Water vapor absorption in the visible and near infrared:
results of field measurements
Rodolfo Guzzi and Rolando Rizzi
Measurements of solar irradiance at the ground have been analyzed to obtain information on absorption
from water vapor in the visible and near infrared. Great care has been taken in evaluating the aerosol optical thickness to obtain results compatible with theory. An automatic procedure is presented that eliminates
the recordings in which modifications of the aerosol optical properties not monitored would seriously influence the determination of those of water vapor. Particular care is paid to assessing the error limits of the
derived spectral attenuation parameters.
visible and near infrared have been performed by several authors using different devices (e.g., monochro-
measured transmittances could be due to a concentration of particles of radius >1 umhigher than predicted
by available model aerosol distributions.
Guzzi, Tomasi, and Vittori5 - 7 made extensive mea-
mators, interference filters) both in the laboratory and
in the actual atmosphere. Laboratory measurements
are generally not fully compatible with field studies
to the infrared over several years and in different meteorological conditions. The amount of precipitable
1.
Introduction
Measurements of water vapor absorption in the
because of the difficulty in constructing path cells long
enough and of producing sufficiently high water vapor
content to simulate real atmospheric conditions.
Howard et al.I measured water vapor band-integrated absorption using a long path cell in which the
total pressure, water vapor, and temperature could be
varied.
The results are expressed in terms of two em-
pirical relations for weak and strong bands. Burch and
Gryvnak2 presented high resolution water vapor
transmission values in the 0.69-1.98 Am range.
Since
the water content along the path was obtained by
heating the sample to 443 K, their results are not easily
transferable to real atmospheres.
Atmospheric attenuation studies started with the
3
work by Fowle who analyzed the data at his disposal
in terms of water vapor content. His data were subsequently reprocessed by Eldridge 4 who computed a set
of (error function) absorption coefficients and pointed
out that the lack of agreement between computed and
surements of atmospheric transparency from the visible
water along the light path was simultaneously measured
using an infrared hygrometer. When analyzing the data
obtained in very clear conditions during anticyclonic
events, they found an unexpected increase in spectral
attenuation
going from morning to noon which could
not be accounted for by the variation of precipitable
water. This effect was explained (Guzzi et al.5 and
Vittori et al. 6) in terms of variation of particulate matter
optical properties while water vapor absorption coefficients were presented by Tomasi et al. 7 In the latter
paper it was shown that residual attenuation by water
vapor, after elimination of particulate matter effects,
was present even in wavelength regions considered most
transparent.
Fraser. 8
Similar conclusions were reached by
More recently Tomasi 9 has reprocessed a subset of
the data measured by Guzzi, Tomasi, and Vittori to
discriminate between water vapor absorption and
water-dependent particulate-matter attenuation.
Seasonal sets of atmospheric optical thicknesses were
selected presenting at each wavelength a linear correlation with precipitable water vapor. The dependence
of optical thickness on precipitable water is expressed
as the sum of a nonselective absorption coefficient of
Rodolfo Guzzi is with Istituto di Fisica Generale, Ferrara, Italy, and
R. Rizzi is with UniversitA degli studi di Bologna, Dipartimento di
Fisica, 40127 Bologna, Italy.
Received 7 June 1983.
0003-6935/84/111853-09$02.00/0.
©)1984 Optical Society of America.
atmospheric water vapor and a coefficient expressing
the variation of particulate-matter optical thickness
with precipitable water. His technique gives absorption
coefficients smaller than previously reported using the
same set of data. His data are, however, rather coarsely
spaced in the wavelength interval.
1 June 1984 / Vol. 23, No. 11 / APPLIED OPTICS
1853
In some of the papers mentioned above the spectral
dependence of optical thickness on precipitable water
l
|
|
l
l
x
T
-
1000
is assumed to follow a square-root law within the bands
while, in the so-called windows, a Lambertian law is
Rcy a(I-*o)A5o
0.8(H 20)
used. Moskalenko 0 performed a series of laboratory
QaTH
2
)
measurements using a long path cell. Large thicknesses
of water vapor could be obtained in conditions of null
aerosol attenuation. He expressed his transmission
data using a power dependence of optical thickness on
precipitable water. His results show that no sharp
D
Z 500
distinction exists between windows and bands as far as
the precipitable water dependence is concerned, while
the absorption coefficients changed considerably going
from weak to strongly absorbing regions.
Koepke and Quenzel 1 performed a single measurement of spectral solar irradiance from the ground to
determine the water vapor attenuation coefficient.
Instead of using either the weak or strong approximation, they use Moskalenko's data for the dependence of
optical thickness on precipitable water. They found
no agreement with Moskalenko's spectral attenuation
data at wavelengths smaller than 1 zm.
In light of the previous discussion it is still unclear
C
0.70
080
090
1pO
Fig. 1. Example of strip-chart recording giving deflection of the
recorded D (arbitrary units) versus wavelength in microns on 24 Aug.
1971. At the beginning of the reading air mass is 1.73, relative humidity is 43, screen temperature is 24aC, ground pressure is 1015
mbar, and precipitable water is 14-mm (STP). The curve labeled Do
is the extra atmospheric deflection of the instrument.
whether a Lambertian law is effectively followed by
water vapor absorption in the most transparent regions
of the visible spectrum, and the absolute values of the
windows to more strongly absorbing
attenuation coefficients themselves show variations that
tered when estimating the aerosol optical thickness. An
cannot be explained solely in terms of experimental
errors (which are in general not sufficiently defined).
This uncertainty is also reflected in the computer code
LOWTRAN.
12
In all versions of this computer code,
which is widely used in transmission and emission
modeling, water vapor absorption is reported as being
insignificant over visible and near-infrared window
regions.
Guzzi et al. 13,14 computed solar spectral irradiance
at the ground using the absorption coefficients taken
from Tomasi et al. 7 and Fraser8 and compared it to
experimental values. They found that the model underestimates the measured data and that the main
reason was due to the incorrect parametrization of the
aerosol extinction properties, which were accounted for
by using the Angstrom formulation.
Rizzi et al.,15 extensively using inversion techniques
to determine aerosol size spectra, found that the aerosol
optical depth must be computed with a standard deviation of <7% to obtain reliable size distributions.
These and other findings show that the correct estimation of water vapor window absorption in the visible
and near infrared is important when dealing with remote measurements both from the ground and from
space not only in atmospheric attenuation studies but
also in all satellite applications where adequate atmospheric corrections are needed.
This paper concerns the reduction of the data measured by Guzzi, Tomasi, and Vittori in the visible and
near-infrared regions, some of which have not yet been
processed. Our main concerns are to verify whether a
Lambertian absorption law is applicable to the most
transparent regions and to check the dependence of
optical thickness on precipitable water when going from
1854
APPLIED OPTICS/ Vol. 23, No. 11 / 1 June 1984
regions.
As
pointed out by several authors, problems are encounautomatic method is proposed to eliminate the recordings in which modification of the aerosol optical
properties not monitored would seriouslyinfluence the
determination of those of water vapor. Particular care
is taken in evaluating the errors associated with the
measured data to correctly assess the error limits of the
derived quantities.
II.
Data Set and Reduction
Data presented in this paper were measured from 22
Sept. to 9 Oct. 1971 at Buda in the Po Valley by Guzzi,
Tomasi, and Vittori and partially reported in Refs. 5-7.
The computations are made at wavelengths in which
evident relative maxima and minima in the deflection
curve of the original recordings are found. Although
the experimental apparatus was assembled to obtain a
resolution of better than 4.7 cm-1 in the spectral range
considered,5 the actual angular velocity of the prism
used in the set of measurements presented in this paper
allowed us to detect only broad features in the absorption bands. It follows that the computations in those
regions are done mainly to check the consistency of the
results compared with other reductions.
In Fig. 1 a recording of the solar spectrum between
0.65 and 1.10 m is shown as an example of those ana-
lyzed in this paper. Since the readings were made on
a strip-chart recorder, the first step has been to digitize
all the analog data at our disposal. Simultaneous
measurements of precipitable water vapor (pwv), performed with an infrared hygrometer (Tomasi and
Guzzi16 ), are associated with each recording.
Reduction of the data to obtain the total optical path
at wavelength X is made using the Bouger-Lambert
law
apt for describing cases of intermediate absorption (0.5
DQ() = R(X)Fo(X) exp[-a(Q)]TR(X)Tmg (X)To 3 (X),
where D(X) is the recorded deflection; o(X) is the sum
of particulate matter (pm) extinction optical path and
water vapor (wv) absorption optical path; TR(X),
Tmg(X), and To3 (X) are, respectively, the known spectral
transmittances due to molecular scattering, mixed
gases, and ozone absorption; FO(X)is the extra atmospheric solar irradiance; and R (X)is the electrical response of the instrumental set.
Do(A) = R(X)Fo(X) has been computed by the Lang-
ley plot method, by extrapolating to zero air mass the
logarithm of the deflection D (X)with respect to the air
mass. The computation was performed in the most
transparent regions using a subset of data taken in the
first hours of the morning, to avoid the effect of increasing atmospheric turbidity, for days with a visual
range >20 km. The technique has been described in
a previous paper.5 The results obtained for several
clear days have been averaged and the standard deviation associated with each Do(X) is found to be ade1
quately described by the relation
1 the curve Do(A) is also drawn.
0.07 Do. In Fig.
/ Do =
Once the Do are computed the pm and wv optical
path can be determined by (the dependence on wavelength will be omitted from now on for convenience)
aX) = n DoTRToaTmg
(1)
D
The transmittance functions appearing in Eq. (1)
were computed with the computer code LOWTRAN 5
using the average value of vertical ozone content measured at Vigna di Valle for the days the measurements
were taken.
A general law describes the optical path of wv as given
by Goody1 7 :
a
=
ifMW1
+
(2)
I) 1
where S is the average line intensity in the spectral interval centered at X, a is the mean halfwidth, is the
mean line spacing, w is the pwv along the vertical, and
mw is the air mass computed for a given water vapor
vertical distribution.
Equation
(2) can be approximated
in the cases of
absorption bands (strong absorption) and absorption
in the windows (weak absorption) by
w = al(MWW)1/2
SwMW
>> 1,
Swms
7ra
usually determined using a set of data taken on days in
which the measured pwv lie in a certain range. The
water vapor transmittance computed using Eqs. (2) and
(3) are, in the stated pwv range, in very good agreement.
However Eq. (3) does not describe the natural variation
of the absorption law from, for example, weak to intermediate or from intermediate to strong which is embedded in Eq. (2) as pwv content along the path varies.
Therefore, the use of the power law is based on the
consideration that in the actual atmosphere and in a
given spectral range the naturally occurring variations
of pwv do not require the use of a water vapor depen-
dence of the parameters a and b. Another difference
between transmittances computed by Eqs. (2) and (3)
is that, while the weak- and strong-line limits are contained in Goody's formulation, they must be fixed a
priori by limiting the value of b when using Eq. (3). A
X2 fit performed on some sets of data using Eq. (3) as
distribution to be fitted may lead to values of b outside
the permissible range. A x2 fit of the same set of data
using the Goody absorption law would not produce such
an evident result although the value of the determined
X2 would induce caution when analyzing the results.
This final consideration has led us to use the power-law
dependence to analyze the data to be able at a given
wavelength to eliminate from the set of available recordings a subset that eventually would clearly exhibit
physical mechanisms other than water vapor absorption.
The optical path afis therefore written as
= a(wmw)b + Tama,
(4)
where ma is the air mass for pm.
The aerosol term in Eq. (4) is usually greater than the
wv term in regions of weak absorption and of the same
order of magnitude in strongly absorbing regions. A
least-squares fit of the measured data to obtain a, b, and
ra would almost certainly lead to incorrect results, due
to the great natural variability of the pm optical properties. More information is necessary to eliminate or
normalize the aerosol optical thickness term. In our
case measurements of the visual range R were also
made.
Visual range measurements can be used to estimate
using Koschmieder's formula:
<< 1,
where a, and a2 are absorption coefficients. These
relations have been used by Gates and Harrop1 8 to
compute the absorption of water vapor in the near and
far infrared.
Moskalenko1 0 proposed a more general approxima-
/3H(O.55) = 3.912
R
0.55),
(5)
where 13m is the extinction coefficient due to molecular
scattering.
H (0.55) can also be computed:
1H(O.55) =
Cn(r)irr 2 Qe(0.55)dr = C3N(O.55),
(6)
where Qe is the efficiency factor for extinction, n (r) is
tion based on the following power law:
aw = a(wmw)b.
Strictly speaking, the parameters a and b are also a
function of the pwv optical path. In fact a and b are
the horizontal aerosol extinction coefficient at 0.55 um
7ra
as.= a(m.w)
b < 1).
(3)
It contains the weak- and strong-line limits and is also
the normalized differential aerosol size distribution at
radius r, and C is the pm concentration along the path.
Therefore ON is the normalized extinction coefficient.
1 June 1984 / Vol. 23, No. 11 / APPLIED OPTICS
1855
From Eqs. (5) and (6) it is possible to express C as
C =- 3
H(0.55)
/ N(O.55 )
(j)
(1) The data points of some recordings are in gen-
erally good agreement with the curve [Eq. (11)], that
*
(7)
The optical thickness along a vertical path can be
computed by
is,
(a) the percentage deviation
iJTa
TV(X)
= f n(r,h) f 7rr2Qedrdh,
(8)
where n(r,h) is the height-dependent pm size distribution function. Assuming that the size distribution
is independent of height, Eq. (8) can be approximated:
T,( ) = CH
f n(r)7rr 2 Qedr = CHON(X),
(9)
where H is the scale height for pm.
Inspection of Eqs. (7) and (9) allows us to write
H/3 H(0.55)N(X)
i(XR)
lN(O.55)
The applicability of the preceding relation is certainly
linked to the validity of the simplifying assumptions on
the pm vertical distribution and, above all, on the validity of Eq. (5). Assuming that the Koschmieder
relation holds and that the normalized size distribution
and scale height remain unchanged between two visual-range estimations R and R0 we can write the ratio
F(R,RO) as
3
'rvGN,R)
F(RR.) r(,R
0) =-O H
Tv
(,Ro,)
A
wavelengths is defined in which weak or null wv ab-
sorption is expected.
The quantity ro(X) = TV(XR
0 ) is computed using an
unweighted least-squares fit of the measured Ta (X)
belonging to all recordings to the curve
A
RiA o
recordings are retained.
(2) Some recordings show a different behavior:
at
some wavelengths Pij > Pmaxi while at others Pij <
-Pmaxi. These recordings are rejected since the size
distributions were certainly inhomogeneous with the
mean.
(3) In some cases the deviation Pi. is significant
(greater in absolute value than Pmaxi) but either systematically positive or negative at all wavelengths. The
visual-range value for the latter recording is modified
to agree with the average spectral properties using
1 M
Ravi
jRi
where
3.912
partially, by the experimental data, the measured visual
range can profitably be used to normalize the measured
pm optical thicknesses to some standard conditions
defined by a reference visual range R,
The following procedure was used to evaluate the
applicability of Eq. (10) to our data set. A set of
To(,
percentage deviation at all wavelength is <0.15. These
(10)
[where A = (3.912/R 0 ) - Om(0.55)] which is independent of wavelength. When Eq. (10) is verified, at least
G=
for the ith recording has a mean value Pi of <0.04,
computed using all wavelengths;
(b) the absolute value Pmaxi of the maximum of the
(11)
where Ri is the measured visual range during the ith
recording. The reference visual range R0 = 10 km is
selected because it has a value close to the visual range
during the experimental period. The set of values T0 (X)
constitutes a set of optical information relative to a
mean pm size distribution. Tm is the molecular optical
thickness.
The dispersion in the window wavelengths of the data
points around the curve [Eq. (11)] would constitute a
A
[+
A
Ta(XJ)]
To(X)
here M is the number of selected wavelengths used to
compute Ravi'
The latter recordings are then examined for acceptance according to the procedure already outlined in
cases (1) and (2) above. Out of forty-six available re-
cordings the procedure outlined rejected about half of
the data at each wavelength.
The final visual-range value associated with each
accepted recording (which will be referred to as the effective visual range Re) is either the measured value for
recordings belonging to case (1) or the arithmetic mean
computed using M specified window wavelengths for
recordings belonging to case (3).
Inspection of our data Ta (Xi) shows that it is not
strictly true that the function F(Re,R0 ) is independent
of wavelength. Therefore, two procedures were
adopted to select the window wavelengths to be used to
compute the effective visual range Re for each accepted
recording: the first (P1) uses all windows to estimate
Re; the second (P2) defines three regions in the spectral
interval under consideration, centered around the main
absorption bands, and three values of Re are computed
for each recording using all windows belonging to each
region.
measure of the applicability of the Koschmieder formula to our set of data, if the visual range estimations
were free of error (which is certainly not the case).
However, the comparative behavior at several wavelengths allows us also to examine the validity of the
assumptions regarding the pm size distribution. Sev-
Both procedures have been used to check the sensitivity of the retrieved parameters and their errors with
eral cases can be found:
length:
1856
APPLIED OPTICS/ Vol. 23, No. 11 / 1 June 1984
the choice of the window wavelengths. Once the effective visual range(s) is (are) computed, the pm optical
thickness can be scaled according to Eq. (10).
Equation (4) can finally be written for any wave-
,Y,i= a(WiMwi)b + cF(R,,iR.)m.i;
(12)
here mw is the relative air mass for wm computed using
a mid-latitude winter vertical distribution, ma is the
relative air mass for pm computed using a rural plus
tropospheric aerosol model giving a visual range at the
ground of 10 km.
Parameters
a, b, and c are to be
computed using a minimizingtechnique. The retrieved
value of c represents the spectral optical thickness of a
mean pm size distribution leading a visual range of 10
km.
IV.
Results and Discussion
On the basis of previous discussions the steps required to determine the absorption parameters are:
(a) Preprocessing. P1 or P2 is used to compute Re
for all accepted recordings. Trial minimizations are
attempted
at a few window wavelengths
to check
whether parameter b is within physical bounds (0.5 <
b < 1). When b lies outside the range by more than one
standard deviation the minimization is attempted after
eliminating the recordings one by one, starting with one
111. Minimization Technique
The observations o-iare treated as statistically independent and the expression for x 2 takes the form, at
each wavelength,
X2 =
2
i (a- -ai)
1
(13)
2i
where i is the standard deviation associated with each
oi. The i are computed according to the Gaussian
formula applied to Eq. (1):
2+1 2+1
=A28
AO +TR
+
Y
83+
1
-2
12
.m + D2 D,
where u0, A1R, 03, 1,mg,and D are standard deviations
associated, respectively, with the determination of Do,
TR, To,, Tg, D. AOincreases quasi-linearly with Do;
the errors in transmittance computations are assumed
to be equal to 0.005 (i.e., they affect the third significant
digit) while the estimated error in the deflection D is 5
mm.
To determine the spectral parameters a, b, and c, the
quantity x2 is minimized. The software package used
for the minimization belongs to the CERN Computer
Library.19 Only a brief description will be given of the
various methods adopted.
The process is started using a Monte Carlo technique.
The method by Nelder and Mead followswhich is reasonably fast when far from the minimum; it also estimates the diagonal elements of the covariance matrix
(the parameter errors). The algorithm used to find the
true minimum is Fletcher's switching method based on
Davidon, and the Fletcher and Powell algorithm. The
latter method is extremely fast and stable near the
minimum; it estimates the full covariance matrix which
is used as the starting point to compute true positive
and negative errors for each parameter separately taking into account the actual shape of the x2 curve near
the minimum.
If the function x2 is correctly normalized, that is, the
Ai are standard deviations, the computed parameter
errors are one standard deviation error for the parameters one by one. When the A? cannot be interpreted
as true variances but simply as relative weights, the
parameter errors resulting from such a fit are proportional to the unknown overall normalization factor.
taken at the lowest visual range until b is within permissible bounds.
(b) Processing. P1 or P2 is used to determine Re
and minimization is performed at all wavelengths.
During preprocessing, all the data recorded at visual
ranges <7 km were rejected. The result is not dependent on the previous Re computation since it was obtained using both P1 and P2.
The final processing of the data was performed several times to investigate the effect of different hypotheses on the final results. In particular, attention was
paid to the aerosol normalization in the 10-km visual
range procedure and to the effect of different error estimates on parameter value and derived parameter errors.
Some features were common to all the derived solutions. Regions of null absorption are clearly found
quite independently from the determination of Re.
Absorption coefficients in strongly absorbing regions
are only slightly affected by the aforementioned choice.
Some variations are observed in the value of the b parameter when P1 or P2 is used. In all cases, however,
the results are compatible since they are within one
standard deviation of the final results. The latter are
obtained using P2. The window wavelengths used for
the determination of Re are [the wavelength number
(wn) of Table I is used to identify the wavelengths]
(1) a band: 1 <wn < 10; window wn: 1,3,4,5,11,
12.
(2) 0.8 um band: 11 <wn < 25; window wn: 11,12,
26, 27, 28, 29, 30.
(3) poT band: 26 <wn <46; window wn: 26,27,28,
*29, 30, 42, 43, 44, 45, 46.
In some computations, previously determined wv
absorption coefficients in the windows were used to
improve the values of
Ta
(X) from Eq. (4). These latter
values were used to compute a new set of effective visual-range values, using the procedure already outlined.
The new minimization performed at all wavelengths
produced wv absorption coefficients that lie within the
error limits specified in Table I where a complete set of
results is shown. The parameter a is expressed in units
of cm-b. The parameter errors Aa and Ab are the
largest of the reduced positive and negative errors
computed at each wavelength. The statistical approach
used in computing the latter quantities is outlined in the
Appendix.
In Figs. 2-4 transmittance computed using present
data and LOWTRAN 5 are compared. The equivalent
sea-level absorber amount is 2.76 cm (a slant path from
1 June 1984 / Vol. 23, No. 11 / APPLIED OPTICS
1857
Table 1. Parameters a and b and the AssociatedErrors are Written with a Number of Digits Which Exceeds by One the Number of Significant Digits
wn
M(m)
v(cm-')
a
Aa
b
Ab
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
0.6821
0.6974
0.7010
0.7062
0.7125
0.7194
0.7207
0.7233
0.7289
0.7447
0.7457
0.7561
0.7734
0.7776
0.7825
0.7918
0.8000
0.8071
0.8163
0.8220
0.8230
0.8261
0.8292
0.8418
0.8489
0.8503
0.8764
0.8807
0.8857
0.8905
0.9001
0.9038
0.9074
0.9112
0.9137
0.9200
0.9350
0.9398
0.9443
0.9690
0.9737
0.9780
1.005
1.030
1.043
1.048
14661
14340
14265
14160
14035
13900
13875
13825
13720
13630
13410
13225
12930
12860
12780
12630
12500
12390
12250
12165
12150
12105
12060
11880
11780
11760
11410
11355
11290
11230
11110
11065
11020
10975
10945
10870
10650
10640
10590
10320
10270
10225
9950
9710
9590
9540
0.037
0.024
0.034
0.019
10-4
0.183
0.031
0.107
0.083
0.012
0.010
0.007
0.006
0.006
0.032
0.010
0.016
0.016
0.005
0.014
0.010
0.010
0.011
0.011
0.014
0.010
0.014
0.017
0.015
0.013
0.012
0.014
0.022
0.010
0.011
0.009
0.009
0.009
0.018
0.034
0.016
0.020
0.016
0.038
0.017
0.049
0.016
0.028
0.042
0.044
0.005
0.006
0.005
0.004
0.006
0.89
1.0
0.866
1.0
0.16
0.13
0.083
0.088
0.50
1.00
0.72
0.90
0.20
0.15
0.13
0.17
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.67
0.625
0.582
0.534
0.601
0.522
0.47
0.50
1.0
1.0
1.0
1.0
0.37
0.21
0.26
0.23
0.22
0.24
0.19
0.17
0.063
0.082
0.047
0.089
0.074
0.20
0.13
0.21
0.21
0.16
0.15
0.65
0.69
0.615
0.670
0.49
0.67
0.500
0.684
0.535
0.50
0.50
0.18
0.16
0.082
0.072
0.15
0.14
0.011
0.025
0.039
0.14
0.11
1.0
0.18
ground to space at a zenith angle of 65° using a midlatitude winter wv model).
The agreement between the plotted data is quite
satisfactory in the and poT bands, once account is
taken of the difference in spectral resolution between
the two sets of data.
Transmittances
in the 0.8-Mm
band computed with our coefficients are consistently
smaller than LOWTRAN'S.
Regions of complete transparency to wv are found at
wn = 5, 10, and 42-45. At some spectral ranges (wn =
11, 12, 26-29, and 46) the absorption coefficient a is
close to zero and the associated error is greater than the
parameter value itself so that these regions can be re-
garded as completely transparent. Absorption in the
window is evident at wn = 1-4 and 13-17, and differences between our results and those of Tomasi et al. 7
and Tomasi9 are within the error limits. The increase
in the coefficients around 0.70 Am is similar to that
1858
APPLIED OPTICS/ Vol. 23, No. 11 / 1 June 1984
l0-4
0.002
0.002
0.016
0.041
0.034
0.041
0.038
0.077
0.241
0.171
0.253
0.142
0.180
0.098
0.064
0.009
0.003
0.0005
0.002
10-4
0.156
0.092
0.197
0.183
0.218
0.099
0.826
0.522
0.654
0.175
0.220
10-4
10-4
10-4
l0-4
0.002
observed in Tomasi et al. 7 The value of b is close to 1
except at wn = 3 in which a slight departure from the
Lambertian law is found.
The variation of the parameter b is evident going
from regions of weak to strong absorption.
As noted in Sec. I, Moskalenko 1 0 has found, in the
range of our interest, a value of b = 0.53 and Koepke and
Quenzel"l have determined their absorption coefficients
using the same value of b. This means that a graphical
comparison between our data and those of Koepke and
Quenzel would need a different set of transmittance
computations at varying optical depths to be made and
plotted. The information content of such plots is certainly less than a direct inspection of Table I of the
Koepke and Quenzel paper.
The main differences are found around 0.735,4m in
which no absorption is evident in our recordings, in the
region from 0.77 to 0.79 Am at which absorption is found
.
100
0,70
I
.
I
.
.
.
.
.
0,7
I I
I--
--1;--I
'
X '--
A(YM
0080
0.4
TO
(%1
0.3
I
.
.
I
I
I
.
.
,
I
1,
,.
.
14000
,
.
.
.
13000
v(cn-'
Fig. 2. Water vapor transmittivity in percent vs wave number v(a
band). Precipitable water is 50-mm (STP). The dashed line is the
transmittivity computed using the data in Table I. For comparison
transmissivity
computed using LOWTRAN 5 is drawn.
0.21-
0.1
I
I
I
I
I_
11000 v(cm')
13000
5000
Fig. 5. Retrieved values of c (dots) as a function of wavelengths X
and associated one standard deviation. Mean (relative to 10-km visual range) particulate-matter
optical thickness To is also drawn
(triangles).
I
I
I
.
I
1
111
.
.
.
..
......
......
.
005
Fig. 3.
Same as Fig. 2 for the 0.8 H2 0 band.
0.0
50
0.15
I
,..,i,
0.1
11000
Fig. 6.
10000
v (crrf')
Fig. 4.
1
10
w(cm STP)
Relative transmissivity T as a function of water vapor
content at wn = 32 and 35.
Same as Fig. 2 for the par H2 0 band.
in our data, and at wavelengths around 0.980 Am which
is transparent in our computation.
The retrieved values of c are drawn in Fig. 5. The
values used to normalize the aerosol contribution are
shown in the same figure. The two sets of values com-
puted with different procedures are in good agreement.
A weighted least-squares fit has been applied to the c
values,assuming the relationship c = fX-a to find 3and
a. The values obtained are = 0.26 and a = 0.76.
The x2 minimization using all accepted recordings
has been performed, with Goody's model as the fitting
function, at wavelengths wn = 32 and 35, which are
characterized, respectively, by medium and strong absorption:
2
i
[a- Plimwi
(I +
2 p2Wimwj)-"
2 2
]
where p1 and P2 are defined by inspection of Eq. (2). In
Fig. 6 the quantity
T
Tm(X) - Tg(X)
Tg (X)
is plotted, where Tg and Tm are transmissivities
com-
puted with the optical path given by Eqs. (2) and (3).
Since the wv optical path in our recordings ranges between a minimum of 0.9 and a maximum of 4.7 cm, the
value of Tp in that region is smaller than 0.015. Within
the interval from 1 to 15 cm, which covers most values
normally encountered-in the atmosphere (Fig. 6), the
maximum value of Tp for the strong region is 0.035; in
1 June 1984 / Vol. 23, No. 11 / APPLIED OPTICS
1859
the region of intermediate absorption the discrepancy
between Tm and Tg increases reaching a value of Tp =
-0.09 with an optical path of 15 cm [Tg (15 cm) =
0.603].
We wish to add some information on unexpected
absorption which is clearly evident in all the recordings
in the ranges 11,770-11,680 and 11,550-11,470 cm-.
No dependence on wv is found in the two regions but,
as seen in Fig. 3, the absorption is relevant and cannot
ranges from a minimum of 0.80 to a maximum of 1.30.
In general there is a decrease of x2 as X increases.
Taking S as the variance of the fit and Mi as variances
associated with the data, the reduced x2 can be expressed by
2=e
xv =
X
1
,
-,
1
E 1
be attributed to liquid water absorption since it is also
evident in recordings taken at the visual range >20
km.
The M are characteristic of the dispersion of the data
around the parent distribution and are not descriptive
The data at our disposal do not allow us to assess
whether the weak absorption in the windows is a con-
is characteristic of both the spread of the data points
and the accuracy of the fit. Since the fitting function
is considered a good approximation to the parent
tinuum caused by accumulated contribution by distant
strong absorption lines. However, the region of nearly
complete transparency found at wn = 5-10 indicates
that the eventual continuum may not extend at wavelengths smaller than 0.74 m (nothing can be said about
the wavelength range smaller than 0.68 m). The computed value for wn = 30 is affected by a large error and
no conclusions can be drawn in this respect.
of the fit. The estimated variance of the fit S 2 , however,
function, the values of x2 can be interpreted
Conclusions
V.
1-i = Jgi,
A set of measurements of spectral extinction of solar
radiation is analyzed to determine the magnitude and
associated error of the water vapor absorption coefficients in the range from 0.68 to 1.05 Am.
A power-law relationship describes the dependence
of the optical path on water vapor content along the
path.
The role of particulate matter in extinguishing solar
radiation is relevant. Instead of trying to eliminate,
from the measured optical paths, the contribution due
to aerosol, the latter is described in terms of mean particulate matter conditions. In this way, recordings are
eliminated in which the aerosol optical properties are
sensibly different from the means and would seriously
influence the determination of those of water vapor.
The water vapor absorption coefficients, the mean
aerosol optical depth, and associated estimated true
errors are computed using a weighted least-squares
fit.
The procedure adopted to determine mean optical
properties has been found to be quite successful; also
the retrieved parameters show weak dependence on
aerosol normalization. The proposed methodology
appears to be applicable to any spectral measurement
in real atmospheres.
Appendix
If Eq. (12) in the text is a good description of the
phenomena under study and if the error estimates are
close to the real values, the computed minimum value
of X2 at any wavelength must be close to the number of
degrees of freedom v, which can be computed once the
number of experimental points and the type of function
to be minimized are known.
It is found that the value of the reduced x 2 ,
Xv =
1860
X2/v,
APPLIED OPTICS/ Vol. 23, No. 11 / 1 June 1984
as being
caused mainly by the somewhat incomplete specification of the errors entering into the computation of Mi.
Therefore, it is possible to obtain an estimate of a
wavelength-dependent overall normalization factor fj
for the experimental error at any wavelength which is
found to be
f
= aXj
at any wavelength. The term fj ranges, therefore, between a maximum value of 1.14 and a minimum of
0.89.
The final parameter errors, which will be called reduced errors, are obtained by multiplying by fj those
computed during the fit. The correctness of this procedure depends on the assumption that the fitting
function is a good approximation to the parent one.
This hypothesis can be tested by doing a minimization
using the normalized data error estimates M'. These
tests showed that the parameter values agree (as ex-
pected) to three significant digits with that computed
using the original variance estimates; there is also a two
significant digit agreement between the reduced parameter errors (from the first fit) and the parameter
errors coming put of the fit using the normalized errors
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Non-Invasive Assessment of Visual Function, OSA
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Amsterdam L. Roos, FOM Inst. for Atomic & Molecular Physics, Kruislaan 407,NL-1098 SJ Amsterdam, The Netherlands
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2nd European Conf. on Atomic & Molecular Physics,
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A
Synthesis of Bayesian & Frequentist Approaches, U.
May
Florida, Gainesville NSF, Math. Sciences, Wash.,
D.C. 20550
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Optical Remote Sensing of the Atmosphere, OSA
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OSA/IEEE Lasers & Electro-Optics Conf., Baltimore
Mtgs. Dir., OSA, 1816 Jefferson P., N. W., Wash., D. C.
20036
Top. Mtg., Lake Tahoe OSA, Mtgs. Dept., 1816 Jefferson P., N. W., Wash., D.C. 20036
continuedonpage1880
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1861