# Amplitude scaling in seismic inversion

I've been testing how amplitude scaling subsequently affects the results of gapped/non-gapped deconvolution and an impedance inversion algorithm called BLIMP (Ferguson and Margrave, 1996). Indeed, scaling the amplitude of the input seismic data (whether linearly or non-linearly) greatly affects the final result of both deconvolution and inversion. I am confused by this.

Let's look at the convolution model in its full mathematical form:

$$s(t) = [R(t) \ast W(t) + N(t)] \times A(t)$$

where $s(t)$ is the observed seismic trace, $R(t)$ is the reflectivity time series (or perhaps impulse response if you want a more realistic/complicated situation), $W(t)$ is the source wavelet, N(t) is the additive, random noise component, and finally $A(t)$ is the amplitude. Also, note that $\ast$ denotes convolution and $\times$ denotes multiplication. I understand the convolution model in this mathematical context, but I am failing to transfer that knowledge to grasp how amplitude could so greatly affect deconvolution or inversion calculations.

If anyone has any papers, personal insight, or both to provide, please do so.