Here, \(I\) refers to \(I(\vec{r},v)\) and \(I'\) refers to \(I(\vec{r}+\delta \vec{r}, v+\delta v)\); where “\(I\)” refers to the intensity of light from \(^{13}CO\) emissions, “\(\vec{r}\) refers to the position of the source, and ”\(v\)" refers to the radial velocity at which the source is moving.

Note that the average of all the mixed terms is treated as being close to zero, since we’re expecting the noise intensity to not change too much with changes in position and radial velocity (i.e. \(\left< (I_{noise} - I'_{noise}) \approx 0 \right>\)).