REMOTE SENSING OF ENVIRONMENT 12:141-149 (1982)
141
Irradiance Measurement Errors Due
to the Assumption of a Lambertian Reference Panel
D. S. KIMES AND J. A. KIRCHNER
Earth Resources Branch, Code 923, NASA / Goddard Space Flight Center, Greenbelt, Maryland 20771
Total and diffuse global spectral irradiances, which are often required field measurements in remote sensing, are
commonly obtained by measuring the radiance from a horizontal reference panel with assumed Lambert/an properties.
A technique is presented for determining the error in diurnal irradiance measurements that results from the
non-Lambertian behavior of a reference panel under various irradiance conditions. Spectral biconical reflectance factors
of a spray-painted barium sulfate panel, along with simulated sky radiance data for clear and hazy skies at six solar
zenith angles, were used to calculate the estimated panel irradiances and true irradiances for a nadir-looking sensor in
two wavelength bands. The inherent errors in total spectral irradiance (0.68 #m) for a clear sky were 0.60, 6.0, 13.0,
and 27.0% for solar zenith angles of 0 °, 45 °, 60 °, and 75 °. The technique can be used to characterize the error of a
specific panel used in field measurements and thus eliminate any ambiguity of the effects of the type, preparation, and
aging of the paint.
Introduction
In remote sensing applications the total
and diffuse global spectral irradiances (W
m -9 #m - t impinging on a horizontal
surface) are often required field measurements. These measurements are used both
to calculate reflectance factors of targets
for interpreting remotely sensed data
(Kimes et al., 1980) and to characterize
irradiance conditions for applications in a
number of disciplines. A common method
of obtaining an estimate of the total global
^T
spectral irradiance (Ex)
is to measure the
spectral radiance (Lx) from a horizontal
reference panel of known hemispherical
spectral reflectance (0x) and assumed
Lambertian properties. The equation used
is
An estimate of the diffuse global spectral
irradiance (E^D
x ) is obtained by measuring
©Elsevier Science Publishing Co., Inc., 1982
52 Vanderbilt Ave., New York, NY 10017
the spectral radiance from a shaded panel
(L~,) where only the direct solar flux is
blocked by a sun shade
1~ = L'x ( ~r/px ).
AS
The direct solar irradiance (E x) is calculated as
AS
AT
Ex= Ex -/~.
(1)
A problem with all field measurements
which use a reference panel is that errors
are introduced by assuming that the panel
has Lambertian reflectance properties. In
this study, the errors in measurement of
the above irradiances due to the
non-Lambertian behavior of a painted
barium sulfate panel were explored for
two simulated irradiance conditions and a
range of solar zenith angles. A simulation
approach was taken because errors of this
type are very difficult to measure directly
in the field.
0034-4257/82/020141 +9502.75
142
D. S. KIMES AND J. A. KIRCHNER
Theory
direction of 0 ° to a given direction; and
An extremely useful tool in researching
anisotropic radiation problems is the concept of the bidirectional reflectance distribution function, which describes the
unique scattering properties of a surface
independent of the anisotropy of the
incident radiance distribution. This conceptual function cannot be measured in
practice, however, so it is replaced by the
nearest possible a p p r o x i m a t i o n ~ t h e
biconical reflectance factor. The terminology has been presented by Kriebel (1977)
and Nicodemus et al. (1977), among
others, and is briefly described below.
The spectral biconical reflectance factor, Rx(f~i,f~,), is the radiant flux reflected into a finite solid angle divided by
the radiant flux reflected by a perfectly
reflecting Lambertian panel into the same
finite solid angle under identical irradiance conditions. Assuming that the incident radiance is constant over each finite
solid angle ~i, then (symbols follow
Kriebel, 1977)
a~t( ~'~i, ~'~r) =
ferf~ yx, r(O~, ~,,; O,, ,~,)cos O~d r , cos Ord~2r
Or,4r)
is the bidirectional reflectance
distribution function for incidence direction 0i, q5i and reflectance direction Or, Or"
A perfectly reflecting Lambertian panel
would have a constant Rx of 1.0.
From knowledge of the spectral biconical reflectance factors (Rx) of a barium
sulfate reference panel and of the anisotropic incident radiances for the entire
global hemisphere,/~ may be calculated
as:
^T
E x = LA.r(~r)~
S2,
where Lx,r(~2,) is the reflected radiance
from a panel into solid angle fir, Lx.i(f~i)
is the incident radiance from solid angle
~2i, P/= f~2 cos Oi df~ i , and the summation
is taken over all sources of incident radiation. E^Dx is calculated via Eq. (2) except
that the direct solar sources are omitted,
a n d / ~ is calculated by Eq. (1).
The true irradiance (E T) is calculated
as a summation over all sources of incident radiation:
1 f, f~ cosOid~icosO, d~r
E~'= EL~,.,(~2i)P
where
~i and f~r are the finite solid angles of
incidence and reflection, respectively;
df~
is equal to sinOd0dcp, where 8
is measured from the surface
normal to a given direction, and
0 is the azimuth angle measured from a defined surface
~.
The true Ex° and E~ irradiances are
calculated in a similar fashion except that
the direct solar and diffuse solar sources
are omitted, respectively.
For a barium sulfate panel the percentages of error of the measured total,
diffuse, and direct solar irradiances with
respect to the actual irradiances are
143
IRRADIANCE MEASUREMENT ERRORS
calculated, respectively, as:
E T _ ~T
terence tilter was placed between the
source and the panel. Spectral measurements were taken in NASA's Thematic
Mapper bands 3 (0.63-0.69 /zm) and 4
(0.76-0.90 /am). The entire illumination
assembly was mounted on a movable arm.
The angle of incidence (0 i) as measured
from the panel normal was varied from 15
to 70 ° . The extreme angles (e.g.,
< 1 5 °, > 7 0 °) were not measured because of the physical limitations of the
system.
The sensor was a fiber-optic probe assembly that collected the radiation
scattered into a small solid angle (0.0002
sr). The sensor view angle was fixed normal to the panel because this is the most
common mode of measurements in the
field. The tiber optics transmitted the light
to a photomultiplier tube which was oper-
E l ' - ~I' (100),
x _~ x(100),
E°
"S
and
ES- Ex (1001. (3)
Instrumentation and Methods
The sprayed barium sulfate panel used
in this study was prepared as described
by Shai and Schutt (1971).
A goniophotometer (Fig. 1) was used to
measure the relative biconical reflectance
factors of the barium sulfate panel. The
panel was illuminated by a beam from a
tungsten source, which was confined to a
small solid angle (0.0002 sr). An inter-
~
ER OPTIC
•-.11~ P. M TUBE
TURE
,•"•
POINT SOURCE
~'~,~¢y.~. FILTER
" ~ , , N N . LENS
\ " .~.- ~" VARIABLE
X,2,~,, APERTURE
I
I
I
I
I
I
" % ,
f
O -"~1 I
I
I
t
P
m,x
90 °
I
I
SAMPLE
FIGURE 1. Schematic of the goniophotometer as described by Viehmann (1978). The
incidence angle, 8i, is measured from the sample normal, and the sensor was fixed normal to
the sample.
144
D. S. KIMES AND J. A. KIRCHNER
ated in the photoelectron counting mode.
The projected area of the sensor's field of
view upon the sample was always larger
than, and completely included, the projected area of the illuminating beam. The
background noise due to stray light was
always less than 1% of the signal.
Thirteen points on the panel were sampled for each spectral band at incidence
angles (0~) of 15 °, 25 °, 35 °, 45 °, 55 °,
65 ° , and 70 ° . Means and error bounds
(evaluated as twice the standard error)
were calculated.
In any such measurements the flux
[~x.~(~)] received by the sensor is
•
=
ar)e
,
where Ox,~(~2~) is the total flux from the
source and
Pr = f~ COS6~ d ~
(see Young et al., 1980, for the origin of
this equation). In this experiment the
terms 1/~r ~x,i(f~i) and Pr are constant
(viewing angle fixed at nadir). Once the
value of Rx(~ ~, f~r) is determined for a
particular value of i, the above equation
can be solved to determine the incident
flux and thereby determine the absolute
value of Rx(f~ i, ~2r) for other angles i.
The value of ax(~i,~r) is defined
as 1.0 for a surface which is a lossless
Lambertian reflector. In order to establish
that our panel behaves as a Lambertian
reflector for incidence angles near nadir,
the sensor response was measured at angles between 0 ° and 75 ° (excluding the
angle coincident with the sourse) for incidence angles of 0 ° and 15 °. In both cases,
the response was within 1% of that of a
Lambertian surface. Other investigators,
including Hsia and Richmond (1976),
Viehmann (1978), Robinson and Biehl
(1979), and Young et al. (1980), have also
shown Lambertian behavior at near-nadir
incidence angles for sprayed barium
sulfate paints as well as for other types of
reference paints.
The uncertainty introduced by our lack
of measurements beyond 75 ° is small.
The projected solid angle of an entire
hemisphere is or, and the proportion of
the projected area of the hemisphere contributed by the 75 ° to 90 ° interval is only
0.067. Any reasonable deviation from
Lambertian behavior in this region would
consequently have only a small effect on
irradiance measurements. Thus it was
convenient to normalize the sensor responses in this study by dividing by the
15 ° response (closest nadir measurement
available). The results are the biconical
reflectance factors, Rx(~2~, f20r ), assuming
a perfect (lossless) reflector, where ~r° is
in the nadir direction.
The missing Rx(f~i,f~ °) for incidence
angles of 5 ° , 75 ° , and 85 ° were obtained
as follows. The 5 ° value was set equal to
the 15 ° measured value since for angles
this close to nadir Rx(~2i , f~0) is essentially
constant as shown above. The 75 ° and
85 ° values were determined by extrapolation of a cubic equation (constant coefficient fixed at 1.0) which was obtained by
a least-squares fit of the measured data.
These data were compared with other
values from the literature as reported in
the results and discussion section.
Sky radiance distributions were obtained from the atmospheric models reported by Dave (1978). These models
realistically simulated the anisotropic
spectral r a d i a n c e distributions of
cloud-free midlatitude summer terrestrial
atmospheres for various levels of atmo-
IRRADIANCE MEASUREMENT ERRORS
145
TABLE 1. Total global irradiance and diffuse/total ratios for Dave's (1978)
atmospheric models ~ (clear) and #4 (hazy) at six solar zenith angles and two
wavelengths.
TOTALGUaBALIRnAmANCZ(W-em-2-gm -1)
AND
DIFFUSE/TOTALRATIO(IN PARENTHESES)
RED
(~. = 0.678 gm)
IR
(~, = 0.7963 gin)
SOLARZENITHANGLE
CLEAR
HAZY
CLEAR
HAZY
0
0.137
(0.10)
0.118
(0.11)
0.095
(0.14)
0.130
(0.36)
0.110
(0.39)
0.084
(0.44)
0.108
(0.09)
0.093
(0.10)
0.075
(0.12)
0.103
(0.34)
0.088
(0.38)
0.069
(0.43)
0.085
(0.18)
0.o42
(0.25)
0.055
(0.56)
0.o32
(0.69)
0.052
(0.16)
0.034
(0.9.9)
0.o44
(0.53)
0.026
(0.65)
0.030
(0.31)
0.021
(0.78)
0.025
(0.26)
0.018
(0.75)
30
45
60
70
75
spheric aerosols and absorbing gases and
for seven solar zenith angles. Two of the
models, numbers 3 and 4, representing a
clear and hazy atmosphere, respectively,
were used in this study at the selected
wavelengths of 0.6780 g m (red) and
0.7965 g m (photographic IR), and at six
of the seven solar zenith angles, namely,
0 °, 30 °, 45 °, 60 °, 70 °, and 75 °. The total
irradiance and the diffuse/total ratio for
each case are presented in Table 1.
The above reflectance factors and sky
radiance data were used to calculate the
percent irradiance errors Eq. (3). In all
calculations the solid angles of incidence
and reflection were divided into 108 discrete solid angles, where 0 and q~ were
discretized into nine 10 ° and twelve 30 °
intervals, respectively.
Results and Discussion
The spectral biconical reflectance factors for the barium sulfate panel are pre-
sented in Fig. 2. The error bounds (two
times the standard error) on these mean
values varied from 0.003 to 0.012 for the
15 ° and 70 ° incidence angles, respectively, for the red band and were similar
for the IR band. Our measurements fall
within the range of other reported values.
For example, Young et al. (1980) reported
relative reflectance factors of a sprayed
barium sulfate paint for incidence angles
comparable to ours (ranging from 35 ° to
75°). No wavelength was reported. Our
values decreased on the order of 27%
from 35 ° to 75 ° for both bands, while
those of Young et al. (1980) decreased
20% in this range. Viehmann (1978) has
taken spectral biconical reflectance factor
measurements (wavelength 0.633 #m) of
a sprayed barium sulfate panel. His results show that the reflectance factor decreased relative to a normal incidence
angle by 0, 16, 33, and 67% for incidence
angles of 40 ° , 60 ° , 70 ° , and 80 ° from
normal, respectively. By comparison, our
values are conservative. It is believed that
146
D. S, KIMES AND J. A. KIRCHNER
T
f
I
~
i
T
[
T
]
1.0
A
0.9
C
0.8
d
rr
0.7 t
,~ RED BAND
• I R BAND
0
10
20
30
40
®@
50
60
70
80
i
i
90
ZENITH INCIDENCE ANGLE (8i)
FIGURE 2. Spectral biconical reflectance factor, R(£i,£~J), as a
function of incidence angle. The reflectance direction (~2°) is centered
at nadir, Circles indicate extrapolated values.
the variation between studies is largely
due to variations in paint properties.
Figure 3 shows the percentage error of
the total, diffuse, and direct irradiances as
a function of solar zenith angle in the red
band for clear and hazy skies. It is obvious that as the solar zenith angle increases, the error increases for all three
irradiances. The direct irradiance error is
identical for the hazy and clear skies as
would be expected. The diffuse irradiance
error is less for the hazy sky than for the
clear sky in all cases, because for the hazy
sky, the sky radiance is more concentrated around the solar disk and does
not increase at the horizons as it does for
a dear sky. At solar zenith angles near
75 ° , the sky radiance for hazy skies is still
concentrated around the solar disk, but
the remainder of the diffuse light is spread
more evenly throughout the sky, causing
radiances to be relatively higher at nadir
than for the clear sky. Thus, for all solar
zenith angles the sky radiance distribution of the hazy sky versus that of the
clear sky tends to be weighted more towards incidence angles with smaller panel
errors [i.e., Rx(~i,~r0) values approach
1.0 for incidence angles near nadir] and
de-emphasizes the incidence angles near
the horizon, at which the panel errors are
greatest.
The total and direct irradiance errors
for the clear sky follow each other closely,
because the diffuse/total ratio is relatively low for all solar zenith angles (see
Table 1). However, for the hazy sky the
diffuse/total ratio is much higher, and
IRRADIANCE MEASUREMENT ERRORS
147
32
CLEAR SKY
28
= DIFFUSE IRRADIANCE
= DIRECT IRRADIANCE
= TOTAL IRRADIANCE
•
*
24
cc
O
rc
n."
20
I.Z
w
O
16
w
ta
12
0
10
20
30
40
50
60
80
70
S O L A R Z E N I T H A N G L E (8i)
32-
HAZY SKY
t
cg,'222g'2202
'2 :
/
o = DIFFUSE IRRADIANCE
28
/
:
24
nO
t-,¢,,i
I.Z
w
O
o7
w
ta
2O
16
12'
I
0
10
20
30
40
50
60
70
8O
S O L A R Z E N I T H A N G L E (8 i)
FIGURE 3. Percentage error of irradiances as a function of solar zenith
angle for the red band (0.68 #m) and clear and hazy skies. The errors of the
difh~e, direct, and total irradiances were calculated according to Eq. (3).
148
D. S. KIMES AND J, A. KIRCHNER
32
CLEAR
SKY
/
u ~ DIFFUSE IRRADIANCE
• = DIRECT IRRADIANCE
* = TOTAL IRRADIANCE
28
/
/
t
_/.*
24
nO
n"
u.I
I.-Z
w
O
or"
w
o..
2O
16
12
0
10
20
SOLAR
32
30
ZENITH
40
50
ANGLE
60
70
(8i)
HAZY SKY
u
•
*
28
80
•
= DIFFUSE IRRADIANCE
= DIRECT IRRADIANCE
= TOTAL IRRADIANCE
24
or"
O
rr
r,r"
w
I.Z
2O
16
UJ
(..)
rr
W
12
13..
0
10
20
30
40
50
60
70
80
S O L A R Z E N I T H A N G L E (8 i)
FIGURE 4. Percentage error of irradiances as a function of solar
zenith angle for the infrared band (0.80/~m) and clear and hazy skies.
The errors of the diffuse, direct, and total irradiances are calculated
according to Eq. (3).
IRRADIANCEMEASUREMENTERRORS
thus the total irradiance error approaches
the diffuse irradiance error more closely.
The irradiance errors for the IR band
(Fig. 4) are very similar to those for the
red band because of similar anisotropic
sky radiance distributions, diffuse/total
ratios, and biconical reflectance factors.
Summary and Conclusions
Characterizing the errors in irradiance
measurements is important. For example,
in remote sensing studies an understanding of the behavior of target reflectance as a function of solar zenith angle is
valuable to several applications. In these
studies, reference panel errors cause false
diurnal reflectance trends (Kimes et al.,
1980).
This study provides a technique for
determining the error in diurnal irradiance measurements for a reference panel
and various irradiance conditions. The
technique is simple enough that it can be
used to characterize the error of a specific
panel used in field measurements and
thus eliminate any ambiguity o[ the effects of the type, preparation, and aging
of the paint. In addition, the technique
provides a framework for approximating
the true irradiance from panel measurements, as long as the difhtse/total ratio
can be measured and the sky radiance
distribution can be approximated.
We express our appreciation to Walter
Viehmann and Norman Helmold for use
o f the goniophotometer, and to ]ohn
Schutt and Brent Holben for support in
laboratorg measurements.
149
models with aerosols and common absorbing gases. Sol. Energy 21:361-369.
Hsia, J. J. and Richmond, J. C. (1976), Bidirectional reflectometry. Part I. ]. Res. Nat.
Bur. Standards--]. Phys. Chem. 80A:180205.
Kimes, D. S., Smith, J. A., and Ranson, K. J.
(1980), Vegetation reflectance measurements as a function of solar zenith angle.
Photogram. Eng. R e m o t e Sens.
46(12):1563-1573.
Kriebel, K. T. (1977), Reflection Properties of
Vegetated Surfaces: Tables of Measured
Spectral Biconical Reflectance Factors.
Universitat Muenchen, Meteorologisches
Institut, Wissenschaftliche Mitteilung Nr.
29.
Nicodemus, F. F., Richmond, J. C., Hsia, J. J.,
Ginsberg, I. W., and Limperis, T. L. (1977),
Geometrical considerations and nomenclahire for reflectance. National Bureau of
Standards Monograph 160, U.S. Govt.
Printing Office, Washington, D.C.
Robinson, B. F. and Biehl, L. L. (1979),
Calibration procedures for measurement of
reflectance factor in remote sensing field
research. SPIE 196:16-26.
Shai, C. M. and Schutt, J. B. (1971), Formulation procedure and spectral data for a highly
reflecting coating from 200nm to 2300nm.
Doe X-762-71-266, NASA/Goddard Space
Flight Center, Greenbelt, MD, 8 pp.
Viehmann, W. 1978. Unpublished data.
Materials Control and Applications Branch,
NASA/Goddard Space Flight Center,
Greenbelt, MD.
Young, E. R., Clark, K. C., Bennett, R. B.,
and Houk, T. L. (1980), Measurements and
parameterization of the bidirectional reflectance factor of BaSO4 paint. Appl. Opt.
19:3500-3505.
References
Dave, J. V. (1978), Extensive datasets of the
diffuse radiation in realistic atmospheric
Received41une1981;revised18 November1981.