How to estimate the pef $\bf{A_s}$ for the signal?

It is always difficult to estimate the pef for the signal $\bf{A_s}$ because the signal is what we are looking for! It is a chicken and egg story. Fortunately, I have three simple recipes that seem to work well in practice.

The first one has been used by Brown and Clapp (2000) and Guitton (2001) and consists of deconvolving the data pef by the noise pef. It works pretty well but the deconvolution might be unstable when dealing with non-stationary filters Rickett (1999). If $\bf{A_d}$ and $\bf{A_n}$ are the data and noise pef respectively, the estimated signal pef becomes

$$\bf{A_s} = \bf{A_d}/\bf{A_n}.$$ (7)

The second one is a simple technique that requires the noise pef only. I first estimate the pef from the noise and apply it to the data. My hope is to obtain a good model for the signal. Then I estimate the signal pef from this model (Spitz, 2001, personal communication).

The last method consists of estimating the signal with the standard adaptive subtraction scheme. A pef is then estimated from the signal and used inside the hybrid scheme. You might be tempted to repeat this process iteratively in order to improve your signal pef. In my experience, I find that an iterative process might diverge, especially when non-stationary filters are involved.

In the next two sections, I show synthetic and field data examples. They prove that the hybrid adaptive filtering gives a better estimate of the noise when primaries and multiples interfere.