I have some data, in which I desire to know the value $X$, but in which I can only measure:
$$ Y_{(X)} = G_{(X)}*X $$
The form of $G_{(X)}$ is unknown, but $X$ is known to be a relatively well behaved shape (specifically a summation of various normal / Gaussian distributions with different centres on a linearly varying background), and $Y_{(X)}$ is broadly the same well behaved shape (again a summation of various normal / Gaussian distributions with different centres on a linearly varying background). I cannot directly measure X, but I can directly vary it by a linear amount and measure $Y_{(X)}$ again, such that:
$$ Y_{(N.X)} = G_{(N.X)}*N.X $$
where $N$ is a real number greater than $0$.
From inspection of:
$$\frac{Y_{(N.X)}}{Y_{(X)}}$$
it is clear that $G_{(X)}$ is not:
$$m.X+c$$
nor
$$m^2.X+c$$.
Instead the shape of:
$$\frac{Y_{(N.X)}}{Y_{(X)}}$$
more closely resembles a dampened sinusoidal summed with a negative linear variation:
$$\frac{Y_{(N.X)}}{Y_{(X)}}\propto a.e^{-b.X}.c.cos(d.X.\pi)-e.X$$
With a with a variation between ~1.5 and 0.9.