In Radar systems you have both additive noise and speckle. The additive noise is generally the additive thermal noise. In the case of additive noise, you can improve the SNR by simply increasing the transmitted signal power - whether or not you can practically do this for your system is another matter.
In the case of multiplicative noise increasing the transmitted signal power will not improve the SNR because the noise increases in proportion to the transmitted signal power. See for example the reverberation problem in Sonar mentioned in another post.
In Radar, speckle is caused by the presence of lots of individual scatterers within a resolution cell (resolution due the pulse bandwidth and aperture length in Synthetic Aperture systems). Consider viewing a field of grass at 1m resolution - due the presence of a lot of individual blades of grass - the results add incoherently and produce what is known as speckle. If we increase the transmitted power, the grass will simply reflect more of that energy back to the receiver and they will still add incoherently, so you will not see any improvement in Signal to Noise Ratio.
In the example you give, the noise power is proportional to the sqrt of the signal, so the variance of the noise (assuming noise with zero mean) is proportional to the signal level - but the variance is the measure of noise power. Lets say the power in the signal $u=1$ and the variance of $\eta=1$ i.e. $E(\eta^2)=1$, so the SNR $= 1/1$.
If we scale the input signal by $A$ then, under the model you give, the signal power is $A^2$ but the noise power is $E(A\eta^2)=AE(\eta^2)=A$. Thus the new SNR = $A^2/A=A$ so there is some improvement in the SNR. But in the case of additive thermal noise the improvement would have been $A^2$ since the power of the additive thermal noise is unaffected by the signal power i.e. they are independent.
So your example is kind of a hybrid between the two extremes. For your reference another example of multiplicative noise, is phase noise in clocks for local oscillators, ADC etc.