2D polynomial fit when the data is sorted

Hi everyone !

How can I fit an array of data like [80.3 112.4 183.2 ....] in a model like $ f_{m,n} = a m^2 + b n ^2$ where m, n are integers and a,b are the coefficients to fit?

The problem is that the data is sorted. As I cannot know to which integer pair (m,n) each coefficient corresponds, I cannot simply use the 'fit' function. I might even miss some coefficients, for example the first one might be $f_{3,1}$.

In details, what I am actually trying to do is approximating a vibrating structure with a thin plate model. I get the spectrum from a chirp excitation and try to fit the peak frequencies with a thin plate model. The problem is that the first natural frequency is below my sound card's cutoff frequency so I cannot get it. In addition, because of the vibration modes geometry I might miss some if the sensor is placed on a vibration node.

Thank you in advance !

Note : Sorry for the horrible formatting, I couldn't find how to insert mathematical formulas in the question. This question might be a little bit too 'physics like'. If you know a better forum to ask it I can migrate it.