**5. Discussion**

Errors associated with in-situ measurements, data processing, and calibration method selection all contribute to uncertainty in satellite calibration coefficients [20]. Given that calibration uncertainty sources, such as ground reflectance measurements, inherent code accuracy, etc. have been thoroughly discussed in the literature, we focused our analysis on uncertainties caused by aerosol type assumptions, AOD measurements, water vapor content measurements, atmospheric profile measurements, and satellite image misregistration. Furthermore, the wavelength shift occurring in hyperspectral data would also impose an additional influence on the radiometric calibration accuracy, especially near the atmospheric absorption wavelengths, due to gases like oxygen, water vapor, carbon dioxide, etc.

#### *5.1. Uncertainty Due to Aerosol Type Assumptions*

The aerosol type used in the Radiative Transfer Model (RTM) introduces grea<sup>t</sup> uncertainty in vicarious calibrations, especially in situations in which the AOD is large. It was impractical to measure the vertical distribution of aerosol characteristics during the calibration campaign. However, the actual aerosol type in Dunhuang, which is in arid northwestern China, is generally close to the RURAL and DESERT types described in MODTRAN. To evaluate uncertainty due to aerosol type, three additional aerosol types similar and dissimilar to local conditions (i.e., urban, desert, and maritime) were chosen to replace the rural aerosol type used in the original calculations. The radiance was computed again using these three aerosol types, and the results were compared to those derived using the rural aerosol type. The angstrom exponent coefficients derived from CE318 measurements were also used as inputs for MODTRAN and were 0.75 and 0.3519 for the SPARK-01 and -02 satellites, respectively. The resulting differences in radiance can be used to evaluate uncertainty due to aerosol type (SPARK-01: Figure 23; SPARK-02: Figure 24). The average relative differences are listed in Table 4.

**Figure 23.** Relative differences in radiance using different aerosol types for the SPARK-01 calibration for 7 March 2017 using (**a**) the reflectance-based method and (**b**) the irradiance-based method.

**Table 4.** Average relative differences in radiance for the SPARK-01 and -02 satellites.


**Figure 24.** *Cont.*

**Figure 24.** Relative differences in radiance using different aerosol types for the SPARK-02 calibration for 28 February 2017 using (**a**) the reflectance-based method and (**b**) the irradiance-based method.

Both the irradiance-based and improved irradiance-based methods use the diffuse-to-global ratio to minimize the uncertainty associated with aerosol-type assumptions. The average maximum uncertainty of the irradiance-based method used for SPARK-01 on 7 March 2017 is 2%, or less than half of that calculated for the reflectance-based method. The uncertainty in the reflectance-based method due to the aerosol-type assumption increased to 14% for the SPARK-02 satellite on 28 February 2017 because of the relatively large AOD at 550 nm (0.35). The improved irradiance-based method, despite replacing only the downward transmittance features, considerably decreased uncertainty, i.e., by 9%. In reality, the Dunhuang calibration site is surrounded by the Gobi Desert and, thus, the local aerosol is likely fall into the rural or desert types. However, the uncertainty due to aerosol type was conservatively estimated by using the setting of half the difference between the predicted TOA radiance with the urban aerosol type and that with the rural aerosol type.

#### *5.2. Uncertainty Due to AOD Measurements*

AOD is retrieved from CE318 measurements with a total uncertainty of ~0.01–0.021, which is spectrally dependent and features higher errors in the UV bands [36]; this uncertainty is validated using CE318 and Microtops II measurements in Section 2.4. Therefore, an uncertainty of ±0.02 was added to the 550 nm AOD used for the SPARK-01 and -02 reflectance-based calibrations. For the reflectance-based method, the uncertainty was estimated by comparing the predicted TOA radiance using different AOD values in MODTRAN 5. For the irradiance-based methods (Equations (6) and (9)), the uncertainty in the predicted TOA radiance can be attributed to both errors in the directly measured transmittance *e*<sup>−</sup>*<sup>τ</sup>*/*μ<sup>s</sup>* and errors in the retrieved CE318 measurements. In reality, the transmittance values consist of the CE318 direct retrievals divided by the calibration coefficient for each channel. Thus, the retrieved transmittance uncertainty is a combination of calibration uncertainty from the CE318 calibration coefficient and uncertainty due to the process of interpolating measured transmittance in a few bands into the SPARK satellite bands. The former (calibration) uncertainty is estimated to be ~0.01–0.02 (higher in the UV bands) [36], while the latter is 0.5% of the transmittance [37]. It is reasonable to set a relative uncertainty of 0.015 for the measured transmittance in the solar direction because the SPARK satellite spectral range spans from the visible to the near-infrared bands, without UV bands. The transmittance uncertainty in the view direction can be inferred from that in the solar direction by applying the cosine of the view zenith angle. Also, the uncertainty in the retrieved AOD would cause the path radiance and sphere albedo to change with different signs [2]. In order to simplify the calculation, the downward and upward direct transmittances, *e*<sup>−</sup>*<sup>τ</sup>*/*μ<sup>s</sup>* and *e*<sup>−</sup>*<sup>τ</sup>*/*μ<sup>v</sup>* , were replaced with *Tdir*(*θs*) and *Tdir*(*θv*). The transmittance and AOD uncertainties were assumed to be independent; thus, the error propagation equations for the TOA reflectance uncertainties using the irradiance-based (Equation (6)) and improved irradiance-based (Equation (9)) methods can be written as:

$$\begin{array}{c} \Delta \rho^\* = [\left(\Delta \rho\_a - \frac{T\_{\text{dir}}(\theta\_s)}{1 - a\_s} \times \frac{T\_{\text{dir}}(\theta\_v)}{1 - a\_v} \times \rho^2 \times \Lambda s \right)^2 + \\\ [\left(\frac{1}{1 - a\_s} \times \rho \times (1 - \rho \times s) \times \frac{T\_{\text{dir}}(\theta\_v)}{1 - a\_v} \times \left(1 + \frac{\mu\_s}{\mu\_v} \right)]^2 \times \left(\left(\Delta T\_{\text{dir}}(\theta\_s) \right)^2 + \left(0.005 T\_{\text{dir}}(\theta\_s) \right)^2\right)]^{1/2} \end{array} \tag{10}$$

$$\begin{array}{c} \Delta \rho^\* = [\left(\Delta \rho\_d + \frac{\rho \times T\_{dir}(\theta\_s)}{1 - a\_s} \cdot \Delta T(\theta\_v)\right)^2 + \\ \left(\frac{\rho}{1 - a\_s} \times T(\theta\_v)\right)^2 \times \left(\left(\Delta T\_{dir}(\theta\_s)\right)^2 + \left(0.005 T\_{dir}(\theta\_s)\right)^2\right)]^{1/2} \end{array} \tag{11}$$

The uncertainty estimated for SPARK-01 and -02 using the reflectance- and irradiance-based methods is shown in Figure 25. For the reflectance-based method, an AOD uncertainty of 0.02 contributes little (maximum values of 0.6% and 0.7%, respectively, for SPARK-01 and -02) to the total TOA radiance prediction uncertainty. However, the uncertainties for the irradiance- and improved irradiance-based methods appear higher than that for the reflectance-based method. The average and maximum uncertainties are 2.17% and 2.60%, respectively, for SPARK-01 and 1.20% and 1.45% for SPARK-02. The higher uncertainties for the irradiance- and improved irradiance-based methods may be attributed primarily to the direct transmittance uncertainty, which would be partially decreased by the diffuse transmittance uncertainty calculated by MODTRAN, although with the opposite sign.

**Figure 25.** Calibration uncertainties caused by the AOD measurements for SPARK-01 and -02 using the (**a**) reflectance- and (**b**) irradiance- based (or improved irradiance-based) methods, respectively.

#### *5.3. Uncertainty Due to Water Vapor Measurements*

The water vapor content retrieval from the CE318 measurements is expected to have an uncertainty of 10%. Therefore, an uncertainty of ±10% was added to the water vapor content retrievals during the SPARK satellite calibration site overpass. The TOA radiance was computed again with MODTRAN, and the difference represents the calibration uncertainty caused by the water vapor measurement (Figure 26). Large uncertainties are apparent in the water vapor absorption bands near 720, 820, and 940 nm. The uncertainties for the water vapor non-absorption bands are lower than 0.2% and, thus, can be omitted. The reflectance- and irradiance-based methods show similar results (Figure 26). The highest values occur in the 940 nm band, amounting to 4.45% and 4.39% for the reflectance- and irradiance-based (or improved irradiance) methods, respectively, in SPARK-01, and 4.17% and 4.04% in SPARK-02. Due to the low water vapor content in arid areas like Dunhuang, the uncertainty caused by the water vapor measurement is relatively small.

**Figure 26.** Calibration uncertainties caused by the water vapor content measurements for SPARK-01 and -02 using the (**a**) reflectance- and (**b**) irradiance- based (or improved irradiance-based) methods, respectively.

#### *5.4. Uncertainty Due to Atmospheric Profile Measurements*

The vertical distributions of temperature, humidity, pressure, and other atmospheric constituents also influence the TOA radiance prediction. In order to explore uncertainty due to the atmospheric profile, the measured radiosonde data used in MODTRAN 5 were replaced with three atmospheric models (i.e., the Mid-Latitude Summer, MS; Mid-Latitude Winter, MW; and 1976 US Standard Atmosphere, US models). The differences in TOA radiance predicted by the three additional atmospheric models and those by the measured radiosonde data represent the uncertainty due to atmospheric profile measurement, as shown in Figure 27. The irradiance- and improved irradiance-based methods show slightly higher uncertainties due to the atmospheric profile than does the reflectance-based method; however, their uncertainties are less than 1.3% in all bands apart from the water vapor absorption bands near 940 nm and 1135 nm. In addition, the MW model appears to be more similar to the radiosonde measurements, as evidenced by the relatively small difference in the radiances predicted using these two inputs. The MS model is likely to represent the actual conditions, considering the location and season of the calibration experiment. Therefore, the maximum differences, which were derived from replacing the radiosonde measurements with US and MW models, were applied in the calibration uncertainty calculations.

**Figure 27.** Calibration uncertainties caused by the vertical atmospheric profile measurements for SPARK-01 and -02 using the (**a**) reflectance- and (**b**) irradiance-based methods, respectively. "US", "MS", and "MW" refer to the relative difference in predicted radiance derived from replacing radiosonde measurements with these three atmospheric models.

#### *5.5. Uncertainty Due to Image Misregistration Errors*

To locate the calibration site, the OLI image acquired on 28 February 2017 was used to geo-rectify the SPARK satellite images. The first-order polynomial method was applied with nearest-neighbor resampling around the calibration site to retain the raw DN acquired by the sensors. A one-pixel misregistration around the calibration site is reasonable between the SPARK and OLI images. In addition, the border of the calibration site can be seen in the OLI image due to its 30 m spatial resolution and high radiometric resolution. Therefore, a misregistration of up to two pixels was assumed in the computation of the average DNs at the calibration site. The average DNs and minimum and maximum average DNs determined by shifting the 6 × 6 pixel area by up to two pixels in all directions are shown in Figure 28. The difference between the averaged DNs and the shifted averaged DNs reflect the uncertainty due to image misregistration errors. The differences caused by misregistration are <1.5% in the SPARK-01 495–955 nm spectral range and the SPARK-02 459–995 nm range. The differences are large at the ends of the spectral range due to high noise; data in these ranges are not generally used.

**Figure 28.** Averaged DNs from 6 × 6 pixel areas in the calibration site and DNs calculated by shifting the 6 × 6 pixel area by up to two pixels in all directions for SPARK-01 and -02.

#### *5.6. Uncertainty Due to Spectral Wavelength Shift*

Although the central wavelength values were measured for all the 2048 pixels of SPARK-01 and -02 in the laboratory before launch, the wavelength shift may affect the radiometric calibration result in the band near atmospheric gas absorption wavelength. The spectral shifts of Hyperion were estimated to be 0.38–1.39 nm at the 760 nm oxygen band by a spectral fitting algorithm compared with the laboratory spectral calibration [8]. The same method was also applied to TG-1 hyperspectral imager and the spectral shifts were 2–3 nm, with an uncertainty of 0.3 nm [38]. Thus, the spectral fitting algorithm was also used to estimate the spectral shifts for the SPARK satellite. The measured radiance spectrum over the desert was compared with a MODTARN 5-modeled radiance spectrum using the SPARK spectral calibration parameters to derive the spectral shift value. Figure 29a,c show the comparison near the 760 nm oxygen band in the desert (Figure 5) for SPARK-01 and -02, respectively. The MODTRAN 5 spectrum was normalized to match the SPARK-measured radiance level and the spectral wavelength was shifted in 0.1 nm increments. The optimal shifts were estimated to be −0.1 nm for both SPARK-01 and -02. Such a minor spectral shift indicated that SPARK satellites do not undergo an evident spectral shift. Figure 29b,d show the slightly minimized radiance difference after applying a −0.1 nm shift in the SPARK pre-launch laboratory spectral calibration position.

**Figure 29.** *Cont.*

**Figure 29.** Spectral fit result and optimal spectral calibration result with an –0.1 nm shift at the desert in Dunhuang (Figure 5). (**<sup>a</sup>**,**<sup>c</sup>**) are for SPARK-01 before and after spectral shifting; and (**b**,**d**) are for SPARK-02 before and after spectral shifting.

The spectral shift of −0.1 nm was applied to SPARK-01 and -02 to calculate its contribution to radiometric calibration accuracy (Figure 30). As the SPARK satellite uses a prism, the absolute spectral shift for each band will be linear to its FMHW, expressed as:

$$
\Delta\lambda(i) = \frac{FWHM(i)}{FWHM(i|\_{\lambda=760\text{ nm}})} \times \Delta \tag{12}
$$

Considering the additional errors caused by the spectral fitting algorithm itself and laboratory calibration, a ±1 nm spectral shift at the 760 nm band was assumed to further estimate the influence on the radiometric calibration of SPARK satellites. The wavelength position at the 760 nm band was shifted by ±1 nm and the radiance difference was calculated for SPARK-01 and -02. As expected, the

uncertainty is evident near the atmospheric gas absorption wavelengths, e.g., Fraunhofer 430 nm and 685 nm, the 760 nm and 690 nm oxygen bands, and the 720 nm, 820 nm, and 940 nm water vapor bands. The maximum value occurred for the 760 nm oxygen band. The uncertainties are less than 2% in the oxygen bands, and less than 1% in the water vapor bands if a spectral shift of −0.1 nm was ignored during SPARK satellite calibration. However, if the spectral shift was increased to 1 nm, the uncertainties would increase considerably in these atmospheric gas absorption bands (e.g., 8% by the spectral shift of +1 nm for SPARK-01 and >10% by the spectral shift of −1 nm for SPARK-02 (Figure 31). In addition, the uncertainty in the 940 nm band for SPARK-02 is higher than that of SPARK-01 due to the larger water vapor content occurring in the daytime for SPARK-02 radiometric calibration (0.35 g/cm<sup>2</sup> for SPARK-01 versus 0.54 g/cm<sup>2</sup> for SPARK-02).

**Figure 30.** Calibration uncertainties caused by spectral shift of −0.1 nm for SPARK-01 and -02.

**Figure 31.** Calibration uncertainties caused by spectral shift of ±1 nm for SPARK-01 (**a**) and -02 (**b**).

#### *5.7. Total Calibration Uncertainty Estimation*

Calibration uncertainties caused by other sources are relatively constant and have been discussed previously [20,39,40]. The uncertainty due to ground reflectance measurements has been estimated at 2% in field experiments, and this estimation was validated measurements taken on different days during the calibration experiment. The uncertainty in measurement of diffuse-to-global irradiance ratio contributes 2.0% to the total calibration uncertainty [40]. The uncertainty due to ozone measurements with an error of 20% was estimated to be 1.3% [39]. Thus, because the ozone acquired from OMI has the uncertainty of 4%, it is reasonable to set this uncertainty to 0.6% [41]. Although the accuracy of MODTRAN 5 is much improved and comparable to that of the benchmark Line-by-Line Radiative Transfer Model (LBLRTM) [42], this uncertainty is conservatively estimated to be 1%. The uncertainty due to non-Lambertian ground characteristics was estimated at 2% for the Dunhuang calibration site [16]. In total, the overall vicarious calibration uncertainty contains uncertainties and errors caused by atmospheric characterization, surface characterization, radiative transfer calculations, and site-average DN calculations [43]. The uncertainties discussed above associated with the reflectance-based method are summarized in Table 5 for the SPARK-01 and -02 satellite calibrations; those associated with the irradiance-based method (used for SPARK-01) and the improved irradiance-based method (used for SPARK-02) are summarized in Table 6. The uncertainties associated with different methods are shown for each spectral band of both satellites in Figure 32. Total uncertainty statistics are listed for different spectral ranges in Table 7. For SPARK-01, uncertainties of 4.71 ± 0.34% and 4.11 ± 0.21% were estimated using the reflectance- and irradiance-based methods, respectively. For SPARK-02, uncertainties of 8.12 ± 0.29% and 5.86 ± 0.29% were estimated at >456 nm using the reflectance- and improved irradiance-based methods, respectively. The uncertainty is greatly increased in other spectral ranges due to high image noise. As expected, the uncertainties in both the irradiance- and improved irradiance-based methods are lower than that in the reflectance-based method, especially when the aerosol optical depth is large (e.g., in the SPARK-02 results). However, the irradiance and improved irradiance-based methods depend greatly on the accuracy of the direct transmittance measurements (in Section 5.2) and the diffuse-to-global irradiance ratios; it is therefore important to improve the accuracy of these measurements.

**Figure 32.** Total calibration uncertainty estimated for SPARK-01 and -02 using (**a**) the reflectance-based method and (**b**) the irradiance-based method.


**Table 5.** Estimated uncertainty associated with the reflectance-based method.

**Table 6.** Estimated uncertainty associated with the irradiance-based method used for SPARK-01 and the improved irradiance-based method used for SPARK-02.


**Table 7.** Average relative differences for various wavelength ranges.


#### *5.8. Spectral Smile Effect Correction*

The smile effect is a common phenomenon in the pushbroom sensor. It is mainly caused by optical aberrations and misalignments and cannot be completely avoided. Figure 33 shows relative differences of the central wavelength positions of all the 2048 pixels for SPARK-01 and -02 satellites, compared with the average central wavelength, expressed as:

$$
\Delta\eta(i,j) = \frac{\mathbb{C}WV(i,j) - \overline{\mathbb{C}WV}(j)}{\overline{\mathbb{C}W}HM(j)}\tag{13}
$$

where Δ*η* is the relative difference of the central spectral wavelength, *CWV* is the central spectral wavelength for each pixel, *i* and *j* denote the cross-track position and band index, respectively, and *CWV* is the average central spectral wavelength.

The spectral smile is more pronounced in SPARK-02 than in SPARK-01, with the maximum difference even exceeding half of the FWHM, which is nearly of the same magnitude as that in the Hyperion data [8]. The de-smiling technique is always applied to interpolate the raw data from individual spectral positions into the commonly defined spectral central wavelengths. The de-smiling processing would not affect most spectral bands to a grea<sup>t</sup> degree, but it may introduce artificial features near the atmospheric gas absorption wavelengths when deriving ground surface reflectance through atmospheric correction. Thus, it is strongly recommended that the spectral polishing technique be applied after atmospheric correction to remove the spectral artificial features near the gas absorption wavelengths.

**Figure 33.** Central wavelength positions measured pre-launch in laboratory for SPARK-01 and -02.
