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Hopefully this is the right section for my post, if not I apologize.

I'm hoping I can just get some advice to help me get started in the right direction. I am trying to do a mathematical inversion for the following:

[tex] \frac{1}{N(zi)} \frac{dN}{dz}|_{z=zi} = -\frac{2}{zi} - \frac{1}{T(zi)} \frac{dT}{dz}|_{z=zi} - \frac{C}{T(zi)} [/tex]

[tex] N(zi) [/tex] are measurements made at a series of altitudes.

There is the above relation between measurements and temperature [tex] T(zi) [/tex]

So temperature is what I am looking to find from the [tex] N(zi) [/tex] measurements. C is just a system constant.

What I am trying to do is guess an initial temperature vector, then minimize the [tex] \chi^{2} [/tex] between the measurements and the forward model above. So that once chi square is minimized as much as possible, we can determine the temperatures. I am very new to this, but have done some research into optimization and grid search methods. I was looking in the Marquardt Method listed in Numerical Recipes, but I figured I would post here to see if anyone else has any opinions on what would work the best.

If you guys need any more info just let me know. Thanks!