Summary We propose an inverse data processing method in the plane wave domain for free surface multiple attenuation. Traditionally, inverse data processing is carried out using data matrix in space-frequency (x-?) domain (Berkhout, 2006). For flat-lying layers, the data matrix will have a Toeplitz structure, which means that the inversion can be carried out in frequency wave-number (k-?) domain. Here we propose an alternate method of implementing the inversion in a coupled plane-wave domain where the bandwidth of the matrix can be chosen based on the dominant dip resulting in a stable and fast algorithm.
Introduction Seismic data can be conveniently arranged as data matrices as the discrete sampling of the wave-filed (Berkhout, 1982). Berkhout (1982) put forward a feedback model by making use of this data matrix. The data matrix contains the propagation information of the waves. Many methods are introduced based on the matrix and feedback model, such as multiple removal and pre-stack inversion.
Inverse data processing is introduced by Berkhout (2006) for attenuating surface multiples by making use of the feed back model. In the inverse data domain, primary and multiples have a simpler relationship than in the forward data domain. In the inverse data domain, multiples are focused in a very small area around zero time, while primaries are at the negative time. Thus the multiples can be easily attenuated by a simple muting.
As the source-related term can be easily separated from the data, Berkhout also developed this method for wavelet deconvolution in the inverse data space (Berkhout, 2006). Kelamis applied this method to land surface-related multiples elimination using post-stack data (Kelamis, 2006). Luo (2007) applied this method to internal multiple reduction using two approaches based on redatauming and correlation principles. A new approach to time lapse seismic processing has also been proposed by making use of the inverse data space (Berkhout and Verschuur, 2007). All these applications make use of separation features of inverse data processing.
The inverse data processing is based on a very important procedure, namely, matrix inversion. Traditionally, this inversion is carried out in x-? domain. When the layers are flat meaning that all the shots are the same, the x-? matrix will be a Toeplitz matrix. In this case, matrix inversion can be carried out in k-? domain by a scalar inversion resulting in a very fast algorithm (Berkhout and Verschuur, 2006).
Sen et al (1998) proposed a 1D method based on invariant embedding technique for multiple elimination in the plane wave domain. Liu et al (2000) extended this method to the 2D problems in the plane wave domain. In the 2D case, seismic data with dipping layers can be well compressed, resulting in a band limited matrix, which decreases the computation cost. Essentially, the plane wave invariant embedding technique is a tau-p (?-p) domain realization of surface related multiple elimination proposed by Verschuur (1992). Thus a relationship between surface related multiple elimination and tau-p (?-p) transformation can be made.