Heat - Remote Sensing of Temperature

Michael Jacobsen

The problem is to deduce how the temperature changes (we will call this a temperature profile) at one place from measurements at another place.

We restrict ourselves to a 1-D problem. Assume that we can apply a temperature profile (f(t)) to the end of a rod (at the position x = 0). We now define the forward problem that gives the temperature profile at any point by means of a Partial Differential Equation (PDE):

u_t = \kappa^2 u_{xx}, \quad 0 < x < \infty, \quad 0 < t < ... ...0 \le x \le \infty, \\ & u(0,t) = f(t), \quad 0 \le t < \infty

We have assumed that the rod is isolated such that no energy escapes through the top and bottom. Furthermore we assume that the rod has initial temperature 0 everywhere. The solution is

u(x,t) = \frac{x}{2 \kappa \sqrt{\pi}} \int_0^t \frac{f(\... ... \left( \frac{-x^2}{4 \kappa^2 (t - \tau )} \right) d\tau .

Hereby we can find the temperature at any point at any point in time using the above formula. This was the forward problem.

The inverse problem is when we have a temperature profile g(t) at some point x = l <= 0 and wish to find the temperature profile f(t) at end of the rod. This occurs in practice when the end of the rod is inaccessible.


Figure 1: Illustration of forward and backward (inverse) problem.

From the previous we see that the forward problem poses no problems as we have a closed form solution. The inverse is on the other hand much more difficult. One can imagine that many different temperature profiles f(t) will yield the same measurements g(t) at least when we also include measurement accuracy. The forward operation smoothens the temperature profile as we travel down the rod. Therefor the inverse operation must desmooth as we go towards x = 0, but it the operation will also be applied on the noise and errors made in the measurements.

In this case an many others a naive solution of the inverse problem will yield unusable results due to the influence from the noise. Regularization is a technique to stabilize the inverse problem such that some information can be extracted. The applet is constructed to demonstrate this.

The Regularization Applet

Last update of applet 2001-07-31. Needs update

The problem is discretized to have 128 points. Therefor you can use the integers from 0 to 128 as argument for T-SVD. Tikhonov accepts all values greater than or equal to zero. CGLS accepts integers greater than 0.

You can read more details on the algorithms and methods if you follow the links

Instructions

You can change the source (solution) by dragging the points with the mouse. The parameter kappa controls how difficult (ill-posed) the problem is, a value of 1 gives an ill-posed problem while a value of 5 makes a fairly easy problem (well-posed). You can increase the noise level and the number of outliers to hide information.

To see the naive solution select k = 128 for T-SVD or lambda = 0 for Tikhonov. Notice that even with very little noise we get very bad results.

Observe that outliers makes it very hard to find a good solution. Also sources with discontinuities are hard to reconstruct using these regularization techniques.