algorithm can be used to estimate the spatial Fourier coefficients that synthetize the data. For 3D data, MP and MWNI are implemented the f-x-y domain. MWNI (minimum weighted norm interpolation) is a Fourier reconstruction method that is based on regularized inversion. In other words, one estimates the Fourier coefficients that synthetize the spatial data via the solution an inverse problem (Liu and Sacchi, 2004). The procedure is an extension of the adaptive frequencydomain weighted norm scheme proposed by Cabrera and Parks (1991) to extrapolate time series. The Matching Pursuit (MP), on the other hand, uses a Greedy Algorithm (Cormen et al., 2009) to estimate the Fourier coefficients that synthetize the observations. Once the Fourier coefficients have been found via MWNI or MP one can use the inverse Fourier transform to reconstruct spatial data. In this paper, we compare Matching Pursuit with MWNI method in two spatial dimensions. In other words, the regularization is carried simultaneously in receiver and source domain. Minimum weighted norm interpolation The MWNI method entails solving an inverse problem where a wavenumber-domain regularization term is included (Liu and Sacchi, 2004). In general, it minimizes a wavenumber weighted norm that lets us incorporate a prior spectral signature of the unknown wavefield. It is a fast and efficient method because it uses an iterative solver (conjugate gradients) in conjunction with fast matrix-vector multiplications that are implemented via the fast Fourier transform (FFT). The algorithm can be used to perform multidimensional reconstruction in any spatial domain. Thus, for each temporal frequency the algorithm minimizes equation 1 in order to estimate the correct Fourier coefficients. We can call J our objective function:    2 2 2 Wc λ + d TFc = J dec  (1) Where the operator T is the sampling matrix with size NxNy. The elements of the sampling are 1 positions with data, and zero for data bins that are empty. The operator F symbolizes the inverse Fourier transform, it can be performed through the inverse fast Fourier transform A COMPARISON OF 2D SEISMIC DATA REGULARIZATION WITH MWNI AN. ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Fifteenth International Congress of the Brazilian Geophysical Society 2 (IFFT), and c are the complex Fourier coefficients. The term ddec is the binned data with missing samples, its size is Nx-Ny, with zeros in the empty positions. The scalar λ is the trade-off parameter of the problem, and W is a matrix of weights that updates the Fourier coefficients in each iteration. The conjugate gradient method is adopted for the minimization of (1). However, in MWNI the matrix of weights W is a function of the unknowns c. Therefore, we adopt the method of Iterative Reweighted least-squares (Zhou et al., 2014) to update the weights. To summarize it, we have two iteration loops: an internal loop to solve minimization problem (1) via conjugate gradients for fixed W and an external loop to update the weights. The method can be summarized as follows: 1. Initialize the Fourier c coefficients and the weights W; 2. Use conjugate gradients to find c that minimizes (1); 3. Use current Fourier coefficients c to update the weighs W; 4. Go to 2 and repeat until convergence; 5. Use the estimated Fourier coefficients to synthetize spatial data via the inverse Fourier transform.