\[\begin{aligned} S_2 &= \left< \left[ I(\vec{r},v) - I(\vec{r}+\delta \vec{r},v+\delta v) \right]^2 \right>\\ &= \left< \left[ I - I'\right]^2 \right>\\ &= \left<\left[ (I_{signal}+I_{noise}) - (I'_{signal}+I'_{noise}) \right]^2 \right>\\ &= \left< I_{signal}^2 + 2I_{signal}I_{noise} - 2I_{signal}I'_{signal} - 2I_{signal}I'_{noise} + I_{noise}^2 - 2I_{noise}I'_{signal} - 2I_{noise}I'_{noise} + (I')_{signal}^2+2I'_{signal}I'_{noise} + (I')_{noise}^2 \right>\\ &= \left< I_{signal}^2 - 2I_{signal}I'_{signal} + (I')_{signal}^2 + I_{noise}^2 - 2I_{noise}I'_{noise} + (I')_{noise}^2 \right> + \left< 2I_{signal}I_{noise} - 2I_{signal}I'_{noise} - 2I_{noise}I'_{signal} - 2I'_{signal}I'_{noise} \right> \\ &= \left< \left[ I_{signal} - I'_{signal} \right]^2 + \left[ I_{noise}^2 - I'_{noise} \right]^2 \right> + \left< 2I_{signal}I_{noise} - 2I_{signal}I'_{noise} - 2I_{noise}I'_{signal} - 2I'_{signal}I'_{noise} \right> \\ &= \left< \left[ I_{signal} - I'_{signal} \right]^2 \right> + \left< \left[ I_{noise}^2 - I'_{noise} \right]^2 \right> + \left< 2I_{signal}I_{noise} - 2I_{signal}I'_{noise} - 2I'_{signal}I_{noise} - 2I'_{signal}I'_{noise} \right> \\ &= \left< \left[ I_{signal} - I'_{signal} \right]^2 \right> + \left< \left[ I_{noise}^2 - I'_{noise} \right]^2 \right> + \left< 2I_{signal}(I_{noise} - I'_{noise}) - 2I'_{signal}(I_{noise} - I'_{noise}) \right> \\ &\approx \left< \left[ I_{signal} - I'_{signal} \right]^2 \right> + \left< \left[ I_{noise}^2 - I'_{noise} \right]^2 \right>\\ &\approx S_{2(signal)} + S_{2(noise)} .\\\end{aligned}\]