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For an MPD of 0.122 cm, the sole contribution of the finite interferogram length to the resolution yields 4.96 cm 1. The actual FWHM of the ILS can be written as: 1 . 207 (10.35) 2 MPD where is a broadening factor to account for parameters other than the finite interferometer length. The broadening factor is mainly due to the finite divergence in the interferometer, a contribution that varies with the wave number, being larger for higher wave numbers. FWHMILS =
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10.8.2 Apodization
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As stated in the previous sections, the ILS is governed mainly by the finite interferogram length and thus closely resembles a Sin(x)/x function as seen in Fig. 10.11. This function is sometimes considered a disadvantage because it has amplitude oscillations, which are large (21 percent for the first ripple), and which die off very slowly. It is possible to reduce the oscillations of the ILS by using post-acquisition digital processing on the acquired interferogram prior to Fourier transformation. This is referred to as apodization, or removal of feet, since the purpose of the operation is to remove or reduce the oscillations. This improvement of the ILS is gained at the cost of spectral resolution or increased ILS FWHM. The effect of various types of apodizing functions on the ILS FWHM is illustrated in Fig. 10.11. Triangular, Hamming, and Gaussian functions are commonly used apodization functions. Boxcar apodization is the use of no apodizing function. For a given MPD, one can see in Fig. 10.12 that the ILS
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Apodization Boxcar 1 Triangular Gaussian Hamming
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FWHM for boxcar apodization is better than that obtained using triangular, Hamming, or Gaussian apodization by a factor of 1.47, 1.52, and 1.68, respectively.
Radiometric Calibration
Radiometers with no spectral capability, whether they are single-pixel or imaging systems, are extremely difficult to calibrate radiometrically. The main cause for this is that at the detector level, a radiometer integrates the radiant signal over a certain spectral range producing a single output. The integration translates into a loss of information, which limits the interpretation of the measurement. The measurement produced by a radiometer is intended to be the radiance of an object view integrated over a specific spectral range. The actual output of a radiometer, however, is the radiance of the object view integrated over a specific spectral range but weighted by the spectral response of the system. Even after calibrating with a source of known radiance (e.g., a calibration blackbody), the uncertainty of the spectral response of the system induces an uncertainty in the intended measurement. This is because the response of the system is never flat within the spectral band pass, and thus spectrally varying object views of different integrated radiances can produce the same output. In other words, uneven spectral responses are linked with radiometric errors. The SpectRx Spectroradiometer represents an improvement over non-spectral radiometers. First, the addition of many spectral channels contributes to the narrowing of each channel, thus potentially making each of them more uniform. Another advantage is that it may be possible to restore distorted signals. If spectral channels are located in a contiguous manner, spectrally side-by-side, an over-evaluation of a given channel is accompanied by the complementary under-evaluation in neighboring channels. This means that the correct radiance of a feature in the spectrum can be evaluated by spectrally integrating over all channels that responded to the signal. Spectroscopically, this is a known phenomenon. When a spectrometer is operated at a resolution lower than the intrinsic features of an object view, the representation of these features in the spectrum is broader and weaker, but the integrated intensity is correct. Consequently, the radiometric characterization of a FTIR-based spectroradiometer is a straightforward operation that leads to a radiometrically sound calibration. This is due to the intrinsically large number of spectral channels in FTIR-based systems. In fact, the spectral response of a channel is given by the ILS function described in the previous section. An FT-based spectroradiometer has a large number of overlapping and smoothly varying channels.
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