2. Given the set
$$
S_{1}=\left\{(x, y) \| \log _{2}\left(1+x^{2}+y^{2}\right) \leqslant 1+\log _{2}(x+y)\right\}
$$

and the set
$$
S_{2}=\left\{(x, y) \log _{\frac{1}{2}}\left(2+x^{2}+y^{2}\right) \geqslant-2+\log _{\frac{1}{2}}(x+y)\right\} \text {. }
$$

Then the ratio of the area of $S_{2}$ to the area of $S_{1}$ is ( ).
(A) $2: 1$
(B) $4: 1$
(C) $6: 1$
(D) $8: 1$
2. C.

From the problem, we know that $S_{1}$ represents the interior of the circle (including the circumference)
$$
(x-1)^{2}+(y-1)^{2} \leqslant 1 \text {, }
$$

with an area of $\pi$;
$S_{2}$ represents the interior of the circle (including the circumference)
$$
(x-2)^{2}+(y-2)^{2} \leqslant 6 \text {, }
$$

with an area of $6 \pi$.
Therefore, the area ratio of $S_{2}$ to $S_{1}$ is $6: 1$.