Theory of adaptive subtraction

The goal of adaptive subtraction is as follows: given a time series $\bf{b}$ and a desired time series $\bf{d}$, we seek a filter $\bf{f}$ that minimizes the difference between $\bf{f*b}$ and $\bf{d}$ where * is convolution. We can rewrite this definition in the fitting goal  

$$\bf{0} \approx \bf{Bf - d}$$ (1)

where $\bf{B}$ represents the convolution with the time series $\bf{b}$. We can minimize this fitting goal in a least-squares sense leading to the objective function

$$g(\bf{f})=(\bf{Bf-d})'(\bf{Bf-d})$$ (2)

where (') is the transpose. The minimum energy solution is given by  

$$\hat{\bf{f}} = (\bf{B'B})^{-1}\bf{B'd}.$$ (3)

where $\hat{\bf{f}}$ is the least-squares estimate of $\bf{f}$.This approach is very popular but has some intrinsic limitations. In particular $\bf{B}\hat{\bf{f}}$ is by construction orthogonal to the residual $\bf{B}\hat{\bf{f}}-d$. In the multiple attenuation problem $\bf{d}$ is the data, $\bf{b}$ the multiple model and $\bf{B}\hat{\bf{f}}-d$ the estimated primaries. If both signal and noise are correlated, the separation will suffer because of the orthogonality principle.

From now on I will refer to this method as the ``standard approach''.

In the next section I propose improving the adaptive subtraction scheme. This improvement leads to an unbiased matched-filter estimation when both signal and noise are correlated.