PII: S0021-8502(98)00046-9
J. Aerosol Sci. Vol. 29, No. 10, pp. 1199—1211, 1998
( 1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0021-8502/98 $19.00#0.00
DETERMINATION OF AEROSOL OPTICAL THICKNESS FROM
MEASUREMENTS OF SPECTRAL SKY RADIANCE
C. Sánchez Oliveros, F. J Olmo Reyes and L. Alados-Arboledas*
CRYTIS, Departamento de Fı́sica Aplicada, Facultad de Ciencias, Universidad de Granada,
Fuentenueva s/n, 18071-Granada, Spain
(First received 12 September 1997; and in final form 12 March 1998)
Abstract—Simultaneous measurements of spectral optical thickness and path radiance are used to
test the relationship between path radiance and aerosol optical thickness for a given geometry of
illumination and observation. The selected geometry is representative of satellite remote sensing
conditions. The measurements have been accomplished in an urban location with a slightly polluted
atmosphere. Ground-based measurements were made with a device that combines sun-photometer
and sky radiometer functions. The measurement sequence includes measurements of transmission of
the direct sunlight in order to retrieve the aerosol optical thickness in nine spectral bands between
440 and 1030 nm, followed by measurements of the sky radiance in the solar almucantar on both
sides of the sun in the same spectral bands. For this purpose we have used a set of radiances acquired
around 60° solar zenith angle in the antisolar direction of the solar almucantar. The results obtained
show that aerosol optical thickness estimates are possible from path radiance measurements. On the
other hand, it seems feasible to derive the atmospheric correction in a given spectral band from
information acquired at another wavelength. The aerosol optical thickness (radiance) in one spectral
band allows appropriate estimates of aerosol optical thickness (radiance) in another spectral band.
In this sense, we also obtain a relationship between the spectral dependencies of aerosol optical
thickness and path radiance. ( 1998 Elsevier Science Ltd. All rights rserved
1 . I NT RO D UC T IO N
Knowledge of the parameters that determine the optical properties of atmospheric aerosols
is essential for the determination of their climatic effects, development of techniques for
remote sensing of aerosols from space, or the necessary correction of atmospheric effects in
satellite imagery.
During the past decades, remote sensing of the earth from a satellite has grown to become
a fundamental tool for climate and environmental studies. The use of images acquired from
spatial sensors to characterise the earth’s surface requires a correction process of the
atmospheric effects, especially the effect of aerosols. This is due to the difficulties associated
with their radiative effect as well as by the absence of measurements of their concentrations
and characteristics. In fact, due to the short life of tropospheric aerosols, their properties
vary in space as well as in time. Their concentrations and their properties depend on the
intensity of the sources, on the atmospheric processes that affect them, and on the particle
transportation from one region to another. Thus, to determine their characteristics the
measurements must be accomplished with a given frequency in places with different types of
aerosols and for assorted meteorological conditions. Then the remote sensing of aerosols
appears as an important tool considering both their global coverage and their frequency
in time.
In order to use the reflectance observed by a spaceborne sensor to characterise the earth’s
surface properties, we must consider gaseous absorption and scattering by molecules and
aerosols. Generally, the absorption by aerosols is small, and satellite sensor channels have
been selected in order to avoid the molecular absorption bands. Therefore scattering is the
most important effect.
Figure 1 describes the radiance reaching the satellite sensor. It includes the photons
coming from the sun, which do not reach the surface and are backscattered towards space,
* Author to whom correspondence should be addressed.
1199
1200
C. S. Oliveros et al.
Fig. 1. Schematic diagram of the satellite and ground based observation geometry.
¸ . This must be included in our radiative balance, although it is an interference term that
16
does not carry any information about the target. A second contribution is due to photons
scattered by the atmosphere into the path from the surface to the satellite, ¸ . This
D
component must be carefully considered if the surface has a patchy structure. Finally,
¸ represents the radiance reflected by the target that is not absorbed or scattered on its way
S
to the sensor. This signal carries information about the target surface. In this way, the
radiance that reaches the satellite sensor is obtained by the sum of these components:
¸"¸ #¸ #¸ .
16
4
D
(1)
This expression can be written as (Chandrasekhar, 1960)
F
o
¸"¸ # $ ¹
16
6
n
1!so
(2)
¸!¸
16
,
o"
(F /n) ¹ !s(¸!¸ )
$
6
16
(3)
and, finally,
where F is the incoming irradiance at the surface, o the reflectance and s represents the
$
atmospheric albedo. Since the atmospheric albedo, s, is normally small, the atmospheric
correction consists mainly in estimating the upward path radiance ¸ , the radiative flux
16
F and the upward atmospheric transmission ¹ . For surfaces of low reflectance the
$
6
difference ¸!¸ is quite small, therefore an exact determination of o depends mainly on
16
an exact determination of ¸ (Gordon, 1978). To carry out the atmospheric correction,
16
a radiative transfer code can be used. In this way, ¸ , F and ¹ can be estimated from
16 $
6
aerosol optical thickness measured from the ground or directly deduced from satellite data
(Tanré et al., 1988). A different approach uses a surface of low reflectance to obtain directly
the path radiance in one or two channels that can be used to deduce the optical thickness,
and consequently F and ¹ (Kaufman and Sendra, 1988).
$
6
The aerosol optical thickness can be obtained from satellite images in the visible and near
infrared spectral bands (e.g. Kaufman, 1993; Soufflet et al., 1997). Kaufman and Sendra
(1988) have developed a technique to deduce the aerosol optical thickness in different
spectral bands using the radiances observed via satellite. The method which the authors
applied to Landsat TM and NOAA AVHRR images, requires observations over surfaces of
low reflectance. In this way, the radiance observed by the sensor on board the satellite
corresponds substantially to the component dispersed by the molecules and atmospheric
aerosols (path radiance). In their original proposal, the retrieval of the aerosol optical
thickness was based on look-up tables generated through simulations of the atmospheric
conditions by means of a radiative transfer code. In a recent work, Kaufman (1993)
proposed the use of simple relationships to obtain the optical thickness from the radiance
measured by the satellite over a low reflectance surface. These relationships have been
Measurements of spectral sky radiance
1201
obtained through simultaneous observations of downward atmospheric radiance and
optical thickness accomplished from the ground. For the proposal of these relationships,
Kaufman (1993) has used measurements made in different localities with different loads and
type of aerosols. He obtained good results in all cases excluding those cases in which there is
atmospheric dust.
One objective of this work is to measure the relationship between the path radiance ¸ ,
16
which ‘‘contaminates’’ satellite observations of the Earth’s surface, and the aerosol optical
thickness, q . When the observation from space is accomplished over a low reflectance
A
surface, the net effect due to the aerosols prevails over the other contributions. The path
radiance is the result of backscattering of aerosol particles and molecules in the atmosphere
towards space. The downward path radiance, ¸ , which can be measured from the ground,
1$
shows a relationship with the upward path radiance, ¸ , measured by the satellite sensor.
16
This correlation between both radiances is valid for a variety of aerosol optical thicknesses
and a variety of surface’s reflectances. Kaufman (1993) has studied this relationship with the
help of a radiative transfer code. For the case of surfaces of low reflectance, in which the
difference between ¸ and ¸ is small and their correlation thus strong, Kaufman
16
1$
proposes the following relationship, assuming equivalent geometric conditions in both
radiances (Fig. 1):
¸ !0.01
¸ + 1$
, o)0.4.
16
1.7#o
(4)
In this paper, we use simultaneous measurements from the ground of spectral optical
thickness and path radiance in order to test the relationship between path radiance and
aerosol optical thickness for a given geometry of illumination and observation. The selected
geometry is representative of satellite remote sensing conditions (Kaufman, 1993). We check
the feasibility to carry out the atmospheric correction in a spectral band from the information acquired in other specific wavelengths. For this purpose, the aerosol optical thickness
as well as the radiance in one spectral band are used to estimate both quantities in other
spectral bands. We approach this task by analysing the relationship between aerosol optical
thickness (path radiance) values measured in different spectral bands. Additionally, we
analyse the relationship between the spectral dependencies of aerosol optical thickness and
path radiance.
2 . ME ASU RE ME N TS
The purpose of the experimental campaigns carried out at the University of Granada
(Spain) is to obtain a set of measurements of aerosol optical thickness and sky radiance.
These last measurements have been made in the solar almucantar and have been restricted
to solar zenith angles around 60°. In this way, we could measure the scattering phase
function for scattering angles up to 120°.
All the measurements have been carried out from the roof of the Physics Faculty building
of Granada University (36.18° N, 3.6° W, 688 m), in the urban centre of Granada city
characterised by a slightly polluted atmosphere. The measurements cover a full year from
February 1996 to May 1997. The abnormally high frequency of cloudy and rainy conditions
during 1996 winter has restricted the number of observations during this season.
The measurements have been accomplished with a portable LICOR spectroradiometer
LI1800. This device is equipped with an adjustable telescope that can measure both the sky
radiance and the solar direct irradiance in selected spectral bands. The field of view of the
telescope can be regulated to choose values of 0.8, 1.6, 3, 4, 8, 15°. For the present work, we
have used a 3° field of view for the direct irradiance measurements and a 0.8° field of view for
the sky radiance measurements. We have used the measurements made from 440 to 870 nm
in 10 nm steps with a spectral half-bandwidth of 6 nm. All of them were taken under
cloudless conditions. Each experimental sequence includes measurements of the solar direct
irradiance to obtain the aerosol optical thickness in a set of relevant spectral bands. In order
1202
C. S. Oliveros et al.
to avoid exceeding the response range of the device a neutral filter is used. These experimental values are completed with measurements of spectral sky radiance in the solar
almucantar at both sides of the sun. In this way, the symmetry in the almucantar with
respect to the sun’s position is used as a test for the homogeneity of the sky’s conditions
during the measurement process.
For the present work we have exclusively used measurements of optical thickness and sky
radiance in the antisolar direction, i.e. measured in the solar almucantar with an azimuth at
180° from the sun’s position. The choice of this observation geometry and the limitation to
solar zenith angles of about 60° implies scattering angles about 120 that are representative
of a great number of satellite observations (Fig. 1).
Concerning the wavelengths analysed we have focused on the effective wavelengths for
the visible and near infrared channels of the AVHRR radiometer on board the satellites
NOAA-12 and NOAA-14, i.e. 630 and 840 nm. These wavelengths have been completed
with those centred at 440, 520, 560, 620, 670, 780 and 870 nm.
The aerosol optical thickness was derived from the total optical thickness by subtracting
the Rayleigh optical thickness and O and NO absorption optical thicknesses. For this
3
2
purpose we follow a procedure similar to that described by Schmid and Wehrli (1995). By
selecting constant solar and view direction we assure rather constant molecular scattering
and absorption effects.
Periodic calibrations of the experimental device were performed by two different methods
for each analysed wavelength: (1) for the solar direct irradiance measurements used for the
estimation of the optical thickness; (2) for the sky radiance measurements.
For the first calibration, we have obtained the spectro-radiometer calibration constant
from a Langley plot technique following the procedures described by Schmid and Wehrli
(1995). High mountain measurements have been used for this purpose (Sierra Nevada
Range, 37.1°N, 3.38°W, 2690 m. a.m.s.l.). Then the measured aerosol optical thickness with
an error $0.01 to $0.02 was computed using the appropriate calibration constant and
subtracting the contributions due to Rayleigh, O and NO . The O column contents were
3
2
3
taken from measurements performed in El Arenosillo (37°6@N, 6°44@W, Dobson spectrophotometer n.120). The NO column contents were obtained from the mid-latitude model
2
atmospheres in LOWTRAN7 code (Kneizys et al., 1988). Generally, in the Langley method
the optical air mass for the different radiatively active atmospheric components is approximated by the Rayleigh optical air mass. We have proved that the use of a specific
expression for the optical air mass of each constituent reduces the standard deviation in the
computation of the calibration constant. Thus for the optical air mass of O we use an
3
expression that has been obtained under the assumption that O is concentrated in
3
a narrow layer located at an altitude about 22 km (Iqbal, 1983). On the other hand, the use
of the O optical air mass to approximate that of NO does not introduce a significant error
3
2
due to the small contribution of this constituent in the atmosphere. Some authors (e.g.
Schmid and Wehrli, 1995) propose a hybrid expression for the optical air mass of the
aerosol. Nevertheless, in our case the best results are obtained using the same expression
that has been used for the Rayleigh optical air mass (Kasten and Young, 1989).
The second calibration was carried out in the laboratory with a 200 W standard lamp of
known response. An absolute accuracy of the path radiance measurements better than
$5% is expected. The wavelength accuracy is $2 nm and the wavelength repeatability is
$0.5 nm.
3 . AN AL YSIS AN D R ES UL TS
Table 1 shows a summary of the accomplished measurements. The data are grouped
according to different intervals of aerosol optical thickness. We have not corrected the path
radiance for the effects of molecular scattering and the absorption of gases. For this reason,
the data have been interpolated to solar elevation angles of 30°. Second and third columns
include the average optical thickness and the normalised radiance for 630 nm. Another
averaged parameter included in Table 1, is aq which represents the spectral dependency of
Measurements of spectral sky radiance
1203
Table 1. Average results for groups of the aerosol optical thickness
Range of q (630 nm)
A
q (630 nm)
A
¸ (630 nm)
1$
0.06—0.11
0.11—0.13
0.13—0.25
0.095
0.120
0.155
0.0329
0.0351
0.0375
a
q
!1.20
!1.19
!1.13
a !a
q!
q"
!0.15
!0.08
!0.28
a
L
!2.57
!2.43
!2.12
the aerosol optical thickness q . This well-known Angström exponent was calculated as the
A
least-squares fit slope of the logarithm of the aerosol optical thickness as a function of the
logarithm of the wavelength. This computation was made including all the analysed
wavelengths. In addition we have included the difference a !a , where a is the slope
q!
q"
q!
calculated using the wavelength interval 440—630 nm, while a has been computed for the
q"
wavelength interval 630—870 nm. Finally, a represents the radiance spectral dependency.
L
a has been obtained as the slope of the radiance logarithm versus the wavelength
L
logarithm.
Inspection of Table 1 evidences the existence of a relationship between the averaged
optical thickness q and path radiance ¸ for each group. An increase in q leads to an
A
1$
A
increase in atmospheric scattering and thus in ¸ . Kaufman (1993) presents similar results
1$
for a wide range of places, covering four continents.
We can see that for our experimental data set the parameter aq increases slightly as the
aerosol optical thickness increases. This means that as the aerosol optical thickness
increases there is a slight increase of the contribution of larger particles. Non zero values of
the difference aq !aq imply the existence of a curvature in the spectral dependence of the
!
"
optical thickness reflecting a deviation from a power-law size distribution. In our case this
difference presents negative values, i.e. the slope decreases with the wavelength. In this sense,
the effect of two separate particle modes is evident (Kaufman and Fraser, 1983). On the
other hand it can be seen that as the aerosol optical thickness increases, the path radiance
spectral exponent, a , increases, thus indicating a lower spectral dependence. Therefore, this
L
reveals a positive correlation between the spectral dependencies aq and a .
L
3.1. Relationships between optical thickness and path radiance
The relationship between upward path radiance and aerosol optical thickness is
of great interest for two main reasons. The first is related to the capability to
compute the atmospheric correction for radiances acquired by spaceborne devices from
available aerosol optical thickness data. The second reason is connected with the
retrieval of aerosol optical thickness from satellite images that has been acquired over
surfaces of low reflectance. Both topics are related representing a very important task in
climatological and environmental applications of satellite remote sensing. Our measurements could be used to study these relationships considering the correlation between
upward and downward path radiances (equation (4)). Obviously, considering the geometry
selected for both illumination and observation, the results are limited to a satellite geometry
shown in Fig. 1, which in any case could be considered as representative of a great number
of satellite observations.
In this section, we test the correlation between the path radiance and the aerosol optical
thickness for different wavelengths. The results for several of the analysed wavelengths will
allow a direct comparison with previous results obtained by Kaufman (1993).
Kaufman (1993) has shown that the best fits between path radiance and aerosol optical
thickness are obtained using parabolic functions. Due to our narrower range of aerosol
optical thickness, we have found that the best fit for our data set is better represented by
linear functions. An increase in the correlation between path radiance and optical thickness
can be observed for higher wavelengths (Kaufman, 1993). This is the result of the prevailing
effect of the aerosols on the molecules in the scattering process for higher wavelengths.
1204
C. S. Oliveros et al.
The empirical relationships obtained for the range of q analysed are:
A
¸
"0.0618#0.034 * q
,
1$520
A520
r2"0.26,
(5a)
¸
"0.0280#0.082 * q
,
1$620
A620
r2"0.60,
(5b)
¸
"0.0242#0.093 * q
,
1$630
A630
r2"0.65,
(5c)
¸
"0.0219#0.111 * q
,
1$670
A670
r2"0.65,
(5d)
¸
"0.0133#0.146 * q
,
1$780
A780
r2"0.75,
(5e)
¸
"0.0088#0.125 * q
,
1$840
A840
r2"0.84,
(5f )
¸
"0.0075#0.150 * q
,
1$870
A870
r2"0.82.
(5g)
In these expressions the radiance is expressed in terms of apparent reflectance, obtained
by normalising to the available solar flux at the top of the atmosphere in the associated
wavelength interval (n¸ /Fo). The extraterrestrial solar spectrum used has been that of
$
Fröhlich and London (1986) interpolated at 1 nm intervals. The use of these relationships
provides estimates of the normalised radiance with a maximum root mean square deviation,
RMSD about 0.002, that is 8% over the average value. For practical applications in remote
sensing atmospheric correction, this expression will be used in conjunction with equation
(4). Considering the interest in the retrieval of the aerosol optical thickness, from satellite
radiances measured over surfaces of low reflectance, we have obtained the following
functions, applicable to the channels 1 and 2 of the AVHRR sensor (NOAA-12 and 14):
q
"!0.126#7.0 * ¸
,
A630
1$630
r2"0.65,
(6a)
q
"!0.044#6.7 * ¸
,
A840
1$840
r2"0.84.
(6b)
These expressions yield estimates of the aerosol spectral optical thickness from path
radiance measured in the same wavelength with a maximum root mean square deviation of
0.02, that represents 14.5% over the average value.
Figures 2a and b show the relationship between path radiance and aerosol optical
thickness for the wavelengths 630 and 870 nm. These figures display the experimental data
and the fitting function obtained. For 870 nm we include the model developed by Kaufman
(dotted curve). Table 2 includes statistical results concerning the validation of the expression obtained by Kaufman for the closer wavelengths to those used in our study. For this
purpose, we use the root mean square deviation (RSMD) and the mean bias deviation
(MBD). These statistics allow the detection of both the differences between experimental
data and model estimates and the existence of systematic over or under-estimation tendencies, respectively. Our results show the convenience of the functions developed by Kaufman
for the wavelengths coincident with our experimental measurements. The MBDs are
negligible and the RMSDs are below 11%. Kaufman (1993) claimed that his expressions
provide a relationship between path radiance and aerosol optical thickness that includes the
effect of different types of aerosols. Our study confirms their validity for the urban aerosol
that characterises the location of our experiments. Figure 3 shows the relationship developed to obtain the aerosol optical thickness from path radiance for 630 nm wavelength.
3.2. Spectral relationships
As indicated previously, an appropriate procedure for atmospheric correction is the use
of the path radiance (aerosol optical thickness) obtained at a particular wavelength to
estimate the path radiance (aerosol optical thickness) corresponding to other wavelengths.
For these reasons, we have tested both the relationships between aerosol optical thicknesses
at different wavelengths and those of path radiances for different wavelengths. In order to
Measurements of spectral sky radiance
1205
Fig. 2. Relationship between aerosol optical thickness and atmospheric path radiance. ( a) Path
radiance as a function of aerosol optical thickness for 630 nm. (b) Path radiance as a function of
aerosol optical thickness for 870 nm, where we have added comparison with Kaufman function for
872 nm (dotted curve).
Table 2. Validation of Kaufman expressions validation
¸
"¸
(q )
1$520
1$520 520
¸
"¸
(q )
1$670
1$670 670
¸
"¸
(q )
1$780
1$780 780
¸
"¸
(q )
1$870
1$870 870
MBD
RMSD
0.0076
0.0029
0.0015
0.0016
0.0087
0.0036
0.0026
0.0024
(11%)
(9%)
(9%)
(11%)
extend the aerosol optical thickness information to other wavelengths we have also tested
the use of the exponent aq . Thus, we have analysed the relationship between this wavelength
exponent for the aerosol optical thickness and a , which expresses the spectral dependence
L
for the path radiance. On the other hand, considering the apparent correlation between
aq and the aerosol optical thickness, evidenced in Table 1, we have modelled this
dependence.
1206
C. S. Oliveros et al.
Fig. 3. Aerosol optical thickness versus path radiance for 630 nm.
Using data acquired for the solar zenith angle ranging from 75 to 50°, we have analysed
the correlation between aerosol optical thickness for different wavelengths. The following
relationships are obtained for the aerosol optical thickness corresponding to 440, 630, and
840 nm:
q
"!0.010#1.60 * q
,
A440
A630
q
r2"0.89,
(7a)
q
"0.029#1.70 * q
,
A440
A840
r2"0.71,
(7b)
q
"0.046#1.81 * q
,
A440
A870
r2"0.72,
(7c)
"!0.004#0.79 * q
.
A840
A630
r2"0.89.
(7d)
Obviously, the applicability of these expressions is limited to the urban type of aerosol
analysed in this study. By using these relationships we can obtain the aerosol optical
thickness in a wavelength from the aerosol optical thickness measured in another
wavelength without previous knowledge of the aerosol size distribution. These estimates
have a maximum RMSD about 0.03 and a MBD close to 0.02 that represents an overestimation close to 15%. This is an interesting result for the atmospheric correction of satellite
data. In this process, the aerosol optical thickness in a wavelength is calculated from aerosol
optical thickness measured in another wavelength (Kaufman and Sendra, 1988), thus
allowing the atmospheric correction in different spectral bands.
In Figs 4a and b we have plotted equations (7c) and (7d). For the wavelength of 440 nm
we have also included the function proposed by Kaufman (1993) (dotted curve) for the
closest wavelength (441 nm). As can be seen in Fig. 4a, there are some differences between
our model and that proposed by Kaufman (1993). The MBD and RMSD associated with
the use of the expression proposed by Kaufman for 440 nm with our experimental data are
about 0.04. Thus, the use of this equation could imply an overestimation by about 20%.
The atmospheric corrections of satellite imagery can be carried out using directly
measured radiances from space over dark surfaces, for example the oceans (Gordon, 1978),
Measurements of spectral sky radiance
1207
Fig. 4. (a) Aerosol optical thickness at 440 nm versus aerosol optical thickness at 870 nm. Comparison with Kaufman function at 441 nm (dotted curve). (b) Aerosol optical thickness at 840 nm versus
aerosol optical thickness at 630 nm.
or the dense vegetation zones (Kaufman and Sendra, 1988). The atmospheric correction in
different spectral bands requires an estimate of this path radiance in the corresponding
wavelength. In this way, it could be of interest to estimate the path radiance in a wavelength
from path radiance measured in other spectral interval. In our case, we have tested the
correlation between the measured path radiances at the effective wavelengths of AVHRR
visible and near-infrared channels. The best fit corresponds to parabolic functions:
,
¸
"0.010!0.1 * ¸
#10.5 * ¸2
1$630
1$840
1$630
r2"0.90,
(8a)
,
¸
"0.006#0.11 * ¸
#8.5 * ¸2
1$670
1$870
1$670
r2"0.91.
(8b)
1208
C. S. Oliveros et al.
A similar correlation is obtained for a wider spectral range:
,
¸
"0.038!1.12 * ¸
#15.1 * ¸2
1$560
1$870
1$560
r2"0.81.
(8c)
Equations (8a)—(8c) are plotted in Fig. 5, including the comparison with the functions
proposed by Kaufman (dotted curves in Figs 5b and c). The use of equations (8a)—(8c)
provides estimates of path radiances with low values of RMSD. The maximum RMSD
corresponds to the widest spectral interval (equation (8)). For this case, we obtain a RMSD
close to 0.003, which represents about 13% over the average path radiance. Furthermore,
the equations (8a)—(8c) are obtained with determination coefficients, r2, about 25% better
than that obtained when we analysed the relationship between path radiance and aerosol
optical thickness (equations (5a)—(5g)). Thus, for the atmospheric correction of remote
sensing imagery, it could be better to obtain the path radiance in a wavelength from that
measured in another spectral interval than to deduce it from the aerosol optical thickness
measured in the same wavelength. This result is in accordance with Kaufman’s (1993)
previous results. Using the relationship proposed by this author for radiances at 870 and
560 nm we obtain an overestimation below 20%. Figure 5b shows the results for the
radiances at 870 and 670 nm, in this case the use of Kaufman expression implies an
underestimation by about 10%.
We have analysed the spectral dependence of the path radiance a . This coefficient has
L
been computed as the slope of the path radiance logarithm versus the wavelength logarithm. In all cases, the fits computed to obtain this coefficient have determination coefficients, r2, about 0.9. This could be considered as an evidence of the goodness of a power-law
spectral dependence for the path radiance. As evidenced in Table 1 the spectral exponent for
the path radiance a presents a tendency with aerosol optical thickness similar to that found
L
for the Angström exponent aq . Thus, we have analysed the correlation between these
spectral coefficients aq and a . The best results are obtained with the following parabolic
L
expression:
aq!a "!3.6!3.1 * a !0.45 * a2 ,
L
L
L
r2"0.74.
(9)
This expression provides a method to predict the spectral dependence of aerosol optical
thickness, aq , from the spectral dependence of path radiance, a , with an error less than
L
14%. Thus, from measurements of path radiance in different spectral bands we can obtain
information on the aerosol size distribution, considering its relationship with the Angström
exponent, aq. Figure 6 shows that the differences between both spectral dependencies, aq and
a , decreases for lower spectral dependencies in both variables. That is, for stronger
L
contributions of larger particles there is a better accord between both spectral dependencies,
while for larger contributions of smaller particles the difference between the spectral
exponents, aq!a , tends to a value about 1.75.
L
4 . CO N CLU SI ON S
We have analysed a set of simultaneous measurements of sky radiance and optical
thickness affected by aerosols of urban type. These data were obtained with a portable
spectro-radiometer in the range 440—1030 nm with constant illumination and observation
geometry, resulting in a fixed scattering angle of about 120° typical for a great number of
observations carried out from space. We have used these experimental data for testing
different steps in the atmospheric correction of remote sensing imagery. We have checked
the feasibility to compute the optical thickness from the path radiance. We have also tested
the reliability of using the path radiance or the aerosol optical thickness that has been
obtained in a particular wavelength to estimate the values corresponding to other
wavelengths. Our results show that for an urban aerosol typical for the location of our
experiments, the radiance measured over low reflectance surfaces can be used to obtain the
aerosol optical thickness using an empirical model with an error of about 0.03.
Measurements of spectral sky radiance
Fig. 5. (a) Path radiance at 840 nm versus path radiance at 630 nm. (b) Path radiance at 870 nm
versus path radiance at 670 nm. (c) Path radiance at 870 nm versus path radiance at 560 nm.
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C. S. Oliveros et al.
Fig. 6. Relationship between the spectral dependency of the aerosol optical thickness, a , and the
q
spectral dependency of the path radiance, a .
L
The results indicate a high correlation between the aerosol optical thickness measured in
different wavelengths. Consequently, it is possible to extrapolate the aerosol optical thickness to other wavelengths without previous knowledge of the aerosol size distribution.
We have shown that the path radiance presents a spectral dependence of the power-law
type. The corresponding spectral exponent has an acceptable degree of correlation with the
Angström wavelength exponent, describing the spectral dependence of the aerosol optical
thickness. This fact could be used to obtain information about the aerosol size distribution
from spectral radiance observations.
The atmospheric correction requires extrapolation of path radiance from one wavelength
to another. We have shown that for urban aerosols this could be done by empirical
functions. It is also shown that it would be better to obtain path radiances at one
wavelength from path radiances at another spectral interval, rather than deduce them from
the corresponding aerosol optical thickness measured at the same wavelength.
All these results are conditioned on the type of aerosol present in our measurements; thus,
a wider field campaign is now planned. In any case, our results confirm previous developments obtained by Kaufman (1993) using a data set collected from different places around
the world affected by different types of aerosols.
Acknowledgements—This work was supported by La Dirección General de Ciencia y Tecnologı́a from the
Education and Research Spanish Ministry through the project No. CLI95-1840 and the Grant IN94-0563. We
appreciate the comments and suggestions of Yoram J. Kaufman and two anonymous referees.
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