##### 2.b.iii.v0.6

**This is the recommended option.** Tahara and Kato [2009] define virtual channels named gap channels to fill the spectral gaps and introduce the spectral compensation method by constrained optimization. The gap channels to fill the AIRS spectral gaps (AIRS gap channels) are defined by 0.5 cm

^{-1} intervals, and are characterized by a unique SRF, whose shape is a Gaussian curve with a sigma of 0.5 cm

^{-1}. The gap channels to extend the IASI spectral region (IASI gap channels) are defined by the same intervals (0.25 cm

^{-1}) and SRFs as the IASI level 1c channels. The radiances of the missing channels are calculated by regression analysis using radiative transfer simulated radiances with respect to the eight atmospheric model profiles as explanatory variables.

**Equation 7:**
\log I_{i}^{calc} = c_0 + \sum_{k=1}^{K} c_{k}\log I_{i,k}^{sim} ,\ \ (i=hyper\ and\ gap\ channels)
where

I_{i}^{calc} is the calculated radiance of the hyper channel

*i*,

I_{i,k}^{sim} is the simulated radiance of the hyper channel

*i* with respect to the atmospheric model profile

*k*,

c_{k} are regression coefficients, and

*K* is the number of the atmospheric model profiles. Equation 7 introduces logarithm radiances as response and explanatory variables in order to increase fitting accuracy and avoid calculation of negative radiance. The regression coefficients

c_{k} are independent of the hyper channels, and are generated for each observing position of the hyper sounder.

c_{k} are obtained by the least-square method applying a set of validly observed radiances

I_{i}^{obs} in place of

I_{i}^{calc} to Equation 7,

**Equation 8:**
{c_k} = argmin\sum_{i=exist(I_{i}^{obs})}\{\log I_{i}^{obs}-(c_0 + \sum_{k} c_k \log I_{i,k}^{sim})\}^2
Once the regression coefficients

c_{k} are computed, the radiances of the missing channels can be calculated by Equation 7. It might be possible to apply the observed radiances of all hyper channels to Equation 8 to compute

c_{k} and then calculate the radiances of all missing channels at once. However, this yields a large fitting error in practice. In inter-calibration application, the coefficients

c_{k} are computed for each broadband channel spectral region. Equation 7 and Equation 8 use the simulated radiances

I_{i,k}^{sim}. For the radiance simulation, this study uses the following eight atmospheric model profiles:

- U.S. standard without cloud,
- U.S. standard with opaque cloud with tops at 500 hPa altitude,
- U.S. standard with opaque cloud with tops at 200 hPa altitude,
- Tropical without cloud,
- Tropical with opaque cloud with tops at 500 hPa altitude,
- Tropical with opaque cloud with tops at 200 hPa altitude,
- Mid-latitude summer without cloud,
- Mid-latitude winter without cloud.

These profiles include not only clear weather conditions but also cloudy conditions because Equation 7 should be applicable under any weather conditions. As for radiative transfer code, the line-by-line code LBLRTM (Clought et al., 1995) version 11.1 is used with the HITRAN2004 spectroscopy line parameter database (Rothman et al., 2003) including the AER updates version 2.0 (AER Web page). The emissivities of the surface and clouds are assumed to be one. The benefit of this spectral compensation method is that it does not use radiative transfer computation in inter-calibration operation. This not only speeds up the computation but also prevents super channel radiance computation from introducing biases contained in radiative transfer code and atmospheric state fields.