Regularization 3: Crosstalk penalty weights  

The third and final discriminant between crosstalk and signal exploits the inherent predictability of the crosstalk to suppress it. If we have an estimate of the signal, we can directly model the expected crosstalk events on each $\bold m_{i,k,m}$, and construct a model-space weighting function to penalize crosstalk. Unfortunately, unless we employ a nonlinear iteration (see section ), we do not, a priori, have this signal estimate. However, between the seabed reflection and the onset of its first multiple, the recorded data contains only primaries (inter-bed multiples and locally-converted shear waves are generally weak), and it is these strong, shallow events that often spawn the most troublesome crosstalk events. Therefore, we can directly model any pegleg multiple arising from a multiple generator with traveltime less than that of the first seabed multiple.

If we define $\bold M_0$ as an operator that applies a flat mute below twice the zero-offset traveltime of the seabed, and $\bold M_i$ as a similar operator that mutes above the zero-offset traveltime of the $i^{\rm th}$ multiple generator, then  

$$\bold z_{i,k,m} = \bold L_{i,k,m} \bold M_{i,m} \bold M_0 \bold L_0^T \bold d$$ (9)

is a model of the $k^{\rm th}$ leg of the $i^{\rm th}$ order multiple from the $m^{\rm th}$ multiple generator. Each $\bold m_{i,k,m}$ in equation () should ideally contain only the $k^{\rm th}$ leg of the $i^{\rm th}$-order multiple from the $m^{\rm th}$ multiple generator - all other energy is crosstalk. To simulate crosstalk noise in $\bold m_{i,k,m}$, we apply $\bold L_{i,k,m}$ to all multiple model panels $\bold z$ (except $\bold z_{i,k,m}$) and sum:  

$$\bold c_{l,n,q} = \sum_{j=l_0}^p \sum_{k=0}^j \sum_{m=1}^{n_... ...\mbox{if } \; l=0 \ l & \mbox{otherwise} \end{array} \right.$$ (10)

$\bold c_{i,k,m}$ is a kinematic model of crosstalk for $\bold m_{i,k,m}$. It could be used as a traditional multiple model (see section ) and subtracted from the data, but I instead convert each $\bold c_{i,k,m}$ into a weighting function by taking the absolute value. We can write the model residual corresponding to the third model regularization operator:  

$$\bold r_m^{[3]}[i,k,m](\tau,x) = \left\vert \bold c_{i,k,m}(\tau,x) \right\vert \; \bold m_{i,k,m}(\tau,x).$$ (11)

Although the crosstalk weights will likely overlap (and damage) signal to some extent, the signal's flatness and self-consistency between images ensures that regularization operators () and () will ``spread'' redundant information about the primaries from other $\bold m_{i,k,m}$ and other offsets to compensate for any losses. Figure illustrates the application of the crosstalk weights for the primary image and a multiple image. On panel (a), the primary image, the crosstalk is the obviously curving events. On panel (c), the seabed pegleg image, the crosstalk events are multiples from other multiple generators (e.g., R1M and R2M). Notice that in both cases the unwanted multiples are picked cleanly out of the data, leaving the underlying signal intact.

 

crosstalk.gulf
Figure 2 Application of crosstalk weights to real CMP after prestack imaging. Panel (a): primary image, $\bold L^T_0 \bold d$. Panel (b): weighted primary image, $\vert\bold c_0\vert \bold L^T_0 \bold d$. Panel (c): seabed pegleg image, $\bold L^T_{1,0,1} \bold d$. Panel (d): weighted seabed pegleg image, $\vert\bold c_{1,0,1}\vert \bold L^T_{1,0,1} \bold d$. $n_{\rm surf}=4$ in this case. Prominent crosstalk events are labeled on the various panels.