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FIGURE 10.13 Linear relationship between the object view spectral radiance and the power at the detector.
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The ideal radiometer is a linear instrument (i.e., the measured signal for each pixel and each spectral channel is proportional to the radiant spectral power at the detector). This is illustrated in Fig. 10.13. The power at the detector is composed of two contributions, the spectral power from the object view, and that from the thermal emission of the spectrometer itself. Because of this, the response line in Fig. 10.13 does not cross the abscissa at x = 0 but rather at some value corresponding to the thermal emission of the spectrometer. A calibration for such a system thus requires at least two measurements, as illustrated by the s in Fig. 10.13. If the emission of the instrument could be neglected, only one characterization measurement would be necessary. Let us now look at the calibration equations. An uncalibrated measurement can be expressed as: SMeasured ( ) = K ( )(LSource ( ) + M Stray ( )) where (10.36)
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S Measured( ) = measured complex spectrum (arbitrary) K( ) = complex instrument response function (arbitrary cm2 sr cm 1/W) LSource( ) = source spectral radiance (W/cm2 sr cm 1) M Stray( ) = spectral power of the stray radiance (W/cm2 sr cm 1)
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An interferogram always has a certain degree of asymmetry due to dispersion present in the beam-splitter (the wavelength-dependent refractive index) and in the amplification stages of the detectors (the frequency-dependent electronic delays). This asymmetry causes the Fourier transform of the interferogram (i.e., the spectrum) to have an imaginary part. At this point, a phase correction can be applied to the complex spectrum to yield a real spectrum. With the present calibration algorithm, however, the phase correction is not needed and instead we can work with complex spectra.
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A radiometric calibration is a two-step process. First, the two unknowns per detector radiometric gain K( ) and radiometric offset M Stray( ), are determined using an experimental step called a characterisation. Second, the calibration is applied to an un-calibrated measurement to produce a calibrated spectrum. The gain and offset characterisation requires two measurements, one hot blackbody measurement and one cold blackbody measurement, ideally both uniformly filling the field of view. This is referred to as a two-point calibration, as shown in Fig. 10.13. The temperatures of the calibration blackbody measurements are judiciously set in order to minimize calibration error. Applying Eq. (10.17) to the measurements of the hot and cold blackbodies, we obtain the following equations:
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Measured LH ( ) = SH ( )/ K ( ) M Stray ( ) Measured LC ( ) = SC ( )/ K ( ) M Stray ( )
(10.37) (10.38)
where LH( ) and LC( ) are the theoretically calculated spectral radiances using the emissivity of the blackbody and the Planck function at the temperature of the blackbody: Lx ( ) = x ( ) PT ( ) = x ( )
C1 3 C exp 2 1 Tx
(10.39)
where x( ) = emissivity of the x blackbody x = C for the cold blackbody and H for the hot blackbody C1 = 1.191 10 12 W cm2 and is the first radiation constant C2 = 1.439 K cm and is the second radiation constant = wave-number (cm 1) Tx = temperature of the x blackbody (K) The solution to Eqs. (10.18) and (10.19) yields Eqs. (10.21) and (10.22). It is then simple to solve for the two unknowns K( ) and Mstray( ). K ( ) =
Measured Measured SH ( ) SC ( ) LH ( ) LC ( )
(10.40)
M Stray ( ) =
Measured Measured LH ( ) SC ( ) LC ( ) SH ( ) Measured Measured SH ( ) SC ( )
(10.41)
The calibrated spectrum is given by: SCalibrated ( ) = SMeasured ( )K 1 ( ) M Stray ( ) (10.42)
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