Approximating the noise covariance matrix with a PEF

To address the delicate problem of estimating the noise covariance matrix Tarantola (1987), while building on Claerbout and Fomel (1999), I have proposed to use a PEF for the approximation Guitton (2000). The PEF is estimated from a noise model or from the residual of a previous inversion. In this paper, I show that both methods filter the noise components as long as the PEF incorporates enough spectral information for the noise. Because the noise covariance matrix is supposed to filter out the inconsistent part of the data (the coherent noise), I call this method the filtering method. It is based on the following fitting goal:  

$${\bf 0} \approx {\bf A_n(Hm-d)},$$ (1)

where ${\bf A_n}$ is a PEF estimated from the residual or from a noise model, ${\bf H}$ denotes a seismic operator, ${\bf m}$ is the model we seek, and ${\bf d}$ the seismic data. The corresponding least-squares inverse, or the pseudo-inverse of ${\bf m}$, is given by the equation  

$${\bf \hat{m}} = ({\bf H'A_n'A_nH})^{-1}{\bf H'A_n'A_nd},$$ (2)

where (${\bf '}$) is the adjoint operator. The model space is computed iteratively rather than by using the direct inversion described in equation (2). I use two distinct strategies to compute the PEF ${\bf A_n}$ needed in equation (1). With the first strategy, I derive a noise model from which a PEF is estimated and kept unchanged, whereas with the second one, I compute the PEF from the residual. I have slightly modified my inversion scheme described in my earlier report Guitton (2000). I propose the following algorithm when the PEF is estimated from the residual:

1.

Solve the inverse problem for the fitting goal 0 $\approx$ Hm - d.

2.

Estimate a PEF ${\bf A_n}$ from the residual when only coherent noise remains in the residual.

3.

Restart the inverse problem (${\bf m=0}$) for the fitting goal in equation (1).

4.

Iterate with the new PEF.

5.

Reestimate the PEF ${\bf A_n}$ from the residual Hm - d.

6.

Go to step (4).

7.

Stop when the residual has a white spectrum.

The novelty of this algorithm is that the first PEF is not estimated from the data but rather from the residual of a previous inverse problem where no PEF is used in the fitting goal. This has the advantage of isolating the coherent noise more accurately, thus furnishing a satisfying noise model for the PEF estimation. This algorithm can be seen as a two-stage process in which the first stage helps to estimate a first PEF for our inversion and the second stage reestimates the PEF iteratively. Determining when to stop the inversion to compute the first PEF is critical: too many iterations and the coherent noise is fitted, too few and some signal remains in the residual. The filtering method is viable if the operator ${\bf H}$ does not model the coherent noise properly. In other words, the signal is expected to be orthogonal to the coherent noise for proper attenuation.

The sections that follow describe the application of this method with synthetic and real data. For each case, I estimate the PEF from a noise model or from the residual of a previous inversion. My results demonstrate that both methods lead to a proper attenuation of the noise components.