The response of the physical spectel is modeled using a Gaussian function. Binning the data of 2 physical spectels shows that a Gaussian function also provides a satisfactory fit of the instrument’s spectral response for nominal spectels. Fig. 15 compares the response of a physical (oversampled) spectel and a nominal spectel (2 physical spectels binned). The FWHM is multiplied by a factor of approximately 1.215 compared to the physical (oversampled) spectel. A large portion of the calibration spectral data (~ 100 spectels) was tested. The measured average factor increase of the FWHM was 1.21 ± 002. This factor is in line with the expected theoretical factor (factor of 1.21) resulting from the numerical summation of two Gaussian functions, which have a central wavelength, λ0, separated by the theoretical spectral sampling of the instrument. The λ0 parameter for the nominal spectel is found to be simply equal to the averaging of the corresponding parameters in the two Gaussian responses binned.

[Figure: Fig. 15. Comparison of the oversampled and nominal spectral response of the VISNIR channel around 1.4 µm (physical spectel N°498 (out of 1016) in blue and its associated nominal spectel in orange). The retrieved FWHM for the nominal case is very close to the value of the FWHM of the nearby physical pixel multiplied by the theoretical factor of 1.21.]

On the other hand, the response can no longer be modeled using a Gaussian function for larger binning modes of the nominal spectel ×2 (leading to a 72-µm pixel size) and ×4 (leading to a 144-µm pixel size). Fig. 16 shows the effect of larger spectral binning modes on the shape of the response. All derived values of FWHM in Fig. 16 are close (within 5%) to the theoretical values that could be calculated from the proportionality coefficients obtained from numerical summations of Gaussian functions and listed in Table VIII.

[Figure: Fig. 16. Examples of the spectral responses for the four spectral binning modes (oversampling (blue), nominal (orange), binning ×2 (green), and binning ×4 (red)). Column 498 (in terms of the first physical pixel position) corresponds to 1408.5 nm.]

In the case of binning ×2 and ×4, the proposed modeling approach uses a function defined as the absolute difference between two Cumulative Distribution Functions (CDFs) of the normal distribution, normalized by its maximum value. This Gate-Gaussian function is characterized by its center, the Gaussian component's standard deviation, and the Gate function's width. As for the nominal binning, the FWHM for the ×2 and the ×4 binning modes can be estimated from the spectral response of the physical spectel by applying a factor of 2.03 and 3.60, respectively (Table VIII). Fig. 17 shows this methodology applied to the spectral binning modes ×2 and ×4, corresponding to the summation of 4 and 8 physical spectels, respectively. The function used for the fitting matches the experimental data well in both binning cases, leading to a FWHM from the experimental data virtually identical to the FWHM of the fitted function. The standard deviation around the expected theoretical value resulting from the modeling of about 100 spectels is ~5%.

[Figure: Fig. 17. Fitting of the spectral response in the case of binning mode ×2 (blue plot) and ×4 (orange plot) at 1420 nm. A Gate-Gaussian function (representing a gate function with Gaussian edges) reproduces the measured spectral response in these cases. The FWHM ratio (~1.7) is consistent with the theoretical proportionality parameter (1.77) from Table VIII.]

Table VIII. Spectral response models and FWHM proportionality parameters for the MAJIS spectral binning.

| Binning Mode | Function | Proportionality |
| --- | --- | --- |
| Oversampling | Gaussian | 1 |
| Nominal | Gaussian | 1.21 |
| ×2 | Gate Gaussian | 2.03 |
| ×4 | Gate Gaussian | 3.60 |