Which theory is used to calculate the position and energy of a point source?

Question

Consider an empty room with one point source that emits a stationary signal (constant sound, radioactive radiation, ...). The energy nor the position of the point source is known. We send someone in the room to do some intensity measurements on different positions in the room.

With this collected data I want to determine the position and the energy of the point source. Which mathematical theory can be used to do such calculations?

Answer 1

Assuming the data set is in the form of $$\{(E_i,x_i,y_i,z_i)\}_{i=1}^{i=n}$$ And Given the following notations:

• $n$ is the number of measurements
• $E_i$ is the measured energy
• $x_i$,$y_i$ and $z_i$ are the coordinates of the measurement
• $E_*$ is the actual source of energy
• $x_*$,$y_*$ and $z_*$ are the the actual coordinates of the source

We know (from Physics) that that the measured energy decreases in proportion to the distance from the source squared. $$E_*\propto \frac{E_i}{(x_i-x_*)^2+(y_i-y_*)^2+(z_i-z_*)^2}$$ Adding a proportion coefficient $K$ we get this equation $$E_i=KE_*((x_i-x_*)^2+(y_i-y_*)^2+(z_i-z_*)^2)$$ We just got ourselves a model with the 5 parameters: $$\theta=(E_*,x,_*,y_*,z_*,K)$$ The parameters can be approximated using Quadratic regression (Maximum Likelihood approach) or Bayesian inference with $K$ as a nuisance parameter.

Answer 2

It was already pointed out that your measures give something like $m(r)=C/\|r-r_*\|^2$, where $C$ is a constant and $r_*$ is the position of the source. From the point of view of mathematics, you are able from these measurement to compute the gradient $$\nabla m(r)=-2C\frac{r-r_*}{\|r-r_*\|^4},$$ which gives the direction of the source. Computing this gradient at several positions should result in a unique intersection point. Mathematically you should just need to get this gradient in two points. In practice, this is more difficult, because of accuracy issues, errors, etc. But then you should ask this question on a more appropriate StackExchange site.

Of course, once you have located the source, it is easy to obtain the value of the constant.