1. A path radiance estimation algorithm using reflectance measurements in radiometric control areas
Ke-Sheng Cheng
Dept. of Bioenvironmental Systems Engineering
National Taiwan University
9/12/2018

2. Atmospheric conditions (e. g
Atmospheric conditions (e.g. size distribution and concentration of these atmospheric particles) vary over time and, as a result, for remote-sensing applications which require usage of multi-temporal images, the atmospheric effect varies from one image to another.
The dynamic Earth surface processes cannot be truly monitored unless the temporally varying atmospheric effect has been removed.
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3. One major source of atmospheric interference is the upwelled radiance (or the path radiance) which represents the amount of solar radiation scattered by the atmospheric molecules and particles directly towards the sensor.
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4. At-sensor Radiance Modelling
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5. For cases which do not cover extensively wide study areas, the downwelled irradiance, path radiance, and atmospheric transmittances can all be assumed to be spatially homogeneous. Thus, the at-sensor radiance in equation (1) reduces to
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6. Within a study area, the at-sensor radiance in equation (2) is only dependent on the topographical conditions (adjacent obstruction and terrain slope and aspect), which are characterized by σ and F, of individual ground samples.
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7. Methods of Path Radiance Estimation
Dark Object Subtraction (DOS)
Covariance Matrix Method (CMM)
Multi-band Regression Technique (MBR)
Radiometric Control Area Method (RCA)
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8. DOS Method
The basic concept of DOS is to identify very dark features (or features with near-zero reflectance) within the scene. Large and clear water bodies have values of near-infrared digital number at or very near zero because clear water absorbs strongly in the near-infrared spectral wavelength range and the atmospheric scattering effect is relatively insignificant.
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9. From equation (2), radiance received at the sensor consists of only path radiance if the target has zero reflectance. Thus, the minimum radiance from selected dark objects within the scene represents the path radiance.
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10. The DOS method often breaks down when applied to the visible spectral range where minimum reflectance within the scene may still be a few per cent (Schott 1997).
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11. Although no primary solar irradiance reaches the target, downwelled solar radiance from a portion of the sky dome can still reach and be reflected from the target area.
As a result, the minimum radiance from the target area is composed of path radiance and reflected downwelled radiance.
The minimum radiance within the scene thus overestimates the path radiance if the selected dark objects are not near-zero reflectors.
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12. Multi-band Regression Technique
To overcome the limitation of the DOS method, Schott (1997) suggested an MBR technique (also known as the channel correlation method) for visible-band path radiance estimation.
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13. Assume that path radiance of a specific band, say band 1, can be estimated with reasonable accuracy (e.g. near-infrared band using clear water bodies). Let us further assume that reflectance of surface objects in band 1 and another band (band 2) are approximately correlated with zero bias, that is,
r2 = C*r1 + e, (6)
where r1 and r2, respectively, are reflectances in bands 1 and 2, C is the regression coefficient and e is the regression error.
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14. Assuming the path radiance, downwelled radiance, and atmospheric transmittance are all homogeneous over the study area, the at-sensor radiances of band 1 (λ1) and band 2 (λ2) from a horizontal and unobstructed Lambertian surface can thus be, respectively, expressed as
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15. Thus, by regressing the band 2 at-sensor radiance against the adjusted radiance in band 1, the path radiance of band 2 (Lpλ2 ) can be represented as the intercept of such a regression relationship.
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16. Covariance Matrix Method
See details in the paper.
Application of the MBR technique requires finding areas of uniform topographic relief (e.g. areas of horizontal surface) and homogeneous land cover condition for which the multi-band surface reflectances satisfy equation (6). In contrast, the CMM can be applied to local areas that have homogeneous land cover conditions, yet sufficient variation in topographic relief.
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17. RCA-based Method
For local remote-sensing applications, spatial variation of the at-sensor radiance in equation (2) is due to cos σ and F, which represent the topographic characteristics of individual ground samples. Dependence of K0 of equation (2) on the wavelength λ, which can be considered as the central wavelength of a spectral band, indicates that path radiance is band-specific.
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18. If only ground samples of horizontal and unobstructed surfaces are considered, cos σ and shape factor (F = 1) are constant and the at-sensor radiance associated with such ground samples can be expressed as
Lsλ = K0(λ) + rdλK∗(λ),
where K∗(λ) accounts for downwelled radiance and primary radiance. The path radiance K0(λ) can thus be estimated by choosing a few RCAs of the same land cover types within the scene.
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19. Suppose that two RCAs (RCA-1 and RCA-2) of the same land cover types within the study area are chosen. An RCA is an area with spatially homogeneous and temporally stationary land surface conditions. Surface reflectance of ground samples in RCAs can be obtained by taking in situ measurements using a spectroradiometer.
Measurements of surface reflectance should then be integrated over the wavelength ranges of individual spectral bands of space-borne or airborne sensors.
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22. Study area and measurement of surface reflectance
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24. A multispectral radiometer (MSR) was used to measure the surface reflectance of ground control areas. Aided by a pre-calibrated standard reference disk, the MSR is capable of measuring average surface reflectance within a spectral window of varying size (from 1.5 to 2 nm) continuously from 350 to 1050 nm.
The MSR was moved around within each RCA, achieving a few reflectance measurements within individual Formosat-2 ground samples.
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25. RCA reflectance calibration
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26. Figure 4. Illustration of multispectral radiometer (MSR) sampling locations (marked by circles) within each radiometric control area (RCA).
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27. RCA-1 Reflectance Curves
Newly paved area
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28. RCA-2 Reflectance Curves
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29. RCA-3 Reflectance Curves
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30. In order to investigate the general validity of the assumption of proportional relationship between reflectances of different spectral bands (i.e. equation (6)), in situ measurements of surface reflectance were also taken within a grass-covered area (see figure 5(a)).
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34. From each spectral reflectance curve established using MSR measurements in the three RCAs, average reflectance over the wavelength range of each Formosat-2 band was calculated. Then band-specific ground sample-average reflectances, hereafter referred to as the pixel reflectances, were calculated using band-average reflectances from measurement locations that fall within the spatial coverage of individual ground samples.
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35. Correlation between surface reflectances of different spectral bands
An essential assumption of the MBR technique is the validity of equation (6), which requires a linear and unbiased (zero intercept) correlation between surface reflectances of different spectral bands. In other words, theMBR technique can only be applied to areas of certain land cover types whose multi-band reflectances satisfy equation (6).
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37. Asphalt-paved surface has a very good linear and unbiased relationship between reflectance of the near-infrared band and reflectances of other bands, although the relationship degraded slightly between reflectances of the near-infrared and blue bands.
By contrast, correlation relationships between reflectance of different spectral bands are generally poor for grass-covered surface, probably due to less homogeneous grass cover and growth condition over its spatial extent.
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38. Estimates of path radiance
Since the effect of Rayleigh scattering is inversely proportional to the fourth power of the wavelength of incident solar radiation, the effect of atmospheric scattering reduces dramatically with increase of wavelength. Thus, if the sky is not quite hazy, path radiance in the near-infrared band is usually very low and can be ignored or assumed to be zero.
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39. For example, Switzer et al
For example, Switzer et al. (1981) assumed zero path radiance for the near-infrared band and applied the CMM to path radiance estimation of other visible bands.
Both the MBR technique and the CMM require knowing the path radiance of one spectral band a priori.
In this study, path radiance of the near-infrared band was assumed to be zero, and path radiances of the blue, green and red bands were estimated using different methods.
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40. Results of path radiance estimation using different methods.
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41. Locations of pixels of minimum radiance
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42. Due to its nature of overestimation when applied to areas of higher-than-zero reflectance, path radiance estimates of the DOS method can serve as a criterion for investigating the physical correctness of other estimation methods.
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43. For the MBR technique, we assumed zero path radiance for the near-infrared band and established regression relationships between radiance of the near-infrared band and radiance of other visible bands (blue, green and red) using pixels within the three RCAs. Intercepts of individual regression relationships represent estimates of path radiance of the corresponding spectral bands.
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45. Estimation of band-specific path radiances using the radiometric control area (RCA) based method. Intercepts of the regression lines represent estimates of path radiance.
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46. Evaluation on performance of different methods
Qualitative evaluation
The path radiance reduces with increasing wavelength.
The DOS method overestimated band-specific path radiances. Thus, we compare the relative magnitude of path radiance estimates by the MBR technique, CMM and RCA-based method, in comparison with estimates of the DOS method.
Examining the ratio of radiances of different spectral bands, after path radiance was removed.
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47. Line A–A’ in figure 3 consists of 80 pixels of homogeneous forest cover. It traverses across the mountain ridge, with half of the pixels facing northwest and others facing southeast. The ratio of adjusted radiances (i.e. Lsλ – Epλ) of the green band and red band (G/R) was calculated for all pixels along the traverse line using path radiance estimated by each of the four methods.
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48. Raw band-ratio values of the northwest-facing and southeast-facing pixels are significantly different due to the effect of opposite slope aspect. If the path radiance is properly removed, one can expect similar values of band ratio across the traverse line because the effect of slope aspect is subdued.
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49. Band ratio (G/R) values across traverse A-A’
Band ratio (G/R) values across traverse A-A’. Raw: band ratio without correction of path radiance.
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50. The RCA-based method has the lowest and very small standard deviation of band ratios, indicating that the slope aspect effect has largely diminished.
The DOS method may result in abnormally high values of G/R band ratio due to overestimation of path radiance. Similar observation has also been reported by Switzer et al. (1981).
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51. Evaluation on performance of different methods
Quantitative evaluation
a more sophisticated quantitative evaluation was implemented using measurements of aerosol and molecule (Rayleigh) optical depths available from the AERONET and MODIS archive.
See details in the paper.
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