# Fitting nonlinear numerical model

I am reading a scientific paper that fits a numerical model to a timeseries (observations). The numerical model has a few free parameters, some of them linear and some are not, which the authors wish to fit to the observations.

The numerical model produces a timeseries for each set of input parameters. The authors use regression (I think, they don't explicitly say so) to produce a $$\chi^2$$ value for each model run. Then, they report the parameters that minimize the $$\chi^2$$ value as those that best fit the observed timeseries.

So, just as an example, if the observations timeseries are represented by $$O(t)$$, then the model produces $$i={1..N}$$ timeseries $$M_i(t,a_i,b)$$, where $$a_i$$ is a linear model parameter that changes for each timeseries and $$b$$ is a nonlinear parameter which the authors choose to hold constant and do not attempt to fit, but is a free parameter (so in theory it may be changed). The parameter $$a_i$$ producing model timeseries $$M_i$$ that has the lowest $$\chi^2$$ is selected as best fit parameter.

This confuses me, I admit. Here are my questions:

1. As I mentioned, the authors do not explicitly say which method they used, but do report $$\chi^2$$ values. Is there any possibility they did not use regression? Is $$\chi^2$$ minimization used in any other goodness of fit method?
2. If regression is the only option, can the authors even do that? As I mentioned, some of the parameters are nonlinear with the function. I mean, the model is linear with the one fit parameter $$a$$, but the for the other parameter $$b$$ (which they hold constant and do not attempt to fit) it is nonlinear. I am only mentioning it, since for the best of my knowledge you cannot trust $$\chi^2$$ for nonlinear regression since the number of degrees of freedom is unknown. I am not sure if this holds in this case, as they treat $$b$$ as a constant.
3. If this approach does indeed make sense: the authors compare the $$\chi^2$$ of a few different models to say which model best fits the data (that is, after they found the best fit parameter $$a_i$$ for each timeseries). How do you estimate the errors in this case? Using cross validation? (they did not produce any errors with their $$\chi^2$$s).

I would appreciate any help.