Suppose we have a 2-d matrix $Z \in \mathbb{R}^{m\times n}$, it could be some signal, or an image let's say.
Suppose we have this noise model: $N(\mu=0,\sigma=1)$.
If we compute:
$K=Z + N(\mu=0, \sigma=1)$
and then
$L = K+ N(\mu=0,\sigma=2)$.
Can we say that the total noise on $L$ is $N(\mu=0,\sigma=3)$ ?
What exactly is additive on this noise?
The additive property of a noise is called so because the noise is get added to the signal. For example if we denote the input signal by $s$, the noise by $v$ and the output signal by $y$ and assume that the noise is additive, then $$y=s+v$$ in which $y,s,v$ could be matrices, vectors, scalars or higher order tensors. One prominent example of such kind of noises is AWGN .
Now, by substitution we can write $$L=Z+N(\mu=0,\sigma^2=1)+N(\mu=0,\sigma^2=4)$$a mathematical theorem states that the summation of two independent Gaussian random variable with the same average, is also a Gaussian random variable with its variance being the summation of the other two variances. In our case, we can write $$L=Z+N(\mu=0,\sigma^2=5)$$