Error Propagation From Aerosol Retrieval to Atmospheric Correction Due to Adjacency Effect for FORMOSAT-2 RSI Band

1 Department of Information Management/Geographic Information Management and Research Center, Transworld Institute of Technology, Yunlin, Taiwan, ROC * Corresponding author address: Prof. Chien-Hui Liu, Department of Information Management/Geographic Information Management and Research Center, Transworld Institute of Technology, Yunlin, Taiwan, ROC; E-mail: chliu@tit.edu.tw doi: 10.3319/TAO.2007.18.4.827(AA) Homogeneous surface is usually assumed in aerosol retrieval for the dark target method. Retrieved aerosol optical depth can thus be overestimated. This error will also be propagated into retrieved surface reflectance during atmospheric correction, especially for high spatial resolution of 8 m for FORMOSAT-2 (also known as ROCSAT-2) RSI. The error in retrieved surface reflectance during atmospheric corrections induced by retrieved aerosol optical depth error due to adjacency effect over a dark target is studied in the green and red bands of FORMOSAT-2 RSI. The results show that significant errors in retrieved surface reflectance, i.e., larger than 0.01, may occur in most cases except at low contrast for a bright target. Such an error for a dark target is larger than that for a bright target for given contrast and haziness. Relatively, the error of retrieved surface reflectance is enhanced by the introduced error of aerosol optical depth in atmospheric correction for dark targets. The error can be larger than 35% for dark targets and less than 10% for bright targets. Hence, it is suggested that adjacency effect be considered from aerosol optical depth retrieval to surface reflectance retrieval in atmospheric correction except at low contrast for bright targets. (


INTRODUCTION
FORMOSAT-2 (also known as ROCSAT-2) developed by the National SPace Organization (NSPO) of Taiwan was successfully launched on 21 May 2004 (http://www.nspo.gov.tw).Remote sensing instrument (RSI) on board can provide images of high spatial resolution of 2 m in panchromatic band and 8 m in four multispectral bands (Lee et al. 2002) with central wavelengths of 0.484, 0.561, 0.661, and 0.821 µm , respectively (Liu 2005).Its mission is to image Taiwan and the surrounding areas on a daily basis for various applications, such as environmental monitoring, natural resources research, disaster prevention, and rescue work.Since scattering and absorption in the atmosphere can distort the received signal from the satellite, atmospheric correction (AC) is necessary to convert the top-of-atmosphere (TOA) radiance to surface reflectance for analysis of remotely sensed data quantitatively (Kaufman 1989;Liang et al. 2001;Liang et al. 2002) such as leaf area index retrieval (Butson and Fernandes 2004), forest mapping (Pax-Lenney et al. 2001), crop detection (Sakamoto et al. 2005), and water quality monitoring (Pozdnyakov et al. 2005).
To correct the atmospheric effect, the optical characteristics of the atmosphere should be first estimated.Molecular scattering and ozone absorption can be considered to have an invariant effect and be accounted for satisfactorily.The absorption of water vapor can be significant for RSI's near-infrared band.Its accuracy can be improved if radiosonde data are available (Liu et al. 1996); however, climatology data or other satellite output can be applied in practice (Liang et al. 2001).Owing to the nature of temporal and spatial variations, aerosol effect is the most difficult part of AC to correct.Aerosol characteristics, such as aerosol optical depth (AOD), size distribution and single scattering albedo, can be observed and determined from sunphotometer (Holben et al. 2001;Dubovik et al. 2002).Suitable aerosol models can also be chosen by geo-location (Kaufman et al. 1997).The image-based method seems to be the only way to retrieve AOD for operational AC.It has also been developed in numerous previous studies (Kaufman and Sendra 1988;Liang et al. 1997;Ouaidrari and Vermote 1999;Liang et al. 2001;Vermote et al. 2002;Lyapustin et al. 2004).After these parameters are known, retrieval of surface reflectance can be very straightforward (Kaufman 1989).
Currently, the most promising operational algorithm for global and regional AOD retrieval over land is the dark target (DT) method (Kaufman and Sendra 1988).This method has been modified to take advantage of the low opacity of most aerosol types in the mid-IR bands, and the high correlation of surface reflectance between visible and mid-IR bands for the EOS moderate resolution imaging spectroradiometer (MODIS) (Kaufman et al. 1997) and Landsat TM/ETM+ (Liang et al. 1997;Ouaidrari and Vermote 1999).The DT can be identified as pixels with low near-IR signal and high vegetation index (Kaufman and Sendra 1988) for sensors without mid-IR bands, such as SPOT HRV and FORMOSAT-2 RSI.In this case, reasonable reflectance of DT can then be assumed in the visible bands.This is because the TOA signal contains larger atmospheric reflectance and surface reflectances are very low in the visible bands for DTs, the errors of retrieved AOD due to the errors of the assumed reflectances over DTs are expected to be much lower than those over bright targets.DTs are mainly dense vegetation and dark soils (Kaufman et al. 1997).The main sources of errors for the DT method are surface inhomogeneity and sub-pixel water contamination (Chu et al. 2002;Levy et al. 2005).
Classical 1-D radiative transfer (RT) theory, which assumes the surface to be infinite and uniform, is usually used in aerosol retrieval for the DT method.In fact, the interaction of atmospheric scattering and heterogeneous surface can systematically brighten the DT and darken the bright target (BT) at the TOA level (Lyapustin 2001;Lyapustin and Kaufman 2001;Lyapustin et al. 2004).This is caused by the so-called adjacency effect.Thus retrieved AOD from DT will be overestimated.The error is significant even at 1 km resolution (Lyapustin and Kaufman 2001).It is reported that the error of retrieved AOD at 30-m resolution for the red wavelength can exceed 100% over a medium surface at nadir view in clear sky.For FORMOSAT-2 RSI with 8-m spatial resolution, the errors of retrieved AOD can be about 100% and 97% for the blue and red bands, respectively, over a medium surface at a viewing zenith angle of 45° in clear sky (Liu 2005).As can be shown later in this study, error increases when the viewing angle moves to nadir.It is also shown that this error is larger for urban models than that for continental models.This AOD error can further introduce error into surface retrieval in AC.These errors are increased as spatial resolution increases.Therefore, it is suggested that adjacency effect (AE) be taken into account not only in the AC algorithms but also in aerosol retrieval over land even at a sensor resolution of 1 km (Lyapustin and Kaufman 2001).In spite of 1-D RT theory assumed by operational algorithms (Kaufman et al. 1997;Liang et al. 1997;Ouaidrari and Vermote 1999), a new DT method of aerosol retrieval based on 3-D RT theory for the Landsat enhanced thematic mapper plus (ETM+) has been developed (Lyapustin et al. 2004) recently.It simultaneously retrieves the aerosol model and AOD with excellent accuracy ( ±0.02 -0.03) in the Washington-Baltimore area.Surface climatology considering seasonal and geographic variations of regression coefficients between visible and mid-IR reflectances is proposed in this method.It may be helpful for aerosol retrieval from RSI data, since the spectral characteristics of RSI are very similar to those of ETM+ (Liu 2005), even though there is a lack of mid-IR band.
Based on the previous study (Liu 2005), error analysis of retrieved surface reflectance caused by the error of retrieved AOD due to AE for FORMOSAT-2 RSI with its high spatial resolution of 8 m is presented in this paper.The reasons why these errors occur are also explained with their physical mechanisms.For comparison purposes with the results of Lyapustin and Kaufman (2001), some parameters are set to the same.Because of the lower sensitivity of TOA reflectance to AOD for the more absorptive urban aerosol model and the usual absence of feasible AOD to fit ρ TOA for medium contrast (Liu 2005), only the continental aerosol model is discussed in this study.The analysis is performed in the green and red bands.The error with viewing zenith angle is also discussed.

METHODOLOGY
A simple AC model (SACM) (Liu 2003), previously used to estimate the error in AOD due to AE over a DT (Liu 2005), is applied to determine the error of retrieved surface reflectance ρ c in AC for ROCSAT-2 RSI.The SACM is based on the second simulation of the satellite signal in the solar spectrum (6S) (Vermote et al. 1997), which is accurate and has been successfully used in the development of an atmospheric correction model for EOS-MODIS data (Vermote et al. 2002).The gaseous transmission and Rayleigh optical depth are simplified as analytic functions.Lookup tables are compiled to determine Rayleigh and aerosol scattering.The environment function is defined as the probability of scattered photons reaching the sensor from the assumed uniform circular target over an inhomogeneous surface in 6S.Its lookup table is compiled with a radius of 4 m in SACM, i.e., half of spatial resolution of ROCSAT-2 RSI.The rmse values of TOA reflectances ρ TOA simulated by SACM are 0.0008 and 0.0006 in blue and red bands, respectively, for the continental aerosol model, when compared with 6S for a wide range of parameters.The major contribution of SACM is that it not only quite accurately reproduces TOA reflectance, but also does so faster than 6S (Liu 2003).A similar approach is also applied for SPOT HRV data (Liu 2004).Numerical studies have been conducted to estimate the induced errors of retrieved AOD τ a due to the rmse of ρ TOA by SACM.A DT with the reflectance ρ c of 0.01 in the red band is assumed at nadir view and solar zenith angle of 30°.Two contrasts over DT are considered, according to Lyapustin and Kaufman (2001).The low contrast (LC) and medium contrast (MC) denote that DT is surrounded by the background with its reflectance ρ b = 0.06 and ρ b = 0.11, respectively.The results show that the induced errors are only 0.013 and 0.027 in clear sky ( τ a = 0.23) and hazy sky ( τ a = 0.76) for MC, respectively, for the continental aerosol model.Such an error in AOD can only further introduce the errors in surface reflectance by 0.0009 and 0.0018, respectively.Errors in AOD and introduced error in surface reflectance are much less than 0.1 and 0.01.Hence, these errors are neglected in the following discussion.
To better understand both processes of AOD retrieval and its application to AC of ρ TOA , let us simply consider ρ ρ τ ρ ρ TOA TOA a c b = ( , , ) by neglecting its dependence of altitude z, solar zenith angle θ s , viewing zenith angle θ v and viewing azimuth angle φ for given aerosol model.The whole process includes three steps: (1) ρ TOA is simulated with the true AOD τ a and ρ c as well as ρ b by considering AE; (2) the estimated AOD τ a ' over the DT is retrieved by inverting ρ TOA with the assumption of uniform surface ( ρ b = ρ c ), which means τ a ' is retrieved by neglecting AE; (3) the estimated surface reflectance ρ c ' is determined by inverting ρ TOA with τ a ' and ρ b , which represents the error in τ a ' can be potentially propagated into ρ c ' .Hence, the errors in estimated surface reflectance in AC due to the errors in retrieved AOD by neglecting AE can be estimated.Meanwhile, the mean absolute and relative errors in the estimated surface reflectance in AC are determined by considering these errors for different contrasts, haziness and θ v .In numerical computation, it is worth noting that in step (2), AOD is interpolated from the lookup table of ρ τ TOA a ( )(Kaufman and Sendra 1988); in step (3), surface reflectance is determined by computation of the analytic function of gas transmittance as well as interpolation of lookup tables of Rayleigh and aerosol scattering for given θ s , θ v , and φ (Ouaidrari  and Vermote 1999; Liu 2004).
Numerical simulations under different conditions are conducted using the aforementioned processes.Both τ a values of 0.23 and 0.76 at 550 nm are considered.The values of θ s are set to be 30° and 60°, and θ v varies from 0° to 45°.The values of φ are set to be 0° (backscattering) and 180° (forward-scattering).Only DTs, i.e., 0 01 0 06 . .≤ ≤ ρ c (Lyapustin and Kaufman 2001), are considered in the retrieval of AOD; however, targets with ρ c ranging from 0.01 to 0.21 are considered in AC.The values of ρ b vary from 0.01 to 0.21 in aerosol retrieval and AC.As mentioned above, two contrasts over the DT including LC with ρ b of 0.06 and MC with ρ b of 0.11 are considered, according to Lyapustin and Kaufman (2001).

RESULTS AND DISCUSSION
Before estimation of the errors in ρ c for AC, AOD retrieval over the DT is performed under the assumption of homogeneous surface.The errors in the retrieved AOD are reported in green and red bands for different haziness, θ v and contrasts (Table 1a, b).The values of θ s and φ are 30° and 0°, respectively.The mean absolute and relative errors ε A and ε R are deter- mined from the errors over the DT with ρ b ranging from 0.01 to 0.21.As indicated in the previous study (Liu 2005), ε τ A a ( ) increases as the AOD increases and/or the contrast increases.This is because of neglecting stronger scattered radiance due to the homogeneous surface assumption.Tables 1a and b also show that ε τ A a ( ) decreases as θ v increases.This is due to the fact that atmospheric reflectance increases and the proportion of the scattering attributed to adjacent surface decreases when θ v increases.However, compared with the sensitivity to con- trast and AOD, ε τ A a ( ) is less sensitive to θ v .In fact, one can see that ε τ A a ( ) is most sensitive to contrast, which is consistent with the results of Lyapustin and Kaufman (2001), and least sensitive to θ v for the ranges of three parameters discussed.In both bands, ε τ ( ) in the green band can be up to 0.05 greater than that in the red band, which will be discussed later.Hence ε τ A a ( ) can be quite significant especially for MC and/or hazy sky.As for ε τ R a ( ), it increases as AOD decreases and/or contrast increases.ε τ R a ( ) also decreases with θ v .ε τ R a ( ) can be at least 16% for LC in hazy sky at θ v = 45° and up to 105% for MC in clear sky at nadir in the red band.The latter error seems to be reasonable, since this error at 30-m spatial resolution is reported to exceed 100% for the same conditions (Lyapustin and Kaufman 2001).In the green band, the values of ε τ R a ( ) ranging from 15% to 103% are similar to that in the red band.These results show the need to consider AE in aerosol retrieval especially for MC.
The error in retrieved AOD due to AE may cause an error in derived surface reflectance in AC, since surface reflectance is retrieved by inverting ρ TOA with the same parameters given in the simulation of ρ TOA , except for the retrieved AOD.As pointed out by Fraser and Kaufman  (1985) and Kaufman (1989), critical reflectance (CR) is defined as the surface reflectance where ρ TOA stays essentially constant as τ a increases.CR depends on aerosol optical charac- teristics, sun and viewing geometry.At CR, as τ a increases, the amount of increase in atmo- spheric reflectance due to more aerosol backscattering is equal to that of the decrease in surface reflection owing to decreased atmospheric transmittance.The amount of decrease in the surface reflection is weighted by surface reflectance.For ρ c < CR, ρ TOA increases with increasing τ a , and for ρ c > CR, ρ TOA decreases with increasing τ a .As mentioned above, CR has been defined as the surface reflectance where ρ TOA stays essentially constant with varying τ a .To see the performance of CR under a heterogeneous surface assumption, ρ TOA is plotted as a function of ρ c and ρ b for different τ a in the red band of FORMOSAT-2 RSI (Fig. 1).The Table 1.The mean absolute and relative errors (% in parenthesis), i.e., ε A and ε R , of the retrieved τ a due to AE and ρ c in AC for different viewing zenith angle θ v , haziness and contrasts in the (a) green and (b) red bands of FORMOSAT-2 RSI.The retrieval of AOD is only for a DT ( ρ c = 0.01 0.06).The low and medium contrast with a DT surrounded by ρ b = 0.06 and ρ b = 0.11 are denoted by LC and MC, respectively.Clear and hazy skies correspond to the τ a values of 0.23 and 0.76, respectively.θ s and φ are 30° and 0°, respectively.The errors of retrieved ρ c are discussed with respect to DTs and bright targets (BTs), i.e., ρ c = 0.16 ~ 0.21.values of θ s and θ v are 30° and 0°, respectively.CR increases as ρ b increases.In fact CR is not along the intersected line of ρ TOA for various τ a .Figure 2a, b, and c illustrate the cases as in Fig. 1 but for special ρ b in the red band.For ρ b = 0.01 (Fig. 2a), CR locates in the region among 0.09 and 0.14, and for ρ b = 0.21 (Fig. 2c), CR locates around 0.19.For homogeneous surface ( ρ b = ρ c ) (Fig. 2b), CR locates around 0.17.Since the cases for special ρ b in the green band are similar to those in the red band, their figures are omitted; however, quantitative results are reported.CR locates in the region among 0.10 and 0.16 for ρ b = 0.01, around 0.19 for ρ b = 0.21, and 0.17 for ρ b = ρ c in the green band.Therefore, CR increases as τ a increases for low ρ b in both bands.One can then see that CR locates somewhat in a triangular region whose base is at low ρ b and vertex is at high ρ b .This region is called the critical reflectance region (CRR) here.Hence CR will be represented as the critical reflectance inside the CRR in the following discussion.Based on this study, when positive bias occurs on τ a retrieval due to AE over a DT, ρ c will be underestimated for ρ c < CR, and it will be overestimated for ρ c > CR as indicated in Fig. 3.The contour lines show the errors in retrieved surface reflectance δρ c caused by the errors in retrieved AOD δτ a due to AE for different ρ c and ρ b in the red band of FORMOSAT-2 RSI.The values of θ s and θ v are 30° and 0°, respectively.The AOD has been retrieved over a dark target ( ρ c = 0.03) under the assumption of homogeneous surface for different contrasts and haziness.As discussed above about the sensitivity of retrieved AOD error to contrast and haziness, δτ a is largest, i.e., 0.47, for hazy sky and MC (Fig. 3d); and it is smallest, i.e., 0.09, for clear sky and LC (Fig. 3a).For given ρ c , δρ c decreases as ρ b increases for both contrasts and haziness.It is because increased adjacent scattering needs to be compensated by decreased surface reflection.For example, as ρ b increases from 0.01 to 0.21 for ρ c of 0.03, δρ c decreases from -0.005 to -0.015 for clear sky and LC and -0.083 to -0.102 for hazy sky and MC.For given ρ b , δρ c increases as ρ c increases for both contrasts and This is because ρ c increases, the attenuation of surface reflection due to the positive bias δτ a increases.Thus it needs to be compensated by increased ρ c .For example, as ρ c increases from 0.01 to 0.21 for ρ b of 0.15, δρ c increases from -0.014 to 0.006 for clear sky and LC and -0.109 to 0.020 for hazy sky and MC.The values of δρ c range from -0.017 to 0.013 for clear sky and LC and -0.115 to 0.035 for hazy sky and MC in the red band.In the green band (Fig. 4), the behavior of δρ c is similar to that in the red band.The difference of δρ c in both bands is insignificant, i.e., less than 0.01, in most regions of ρ c and ρ b for LC in both skies and MC in clear sky.However, the magnitude of δρ c is more than 0.01 larger than that in the red band in most regions of ρ c and ρ b for hazy sky and MC This is because a larger increase in atmo- spheric reflectance in the green band, in spite of comparable δτ a in both bands for this case.As a result, significant δρ c in AC can be caused by δτ a due to AE, especially over a DT for hazy sky and/or MC.
To better understand these errors of retrieved surface reflectance in general, mean absolute errors in retrieved surface reflectance ε ρ A c ( ) due to δτ a for different contrasts and haziness as a function of ρ c in the red band of FORMOSAT-2 RSI are illustrated in Fig. 5.The viewing direction is in nadir and θ s is 30°.For given contrast and haziness, when ρ c increases, ε ρ ( ) occurs in CRR.It is interesting to note that CRs for both contrasts are the same in clear sky, i.e., 0.13 as well as in hazy sky, i.e., 0.17.As discussed above, it is because CR increases as τ a increases for low ρ b (Fig. 2a).The CR is around 0.09 when τ a changes from 0.2 to 0.6, and it is around 0.14 when τ a changes from 0.6 to 1.0.This is because for low ρ b , a lower value of CR can be enough to balance the increase in atmospheric reflectance and the decrease in surface reflection as τ a increases in clear sky; however, an increase in atmospheric reflec- tance due to the increase of τ a should be compensated with the larger attenuation of surface reflection, i.e., larger value of CR, in hazy sky.Since the behavior of ε ρ A c ( ) due to δτ a as a function of ρ c for different contrasts and haziness in the green band is similar to that in the red band (Fig. 5), its figure is omitted.However, quantitative analysis is reported.For a DT, ε ρ A c

( ) increases as ε τ
A a ( ) increases from clear sky to hazy sky for given contrast and/or from LC to MC for given haziness (Table 1).For a BT in the green band, i.e., ρ c = 0.16 ~ 0.21 (Lyapustin  and Kaufman 2001) ( ) remains relatively stable from clear sky to hazy sky for given contrast for nadir view; however, it also increases as ε τ A a ( ) increases for θ v at 45°.In the red band, ε ρ ( ) for given haziness in both bands.For nadir view, ε ρ A c ( ) ranges from 0.010 for LC in clear sky to 0.111 for MC in hazy sky for a DT and is still greater than 0.01 for BT at MC in the green band.In the red band, it varies from 0.008 to 0.084 for a DT and is also greater than 0.01 for a BT at MC.The values of ε ρ ( ) for DTs are larger than 44% and can be up to 481% for MC in hazy sky in the green band.In the red band, they are larger than 35% and can also be up to 370% for MC in hazy sky.However, they are smaller than 8% for BTs in both bands.Quantitatively, ε ρ A c ( ) can be only negligible for BTs at LC.For a DT and MC in hazy sky at nadir view, ε ρ A c ( ) can be very significant, i.e., 0.111 and 0.084, in green and red bands, respectively.
To see the effect of θ v on ε ρ ( ) in the red band is plotted as a function of ρ c for different θ v and haziness at MC (Fig. 6).The value of φ is 0°.Since the behavior of ε ρ A c ( ) for LC is similar to that for MC, its figure is omitted.For θ v = 45°, the shape of ε ρ A c ( ) vs. ρ c is similar to that for nadir.The value of CR increases as θ v increases for given contrast and haziness.This is because as θ v increases, atmospheric transmittance decreases, and surface reflectance should be increased to balance the increase in atmospheric reflectance.This physical mechanism is quite similar to the increase in CR with increasing τ a for a given contrast, as discussed above (Fig. 5).The values of ε ρ ( ) for θ v =45° are also listed in Table 1.Again, ε ρ A c ( ) of a DT is larger than that of a BT for θ v =45° in both bands, which is similar to the case at nadir view.For a DT, the values of ε ρ A c ( ) range from 0.013 for LC in clear sky to 0.176 for MC in hazy sky in the green band, and they vary from 0.01 to 0.125 in the red band.For a BT, ε ρ A c ( ) is negligible, i.e., less than 0.01, for LC and still larger than 0.01 for ( ) due to the errors of retrieved AOD for different contrasts and haziness as a function of ρ c in the red band of FORMOSAT-2 RSI.The values of solar zenith angle θ s and viewing zenith angle θ v are 30° and 0°, respectively.surface reflectance over a wide range of surface and background reflectances.The mean absolute and relative errors of retrieved surface reflectances are determined and discussed for different contrasts, haziness and θ v .Simulation results show that the positive bias of the retrieved AOD due to AE over a DT can underestimate the surface reflectance of a DT and overestimate the surface reflectance of a BT in both bands.Significant errors of retrieved surface reflectance may occur except at LC for a BT.The error for a DT is larger than that for a BT for given contrast and haziness.For a DT, the error increases with τ a , contrast and θ v .It is most sensitive to contrast and least sensitive to θ v .For a BT, the error increases with contrast in both bands.
It is sensitive to τ a only for MC and θ v ≥ 25° and to θ v only for MC and hazy sky in the green band.However, it is insensitive to τ a and θ v in the red band.Relatively, the error of retrieved surface reflectance is enhanced by the introduced error of AOD in AC for a DT.It is suggested that AE be considered from AOD retrieval to surface reflectance retrieval in except at LC for a BT.
It is worth noting that though global observation of AOD has been successfully done by MODIS, one of the main sources of errors for the DT method is surface inhomogeneity (Chu et al. 2002;Levy et al. 2005).One might consider that this error can be testified by FORMOSAT-2 RSI image in practice together with in-situ measurements and the preflight radiometric calibration of RSI.It can be identified and estimated through rigorous numerical simulation.Otherwise, it will be due to all error sources including those pertaining to pre-launch radiometric calibration coefficients and AE.Furthermore, although the in-orbit radiometric calibration has been performed very limitedly (Liu and Lin 2004), it has not also been suggested and applied by NSPO (An-Min Wu, private communication).
LC in hazy sky and 0.2 for MC in both skies.Even for LC in clear sky, ε τ A a ( ) can be 0.07 at nadir.Although the values of ε τ A a ( ) in both bands differ only less than 0.05, the induced ε ρ A c

Fig. 1 .
Fig. 1. ρ TOA as a function of ρ c and ρ b for different aerosol optical depth τ a in the red band of FORMOSAT-2 RSI.The values of solar zenith angle θ s and viewing zenith angle θ v are 30° and 0°, respectively.
maximum, reaches a minimum value and then increases.The minimum of ε ρ A c relatively stable from clear sky to hazy sky for given contrast.For a BT, a DT in both bands are larger than that for a BT with given contrast and haziness.The result of ε ρ A c ( ) at θ v = 45°w ill be discussed later.The values of ε ρ R c