Regularization 1: Differencing between images  

Corresponding signal events on all $\bold m_{i,k,m}(\tau,x)$ are focused at a single $\tau$ for all x, and by design, have directly comparable amplitudes. Conversely, corresponding crosstalk events on two model panels (e.g. residual first-order multiples on $\bold m_0$ and residual second-order multiples on $\bold m_{1,k,m}$) generally have different residual moveout. While the exact magnitude of the moveout differences depend on the choice of imaging operator, Figure 1 illustrates that they generally are small at near offsets, but more pronounced in the presence of subsurface complexity, and at far offsets/reflection angles.

 

crossdiff2.gulf
Figure 1 Comparison of crosstalk events on primary and first-order multiple images, for my particular choice of multiple imaging operator. ``X'' indicates position of split first-order pegleg on primary image, $\bold m_0$. ``o'' indicates position of the three second-order pegleg events on both $\bold m_{1,0,1}$ and $\bold m_{1,1,1}$. Left panel is with no subsurface dip, right has seabed and target reflector dip of $4^{\circ}$. With no dip, corresponding crosstalk events have little differential moveout. A small amount of dip quickly increases differential moveout.

We therefore conclude that at fixed $(\tau,x)$, the difference between two $\bold m_{i,k,m}$ will be relatively small where there is signal, but large where there is crosstalk noise. We now write this difference as a model residual:  

$$\bold r_m^{[1]}[j](\tau,x) = \bold m_j(\tau,x) - \bold m_{j+... ...), \hspace{0.1in} \mbox{where} \hspace{0.1in} j=[0,p(p+3)/2].$$ (7)

p is the maximum order of multiple included in equation (2). Here I have modified the notation a bit and written $\bold m_j$ rather than $\bold m_{i,k,m}$ because the difference (7) is blind to the order or leg of the pegleg corresponding to $\bold m_j$; it is simply a straight difference across all the model panels.

As mentioned early in this thesis, a central motivation for LSJIMP is the desire to combine information from the multiple and primary images by averaging. In addition to discriminating against crosstalk, equation (7) provides a systematic framework for this averaging. If a signal event on one image is obscured by noise, the noise may not be present on an adjacent image, and equation (7) will attenuate it. This regularization enforces a degree of smoothness and consistency between images.