# 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.

## 1 Answer

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?".