I have a system with a moving sensor, and the measured value is in the form of a gaussian function. Thus, I'd like to model this sytem with a gaussian. Naming Ms the measured value, I'd like to approximate with the following equation: $$Ms = Mp*e^{-\frac{1}2(\frac{s-\bar s}\sigma)^2}$$ Where s is the sensor location and Mp, $\bar s$ and sigma are parameters of our gaussian. Applying a logarithm, we obtain: $$ln(Ms) = ln(Mp) - \frac{1}2(\frac{s-\bar s}\sigma)^2$$ We have 3 unknows so we need 3 equations to solve the system. We can thus arbitrarily chose 3 positions s, named s1, s2 and s3 and we obtain the following: $$ln(Ms1) = ln(Mp) - \frac{1}2(\frac{s1-\bar s}\sigma)^2$$ $$ln(Ms2) = ln(Mp) - \frac{1}2(\frac{s2-\bar s}\sigma)^2$$ $$ln(Ms3) = ln(Mp) - \frac{1}2(\frac{s3-\bar s}\sigma)^2$$ The interesting parameter for me is $\bar s$ since it's the value of s that will maximize the measured value. Thus, I tried to solve the system for $\bar s$ and I obtained: $$\bar s = \frac{((s1^2 - s2^2) * (M2 - M3) - (s2^2 - s3^2) * (M2 - M1))}{((s2 - s1) * (M2 - M3) - (s2 - s3) * (M2 - M1))}$$ Where $Mi = ln (Msi) \forall i$ But after verifying through tests, it doesn't seems to calculte the value $\bar s$ of a gaussian function. So my question is, is my equation correct? If not, would you mind telling me the correct equation.
Thanks for the time you'll take reading this and sorry if my english is not perfect.