INTRODUCTION The motivation is to directly characterize the nonthermal spectral shape and thermal fraction of the radio emission in a spatially resolved fashion, using the existing CACA in-band measurements (TT0L, TT1L, TT0C, and TT1C) in the L and C bands. This direct method is optimal because it should work particularly well in galactic disk regions without any extinction effect and with good S/N. In outer regions, where the S/N is not as good, adaptive binning may be used, a technique that will be explored later. Furthermore, the method can be self-calibrated: in off-disk regions, the thermal fraction should approaches to zero. In principle, a spatially-resolved global fit of an entire galaxy can be carried out with the a physical or phenomenological model (e.g., assuming that the injected cosmic-ray has the same spectral slope across a galaxy). Here only a simple local phenomenological model is illustrated. PROCEDURE We assume that the spectral flux density can be modeled as S(\nu)=S(1)[(1-\eta_L)e^{\nu+\beta ({\rm ln}\nu)^2}+\eta_L \nu^{-0.1}] where ν is in units of νL (the central wavelength of the L-band), ηL is the thermal fraction of the flux density at νL, and the form of the first (nonthermal) term with two free parameters, b and c is as proposed by Williams & Bower (2010). By definitions, the measurements can be linked to the model via TT0L = S(1), TT0C = S(νC), and TT1L/TT0L = sL and TT1C/TT0C = sC (the spectral slopes at νL and νC). Using Eq. [e:s], we can express the slope at any frequency as s=S}{{\rm ln}\nu}=\alpha+2\nu-(0.1+\alpha+2c{\rm ln}\nu)\eta_L\nu^{-0.1}/r where we have defined r \equiv TT0_C/TTO_L= S(\nu_c)/S(1)=(1-\eta_L)e^{\nu+\beta({\rm ln}\nu)^2}+\eta_L \nu^{-0.1}. Apparently, at νL and νL, Eq. [e:s_d] becomes s_L=\alpha-(0.1+\alpha)\eta_L and s_C=\alpha+2\nu_C-(0.1+\alpha+2\nu_C)\eta_L\nu_C^{-0.1}/r. From Eq. [e:s_d_L] we can immediately get \eta_L = {\alpha+0.1} while Eqs. [e:s_d_C] and [e:eta_L] give \beta={2{\rm ln}\nu_C}}{r+\eta_L\nu_C^{-0.1}}={2{\rm ln}\nu_C}}{r+\eta_L\nu_C^{-0.1}} We may now solve for α and β iteratively. To do so, we rewrite Eq. [e:r] for nth iteration as s_n={{\rm ln}\nu_C}{\rm ln}\Big[}{1-\eta_L}\Big]-\beta {\rm ln}\nu_C where ηL and β are evaluated with sn − 1, using Eqs. [e:eta_L] and [e:c]. We may get the initial α value by assuming the β = 0, which requires \alpha_0=}{r-\nu_C^{-0.1}} If α₀ > sL, then we set α₀ = sL and re-calculate β with Eq. [e:c] to make sure that ηL ≥ 0. The iteration can then proceed until Eq. [e:bn] converges.