In view of the difficulties in traditional long-wave infrared imaging spectrometer which is hard to realize a high signal-to-noise ratio and miniaturization as well under the weak remote sensing signal, two types of concentric optical systems, Offner convex grating and Dyson concave grating imaging spectrometers, are designed and analyzed under the common numerical apertures (NA) of 0.19 and 0.33 with spectral range of 8~12 μm. The optimal structures are achieved by optical design software Zemax. The two structures both have a perfect image at NA1=0.19, the size of Offner structure is 245 mm×213 mm×111 mm, value of modulation transfer function (MTF) is higher than 0.38, spectral resolution is 35 nm, smile is lower than 4.8% and keystone is lower than 12.8%. The size of Dyson structure is 308 mm×61 mm×49 mm, value of MTF is higher than 0.48, spectral resolution is 12 nm, smile is lower than 0.047% and keystone is lower than 0.138%. Dyson has an ideal image at NA2=0.33 different from Offner. The size of Dyson structure is 317 mm×88 mm×88 mm, value of MTF is higher than 0.71, spectral resolution is 1 nm, smile is lower than 0.015% and keystone is lower than 0.028%. It is obvious that Dyson structure possesses advantages of larger numerical aperture, smaller volume, higher spectral resolution, higher MTF, lower smile and lower keystone in long-wave infrared wavelengths compared with Offner structure.