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A nonlinear-estimation method

What I have described above represents my first iteration. It can be called a ``linear-estimation method." Next we will try a ``nonlinear-estimation method" and see that it works better. If we think of minimizing the relative error in the residual, then in linear estimation we used the wrong divisor--that is, we used the squared data v2 where we should have used the squared residual $(v - \alpha h)^2$.Using the wrong divisor is roughly justified when the crosstalk $\alpha$ is small because then v2 and $(v - \alpha h)^2$are about the same. Also, at the outset the residual was unknown, so we had no apparent alternative to v2, at least until we found $\alpha$.Having found the residual, we can now use it in a second iteration. A second iteration causes $\alpha$ to change a bit, so we can try again. I found that, using the same data as in Figure 1, the sequence of iterations converged in about two iterations.

 
reswait
reswait
Figure 2
Comparison of weighting methods. Left shows crosstalk as badly removed by uniformly weighted least squares. Middle shows crosstalk removed by deriving a weighting function from the input data. Right shows crosstalk removed by deriving a weighting function from the fitting residual. Press button for movie over iterations.


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Figure 2 shows the results of the various weighting methods. Mathematical equations summarizing the bottom row of this figure are:
\begin{eqnarray}
{\rm left}:\quad\quad&&
 \min_\alpha \ \sum_i \ (v_i-\alpha h_i...
 ...\over (v_i-\alpha_{n-1}h_i)^2 + \sigma^2} \ 
 (v_i-\alpha_n h_i)^2\end{eqnarray} (9)
(10)
(11)
For the top row of the figure, these equations also apply, but $\bold v$ and $\bold h$ should be swapped.


next up previous print clean
Next: Clarity of nonlinear picture Up: Solution by weighting functions Previous: Solution by weighting functions
Stanford Exploration Project
10/21/1998