How to properly fit a non-linear equation to a set of datapoints?

Question

I have a curve of net longwave radiation (QL) data, which are calculated as follows:

QL = a*Ta^4 - b*Ts^4
where a and b are constants, Ta is the air temperature and Ts is the surface temperature

If I plot a curve of QL versus Ta-Ts, what type of equation should I use to fit the data as follows y = f(x) where x = (Ta-Ts)?

Thanks

-20.5   -176.683672
-19.5   -171.0655836
-18.5   -165.8706233
-17.5   -158.9990897
-16.5   -154.2715535
-15.5   -147.5376901
-14.5   -141.2410818
-13.5   -135.3387669
-12.5   -129.3971791
-11.5   -122.0777208
-10.5   -117.475907
-9.5    -111.107148
-8.5    -104.5999237
-7.5    -99.82769298
-6.5    -93.43215832
-5.5    -87.6278432
-4.5    -81.85415752
-3.5    -76.5997892
-2.5    -70.26308516
-1.5    -65.49437303
-0.5    -60.78052134
0.5 -56.32077454
1.5 -51.74037492
2.5 -47.30542394
3.5 -42.92298839
4.5 -38.13260904
5.5 -34.22676827
6.5 -30.49502686
7.5 -26.89383663
8.5 -22.259631

The complete data https://docs.google.com/spreadsheets/d/1e3gNCKQesrGe9ESrEIUcQw3umERzNRt0/edit?usp=sharing&ouid=115727378140347660572&rtpof=true&sd=true:

TS = surface temperature (degrees Celsius);
TA = air temperature (degrees Celsius);
Lin = longwave in (0.8 * 5.67E-8 * (TA+273.15)^4) (W m-2);
Lout = longwave out (0.97 * 5.67E-8 * (TS+273.15)^4) (W m-2);
QL = Lin - Lout (W m-2);

Answer 1

I took your data and did a 4th order polynomial fit. Here's the result:

QL = -58.607 + x*(4.8336 + x*(-0.0772 + x*(-2e-5 + x*8e-5)))

R^ = 0.9999

x = (Ta - Ts)

If you want the equation to be in terms of Ta and Ts instead of the difference you should substitute and do the algebra.

Answer 2

The notation QL=y=f(x) is falacious because QL doesn't depends on one variable only, but depends two independant variables Ta and Ts.

So, one have to write : y=F(Ta,Ts) or equivalently y=g(x,Ta) or equivalently y=h(x,Ts) with x=Ta-Ts and the functions F or g or h.

Any one of those functions can be determined thanks to nonlinear regression if we have data on the form of a table of three columns (not only two columns) for example :

(Ta,Ts,y) to find the function F(Ta,Ts)

or (x,Ta,y) to find the function g(x,Ta)

or (x,Ts,y) to find the function h(x,Ts)

In fact one cannot definitively answer to your question in which something is missing : Either measurements of another parameter or another relationship between the parameters in addition to the relationship x=Ta-Ts.

Of course one can compute (for example) the coefficients A,B,C,... for a polynomial regression of the kind f(x)=A+Bx+Cx^2+... and get a very good fitting :

The coefficients A,B,C are purely mathematical without physical signifiance. The coefficients a and b in f(x)=aTa^4+bTs^4 cannot be derived from the coefficients A,B,C without more physical information as already pointed out.