Multiple attenuation

First consider that any seismic data $\bf{d}$ are the sum of signal (primaries) and noise (multiples) as follows:  

$$\begin{array} {rcl} \bf{d} &=& \bf{s}+\bf{n}, \end{array}$$ (1)

where $\bf{s}$ is the signal we want to preserve and $\bf{n}$ the noise we wish to attenuate.

Now assuming that the multidimensional PEFs ${\bf N}$ and ${\bf S}$ are known for the noise and signal components, respectively (see the following section for a complete description of the PEFs estimation step), we have  

$$\begin{array} {rcl} \bf{Nn} &\approx& \bf{0}, \\ \bf{Ss} &\approx& \bf{0}, \end{array}$$ (2)

by definition of the PEFs. Equations (1) and (2) can be combined to solve a constrained problem to separate signal from noise as follows:  

$$\begin{array} {rcccl} {\bf 0} & \approx & {\bf r_n} & = & {... ...ubject to} &\leftrightarrow& \bf d &=& {\bf s+n}. \end{array}$$ (3)

The noise $\bf{n}$ can be eliminated in the last equation of the fitting goal (3) by convolving with ${\bf N}$. Doing so, equation (3) becomes:  

$$\begin{array} {rcccl} {\bf 0} & \approx & {\bf r_n} & = & {... ...\\ {\bf 0} & \approx & {\bf r_s} & = & {\bf Ss}. \end{array}$$ (4)

Sometimes it is useful to add a masking operator on the noise and signal residuals ${\bf r_n}$ and ${\bf r_s}$ when performing the noise attenuation. It happens for example when we want to isolate and preserve parts of the data where no multiples are present. For instance, a mute zone can be taken into account very easily. Calling ${\bf M}$ this masking operator, the fitting goals in equation (4) are weighted as follows:  

$$\begin{array} {rcccl} {\bf 0} & \approx & {\bf r_n} & = & {... ...\\ {\bf 0} & \approx & {\bf r_s} & = & {\bf MSs}. \end{array}$$ (5)

Solving for $\bf{s}$ in a least-squares sense leads to the objective function

$$f({\bf s})=\Vert{\bf r_n}\Vert^2+\epsilon^2\Vert{\bf r_s}\Vert^2,$$ (6)

where ${\epsilon}$ is a trade-off parameter that relates to the signal/noise ratio. The least-squares inverse for $\bf{s}$ becomes  

$${\bf \hat{s}} = ({\bf N'MN+}\epsilon^2{\bf S'MS})^{-1}{\bf N'MNd},$$ (7)

where (') stands for the adjoint. Soubaras (1994) uses a very similar approach for random noise attenuation and more recently for coherent noise attenuation Soubaras (2001) with F-X PEFs. Because the size of the data space can be quite large, $\bf{s}$ is estimated iteratively with a conjugate-gradient method. In the next section, I describe how both ${\bf N}$ and ${\bf S}$ are estimated.