I am trying to estimate the surface (isochrone), zi(x) for which T(x,z)=0 from noisy measurements of T(x,z) everywhere and 3 almost noise free control points:
Instead of using the T(x,z) data directly I first estimate the gradient: $T_x$ and $T_z$ by finite differences on T(x,z). I now have two pde's to solve:
$ \frac{\partial}{\partial x}T=T_x, \frac{\partial}{\partial z}T=T_z$
Using finite differences again I express the derivation operators as matrices:
$\ D_x T=T_x, D_z T=T_z $
where $T_x$ and $T_z$ are now column vectors made by concatenating all the columns of their respective matrices vertically, and $D_x$ and $D_z$ are matrices that when multiplied with the T vector gives the partial derivative with respect to x and z.
In addition to these pde's I also have equations for the 3 control points:
$ T(xc_i,zc_i)=0$
(Weighted) Least Squares Error (LSE) is a framework that allows me to mix certain and uncertain data.
My (weighted) linear equations are now:
$\ D_x T=T_x $
$\ D_z T=T_z $
$\ \alpha T(xc1,zc1) = 0$
$\ \alpha T(xc2,zc2) = 0$
$\ \alpha T(xc3,zc3) = 0$
where I have introduced $\ \alpha $: a parameter that decides how much the control points are honored relative to the measurements of T(x,z).
Is there some way to estimate $\alpha$ or alternatively express it as a more intuitive expression?
So far I have just tuned this parameter and compared the offsets from the control points iteratively. It would be good to have some other way to set this, e.g. max allowed z offset from control point to resulting isochrone.
Do statisticians have some other way than LSE of combining certain and uncertain data?
Is Bayes such a method? If so what is the a priori in my example?
