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S p e c t R x N I R Te c h n o l o g y
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This algorithm can be extended to more than three points using statistical regression analysis on the polynomial much in the same way as was done for the linear case.
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Radiometric Accuracy
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Radiometric accuracy is the deviation of the measured spectral radiance from the actual object view radiance. This is illustrated schematically in Fig. 10.16. However, even if noise2 affects the radiometric accuracy, it is treated as a separate parameter, the NESR, which is not included in the radiometric accuracy. In general, radiometric accuracy is an arbitrary function, as shown in Fig. 10.16, but is usually described in two parts, one absolute, the other relative. Radiometric error is absolute, if it does not vary with the object view radiance, or it is relative, if the error does vary with object view radiance. Absolute errors are more difficult to estimate than relative ones. In the following paragraphs, we will only discuss relative radiometric errors, converting absolute errors into relative equivalents. This is a more convenient way to predict the system accuracy for a given object view. Three types of errors influence radiometric accuracy. The first type of calibration source errors are deviations in the production of a perfect (i.e., perfectly known) calibration source. These errors include the accuracy of blackbody temperature and the accuracy of its emissivity over the operational spectral range. The second type of error is calibration drift. Calibration drift includes everything that changes the radiometric gain and offset during the time interval between performing calibration measurements and object view measurements. Calibration drift is influenced by many factors, including the ambient temperature, the stability of the electrical gain, the stability of detector responsivity, and optomechanical stability. Finally, the third type of error that influences radiometric accuracy is the spectrometer intrinsic linearity. This includes system parameters such as detector linearity, channel spectrum, and spectral aliasing. Because they are uncorrelated, the contributions from all predicted errors add up in a root-sum-square fashion.
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Calibration source errors are misevaluations of the spectral radiance supplied by the calibration source. The effect of calibration source
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The distinction between noise and radiometric error is somewhat arbitrary. It is assumed that noise is the spectral-element-to-spectral-element uncorrelated intensity variations, which statistically average out with time. Radiometric errors stem from system imperfections and stay in the calibrated spectra even after all visible noise has been washed out.
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Scene Spectral Radiance
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Measured Radiometric Error Actual
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Spectral Power at Detector
FIGURE 10.16 Actual and measured object view spectral radiance versus spectral power at the detector.
Scene Spectral Radiance
Biased
Actual
Spectral Power at Detector
FIGURE 10.17 calibration.
The effect of calibration source errors on radiometric
errors is illustrated in Fig. 10.17. The calibration points on the graph are misplaced and the resulting calibration line is thus skewed. There are two types of calibration source errors. The first is a misevaluation of the radiance supplied by the calibration source. This is known as blackbody error. Calibration source emissivity error and temperature error are of this type. Emissivity is a factor contributing to blackbody errors. To analyze the effect of underestimating or overestimating blackbody emissivity, it is interesting to examine Eqs. (10.21) and (10.22). When everything else is constant, the calculated radiometric gain is inversely proportional to the emissivity used for the blackbody. The calculated radiometric offset is, for its part, directly proportional to the emissivity used for the blackbody. These dependencies translate into a direct proportionality of a calibrated spectrum [Eq. (10.23)] on the emissivity used for the blackbody. In other words, the use of 1 percent-inflated emissivity values leads to 1 percent-overestimated
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0.07 0.06 Radiometric accuracy 0.05 0.04 0.03 0.02 0.01 0 375 575 775 975 1175 1375 1575 1775 2 mm 3.5 mm 5 mm
Temperature of the scene (K)
FIGURE 10.18 Relative radiometric errors at 5000 cm 1, 2860 cm 1, and 2000 cm 1 (2 m, 3.5 m, and 5 m) due to the uncertainty of the calibration source temperature, assuming a blackbody relative temperature accuracy of 0.2 percent and an absolute accuracy of 1 .
calibrated spectra, and thus 1 percent radiometric errors. Good emissivity for available blackbodies in the 2000 to 2860 cm 1 (3.5 m to 5 m) spectral region is typically 0.98 0.01. This is 1 percent emissivity accuracy. The error contribution due to temperature uncertainty is also important. Estimating the error in this case is more difficult, however, since it involves three Planck functions [Eq. (10.20)], one evaluated at TH (the temperature of the hot blackbody), one evaluated at TC(the temperature of the cold blackbody), and one evaluated at TS (the temperature of the object view). For a particular choice of TS, it is possible to find a certain combination of TH and TC to minimize the error. In general, however, we can say that the relative radiometric error due to calibration source temperature uncertainty will always be less than the values displayed in Fig. 10.18, as long as TH > TS > TC. In other words, the values displayed in Fig. 10.18 are upper-limit relative radiometric errors due to calibration source temperature uncertainty. Here, we have assumed a relative blackbody temperature accuracy of 0.2 percent and an absolute accuracy of 1 . The maximum error is 5 percent for the coldest source (TS = 373 K) at the highest frequency ( = 2860 cm 1). This is a very pessimistic value.
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