Amplitude scaling in seismic inversion
You can think of this amplitude term as being part of the wavelet term such that your definition of the wavelet is: "the necessary filter/operator to convert true reflectivity into the arbitrarily scaled seismic data". This scalar depends on the data itself and therefore must be obtained empirically. In fact, it is very likely that the scalar is frequency dependent. Your best bet to obtain the full wavelet, including its scalar, is by using a well where your reflectivity is known and applying the method of White and Simm (2003) or any other similar method.

Anyway, doing any sort of deconvolution or inversion with the wrong wavelet scalar will leak all error from this into the solution. You are pretty much asking: "What is the necessary (for example) acoustic impedance such that when I do this faulty forward modeling, I get the measured seismic data?".