Does the non-linearity of the square inverse planck function violate assumptions in propagation of uncertainty?

[gerritholl]

For the final step in propagating uncertainties from radiance units to brightness temperature units, applying the law of propagation of uncertainties yields very large uncertainties (hundreds of kelvin), in particular due to the component from u(L). However, if I do a naive numerical approximation of the same uncertainty (L±ΔL→T_b), I don't get the same unrealistically large uncertainties. Perhaps I can transform the radiances to a space in which the Planck equation does not contain an exponent and the inverse Planck equation does not contain a logarithm, and see whether that resolves the problem.

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          ╱                                                                            2                                                                
         ╱  ⎛                                                   3  2                  ⎞                                     8  4  2                     
        ╱   ⎜           h                                6⋅fstar ⋅h                   ⎟   2                          4⋅fstar ⋅h ⋅u (Rₑ)                 
       ╱    ⎜─────────────────────── - ───────────────────────────────────────────────⎟ ⋅u (fstar) + ────────────────────────────────────────────────── 
      ╱     ⎜       ⎛           3  ⎞             ⎛           3  ⎞     ⎛           3  ⎞⎟                                          2                      
     ╱      ⎜       ⎜    2⋅fstar ⋅h⎟       2     ⎜    2⋅fstar ⋅h⎟    2⎜    2⋅fstar ⋅h⎟⎟                          ⎛           3  ⎞      ⎛           3  ⎞ 
    ╱       ⎜k_b⋅log⎜1 + ──────────⎟   Rₑ⋅c ⋅k_b⋅⎜1 + ──────────⎟⋅log ⎜1 + ──────────⎟⎟                4  4    2 ⎜    2⋅fstar ⋅h⎟     4⎜    2⋅fstar ⋅h⎟ 
   ╱        ⎜       ⎜          2   ⎟             ⎜          2   ⎟     ⎜          2   ⎟⎟              Rₑ ⋅c ⋅k_b ⋅⎜1 + ──────────⎟ ⋅log ⎜1 + ──────────⎟ 
  ╱         ⎝       ⎝      Rₑ⋅c    ⎠             ⎝      Rₑ⋅c    ⎠     ⎝      Rₑ⋅c    ⎠⎠                          ⎜          2   ⎟      ⎜          2   ⎟ 
╲╱                                                                                                               ⎝      Rₑ⋅c    ⎠      ⎝      Rₑ⋅c    ⎠ 

[gerritholl]

In principle, one would have to use a Monte Carlo based approach. The numerical approach in 67c0fc8 does not actually make any difference for the local linearity assumption.