Multiple elimination theory

The Delft approach Verschuur et al. (1992) is able to remove surface-related multiples for any type of geology, as long as the receiver and source coverage at the surface is dense enough. One of the main advantages of the Delft method is that no subsurface information is required.

In this implementation of the Delft approach, we first create a kinematic model of the water-bottom multiple by convolving in time and space a water-bottom operator. We do this convolution such that the kinematics of all surface-related multiples are accurate. Then, the relative amplitudes of the first-order multiples are correct, but the amplitudes of higher-order multiples are over-predicted Guitton et al. (2001); 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, we iterate only once and hope that the adaptive subtraction step is flexible enough to handle all the multiples at once.

We use non-stationary filtering technology for adaptive-subtraction 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. Therefore, it is possible to estimate adaptive filters locally that will give the best multiple attenuation result. Note that by estimating two-sided 2-D filters 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 multiples ${\bf M}$ and the data ${\bf d}$, we estimate a bank of non-stationary filters ${\bf f}$such that  

$$g({\bf f})=\Vert{\bf Mf -d}\Vert^2+\epsilon^2\Vert{\bf Rf}\Vert^2$$ (1)

is minimum. In equation (1), ${\bf R}$ is the Helix derivative Claerbout (1998) that smooths the filter coefficients across micro-patches Crawley (2000) and ${\epsilon}$ is a constant to be chosen a-priori. Note that ${\bf M}$ corresponds to the convolution with the model of the multiples ${\bf m}$ Robinson and Treitel (1980). Remember that this model of the multiples is obtained by convolving in space and time the input data:

$${\bf m(\omega)}={\bf d(\omega)*wb(\omega)},$$ (2)

where * defines the convolution process detailed in Verschuur et al. (1992), and ${\bf m(\omega)}$, ${\bf d(\omega)}$,and ${\bf wb(\omega)}$ are the multiples model, the data, and the water-bottom operator for one frequency ($\omega$), respectively. In equation (1), the filters are estimated iteratively with a conjugate-gradient method.

The Delft approach is widely used in the industry and is known to give currently the best multiple attenuation results for complex geology Dragoset and Jericevic (1998). However, it has been shown that this method suffers from an 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 ${\ell^1}$ norm in equation (1) Guitton and Verschuur (2002). Another assumption made in equation (1) 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 from the data.