Benoı̂t Molineaux, Pierre Ineichen, and Norm O’Neill, "Equivalence of pyrheliometric and monochromatic aerosol optical depths at a single key wavelength," Appl. Opt. 37, 7008-7018 (1998)

The atmospheric aerosol optical depth (AOD) weighted over the
solar spectrum is equal to the monochromatic AOD at a certain
wavelength. This key wavelength is ∼0.7 μm, which is
only slightly influenced by air mass and aerosol content. On the
basis of this result, simple relations are proposed to predict
monochromatic AOD from pyrheliometric data and vice versa. The
accuracy achieved is close to ±0.01 units of AOD at ∼0.7
μm, estimated from simultaneous sunphotometer data. The
precision required for the estimation of the precipitable water-vapor
content is approximately ±0.5 cm.

Alcide di Sarra, Damiano Sferlazzo, Daniela Meloni, Fabrizio Anello, Carlo Bommarito, Stefano Corradini, Lorenzo De Silvestri, Tatiana Di Iorio, Francesco Monteleone, Giandomenico Pace, Salvatore Piacentino, and Sergio Pugnaghi Appl. Opt. 54(10) 2725-2737 (2015)

Francisco J. Olmo, Alberto Cazorla, Lucas Alados-Arboledas, Miguel A. López-Álvarez, Javier Hernández-Andrés, and Javier Romero Appl. Opt. 47(34) H182-H189 (2008)

Alcide di Sarra, Daniele Fua, Marco Cacciani, Tatiana Di Iorio, Patrick Disterhoft, Daniela Meloni, Francesco Monteleone, Salvatore Piacentino, and Damiano Sferlazzo Appl. Opt. 47(33) 6142-6150 (2008)

References

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The polydispersions are all normalized to
δ_{
a
}(0.7 μm) = 1. The
proportions of single-mode models constituting the multimodal models
are given as fractions by number. The coefficients s, t,
u, and y were obtained from least-squares fitting to
Mie simulations for all models except 13–15, which were obtained from
simulations with modtran and a relative humidity of
<50%. A relative humidity of ∼75% was assumed for models
8–12.
(–) indicates that the corresponding
numbers are dimensionless.
Ref. 16.
Mix of 2.27 × 10^{-6} ×
(model 1) + 0.93876 × (model 2) + 0.06123 ×
(model 3).
Mix of 1.66 × 10^{-7} ×
(model 1) + 0.5945 × (model 2) + 0.4055 × (model
3).
Ref. 15.
Mix of 0.000125 × (model 9) +
0.999875 × (model 10).
Mix of +0.000125 × (model 11) +
0.999875 × (model 12).
Mix of 0.99 × (model 10) + 0.01
× (model 13).

Table 2

Coefficients of Eq. (19) for a Selection of the
Aerosol Models Presented in Table 1

Standard deviation between the value of
λ^{
*
}
deduced from simulations and Eq. (19).
Relative error in the monochromatic aerosol
optical depth retrieved from pyrheliometric data, calculated as in Eq.
(20), assuming the aerosol size distribution is known a
priori and the error made in estimating λ
^{
*
}
is equal to the
standard deviation of the previous column.
Absolute error in the monochromatic aerosol
optical depth retrieved from pyrheliometric data if the aerosol model
is not known a priori and the coefficients of the
continental model are used for all aerosol models. This error was
calculated from simulations over the range 1 <
m_{
R
} < 5 and 0 <
δ_{
a
}(0.7 μm) < 0.3.
Ref. 16.
Ref. 15.

Table 3

Comparisons between the AOD at ∼0.7 μm
Estimated from Panchromatic and Spectral (MFRSR) Data over a
9-Month Period

The urban, rural, and maritime are those
of Shettle and Fenn (see Table 1) modeled as in Eq. (19b) with
the coefficients of Table 2. Here λ = 0.7 or 0.9 means the
wavelength at which spectral and panchromatic AOD’s are equal is
assumed to remain constant, 0.7 or 0.9 μm.
Time interval in minutes between each data
point (see text).
Calibration of the MFRSR
sunphotometer: (i) Const. means the Langley calibration
constants were assumed to remain unchanged over the period of
measurements, (ii) linear means the calibration constants obtained
from Langley analysis were fitted to a linear decrease over the period
of measurements, as in Fig. 4, and (iii) adjust means the Langley
calibration constants were adjusted according to the comparisons made
with another sunphotometer (see text).
The columnar precipitable water-vapor content
was estimated from either (i) sunph. (sunphotometer),
extinction in the 0.937-μm band compared to that in the 0.869
μm band, or (ii) meteo, from ground-based ambient
temperature and relative humidity (Ref. 32) and
(iii) with the assumption that the precipitable water-vapor content
remained constant over the 9-month period, 0.5, 1.0, or 1.5 cm.
Average AOD estimated from spectral data at a
wavelength of 0.7 μm.
Mean bias, root mean square difference, and
standard deviation between the AOD at the key wavelength, ∼0.7
μm, estimated from panchromatic data
(δ_{
aλ
}′) and spectral data
(δ_{
aλ
}): MBD =
(1/n)${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′
- δ_{
aλ
i
}), RMSD =
[${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′ -
δ_{
aλ
i
})^{2}/(n -
2)]^{1/2}, SD =
(${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′
- δ_{
aλ
i
})^{2} -
[${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′ -
δ_{
aλ
i
})]^{2}/n/(n
- 2))
^{1/2}.

Tables (3)

Table 1

Coefficients of Eqs. (7) and (13) for Several
Single-Mode and Multimodal Log-Normal Aerosol
Modelsa

The polydispersions are all normalized to
δ_{
a
}(0.7 μm) = 1. The
proportions of single-mode models constituting the multimodal models
are given as fractions by number. The coefficients s, t,
u, and y were obtained from least-squares fitting to
Mie simulations for all models except 13–15, which were obtained from
simulations with modtran and a relative humidity of
<50%. A relative humidity of ∼75% was assumed for models
8–12.
(–) indicates that the corresponding
numbers are dimensionless.
Ref. 16.
Mix of 2.27 × 10^{-6} ×
(model 1) + 0.93876 × (model 2) + 0.06123 ×
(model 3).
Mix of 1.66 × 10^{-7} ×
(model 1) + 0.5945 × (model 2) + 0.4055 × (model
3).
Ref. 15.
Mix of 0.000125 × (model 9) +
0.999875 × (model 10).
Mix of +0.000125 × (model 11) +
0.999875 × (model 12).
Mix of 0.99 × (model 10) + 0.01
× (model 13).

Table 2

Coefficients of Eq. (19) for a Selection of the
Aerosol Models Presented in Table 1

Standard deviation between the value of
λ^{
*
}
deduced from simulations and Eq. (19).
Relative error in the monochromatic aerosol
optical depth retrieved from pyrheliometric data, calculated as in Eq.
(20), assuming the aerosol size distribution is known a
priori and the error made in estimating λ
^{
*
}
is equal to the
standard deviation of the previous column.
Absolute error in the monochromatic aerosol
optical depth retrieved from pyrheliometric data if the aerosol model
is not known a priori and the coefficients of the
continental model are used for all aerosol models. This error was
calculated from simulations over the range 1 <
m_{
R
} < 5 and 0 <
δ_{
a
}(0.7 μm) < 0.3.
Ref. 16.
Ref. 15.

Table 3

Comparisons between the AOD at ∼0.7 μm
Estimated from Panchromatic and Spectral (MFRSR) Data over a
9-Month Period

The urban, rural, and maritime are those
of Shettle and Fenn (see Table 1) modeled as in Eq. (19b) with
the coefficients of Table 2. Here λ = 0.7 or 0.9 means the
wavelength at which spectral and panchromatic AOD’s are equal is
assumed to remain constant, 0.7 or 0.9 μm.
Time interval in minutes between each data
point (see text).
Calibration of the MFRSR
sunphotometer: (i) Const. means the Langley calibration
constants were assumed to remain unchanged over the period of
measurements, (ii) linear means the calibration constants obtained
from Langley analysis were fitted to a linear decrease over the period
of measurements, as in Fig. 4, and (iii) adjust means the Langley
calibration constants were adjusted according to the comparisons made
with another sunphotometer (see text).
The columnar precipitable water-vapor content
was estimated from either (i) sunph. (sunphotometer),
extinction in the 0.937-μm band compared to that in the 0.869
μm band, or (ii) meteo, from ground-based ambient
temperature and relative humidity (Ref. 32) and
(iii) with the assumption that the precipitable water-vapor content
remained constant over the 9-month period, 0.5, 1.0, or 1.5 cm.
Average AOD estimated from spectral data at a
wavelength of 0.7 μm.
Mean bias, root mean square difference, and
standard deviation between the AOD at the key wavelength, ∼0.7
μm, estimated from panchromatic data
(δ_{
aλ
}′) and spectral data
(δ_{
aλ
}): MBD =
(1/n)${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′
- δ_{
aλ
i
}), RMSD =
[${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′ -
δ_{
aλ
i
})^{2}/(n -
2)]^{1/2}, SD =
(${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′
- δ_{
aλ
i
})^{2} -
[${\sum}_{i=1}^{n}$(δ_{
aλ
i
}′ -
δ_{
aλ
i
})]^{2}/n/(n
- 2))
^{1/2}.