The concept is described of deducing the temperature profile of a planetary atmosphere from orbiter measurements of the planet's IR limb radiance profile. Expressions are derived for the weighting functions associated with the limb radiance profile for two infrared tranmission models. Analysis of the weighting functions for the Martian atmosphere indicates that a limb radiance profile in the 15 micron CO2 band can be used to determine the Martian atmospheric temperature profile from 20 to 60 km

Ohring, G.

English

NASA Technical Reports Server

To: National Aeronautics and Space Administration
Headquarters
Washington, B.C.
20 October 1971
GCA CORPORATION
GCA TECHNOLOGY DIVISION
Bedford, Massachusetts
A STUDY OF PLANETARY METEOROLOGY
Prepared by: George Ohririg
Quarterly Progress Report No. 2
Covering the Period
1 July 1971 - 30 September 1971
Contract No. NASW-2221
https://ntrs.nasa.gov/search.jsp?R=19720006179 2020-03-11T21:48:47+00:00Z
TABLE OF CONTENTS
Section Title Page
SUMMARY ii
1 INTRODUCTION 1
2 DISCUSSION 5
2.1 Goody Random Model 6
2.2 Bartko-Hanel Transmission Model 9
2.3 Weighting Functions and Instant 12
Inversion
3 FUTURE PLANS 18
REFERENCES 19
SUMMARY
The concept is described of deducing the temperature profile of
a planetary atmosphere from orbiter measurements of the planet's IR
limb radiance profile. Expressions are derived for the weighting
functions associated with the limb radiance profile for two infrared
transmission models. Analysis of the weighting functions for the
Martian atmosphere indicates that a limb radiance profile in the 15ja
CO- band can be used to determine the Martian atmospheric temperature
profile from 20 to 60 km.
ii
1. INTRODUCTION
The primary objectives of the current research program are:
(1) To study the meteorology of the planet Jupiter, with particular
emphasis on the deduction of the atmospheric temperature profile through
analysis of available infrared and microwave observations of the planet's
emission spectrum.
(2) To develop and apply inversion techniques for inferring atmo-
spheric temperature and constituent profiles of planetary atmospheres.
During the first quarterly period, work was performed on the first
objective listed above. This work involved a preliminary analysis of
the Jovian infrared emission observations and an estimate of the tempera-
ture profile based upon this analysis. A complete discussion was pre-
sented in Quarterly Progress Report No. 1. During the second quarterly
period, work was performed on the second objective listed above, and,
in particular, on the concept of measuring the IR limb radiance profile
of a planetary atmosphere from an orbiter to infer the temperature
profile. This work is reviewed in this quarterly report.
The inference of atmospheric temperature and constituent profiles
from measurements of the spectral distribution of the emitted thermal
radiation from a planetary atmosphere has received a good deal of atten-
tion recently. Such experiments have already been successfully flown
in the Earth orbiting NIMBUS satellites (see, for example, Conrath
et al., 1970), are being flown on the current Mariner flight to Mars,
and will be flown on future flights to Mars and Venus. Another tech- ,
nique, based upon scanning the limb of the planet rather than scanning
in wavelength, can also provide valuable information on the vertical
profiles of temperature and constituents of a planetary atmosphere.
This concept is just beginning to receive attention (reference).
This technique has certain advantages over the spectral scan, and
can be used to complement spectral scans.
\
The basic geometry of the limb, or horizon, radiance observation
is shown in Figure 1. A high angular resolution (~0.1 degrees for orb-
3
ital altitude of 10 km) radiometer on a planetary orbiter intercepts
radiation emanating from an atmospheric path that is tangent to the
planetary surface at height h, which is termed the tangent height. In
the figure, R represents altitude. The wavelength interval to which
the radiometer is sensitive is determined by the nature of the experiment.
If the temperature profile is the object of the experiment, the wave-
length interval will lie in an atmospheric absorption band of a consti-
tuent whose distribution is known - e.g., the 15u CO* band would be a
good choice for Earth, and, possibly, Mars and Venus. If the water vapor
profile is the object of the measurement, then an infrared water vapor
band would be selected. The radiation received by the instrument is a
function of the distribution of temperature and absorbing gas along the
ray path. If the distribution of absorbing gas is known, the temperature
above the tangent height can be inferred from the radiance observations.
If the temperature distribution is known, we can infer the distribution
i<0o
00
c
•H
<D
•H
3
•J
0)(-1
00
of an absorbing gas. By scanning in tangent height from top to base
of the atmosphere, it is thus possible to infer the vertical tempera-
ture structure or vertical distribution of absorbing gas. The radiance
versus tangent height curve obtained through such a scan is called a
horizon radiance profile. The geometry of the horizon radiance obser-
vation technique insures that one is receiving information on temperature,
for example, from the atmosphere above a particular level — the tangent
height. It will be shown below that for a gas with constant mixing
ratio — such as C02 in the atmosphere of Earth, and probably in the
atmospheres of Mars and Venus (at least to great heights) — the weight-
ing functions for the horizon radiance observation are sharply peaked
at the tangent height. This means that not only is the vertical slice
of atmosphere under view limited at its lower bound by the tangent height
being observed, but its upper bound is also effectively limited by the
shape of the weighting function. In other words, for a particular tan-
gent height h, most of the radiation comes from a rather narrow vertical
interval between h and h + AZ. Hence, the mean temperature of this
interval could easily be inferred by an 'instant inversion' technique.
Below we develop the technique for two infrared transmission models:
the Goody random model, and the transmission models used by Bartko and
Hanel (1968) for calculations of temperature in the upper atmosphere
of Venus. We then compute weighting functions for the atmosphere of
Mars using the Goody random model.
2: DISCUSSION
The intensity of radiation received by an infrared radiometer with
narrow field of view whose line of sight is tangent to the planetary
atmosphere at the tangent height, h of Figure 1, is given by
B' f idx - f B | jdx +Ox . / ox
where B is the Planck intensity integrated over the spectral interval
of the radiometer, and T is the spectral transmittance of the atmosphere
from the point x to the radiometer. From the geometry, the following
relationship between x and height z exists
(R + H)2 + x2 = (R + z)2 (2)
2 2
The terms h and z can be neglected in comparison with the outer
terms — since we are interested in the lower part of the atmosphere
(Z < 100 km) - so that
x =V2R<Z - h) (3)
and
Thus Equation (1) can also be written as
°° h
J B dz dz + / B 9z dz (5)
Expressions will be derived first for the weighting functions sr—
oz
which determine where in the atmosphere the radiation received by the
sensor originates. We assume that the observations are made in a CO,,
band for purposes of temperature inferences.
2.1 Goody Random Model
If it is assumed that the transmittance is given by the Goody ran-
dom model, then
->"i/2
exp
-]
J (6)
where k is the mean line intensity, 6 is the mean line spacing, m
_2
is the effective amount of absorbing gas in g cm along the path, and
a is the effective half-width along the path. For the Martian and
Venusian CCL amounts and pressures, and the long paths of concern here,
km/na is much greater than unity. Thus, Equation (6) can be written as
exp ( 1/2 (7)
Following Rodgers and Walshaw (1966), the effective absorber amount
and half-width are
in = / *(0)dm (8)
and _
/ a dm
a . (9)
/ dm
where $(0) is a correction for the effect of temperature on the line
intensities and is given by
loge 9(6) . a(e - 260) + b(e - 260)2, (10)
where 6 is temperature and a and b are constants for the band.
For the present purpose of determining the CCL transmittances, an iso-
thermal atmosphere is assumed so that
- - r
m a = 9(6) I a dm
w \ ^  /
Since
. 1/2
(12)
where p is pressure and the subscript s refers to standard tempera-
R 1/2 1ture and pressure, and dm = p dx = p c — r-jr- dz, (13)
UUo fc s * \. -L / ^2
 (z - h)
where p is the carbon dioxide density, p is the atmospheric density,
\s vo
and c is the C02 mass fraction, Equation (11) can be written as
1/2 1/2
m a = 9(6) a. ( ~ H f ) ( f ) c / P P
< Z -
For an isothermal atmosphere both pressure and density have the same
scale height H. Thus, Equation (14) becomes
1/2
- I ( ~
ps
where the subscript o refers to surface values on the planet.
.2 2 (z - h)With the substitution, t
(15) becomes
(2H)
H , the integral in Equation
1/2
~
/ •-'dt
The integral in Equation (16) is equal to y^ 1 - erf / I,
is the error function, and / =v2(z-h)/H. Thus
(16)
where erf
m a = V
1/2
(n R H)1/2
•cp0 e
-2h/H 1 + erf (17)
where the negative sign in front of the error function is to be used
for anterior values of z (to the right of the y axis in Figure 1) and
the positive sign for posterior values of z. The expression for the
transmittance from the point z to z = <» can now be written as
T(Z, h) = exp \ - A e-h/H 1. + erf J (18)
where
1/2 1/2
c
 P, (19)
Differentiating with respect to z yields
i- , ,.1/2-]
TT (z, h) = ± -JS50. exp - Ae (l + erfy V." )
d2
 Y2(z - h)H V7 |_ ^ '_ -
x / ^1/2
r i + prf-i/ifc-LJll 1 -<2z-h)/H (20)
where the upper signs are to be used for anterior values of z and the
lower signs for posterior values of z.
2.2 Bartko-Hanel Transmission Model
The Bartko-Hanel (1968) transtnittance for a spectral interval can
be written as
r * niT = exp I - (mu ) J (21)
*
where m and n are constants for the spectral interval, and u , the effec-
tive absorber amount, .is given by
3/2
where u is path-length of absorbing gas in units of atmo-cm of C09, T£ S
and p are standard temperature and pressure, respectively, and 7 is
S
a constant for a spectral interval. Expression (22) is for a constant
temperature and pressure path. We assume an isothermal atmosphere and,
hence, a constant temperature for purposes of evaluating transmittances.
]t
The pressure does vary over the path and in this case expression (22) p
is replaced by p , where the bar represents an average over the
atmospheric path.
—r n , p
p = / p du// du (23)
9
From the definition of the path length,
du, _^_ dx, _«._ (f) ^^ (24)
p
 co2 " co2 <
z
 -
 h)
where the prime represents conditions of standard temperature and pres-
sure, and c is the C0» mass fraction.
The effective absorber amount can then be written as
u* = f (T) Ps" J .Pk du (25)
where „ 3/2
T ~ " T ] (26)
or
* • _ r k dz ,0-.v
u = B / p p 7-77 (27)
J
 (z - h)L/
where _,
Ps c / D V l / 2(f) (28)
Under the assumption of an isothermal atmosphere (for transmission
calculations only), both pressure and density have the same scale height
H. Thus,
00
u * = B p k p r exp[~-(k + 1) Z/H] dZ
 1/2 (29)
^
 L J
 (z - h)L/^
where the subscript o refers to the planet's surface.
10
If we let
t2 = (k + l) (« - h) (30)
H
the integral in (29) becomes
CO „/ •-'exp -(k + l)h/H / e dt (31)
•/(z-h)(k+1)
H
The integral in Equation (31) is equal to vit (1 - erf a), where
2
a = /(z-h)(k+l)/H (32)
Thus,
*
u = B p. k /itH \
 1/2
 exp ["-(k+l)h/Hl (1+ erf a) (33)
po Vk+lj L J
where the negative sign in front of the error function is to be used
for anterior values of z, and the positive sign for posterior values
of z. The expression for the transmittance from the point z to z = o
along the tangent path can be written as
= exp - [mE exp [-(k+l)h/Hj (1 + erf a)J (34)
where , ,-
Differentiating with respect to z yields
11
1/2
1 W-+- I I
= +
n-l 2( r • ~\ — }|m E exp [-(kH-l)h/Hj (1 + erf a)| e"a (36)
2.3 Weighting Functions and Instant Inversion
The weighting function is defined as
with A and P standing for anterior and posterior, respectively.
We have used the random transmission model expression (Equation 20)
to compute weighting functions for the 15[i CO- band for the Martian
atmosphere. The weighting functions for tangent heights of 0 to 60 km,
at 10 km intervals in tangent height, are plotted in Figure 2. In these
computations it has been assumed that the Martian scale height is 10 km,
-5 -3the surface pressure is 5 mb, the surface density is 1.2x10 g cm ,
the temperature of the Martian atmosphere is 200 and the band constants
are those given by Rodgers and Walshaw (1966). It is obvious from Fig-
ure 2 that for tangent heights greater than 20 km most of the energy
emanates from a rather narrow vertical thickness of atmosphere, of
order 5 km. Thus, observations of 15 (j band radiance at 5 km or greater
tangent height intervals are almost independent of each other and could
be used to infer, quite directly, mean temperatures with 5 km vertical
resolution. A rather simple technique, "instant inversion," is developed
below.
12
1H9I3H
13
From Equation (1) or (5) it can be seen that the intensity of
radiation received by the radiometer can be written as
1
I(h) = / B dT (38)
T(- °°,h)
where t(-<»,h) represents the transmittance of the entire atmospheric
path tangent to the Martian atmosphere at height z = h. The weighting
functions for tangent heights greater than 20 km are such that most of
the energy emanates from a rather narrow vertical interval immediately
above the tangent height. Thus, the temperatures above this narrow
vertical interval do not contribute significantly to the observed emis-
sion. It may, therefore, be assumed that B in the integral of Equa-
tion (38) may be approximated by B(h), which represents the integrated
value of the Planck function for the average temperature of the layer
immediately above the tangent height. Thus,
Kh)
The term [1 - i(-°°, h)] may be called the emittance, e(h). Thus, one
can infer B(h) from
(40)
It is a simple matter to reclaim T(h) from the integrated Planck
function. Hence, the vertical distribution of temperature of the Mar-
tian atmosphere can be obtained from observations of I as a function of
tangent height h.
From Equation (18),
T(-OO, h) = exp (- /2 Ae~h/H) (41)
The emittance of the Martian atmosphere as a function of tangent height
is shown in Figure 3.
For purposes of deriving and illustrating the method, we have
assumed a model isothermal atmosphere to calculate the variation of
emittance with tangent height. Although the emittance varies with
temperature, this is a second order effect compared to the variation of
the Planck function with temperature. Hence, Equation (40) can be used
to infer temperatures at different levels even though e(h) is computed
for an isothermal atmosphere. The effect of uncertainties in the emit-
tance profile upon inferred temperatures can easily be examined.
Preliminary analysis of experiment feasibility indicates that it
is within the state of the art. The desired vertical resolution at. the
tangent height is about 5 km. For a Martian orbiter at an altitude of
1000 km, this vertical resolution can be. achieved with 0.1 degree vertical
field of view. A similar successful measurement of the Earth's horizon
radiance profile in the 15^ (XL band (615 cm to 715 cm ) from a
suborbital rocket flight utilized a radiometer with 0.025 degree vertical
field of view (McKee et al, 1967). Expected Martian radiances in the
15^ band are of the order of 1/10 of Earth radiances (assuming
stratospheric temperatures of 150K on Mars and 230K on Earth). The wider
field of view suggested for the Martian experiment helps to compensate
15
OIGAI68-IOS
O.I ~
0 10 20 3O 40 50 60
TANGENT HEIGHT, h (km)
70 80
Figure 3. Emittance of the 15(.i CO band versus tangent
height for the Martian atmosphere.
16
for the lower signal levels. Scan times of the order of 1 second for
a complete vertical scan, as in the successful Earth experiment, are
within the state of the art. The effective horizontal resolution of
an individual temperature inference would be of the order of 200 km.
This analysis shows that the orbiter based horizon radiance tech-
nique is potentially a powerful tool forimeasuring the vertical tem-
perature structure of the Martian atmosphere above heights of about
j :,
20 km. It would thus complement an orbiter infrared spectrometer ex-
perment at 4.3^  or at 15u which could be used to infer temperature pro-
files for the lowest 20 km of the Martian atmosphere. If one wanted
to observe the temperature profile of the lowest 20 km of the Martian
atmosphere with the limb radiance technique, one has only to slide the
spectral interval of the radiometer to a region of slightly lower C0?
absorption. More sophisticated inversion techniques can also be used.
17
3. FUTURE PLANS
Plans for the next quarterly period include the following:
(1) Analysis of feasibility of deriving the temperature profile
above the clouds of Venus with a limb radiance experiment.
(2) Analysis of the feasibility of deriving water vapor profiles
for Mars and Venus (above the clouds) with a limb radiance experiment.
(3) Continuation of analyses of earth-based observations of the
Jovian emission spectrum for the purpose of deducing the Jovian atmo-
sphere temperature profile.
18
REFERENCES
Bartko, F., and R. Hanel, 1968: Non-gray equilibrium temperature dis-
tributions above the clouds of Venus. Ap. J., 151, 365-378.
Conrath, B., R. Hanel, V. Kunde, and C. Prabhakara, 1970: The infrared
interferometer experiment on Nimbus 3. J. Geophys. Res., 75,
5831-5837.
McKee, T., R. Whitman, and R. Davis, 1968: Infrared horizon profiles
for summer conditions from Project Scanner. NASA Technical Note
TND-5068, 31 pp.
Rodgers, C.D. and C. 0. Walshaw, 1966: The computation of infrared
cooling rates in planetary atmospheres. Quart J. Roy. Met. Soc.,
92, 67-92.
19


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