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Model Regularization 1: Differencing across multiple order

We can write the model residual corresponding to the model regularization operator which differences between adjacent $\bold m_i$ at a fixed (t,x).  
 \begin{displaymath}
\bold r_m^{[1]}(\tau,x,j) = m_j(\tau,x) - m_{j+1}(\tau,x) 
 \hspace{0.1in} \mbox{where} \hspace{0.1in} j=[0,p(p+3)/2].\end{displaymath} (5)
p is the maximum order of multiple included in equation (2). Here we have modified the notation a bit and written $\bold m_j$ rather than $\bold m_{i,k}$. This is because the difference (5) 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.

By design, signal (non-crosstalk) events on the $\bold m_{i,k}$ in equation (3) are assumed physically invariant for all i and k - everything is a ``copy of the primary''. Again, this is the crucial fact underlying LSJIMP. Minimally, multiples provide a redundant constraint on the amplitude of the primaries; where no data is recorded (missing traces, near offsets), they provide additional information about the primaries. Equation (5) is a systematic way in which to exploit the multiples' redundancy, and to integrate any additional information that they might provide.


next up previous print clean
Next: Model Regularization 2: Differencing Up: Regularization of the Least-Squares Previous: Regularization of the Least-Squares
Stanford Exploration Project
7/8/2003