The Effect of Frequency Division on the Hessian

The effect of frequency selection on the Hessian matrix is now investigated. For ease of discussion, I restrict the attention to sources  $s=1,\ldots,S$   that are used to form one supergather. The Hessian can be identified as

$$\tilde{\bf {H}} = \sum_{j=1}^{n_\omega} \tilde{\bf {L}}^\dagger \tilde{\bf {L}}$$ (6.1)

$$= \sum_{s=1}^{S} \sum_{j=1}^{n_\omega} N_s(j) \vert W(j)\vert^2 \underline{{\bf {L}}}_s^\dagger \underline{{\bf {L}}}_s .$$ (6.2)

Here, equation A.2 follows from equations 2.8, 2.4, 2.22,  2.21, and the fact that $N_s^2(j)=N_s(j)$ , as $N_s(j)\in \{0,1\}$ ; equation 2.21 ensures that all cross terms in equation A.1 when expanded by plugging in equation 2.8 will vanish.

In contrast, the Hessian in the standard case is

$${\bf {H}} = \sum_{s=1}^{S} \sum_{j=1}^{n_\omega} \vert W(j)\vert^2 \underline{{\bf {L}}}_s^\dagger \underline{{\bf {L}}}_s,$$ (6.3)

which is equation A.2 lacking the binary encoding function $N_s(j)$ .

Comparing equations A.2 and A.3, we see that the encoded Hessian $\tilde{\bf {H}}$ consists of a subset of terms in the standard Hessian ${\bf {H}}$ .