In my implementation of the Delft approach, I first create a model of the multiples by autoconvolving in time and space the shot gathers. I do this convolution once such that the kinematics of all surface-related multiples are accurate. Doing this, the relative amplitudes of the first-order multiples are correct, but higher-order multiples are over-predicted, amplitude-wise Guitton et al. (2001b); Wang and Levin (1994).
Once a multiple model has been estimated, it is adaptively subtracted from the data. Note that as pointed out by Berkhout and Verschuur (1997), this first subtraction step should be followed by more iterations. The goal of the iterative procedure is to better estimate and eliminate higher-order multiples Verschuur and Berkhout (1997). In this paper, I iterate only once and hope that the adaptive subtraction step is permissive enough to handle all the multiples at once.
Keeping the adaptive-subtraction procedure, I use the non-stationary filtering technology developed at the Stanford Exploration Project to perform the multiple attenuation step Rickett et al. (2001). The main advantage of these filters is that they are computed in the time domain and thus take the inherent non-stationarity of the multiples and the data into account very efficiently. Therefore, I can locally estimate adaptive filters that will give the best multiple attenuation result. Thereby only one iteration of the Delft approach should be needed. Note that I am estimating two-sided 2-D filters, which gives a lot of degrees of freedom for the matching of the multiple model to the real multiples in the data.
Thus, given a model of the multiple and the data , I estimate a bank of non-stationary filters such that
The Delft approach is widely used in the industry and is known to give the best multiple attenuation results for complex geology Dragoset and Jericevic (1998). However, it has been shown that this method suffers from the approximation made during the adaptive filtering step. For instance, when ``significant'' amplitude differences exist between the primaries and the multiples, the multiple model might be matched to the primaries and not to the multiples. A solution to this problem is using the norm in equation (5) Guitton and Verschuur (2002). Another assumptions made in equation (5) is that the signal has minimum energy. Spitz (1999) illustrates the shortcomings of this assumption and advocates that a pattern-based method is a better way of subtracting multiples to the data. In the next section, I present such a methodology. This technique uses non-stationary multidimensional prediction-error filters to estimate the pattern of both the primaries and the multiples. Then, multiples and primaries are separated according to their multivariate spectra Claerbout and Fomel (2001).