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Next: Signal-noise separation with weighted Up: Guitton: Weighted PEF estimation Previous: Introduction

Theory of PEF estimation

This section is largely inspired by chapter 7.5 in Claerbout (1992). I show that the proper way of dealing with amplitude problems with seismic data for the PEF estimation is to weight the residual and not the data itself.

I will now describe the theory of PEF estimation for a three coefficients filter $\bold{a}=(1,a_1,a_2)'$ from a time series $\bold{y}=(y_0,y_1,y_2,...,y_6)'$. Helical boundary conditions Claerbout (1998) make this analysis valid for multidimensional PEFs as well. What we want is to minimize the energy of the residual $\bold{r}$ according to  
 \begin{displaymath}
\bold 0
\quad \approx \quad
\bold r =
\left[ 
\begin{array}
...
 ..._2 \\  
 y_3 \\  
 y_4 \\  
 y_5 \\  
 y_6 \end{array} \right] \end{displaymath} (1)
which can be rewritten
\begin{displaymath}
\bold 0 \approx \bold r = \bold Y\bold K\bold a +\bold r_0\end{displaymath} (2)
where $\bold Y$ is the convolution matrix with $\bold y$, $\bold K$ is a masking operator and $\bold r_0$ the first column of the convolution matrix $\bold Y$. The next step is to find $\bold a$ such that
\begin{displaymath}
f(\bold a)=\Vert\bold r\Vert^2=\Vert\bold Y\bold K\bold a +\bold r_0\Vert^2\end{displaymath} (3)
is minimum. We can solve this problem with an iterative solver such as a conjugate-direction method Claerbout and Fomel (2002). The least-squares inverse of ${\bf a}$ becomes  
 \begin{displaymath}
{\bf \hat{a}}=-{\bf (K'Y'YK)^{-1}K'Y'r}_0.\end{displaymath} (4)
It is common practice to weight the data before doing any processing in order to correct for amplitude anomalies. Doing so, the fitting goal in equation (1) becomes  
 \begin{displaymath}
\bold 0
\quad \approx \quad
\bold r =
\left[ 
\begin{array}
...
 ...3y_3 \\  
 w_4y_4 \\  
 w_5y_5 \\  
 w_6y_6 \end{array} \right]\end{displaymath} (5)
where $\bold w=(w_1,w_2,...,w_6)'$ are weights for a particular point of the time series $\bold y$. This is the wrong approach to correct for amplitude anomalies but it has practical values. First, equation (5) keeps the Toeplitz structure of the Hessian in equation (4), allowing fast computation of ${\bf \hat{a}}$. Then, the PEF can be easily estimated in the Fourier domain. However, inverse theory teaches us that we should be weighting the residual instead such that  
 \begin{displaymath}
\bold 0
\quad \approx \quad
\bold{Wr} =
\left[ 
\begin{array...
 ...3y_3 \\  
 w_4y_4 \\  
 w_5y_5 \\  
 w_6y_6 \end{array} \right]\end{displaymath} (6)
with
\begin{displaymath}
\bold W=
 \left[ 
 \begin{array}
{ccccc}
 w_2 & 0 & 0 & 0 & ...
 ... & 0 & 0 &w_5& 0 \\  0 & 0 & 0 & 0 &w_6\\  \end{array} \right].\end{displaymath} (7)
The difference between equations (5) and (6) is that in the first case, we weight the data points and that in the second case, we weight the regression equations. In equation (6), we weight each row independently and leverage the PEF estimation globally. With the weighted residual, the Hessian loses its Toeplitz structure making the estimation of $\bold a$ less computationally efficient. In addition, the estimation of $\bold a$ in the Fourier domain becomes much more difficult.

It might not be clear why one method is better than the other. The choice of a weighting function can also be tricky. In the next section, I present guidelines and results on how to choose $\bold W$and what it changes for two signal/noise separation examples.


next up previous print clean
Next: Signal-noise separation with weighted Up: Guitton: Weighted PEF estimation Previous: Introduction
Stanford Exploration Project
7/8/2003