# AIRS Frequency Calibration Notes

Tue, Nov 22, 2016## 1 First Testing

### 1.1 Spectral Calibration Approach

The basic idea is to interpolate the observed spectra, say using the first-derivative of some fit to the BT spectrum, or

\begin{equation} BT = BT_o + \left(\frac{\partial BT}{\partial \nu}\right) \delta \nu \end{equation}where \(BT_o\) is the observed spectrum, \(BT\) is the spectrum on the desired fixed ν grid, and \(\delta \nu\) is the frequency offset of the observed data from the fixed grid.

Yibo generalized this equation by including two empirical parameters, \(a\) and \(b\)

\begin{equation} BT = BT_o + \left(a (\frac{\partial BT}{\partial \nu}) +b\right)\delta \nu. \end{equation}that are determined by least-squares fitting to simulated data. This is needed since it is difficult to derive a perfect expression for \((\partial BT/\partial \nu)\) from the data, especially since AIRS is not "Nyquist" sampled.

Yibo's approach for evaluating \((\partial BT/\partial \nu)\) is

\begin{equation} \frac{\partial BT}{\partial \nu} \equiv \frac{BT_o - BT_s}{\delta \nu} \end{equation}where \(BT_s\) is a spline fit to \(BT_o\) evaluated at the desired \(\nu\) grid. Plugging this into Eq. (2) gives you the existing JPL C-code algorithm,

\begin{equation} BT = (1-a) BT_o + a BT_s + b (\delta \nu) \end{equation}We have implemented a second approach, where we instead evaluate the derivative of the spline fit to the observed spectrum analytically, ie

\begin{equation} \left(\frac{\partial BT}{\partial \nu}\right) \equiv \left(\frac{\partial BT}{\partial \nu}\right)_{\text{Spline}} \end{equation}This approach might be less noise sensitive than Jibo's approach?

### 1.2 Approach

- Using Howard Motteler's algorithm for converting AIRS to CrIS ILS
- Deconvolve AIRS, reconvolve with shifted AIRS ILS functions (instead of CrIS ILS)
- Slightly better, especially in the longwave
- More computationally expensive?

Figure 1: Calibration error using deconvolution/reconvolution vs spline interpolation. Data set is the RTA regression profiles.

### 1.3 Early Issue: Yibo's \(a,b\) are for 2378 Channel Set

Howard re-fit for \(a,b\) using 49 RTA regression profiles and tested
against the same profiles (not independent) on **L1c channel set**.

Figure 2: Calibration error for JPL a,b approach vs spline for 49 regression profiles.

### 1.4 New \(a,b\) Coefficients vs Old

Figure 3: \(a\) Coefficients

Figure 4: \(b\) Coefficients

Quite similar. Can be re-done using cloudy simulation data.

### 1.5 Single Spectrum Experiments: No Noise!

Figure 5: Sample simulated spectrum used for single scene testing.

Figure 6: Calibration errors (JPL approach).

Figure 7: Same as above with fill channels removed.

Figure 8: Same as above but also with cleaned channels removed.

### 1.6 Single Spectrum Experiments: With Noise

Figure 9: Single spectrum calibration errors (JPL approach) but with simulated noise.

Figure 10: Same as above with fill channels removed.

Figure 11: Same as above but also with cleaned channels removed.

Figure 12: Comparison of (Obs-Cal BT) versus NeDT noise added to simulated spectrum.

### 1.7 Single Spectrum Experiments: \(\nu\) Calibration Noise

Figure 13: Comparison of scene noise to calibration error minus noise.

Figure 14: Same as above with scene noise removed.

Figure 15: Calibration error (minus noise) compared to calibration error for case where no noise was added to the scene.

Appears that calibration is fairly insensitive to noise?

### 1.8 Comparison of Yibo vs Analytic Spline Derivative

- These results used a full granule
- Differences are smaller than absolute errors
- Statistical comparisons with truth too identical to call a winner

Figure 16: Yibo-Error is calibration error for granule average, using JPL calibration approach. Yibo-LLS is the difference between Yibo Error and calibration using spline analytic derivatives.

### 1.9 Results Using a Full Granule: Yibo approach

Figure 17: Full granule mean calibration errors (with noise) for JPL approach.

Figure 18: Same as above with standard deviation of calibration error.

Mean NedT - (Std Obs-Calc)

Figure 19: Mean NedT noise for graunle minu Std of (Obs - Calc) of calibration error.

- Calibration noise is detectable in the shortwave where curve goes below zero.

### 1.10 Conclusions

- Have derived \(a,b\) on L1c grid
- Yibo approach looks good, although fitting and testing with 1 micron only shifts.
- Probably re-fit \(a,b\) with simulated cloudy data and more shifts
- Analytic spline not worth the computation? (NOT CORRECT, always computed when fitting for splines.)
- Create a test day that cycles through full mission shifts, with Doppler?