Vertical resolution of temperature profiles obtained from remote radiation measurements

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VERTICAL RESOLUTION OF TEMPERATURE

PROFILES OBTAiNED FROM REMOTE

RADIATION MEASUREMENTS

BARNEY J. tONRATH

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{NaSA-TM-X-66009) VERTICAL RESOLOTION OF

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TEMPEBATOBE PROFILES OBTAINED FBOM REMOTE
■RADIATION MEASUREMENTS B.J, Conrath (NASA)
i Nov. 1971 31 p CSCL 04A Unclas

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Preprint

VERTICAL RESOLUTION OF TEMPERATURE PROFILES
OBTAINED FROM REMOTE RADIATION MEASUREMENTS

by

Barney J. Conrath

November 1971

Laboratory for Planetary Atmospheres
Goddard Space Flight Center, Greenbelt, Maryland

VERTICAL RESOLUTION OF TEMPERATURE PROFILES
OBTAINED FROM REMOTE RADIATION MEASUREMENTS

by
Barney J. Conrath
Laboratory for Planetary Atmospheres
Goddard Space Flight Center, Greenbelt, Maryland
ABSTRACT
The Backus- Gilbert theory, originally developed for analysis of inversion
problems associated with the physics of the solid earth, is applied to the problem
of the vertical sounding of the atmosphere by means of remote radiation meas-
urements. An application is made to spectral intervals 2.8 cm-^ wide in the
667 cm~^ band of CO 35 and tradeoff curves are presented which quantitatively
define the relationship between intrinsic vertical resolution and random error
in temperature profile estimates. It is found that for a 1-2K random error with
state-of-the-art instrumentation, the intrinsic vertical resolution ranges from
-^0.5 local scale height (l.s.h.) in the lower troposphere to > 2 l.s.h, in the upper
stratosphere with -^1 l.s.h. resolution in the vicinity of the tropopause. These
values are somewhat smaller than the widths of the radiactive transfer kernels
at similar levels. Increasing the number of spectral intervals from 7 to 16 is
found to produce only a marginal improvement in vertical resolution.

VERTICAL RESOLUTION OF TEMPERATURE PROFILES
OBTAINED FROM REMOTE RADIATION MEASUREMENTS

INTRODUCTION

One of the most significant contributions of the meteorological satellite
program to date has been the development of techniques for the remote vertical
sounding of the atmosphere. The Nimbus 3 and Nimbus 4 satellites carried
instruments which provided measurements which could be used in the determina-
tion of vertical profiles of temperature, humidity, and ozone [ Hanel and Conrath,
1969; 1970; Wark and Hilleary , 1969; Ellis, et al. , 1970]. Extensive use has been
made of these data in the development and application of atmospheric profile in-
version techniques [see, for example. Smith , et al., 1970; Wark , 1970; Conrath,
et al., 1970; Prabhakara, et al ., 1970]. In addition to the obvious applications in
terrestrial meteorology, remote vertical soimding techniques also provide a means
for studying the atmospheres of other planets. Instrumentation has been developed
for this purpose and is currently being flown on the Mariner 9 spacecraft for the
1971 Mars orbital mission [Hanel, et al. , 1970].

One of the basic problems in the study of atmospheric profile inversion
techniques concerns the vertical resolution and precision of the retrieved pro-
files. It has long been recognized that attempts to obtain better vertical resolu-
tion from a given set of data will result in an increase in sensitivity of the re-
trieved profiles to random noise. Therefore, in the analysis of a particular set
of data there should be a tradeoff between the scale of features resolvable in an

estimated profile and its stability against noise. Up to the present time, no
direct quantitative formulation has been given. Now, however, the recently
developed theory of Backus and Gilbert [Backus and Gilbert , 1967; 1968; 1970;
Backus, 1970a; 1970b; 1970c] can be employed in an analysis of the problem.
Although the theory was developed for applications to inverse problems en-
countered in the physics of the solid earth, it is of quite general validity and can
be easily adapted to the atmospheric profile inversion problem.

It is '■he purpose of the present paper to demonstrate the applicability of the
work of Backus and Gilbert to atmospheric profile inversion and to make a spe-
cific application of the techniques to the problem of temperature profile estima-
tion from measurements in the 667 cm-^ (15 jj.m.} CO absorption band in the
terrestrial atmosphere.

A brief review of the formulation of Backus and Gilbert is first presented.
Tradeoff curves are then calculated, using selected spectral intervals in the
667 cm"* band. These curves quantitatively relate the vertical resolution of
the retrieved profiles and their sensitivity to noise. The effect of varying the
number of spectral intervals is examined. The averaging effect implicit in in-
version calculations is studied in detail using the so-called "averaging kernels."
Finally, conclusions are given concerning the intrinsic limitations of measure-
ments obtained in the 667 cm"* CO 3 band.

THE METHOD OF BACKUS AND GILBERT

The work of Backus and Gilbert has been confined primarily to the geophy-
sical literature, so the techniques are generally unfamiliar to those working in
atmospheric physics. Therefore, a brief review of the concepts of the method
will be presented in this section. For a more thorough treatment of the theory,
the reader is referred to the original literature, in particular, Backus and Gil-
bert [1967; 1968; 1970]. In addition, an excellent summary of the subject is
given by Parker [1971] .

Tj^ically, in the remote sensing of atmospheric temperature profiles,
measurements of radiances in a finite number of spectral intervals within an
atmospheric absorption band are employed. For a non-scattering atmosphere
in local thermodynamic equilibrium, a theoretical expression for the radiances
can be written in the form

. : 3t.(x)

li = B.(TJr.(0) + J BjT(x)] -^p dx; i = 1, 2, •••m. (1)

In this expression, the independent variable x can be any monotonic function of
the atmospheric pressure. In the following discussion, x = - In P/p is employed
where P is the surface pressure. Thus, x can be regarded as height expressed

s

in terms of a local scale height (l.s.h.). In the i-th spectral interval, B .(T) is
the Planck function for temperature T, and r^ (x) is the atmospheric transmit-
tance between level x and x^, the effective top of the atmosphere. The term

B. (T ) T . (o) is the contribution from the planetary surface, here taken to be a
blackbody at temperature T,. The surface temperature and hence the boundary
term, is usually specified from measurements in highly transparent spectral

intervals.

The inversion problem then is to estimate the temperature profile T(x), given
the atmospheric transmittances r . (x) and measurements of the radiances I .
(i = I5 2, . . . m). Let T" (x) be a reference profile in the neighborhood of the
actual profile T(x). If the Planck function B, [T{x)] is expanded in a Taylor
series at each level x about the reference profile T°(x), tr-ancated after the linear
term, and substituted into (1), there results

t

AI. =-. K.(x) AT(x)dx; i = 1, 2, ■ • • m. (2)

The radiance difference is Ai =l - 1° where 1° is calculated from (1) using

i i 1 >■

the reference profile, and At(x) = T(x) - T°(x)o The quantities K. (x) (i = 1, 2,
. . , m) are given by

. , ^ dBjT°(x)] 3r,(x)

and will be referred to as the radiative transfer kernels. For a finite number of
measurements m, the set of linear integral equations (2) will possess either no
solution or an infinite number of solutions for AT(x)o The effects of this non-
uniqueness must be considered in examining the vertical resolution obtainable
with a given set of measurements »

Only estimates of AT(x) which are linear in the quantities obtained from
measurements Al . (i = l, 2, . . . m) will be considered. Such estimates can be
written in the form

AT(x) = Y" a.(x) AI^ (4)

where there will generally be a different set of coefficients a . (x) (i = 1, 2, . . . m)
for each value of x. If the expression for I . given by (2) is substituted into (4),
a relationship between the estimate A T(x) and the actual profile AT(x) is ob-
tained

AT(x)

A(x, x') AT(x')dx' (5)

where

A(x,x') =^ a.(x) K.(x') (6)

i = l

Thus, froni (5), the estimate AT(x) at a given level x can be regarded as a
weighted average of the true profile with the weighting determined by the
"averaging kernel" A(x, x').

The vertical resolution of a particular linear estimate at the level x is
determined by the behavior of A(x, x'). Ideally one would like for A(x, x') to be
a Dirac delta function. However, for a finite number of terms in (6) this is not
possible, and A(x, x') will have some finite spread about each level x. A useful
measure of this spread has been given by Backus and Gilbert in the form

s(x) = 12 (x-x')2 A2 (X, x') dx' (7)

The normalizing factor 12 is chosen such that when A(x, x') is a rectangular
function of width l centered on x and satisfying

A(X, x') dx' ::= 1 (8)

then s(x) = ^. Other measures of the spread of A(X5 x') can be defined and
several have been given by Backus and Gilberto However, only (7) will be em-
ployed here.

Another parameter useful in characterizing the behavior of A(X5 x') is the
"center" defined as

c(x) = ' x' A2(x, x') dx7 'a2(x, x')dx' (9)

and the "resolving length" of A(Xs x') can be defined as the spread about its
centers i.e.,

w(x) = 12 ' [c(x) - x'] 2 A2 (x, x') dx'. (10)

Thus, s(x) contains contributions both from the resolving length w(x) of A(x, x')
and from possible displacements of c(x) from Xo

The maximum vertical resolution obtainable from a given set of radiance
measurements using a linear estimation can be found by deriving the set of

9

coefficients aj(x) (i = Ij 2, . . . m) which minimizes s(x) subject to the constraint
that the averaging kernel be unimodular, i.e., that is, satisfy (8). The requirement
that A(x, x') be unimodular insures that (5) will represent a well defined average.

Up to this point, consideration has not been given to the effects of measure-
ment errors. In practice, instrument noise will result in an imprecise determina-
tion of AI. . This noise contaminated value can be written

AI. = AI . + e

(11)

where e . is the unknown error in the measured value of I . The variance of the

1 i

temperature error a^ (x) incurred at level x due to random measurement errors
can be found from (4) to be

a^ (x) = a(x) • E • a (x)

where a(x) is the vector whose elements are the coefficients a. (x) (i = 1, 2, . , .
m), and E is the covariance tensor for the measurement errors.

Ideally one would like to be able to choose a (x) such that both the error
variance ^^ (x) and the spread s(x) are minimized. This cannot be done, but it is
possible to minimize a linear combination of s(x) and a^ (x). Such a linear
combination can be written

Q(x) = qs(x) + (1 -q) r a2 (x). (13)

10

The coefficient r insures that both terms have the same physical dimensions,
but its numerical value is of no fundamental importance. By varying the
parameter q between zero and unity, the emphasis can be shifted from minimiza-
tion of the error to minimization of the spread. Thus, there is a tradeoff be-
tween vertical resolution and accuracy, and the best choice for q must be de-
termined by the particular application.

In carrying out the minimization of Q(x)5 it is convenient to introduce the
vector m whose components are given by

u , = I K^(x) dx;i = l, 2, ••■m

'0

(14)

and the tensor S (x) whose components are

S..(x) = 12 ! ' (x -x')2k.(x') K.(x') dx'; i, j = 1, 2, • • -m. 0-^)

If further, a tensor W (x; q) is defined by

W(x; q) = q S(x) + (1 - q) r E (16)

then the problem^ becomes one of minimizing

Q(x; q) = a • W • a (17)

subject to the constraint

11

a • u = 1. (18)

A straightforward minimization calculation gives

W-^(x; q) • u
a(x; q) = (19)

u • W-l (x; q) • u

Thus, for a particular value of q, the coefficients for a given level x can be cal-
culated using (19). The resulting averaging kernel A(x, x'; q), the spread s(x; q),
and the error cr^(x;q) can then be obtained from (6), (7), and (12). By varying q,
the error a^ can be obtained as a function of spread s. The resulting relation-
ship is called the "tradeoff curve" and can be used to analyze the tradeoff between
vertical resolution and errors in the estimated temperature due to random errors
in the measurements. One such curve is obtained for each atmospheric level
considered.

The Backus and Gilbert formulation provides a means of examining the
intrinsic vertical resolution of temperature profiles obtained from infrared
radiance measurements by means of linear inversion. In addition, the formula-
tion provides inversion coefficients which can be employed in the actual reduction
of data. However, only the analysis of vertical resolution will be considered
further in the present paper.

12

APPLICATION TO THE 667 CM"^ BAND

The formulation reviewed in the preceeding section has been applied to an
analysis of the vertical resolution of temperature profiles inferred from
measurements in the 667 cm"^ {15/j-m) CO^ band. The sixteen 2.8 cm"^ wide
spectral intervals employed in the study are listed in Table 1. The kernels for
a subset of seven spectral intervals denoted by asterisks in Table 1 are shown
in Figure 1.

In order to characterize the behavior of the ith kernel, a mean atmospheric
level X and a width d about the mean were defined, analogous to the center (9)

i i

and resolving length (10) of the averaging kernel, i.e.,

X. .= I ' xK2 (x) dx / \f (x) dx (20)

d. =12u-2 (x-x.)2 k2 (x) dx. (21)

T^f^ foofnT' 11"^ in t9.T\ wTiavo n is crWf^rt hv (TA\. is to nrnvide normalization
of the kernel. The mean levels are given in Table 1 in terms of pressure, along
with the width expressed in terms of local scale heights.

In calculating tradeoff curves, it was assumed that no correlation existed
in the measurement noise of the various spectral intervals and that the noise
variance was the same for all intervals. These assumptions are generally valid
for measurements obtained in the portion of the spectrum being considered

13

here. In this case the error covariance tensor becomes

E--al 1 (22)

e

where 1 is the unit tensor, and o-^ is the variance of the measurement noise.
Relation (12) then reduces to the form

°'t(^) 1/2

-I— = [a(x) • a(x)] (23)

Thus, the tradeoff curves can be expressed as o-^{x)/cr^ vs. s(x), and can be
applied to measurements from an instrument with arbitrary noise level.

Tradeoff curves were calculated for many atmospheric levels. Examples
are shown in Figure 2 to illustrate the behavior of the tradeoff curves in the
troposphere, near the tropopause, and in the stratosphere, tn each case two
curves are shown, one for the full set of 16 spectral intervals listed in Table 1,
and one for the 7- interval subset denoted in the table by asterisks. The results
shown were obtained using a midlatitude temperature profile. Similar calcula-
tions were carried out with a tropical temperature profile, but the results were
not substantially different from those shown in Figure 2.

The gross behavior of the tradeoff curves is the same for all levels. The
random error decreases monotonically with increasing spread. The decrease is
rapid at small values of the spread, followed by a rather abrupt leveling off with
increasing spread. The end point of the curve corresponding to minimum spread
and maximum random error corresponds to a choice of q = 1 in (13) while the

14

opposite end of the curve for which o-^ is a minimum and s is a maximum
corresponds to q = 0, In the latter case, the inversion coefficients a are inde-
pendent of atmospheric level, and the random error is the same for all levels.
The steep portion of the tradeoff curve shifts in the direction of increasing
spread as the atmospheric level increases, so for a given value of ^^/o- ^, the
spread is greater for upper atmospheric levels than for lower levels.

Comparison of the tradeoff curves for 7 and 16 spectral intervals in Figure
2 indicates the degree of improvement which can be expected by increasing the
number of spectral intervals. The minimum random error corresponding to the
maximum spread is a factor of 1.45 higher for the 7- interval set than for the
16- interval set, reflecting the change in redundancy by a fector of approximately
2 in this case, hi the steeper portion of the curves which is of greater
practical interest, use of the 16~interval set results in only a small reduction
in the spread for a given value of a^/r^.

From the inversion coefficients a(x) calculated for a given choice of q, the
corresponding averaging kernel A(x, x') can be obtained using (6). It is instruc-
tive to examine the behavior of A(x, x') for typical cases. Figure 3 shows aver-
aging kernels for the 7 spectral interval set at the 49 mb level, with each curve
corresponding to a selected point on the tradeoff curve shown in Figure 2f. The
value of s and ct^/o-^ for each curve is indicated in Figure 3. The averaging
kernel in Figure 3a corresponds to minimum random error and maximum
spread while at the opposite end of the tradeoff curve is the averaging kernel in

15

Figure 3f corresponding to minimum spread and maximum random error. The
remaining curves in Figure 3 show the evolution of the averaging kernel in pass-
ing from one end point of the tradeoff curve to the other. Figure 4 shovi^s aver-
aging kernels for several tropospheric and stratospheric levels. In each case
a /a- ^4 K/(erg/sec cm^ ster cm~^) except for the 49 mb level. In the case of
the 49 mb level, the maximum value of ^■^/<^, reached by the 7- interval trade-
off curve is 1.6 K/(erg/sec cm^ ster cm"^) (see Figure 2h), and the correspond-
ing averaging kernel is shown. Typical noise levels obtainable for measurements
in this spectral region are 0.25 to 0.5 erg/cm^ sec ster cm-^ , so a choice of
cr^/o-^ = 4 K/(erg/sec cm^ ster cm"^) corresponds to a random error in the
estimated temperatures of 1 to 2 K which is an acceptable level for most appli-
cations. Comparison of the averaging kernels of Figure 4 with the radiative
transfer kernels of the original set of integral equations (Figure 1 and Table 1)
shows that for comparable atmospheric levels the averaging kernels are narrower,
although the 10 mb averaging kernel closely approaches the radiative transfer
kernel for spectral interval 1 (667.5 cm"^) in width.

Since the spread s(x) contains contributions both due to the width of the
averaging kernel and the noncoincidence of its center with the level to which it
pertains, it is instructive to examine the center and width (as measured by the
resolving length) separately. The center, calculated using (9) is plotted as a
function of atmospheric level in Figure 5, and the resolving length, calculated
from (10), is given in Figure 6 for a value of o-^/a^ = 4 K/(erg/sec cm^ ster cm'^).

16

The 16 spectral interval set was employed in this example. The center lies very
close to the appropriate atmospheric level from the surface up to about 10 mb,
but remains almost constant above that level. The abrupt increase in resolving
length above 50 mb reflects the fact that most of the information on this region
is coming from the 667.5 cm"^ interval for which the radiative transfer kernel
is quite broad (see Table 1 and Figure 1).,

The curves of Figures 5 and 6 indicate that little intrinsic information on
the temperature profile above 10 mb is contained in measurements in this set of
spectral intervals. The vertical resolution becomes very poor, and since the
center of the averaging kernel does not move much above 10 mb, temperature
estimates for the upper stratosphere will merely reflect the behavior of the
temperature profile at lower levels.

Vertical resolution is best in the lower troposphere where the radiative
transfer kernels are least broad. The vertical resolution in the vicinity of the
tropopause (100 - 200 mb) is particularly interesting because of the relatively
fine scale structure in the temperature profile which frequently exists there.
From figure 6 it is found that the resolving length in this region is 0.8 - 1.0 l.s.h.
which is considerably broader than the scale of the structure one would like to
resolve. Some improvement in resolution can be gained by allowing the random
error in the inferred temperatures to become larger. However, due to the steep-
ness of the tradeoff curves, only a slight improvement in resolution can be ob-
tained before the errors become intolerably large.

17

SUMMARY AND CONCLUSIONS

The theory of Backus and Gilbert, developed for applications to inverse
problems of the physics of the solid earth, has been shown to be useful for the
analysis of remote vertical atmospheric sounding systems. Tradeoff curves
obtainable from the theory can be employed to define the limitations of measure-
ments in a given set of spectral intervals. The curves can be used for the evalu-
ation of the intrinsic information content of the measurements within the frame-
work of linear inversion.

Analysis of 2.8 cm-i wide spectral intervals in the 667 cm-i CO2 band
indicates that for random temperature errors limited to 1-2 K with present
state-of-the-art instrumentation, the intrinsic vertical resolution (measured in
terms of the resolving length (10)) ranges from '^0.5 l.s.h. in the lower tropo-
sphere to >2 l.s.h. in the upper stratosphere. The resolving length in the vicinity
of the tropopause is -^1 l.s.h. Comparison of the kernels of the original set of
radiative transfer equations with the averaging kernels corresponding to 1-2 K
random temperature errors indicates that linear inversion can provide some-
what better vertical resolution than the width of the radiative transfer kernels
along would indicate. However, the resolution is still too coarse for the direct
retrieval of certain fine scale features of interest, such as those required for an
accurate determination of tropopause temperature and height.

Attempts at improving the vertical resolution by picking points on the
tradeoff curves corresponding to smaller spread will generally be imsatisfactory.

18

Because of the steepness of the tradeoff curves in the region of interest, a small
improvement in vertical resolution results in a large increase in random tem-
perature error. Going to a large number of spectral intervals does not greatly
improve the resolution. It was found that increasing the number of spectral
intervals from 7 to 16 gave only marginal improvement in resolution at a given
random error level. The only practical way to obtain resolutions significantly
better than those indicated by the resolving lengths in Figure 6 with this set of
spectral intervals is to supplement the intrinsic information content of the mea-
surements with additional a priori information on the profile. This may be ac-
complished for example through a careful selection of the reference profile
T°(x) or through a full statistical estimation approach such as that of Rodger s
[1966] and Strand and Westwater [1968].

The analysis indicates that little information can be obtained on the tem-
perature profile above the 10 mb level with the set of spectral intervals con-
sidered. Temperature estimates obtained for levels above 10 mb will in fact be
primarily dependent on the behavior of the profile at lower levels. Howeverj this
situation can be improved through the use of measurements at higher spectral
resolution which results in kernels which peak at higher atmospheric levels.
Measurements of this type have been obtained successfully by Houghton and his
colleagues, [ Ellis, et al. , 1970].

Only the diagnostic properties of the Backus- Gilbert theory have been ex-
plored here. The theory also provides a method of inversion, and this aspect is
a subject of current study.

19

REFERENCES

Backus, G. E., 1970a: Inference from Inadequate and Inaccurate Data, L Proc.

Nat. Acad. Sci ., 65 , 1-7.
Backus, G. E., 1970b: Inference from Inadequate and Inaccurate Data, IL Proc.

Nat. Acad. Sci. , 65, 281-287.
Backus, G., 1970c: Inference from Inadequate and Inaccurate Data, III. Proc.

Nat. Acad. Sci., 67, 282-289.
Backus, G. E., and J. F. Gilbert, 1967: Numerical Applications of a Formalism

for Geophysical Inverse Problems. Geophys. J. R. Astr. Soc , 13, 247-276.
Backus, G. E., and J. F. Gilbert, 1968: The Resolving Power of Gross Earth

Data. Geophys. J. R. Astr. Soc , 16, 169-205.
Backus, G. E., and J. F. Gilbert, 1970: Uniqueness in the Inversion of Inaccurate

Gross Earth Data. Phil. Trans. R. Soc. London, A266 , 123-192.
Conrath, B. J., R. A. Hanel, V. G, Kunde, and C. Prabhakara, 1970: The Infrared

Interferometer Experiment on Nimbus 3. J. Geophys. Res. 75, 5831-5857.
Ellis, P. J., G. Peckham, S. D. Smith, J. T. Houghton, C. G. Morgan, C. D.

Rodgers, and E. J. Williamson, 1970: First Results from the Selective

Chopper Radiometer on Nimbus 4. Nature , 228 , 139-142,
EEinel, R. A., and B. J. Conrath, 1969: Interferometer Experiment on Nimbus 3:

Preliminary Results. Science, 165, 1258-1260.
Hanel, R. A., and B. J. Conrath, 1970: Thermal Emission Spectra of the Earth

and Atmosphere from the Nimbus 4 Michelson Interferometer Experiment.

Nature, 228, 143-145.

20

Hanel, R. A., B. J. Conrath, W. A, Hovis, V. Kunde, P. Do Lownian, C.

Prabhakara, and B. Schlachman, 1970: Infrared Spectrometer Experiment

for Mariner Mars 1971. Icarus , 12^, 48-62.
Parker, R. L., 1970: The Inverse Problem of Electrical Conductivity in the

Mantle. Geophys. J. R. Astr. Soc , 22 , 121-138.
Prabhakara, C, B. J. Conrath, R. A., Hanel, and Eo J. Williamson, 1970: Remote

Sensing of Atmospheric Ozone Using the 9.6 Micron Band. J. Atmos. Sci. ,

27 , 689-697.
Rodgers, C. D., 1966: Satellite Infrared Radiometer; A Discussion of Inversion

Methods. Univ. of Oxford Clarendon Lab. Mem. 66.13, 25 pp.
Smith, Wo L., H. M. Woolf, and W. Jo Jacob, 1970: A Regression Method for

Obtaining Real-Time Temperature and Geopotential Height Profiles from

Satellite Spectrometer Measurements and its Application to Nimbus 3 "SIRS"

Observations. Monthly Weath er Review, 98, 582-603.
Strand, O. N«, and E. R. Westwater, 1968: Minimum RMS Estimation of the

Numerical Solution of a Fredholm Integral Equation of the First Kind.

SIAM J. Numer. Anal., 5, 287-295.
Wark, D. Q., 1970: SIRS: An Experiment to Measure the Free Air Temperature

from a Satellite. Applied Optics, 9, 1761-1766.

21

FIGURE CAPTIONS

Figure 1. Radiative transfer kernels for the 667 cm"^ COj absorption band.
The numbers labeling the curves refer to spectral intervals listed in
Table 1.

Figure 2. Tradeoff curves for the 667 cm- ^ COj absorption band, a^ is the
standard deviation of the temperature error due to a random measurement
error with standard deviation a^ . The atmospheric pressure level to which
each curve pertains is indicated. The solid curves are for a set of 16 spec-
tral intervals while the broken curves are for a 7-interval set.

Figure 3. Averaging kernels for the 49 mb level corresponding to selected
points along the tradeoff curve shown in Figure 2f. The values of a^ /a^
and the spread s are indicated in each case.

Figure 4. Averaging kernels for selected atmospheric levels. The level to
which each kernel pertains is indicated by both a label and a broken line.

Figure 5. The averaging kernel center plotted as a function of atmospheric
pressure level (solid curve). If the center of the kernel exactly coincided
with the level to which the kernel pertained, the broken line would result.

Figure 6. The resolving length plotted as a function of atmospheric level. The
resolving length provides a measure of vertical resolution.

22

Table 1

Characteristics of 667 cm

-^ CO2 Band Radiative Transfer Kernels

Interval

Central

Frequency

(cm-i)

Mean Level
(mb)

Width
(T^cal Scale Heights)

1*

667.5

13.1

2,42

2*

677.5

61.2

1.55

3

682.5

66.6

1.60

4

687.5

74.0

1.52

5

692.5

76.2

1.53

6*

697.5

92.7

1.78

7*

702,5

153.3

1.95

8*

707.5

286.9

1.65

9

712.5

403.3

1,47

10*

727o5

564.7

1,19

11

732.5

547.0

1,30

12

737.5

575.2

1.20

13

742,5

609.1

1,12

14*

747.5

692.9

0,92

15

752.5

740.8

0,74

16

757.5

798.6

0.59

"^Spectral intervals employed in 7- interval set.

23

.a

E

X

:d

CO

</)

a:
a.

dB dt

— — (NORMALIZED)

Figure 1. Radiative transfer kernels for the 667 cm' CO, absorption band. The
numbers labeling the curves refer to spectral intervals listed in Table 1.

500.

200.

100.

50.

20.
10.
5.0

2.0
1.0
0.6

0.2

781 mb

0.1

±

_1 I L^i_l_l

LiaaA.

I I I I I I 1 1

0.1 0.2 0.5 1.0 2.0 5.0 10. 20.

SPREAD (LOCAL SCALE HEIGHTS)

_L_J LX-1_Uj

50. 100.

500. -

200.

E

100.

50,

20.

10.

5.0

2.0

1.0

0.5

0.2

0.1

525 mb

±

I I I I I I ii

Figure 2(a).

0.1 0.2 0.5 1.0 2.0 5.0 10. 20.
SPREAD (LOCAL SCALE HEIGHTS)

Figure 2(b).

50. 100.

4^

0.1 0.2 0.5 1.0 2.0 5.0 10. 20.

SPREAD (LOCAL SCALE HEIGHTS)

50. 100.

0.1 0.2 0.5 1.0 2.0 5.0 10. 20.

SPREAD (LOCAL SCALE HEIGHTS)

50. 100.

■ igure

2(c).

Figure 2(d).

Ol

500.

200.

100.

50.

20.
10.
5.0

2.0
1.0
0.5

0.2

108 mb

0.1

±

±

_i__Lj_lj_l!

J I I I I I I

0.1 0.2 0.5 1.0 2.0 5.0 10. 20.

SPREAD (LOCAL SCALE HEIGHTS)

Figure 2{e).

50. 100.

500

200. -

100

* 2.0

0.5 1.0 2.0 5.0 10. 20.

SPREAD (LOCAL SCALE HEIGHTS)
Figure 2(f).

50. 100.

to

CI

27

QO

-

500.

—

200.

—

100.

—

10 mb

50.

—

20.

-

10.

-

5.0

-

2.0

-

1.0

—

I

0.5

L

^^^

-

^^^^~~~^-^^— _

0.2

n 1

1 , , L ,mI I

1 1 1 1 1 1 1 1 1 1 1 1 1 M 1

0.1 0.2 0.5 1.0 2.0 5.0 10. 20. 50. 100.

SPREAD (LOCAL SCALE HEIGHTS)

Figure 2(g).

Figure 2. Tradeoff curves for the 667 cm' CO- absorption band.
aj is the standard deviation of the temperature error due to a
random measurement error with standard deviation a . The atmo-
spheric pressure level to which each curve pertains is indicated.
The solid curves are for a set of 16 spectral intervals while the
broken curves are for a 7-interval set.

.1

.2

-

.5

-

1

-

2

-

5

-

10

-

20

-

50

-

100

-

200

-

500
nnn

-

(a)

^1=0.42

e

S=7.i

.1
,2

_

(b)

.5

-

^=0.43
S=4.9

i

-

2

-

5

-

10

- .

20

-

\^^

50

^^^--.^

100

-

J

200

-

<^

500

-

1

^

1.0

.1

.2

.5
1
2

5

10
20

. 50
100
200

500
1000

(c)

-

V.

^=0.48
S=3.1

;

^^

>

1.0

.2

- \

.5

-

2

_

J3

t-

5

-

UJ

10

-

(/)

ijj

?0

^

fr

a.

50

-

100

-

200

-

500

-

luOO

.5
1
2

5

10
20

50
100
200

500
1000

(e)

i

^=0.73

-

'>

S=2.0

1.0

.1

.2

.5
1
2

5
10
20

50
100
200

500
1000

(f)

-

v^

^=5.8
S = 1.4

c

>

> , 1

1

Figure 3. Averaging kernels for the 49 mb level corresponding to selected points along the tradeoff
shown in Figure 2f. The values of a^/a^ and the spread s are indicated in each case.

A

curve

1.0

00

.2

-

\ (a)

.5

r

\^ 10 mb

1

-

\^

2

-

^^^^^.^^

5
10

-

^^^--^^^^

^\

20

-

^___^

50

-

'

100

-

c;^^

200

-

^

500

-

> ,

.1

,2

-

(c)

5

-

108 mb

2

-

5

-

10

-

20

-

50
100

-

-,___^

"*"

■~s

200

-

'

"

500

-

'f^

1

1.0

.1
.2

(d)

.5

L

160 mb

1

r

2

-

5

-

10

-

20
50

"

V^

100

~

-______^

200

_____II^^

500 -

f

i

.2

-

(e)

.5

-

238 mb

1

-

2

-

5
10

_

20

-

50

-

"V.^^^

100

-

__^

?nn

-

— — -.^^^

500

r-'--''""''':^'^^^

.1

.2

-

(fl

.5

-

525 mb

1

-

2

-

5

-

10

-

20

-

,

50

-

100

-

200

-

V,

500

-

~.

Figure 4. Averaging kernels for selected atmospheric levels. The level to which each kernel
pertains is indicated by both a label and a broken line.

CO

30

1000 500 200 100 50 20 10

CENTER (mb)

Figure 5. T hs ovsroQing ksrns! C6nt6r ^^'ottcd qs q function of
atmospheric pressure level (solid curve). If the center of the
kernel exactly coincided with the level to v/hich the kernel per-
tained, the broken line would result.

31

5 —

10

-Q

E

LU

20

a:

3

Crt

CO

UJ

50

CL

CO

h-

X

UJ

X

o

CO

_J
<
o
o

100

200 —

500 —

1000

12 3

RESOLVING LENGTH (LOCAL SCALE HEIGHTS)

Figure 6. The resolving length plotted as a function of atmo-
spheric level. The resolving length provides a measure of
vertical resolution.