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https://ntrs.nasa.gov/search.jsp?R=19730018603 2020-06-09T09:15:05+00:00Z

1. REPORT NO.

CR-112311

2. GOVERNMENT ACCESSION NO. 3. RECIPIENT'S CATALOGUE NO.

4. TITLE AND SUBTITLE

Aerosol Physical Properties from Satellite Horizon Inversion

5. REPORT DATE

JULY 1973

7. AUTHOR(S)

6. PERFORMING ORGANIZATIONCODE

DSR-73734

C.R. Gray, H.L. Malchow, D.C. Merritt,R.E. Var, and C.K. Whitney

8. PERFORMING ORGANIZATIONREPORT NO.

AER-24-F10. WORK UNIT NO.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Massachusetts Institute of Technology, Dept. of Meteorologyand the Charles S. Draper Laboratory, Inc.Cambridge, Massachusetts

11. CONTRACT OR GRANT NO.

NAS 1-11092

12. SPONSORING AGENCY NAME AND ADDRESS

National Aeronautics and Space AdministrationWashington, D. C. 20546

13. TYPE OF REPORT ANDPERIOD COVERED

Contractor Report

14. SPONSORING AGENCY CODE

15. SUPPLEMENTARY NOTES

16. ABSTRACT

An investigation is made into the feasibility of determining the physicalproperties of aerosols globally in the altitude region of 10 to 100 km froma satellite horizon scanning experiment. The investigation utilizes a horizoninversion technique previously developed and extended herein. Aerosolphysical properties such as number density, size distribution, and the realand imaginary components.of the index of refraction are demonstrated to beinvertible in the aerosol size ranges (0.01—0.1ju), (0.1—1.0/1), (1.0— 10/u).Extensions of previously developed radiative transfer models and recursiveinversion algorithms are displayed.

17. KEY WORDS [SUGGESTED BY AUTHOR(S)]

Aerosols, Ozone, Horizon Inversion,Remote Sensing, Scattering

18. DISTRIBUTION STATEMENT

Unclassified — unlimited

19. SECURITY CLASSIF.(OF THIS REPORT)

Unclassified

20. SECURITY CLASSIF.(OF THIS PAGE)

Unclassified

21. NO. OF PAGES

211

TABLE OF CONTENTS

Section

1.6 INTRODUCTION. . 1

REFERENCES 5

2.0 RADIANCE INVERSION AND SIMULATION 72.1 AEROSOLS - IMPLICATIONS OF INVERSION 7

2 .1 .1 Introduction 7

2 .1 .2 (0. 1 - 1. O j u ) Aerosol Inversion Simulations . . . . 8

2. 1.3 (0.01 - 0. I/Li ) Aerosol Inversion Simulations. ... 132 .1 .4 (1.0 - 10/j ) Aerosol Inversion Simulations 14

2.1.5' Modeling Errors 152.2 STELLAR OCCULTATION SIMULATION 44

2 . 2 . 1 Introduction 44

2 . 2 . 2 Sensitivity 44

2 . 2 . 3 Selected Simulation Results 452 . 2 . 4 Conclusion . 45

2. 3 HORIZON PROFILE PARAMETER SIMULATION 49

2 . 3 . 1 Introduction 49

2 . 3 . 2 Discussion of Results 50

REFERENCES 54

3. 0 MODELING 55

3.1 FILTER DEVELOPMENT 553.1.1 Introduction 55

3. 1.2 Comparison of Techniques 55

3.1.3 Filter Reviey;:. 573. 1.4 Sensitivity Review 59

3.1.5 Stellar Occultation 60

3.2 RADIATIVE TRANSFER MODELING 62

3.2.1 Introduction 62

3 . 2 . 2 The Computational Problem 62

3 . 2 . 3 Codes Applicable to Related Problems 63

3 . 2 . 4 The Radiative Transfer Codes 64

111

TABLE OF CONTENTS (Cont)

Section Page

3.3 AEROSOL MODELING . 71

' 3/3. 1 Introduction : / . • • • • • • - . ." ."". ."."". . ' ' . 71

3. 3. 2 Choice of Models . . . 71

3.3.3 Error Sources 74

3.3.4 Models of the Extinction, Scattering andAbsorption Cross Sections 83

REFERENCES 90

4.0 SUMMARY • • • 91

4. 1 CONCLUSIONS —AEROSOL PHYSICAL PROPERTIES . . 91

4. 2 RESEARCH EXTENSION . . . . . . . . . 92

REFERENCES 94

5.0 APPENDICES 95

5.1 THEORETICAL HORIZON PROFILES . 95

5.2 MODELED AEROSOL FUNCTIONS 183

5. 3 MULTIPLE SCATTERING CODE C.K.W. . . . . . . . . 191

5.3.1 Introduction 191

5.3 .2 Single Scattering . .. . . . . 192

5.3 .3 Forward Scattering on the Sun-Path. . 197

5.3 .4 Forward Scattering on the Detector Path . . . . . 201

5.3 .5 Full Multiple Scattering . . . 203

REFERENCES : . 211

IV

1.0 INTRODUCTION

This report represents an extension and refinement of programs arid,

techniques developed for determining the vertical distributions of various

atmospheric constituents (aerosols, ozone, and neutral atmospheric density

i.e. Rayleigh scattering air molecules) from radiance measurements of

the earth's horizon (Newell and Gray, 1972). Surveyed in the previously

reported . study were the techniques required for determining selected

constituent distributions and the observability conditions and instrument

characteristics necessary for the successful measurement inversion. This

report extends the radiance inversion technique to the determination of

aerospl physical characteristic distributions. In the previous study

stratospheric aerosols were found to exhibit gross extinction features so

that they could be observed by the proposed technique. Because of increasing

interest in these particles as products of various pollution sources, it was

decided to investigate further (Grayet al, 1972) the potential of the horizon

inversion technique for determining other physical characteristics of the

stratospheric and mesospheric aerosols besides total extinction. Thus an

analysis of the capability of the technique to deduce information about the

particulate size distribution and index of refraction was undertaken.

These investigations were centered around the refinement and extended

development of the techniques previously employed. Here the state vector

in the inversion computer code was extended to include the desired aerosol

characteristics. The radiative transfer simulation was refined to include

arbitrary (noncoplanar) azimuth angles and opaque (thick) clouds. These

refinements were tested for accuracy against other less efficient codes.

To supplement these activities an efficient empirical aerosol model was

developed from data generated by a more complex Mi'e code computation

of the aerosol optical properties. With the empirical aerosol model,

quantities such as the partial derivatives of the angular scattering function

with respect to the real part of the index of refraction can be easily computed.

These refined techniques and aerosol models were then used to simulate

an aerosol inversion procedure for obtaining the vertical distributions of

aerosols and their gross physical properties.

The measurement technique used here is to scan the earth's sunlit

horizon as shown in Figure 1.0-1 with a multiwavelength photometer and

to record the radiant intensities measured at a predetermined sequence of

angular positions. These measured intensities are then processed as shown

in the flow chart Figure 1.0-2. _

The estimation procedure begins with an input of the geometry which,

includes primarily the sun direction and the measurement platform altitude.

The geometric data along with an a priori estimate of the atmospheric

state (constituent densities and aerosol characteristics) is used to compute

a theoretical prediction of the intensity profile and its partial derivatives

with respect to state elements. This is done with a radiative transfer

simulation for the first selected wavelength and scan angle. The theoretical

partial derivatives, theoretical intensities, and the measured intensity are

then fed to the filter equations where an optimal estimate of the atmospheric

state is produced based upon the difference between measured and predicted

intensities, the state covariance, and the measurement noise. The optimal

state estimate is then fed back through the intensity calculations and a

new intensity is computed and compared to the next measurement. The

loop continues until all wavelength channels and scan angles have been

sampled. The result is an optimal estimate, in a minimum variance sense,

of the atmospheric state including aerosol number density, size distribution

parameters, and index of refraction. ,

Section 2 of the report discusses the aerosol inversion capability of

the scattered sunlight experiment, stellar occultation inversion, and radiance

effects from a horizon profile parameter study. The aerosol inversion

simulations show the relative invertability of the various aerosol

characteristics under a range of conditions including different altitudes

and assumed a priori uncertainties. The efficacies of several wavelength

bands are compared. Three aerosol size ranges are considered, (0.01-0.1//),

(0.1-1.0/v) and (1.0-10A/). The effects of changes in variables such as season,

latitude, satellite altitude, solar zenith and azimuth angles, and albedo on

horizon profiles are investigated.

2

Section 3-details'the filter and radiative transfer model development

and inversion procedure refinement work that was carried out to make

aerosol inversion possible. Included are discussions of the augmented state

filter, the refined radiative transfer simulation, and the empirical aerosol

models.

Section 4 provides a review of our conclusions and recommendations

on the invertibility of aerosols,'model development and satellite data

requirements.

Section 5 appends supplementary information.

Appendix 1. A parameter study of horizon profile intensity variability

as a function of season, latitude, albedo, sun azimuths, and zenith and

wavelength is illustrated.

Appendix 2. The regression formulation of the aerosol model used

for the size ranges (0.01-O.i/t/), (0.1-1.6/u) and (1.0-10.0//).

Appendix 3. The multiple scattering code C.K.W. is discussed.

Fig. 1.0-1 Limb Scan Geometry

InitializeComputation:

GeometryState EstimateCovariance

Use EstimatedState Elementsto ComputePredictedIntensities

Use EstimatedState Elementsto ComputePartialDerivativesof Intensities

MeasuredIntensity

UpdateStateEstimateandCovariance

FinalEstimate ofConstituentDistribution

Fig. 1.0-2 Atmospheric Data Inversion Procedure

REFERENCES

SECTION 1

1) Gray, C. R., H. L. Malchow, D. C. Merritt, and R. E. Var, "Aerosol

Monitoring by Satellite Horizon Scanning", J. Spacecr. Rockets, 10,

71-76, January, 1973.

2) Newell, R. E., and C. R. Gray, "Meteorological and Ecological

Monitoring of the Stratosphere and Mesosphere", NASA Contractor

Report, NASACR-2094, August, 1972.

2.0 . RADIANCE INVERSION AND SIMULATION

2.1 AEROSOLS - IMPLICATIONS OF INVERSION

2.1.1 Introduction

Using a Kalman-Bucy filter to invert horizon profile data produces

both an estimate of the atmospheric state and a covariance matrix describing

the accuracy of the estimates. By propagating the covariance matrix through

a proposed measurement schedule it is possible to predetermine which

parameters will or will not be affected by the measurements, and what

linear combinations of. parameters will be well known. A function called

sensitivity has been defined which is a measure of the amount of information

obtained about a particular state element in a measurement. The sensitivity

curves are computed very rapidly, but fail to predict quantitatively the

final variances and covariances of the parameters which are obtained by a

covariance propagation. They do, however, qualitatively predict which

parameters will have small variances at the end of a measurement schedule,

and so they are used extensively in the following discussion.

A sensitivity analysis was applied to the problem of determining which

aerosol parameters are observable from a horizon scan experiment. It

has been demonstrated (Newell and Gray, 1972) that aerosol extinction per

unit volume is readily recovered from limb scan data at all altitudes between-3

10 and 85 km with an idealized instrument having a noise value of 10

times the maximum horizon signal and between 10 and 100 km for a noise-4 ' .

value of 10 . The sensitivity study, therefore, is aimed at determining

the observability of four different parameters, namely aerosol number

density (/°), size distribution parameter (a), real index of refraction (n),

and imaginary (complex) index of refraction (n1).' The size distribution

parameter (a) is the radius exponent where the power law size distribution

is of the form n ( r ) = Ar . The results are broken into three groups

representing three assumed aerosol size ranges: 0.01-0.1/u, 0.1-1.0/y, and

1.0-10/c/. For these size ranges solar zenith angles were varied from +60

to -30 with resulting scattering angles varying from approximately 50

to 140 . Constituent results based on these variable conditions are developed

in Figures 2.1-1 to 2.1-12. Analysis shows how the invertability of the

parameters is affected by 1) the geometry 2) the instrument noise, and 3)

the statistics of the initial estimates of the aerosol parameters. This

analysis was conducted for spherical aerosols and nonspherical aerosol

shape factors which have a spherical equivalent.

2.1.2 (0.1-1.0/u) Aerosol Inversion Simulations

For this aerosol size range (0.1-1.OAT), .as is also the case for the

other size ranges, the aerosol number density is initially adjusted so that

extinction at 5500 A is equal to the extinction used as the standard state

(Newell and Gray, 1972) based on Elterman, 1966andl970, given the initial

estimate that a = 3, n = 1.5, and n1 = 0. Because of the proportionality

between extinction and cross section for a given number density there will

be an increased number density for the (0.01-0.1//) range over the (0.1-1.0/t/)

range and conversely a decrease for the (1.0-10//) range. For the range0 0

(O.I - I .OA-) the number density runs from approximately 10 /cm at ground

to 10"5 /cm3 at 100 km.

For all of the three aerosol size ranges the instrument white noise_ n o

was assumed to be 10 u watts /cm (RMS). It has been shown previously

(Newell and Gray, 1972) that varying the noise simply varies the maximum

altitude from which information can be retrieved. The maximum useful

altitude can be approximated by observing the intensity versus altitude

curves of Section 2.3 and noting at what altitude the intensity reaches the

noise level i.e. S/N = 1. At a few kilometers above this point the initial

estimates of constituents are as good as the filtered values and no significant

changes are affected in the.estimates.

Sensitivity curves were produced fora number of different scattering

angles and wavelengths with an instrument having a finite field of view.

Sensitivity is a function derived from the filter equations, which indicates

the relative information content from constituent to constituent and altitude

to altitude for every measurement condition i.e. (wavelength and tangent

height). Given a measurement, the information content for a particular

constituent and altitude (state element) is reflected in a decrease in the

variance associated with that element and it is natural to look at the

covariance update equation for the definition of sensitivity:

P(nH-l) = P ( m ) - K ( m + l ) B ( m + l ) P ( m ) (2.1-1)

the term of interest here is K(m+l)B(m+l)P(m), which is the decrease in

the covariance for a particular measurement condition. Normalizing by

P(m) yields the sensitivity matrix K(m+l)B(m+l). .Of particular interest

in the sensitivity matrix are the diagonal elements which indicate the relative

decrease in variance of the elements of the state vector. To isolate the

effects of the measurement vector on the sensitivity, the covariance is'

assumed to maintain its initial value. With this assumption the sensitivity

of the jth element of the state vector is

s . ( m ) =3h(x ,m)

dx.± ,

\2Pii + R

2.1-2

In choosing an instrument's wavelengths it is desirable to have as

many linearly independent measurements as possible. That is, each

measurement should be sensitive to a different combination of parameters.

The functions which relate extinction and phase function to aerosol

parameters change slowly with respect to wavelength. For this reason a

wide spread of wavelengths over the region 2500 to 7000 A is best suited

for the inversion (see Figure 2.1-1 or 2.1-4). Thus the wavelengths 3000,

4000, 5000, 6000, and 7000 A, which span the visible region with a minimum

of wavelength channels necessary for the inversion, have been arbitrarily

selected as working values. In this case the actual wavelengths chosen

are not as important as the separation between the wavelengths and in an

actual experiment more wavelengths would be run for greater accuracy. "

The total scattered sunlight radiance at the horizon is a composite

of the radiance contributed by aerosols and Rayleigh scatterers. For each

scattering constituent the radiance contribution is proportional to the product

of its scattering cross section and angular scattering function (phase

function). If the scattering cross .sections for both aerosol and Rayleigh

scatterers are equal then the relative contribution of each constituent to

the horizon radiance depends only upon the relative values of the phase

functions at a given scattering arigle. For most scattering angles away

from the forward region the aerosol phase function will be smaller than

the Rayleigh function thus reducing the relative contribution of aerosols to

the radiance, however this effect can be partly offset by an increase in the

relative scattering cross section of aerosols at longer wavelengths.

Both' the effects of changes in wavelength on cross section and changes

in scattering angle on the phase function can.be observed by examining the

sensitivity curves of Figures 2.1-2 and 2.1-4. In Figure 2.1-2 it can be

seen that the aerosol number density sensitivity "AEROSOL" increases at

longer wavelengths while the "NEUTRAL" molecule scattering decreases.

This is the cross-section effect. Figure 2.1-4 differs only by a change in

solar zenith from Figure 2.1-2. The scattering angle for Figure 2.1-4 is

less favorable (i.e. less forward, w l l O versus ~ 50 ) for aerosols.

Therefore, the aerosol sensitivity is lowered relative to Figure 2.1-2 while

the neutral molecule scattering sensitivity increases.

The most serious .problem encountered is in the uncertainty of the

initial statistics associated with the initial estimate. The updating of

estimates by the filter inversion routine depends in part on the RMS values

of the initial estimate. If a parameter is well known it will not receive

much of an update and its variance will decrease relatively slowly. This

is particularly critical in the case of the aerosol parameters where there

is a large sensitivity to aerosol extinction per unit volume, however, the

particular aerosol parameters which receive these updates are determined

partly by their initial variances.

10

Sensitivity computations in the (0.1-1.0/v) size range were made with

initial RMS error values of 600% for (P), 1 for (a), 0.05 for (n), and 0.005

for (n1) along with realistic estimates of neutral density and ozone (Malchow,

1971) RMS values as a function of altitude (Figure 2.1-3). Curves run

with these values showed a large sensitivity to aerosol number density

and only a small additional sensitivity to (n1) below 30 km. Other runs

with 100% RMS error on (P) (Figure 2.1-1) showed all parameters except

(n) to be readily observable at all altitudes. Thus varying the initial RMS

estimates varies the final RMS estimates and the sensitivity, and until

realistic data are collected about these parameters the covariance matrices

associated with the parameter estimates will be unreliable.

The sensitivity curves indicate a potential problem with regard to ob-

servability. Since the wavelength dependence of sensitivity is basically the

same for both number density(/°) "AEROSOL" and size distribution(o) "ALPHA"

there is the possibility that no two measurements are linearly independent in

these two parameters, therefore making it impossible to estimate either in

one . This similarity in wavelength dependence occurs because increasing

either (a) or (/>) has the effect of lowering the wavelength dependence of

the total received signal by increasing the effective aerosol to Rayleigh

scattering ratio. A complete propagation of the covariance matrix was

made to determine if this was the case. The results indeed showed a.

high correlation (0.6 to 0.8) between the two parameters but also showed a

significant factor of five decrease in variance for both parameters in a

coarse inversion (only three wavelengths and a four kilometer separation

of tangent heights). The final variances were VQ - ±0.2 and crp = ±20%

which indicates that the two quantities are separable and observable.

Covariance propagations were performed for each of the particle size

ranges. The results of these propagations show numerically how the initially

assumed variances of the aerosol parameters are reduced by the information

gained in a measurement sequence.

11

Initial values of the aerosol parameter RMS uncertainties were chosen

to reflect the range of values occurring in the literature (Newell and Gray,

197,2, and Malchow, 1971). Number density (.ft) was assumed to vary over

.one order of magnitude ±3cr, thus the ap used was 166%. This is consistent

with _the large observed, variations in the number density-of stratospheric .

aerosols related to volcanic activity. The size parameter (en) was assumed

to cover the entire observed range (Xa<6, thus aQ = 1. The real part of

the .index of refraction ranges from that of water (1.33) to fused silica

(1.65) thus yielding a ±3a range of 0.32 and a a of 0.05. Finally the complex

part of the index of refraction was assumed to be as large as 0.015 (+3cr)

which is felt to be conservative (Volz, 1973) with a a , value of 0.005.

Both the real and complex parts of the index of refraction are assumed to

be wavelength independent. All initial RMS values are assumed constant

with altitude. ,

Figure 2.1-13 shows graphically the reduction in RMS uncertainty

resulting from a covariance propagation for the (0.1-1.0/v) particle size range.

The solid line indicates the assumed initial value of a particular quantity,

and the error bars show the final RMS uncertainty. Numerical values

corresponding to the error bars are listed in Table 2.1-1. Considerable

reduction of the number density uncertainty occurs for all altitudes although

it is especially prominent at 10 and 20 km where the aerosol turbidity is

at a maximum. A similar pattern of error reduction is displayed for the

size parameter (a). Uncertainty in the complex part of the index of refraction

is reduced, however, the improvement is not as striking as in the case of

the other parameters. In general the covariance propagation for this size

range shows that substantial reductions in the initial uncertainties of aerosol

number density, imaginary index, and size parameters are effected by the

inversion. The uncertainty in the real part of the index of refraction is

however reduced only slightly. This result is consistent with the sensitivity

curves in Figures 2.1-1 and 2.1-2 which show little sensitivity to the real

index of refraction at any altitude.

12

2.1.3 (O .OI -O . IA / ) Aerosol Inversion Simulations

The aerosol number density was adjusted to preserve the standard

extinction versus altitude curve for 5500 A with a = 3, n = 1.5, and n1 = 0.7 3For this range the density varied from 1.4x10 /cm at the ground to 5.5x

- 2 310 /cm at 100 km. Since the phase function for this size range varies

less with angle than in the (0.1-1.0^) range only a few zenith angles were

run. The results of these runs show that the constituent sensitivities did

not significantly change with zenith angle (see Figures 2.1-6 and 2.1-7).

The most notable difference between this size range and the two others

is the lack of sensitivity to (n1). For these small particles only number

density and the size distribution parameter (a) are observable because the

partial derivatives of the scattering phase function with respect to (n) and

(n') for this size range, are near zero. A change from 300% to 100%

uncertainty in number density enhances the (a) sensitivity rather than

increasing (n1) sensitivity (see Figure 2.1-6 and 2.1-5 respectively).

Analysis of the two sensitivity curves for (/>) and (a) indicates a

potential problem in distinguishing between the two parameters. The fact

that the sensitivities for both parameters have the same wavelength

dependence implies that the measurement equations at different wavelengths

are not totally linearly independent (see Figure 2.1-8). However a coarse

covariance propagation was run (five wavelengths and four kilometer tangent

height separation) which indicates that the two are separable. The

correlations were indeed high (0.6 to 0.8) between the parameters but there

is a factor of two decrease in number density variance and a factor of ten

decrease in size distribution variance from the initial values.

The two parameters ( ft and a ) are highly observable from 10 to 100

km or until instrument noise becomes dominant. Typically an instrument_3 '

noise of 10 times the maximum horizon signal restricts the altitude-4sensitivity range to 85 km while a noise value of 10 extends the altitude

sensitivity to approximately 100 km. Thus the interference that comes

13

about from increased uncertainties in neutral density at higher altitudes

does not reduce the observability of the two aerosol parameters significantly.

Figure 2.1-14 illustrates the RMS error reduction for the particles

in this size range. The numerical values are listed" in Table 2.1-II. The

results are generally similar to those obtained for the (0.1-1.0/u) size range

except that there is a smaller reduction of the number density uncertainty.

Somewhat surprising, in view of the near zero sensitivity for (n1) shown in

Figure 2.1-6, is the substantial reduction of the uncertainty in (n1) after a

covariance propagation. This indicates that the factor causing the error

reductionisa buildup of correlations between (rt1) and the other parameters

as opposed to a partial derivative effect which dominates the sensitivity

results. When the sensitivity and covariance propagation results differ

significantly as in this case it is important to consider the differences

between the abbreviated covariance propagation used to produce these

working numbers and a real covariance propagation used in an actual

inversion. To insure the ultimate convergence of the state vector estimate

it is often required to reduce the covariance update by means of a numerical

multiplier. This results in a smaller reduction of the initial covariance

after a given number of reiterative updates than would' be indicated by the

unweighted propagation used here.

2.1.4 (1.0-10/t/) Aerosol Inversion Simulations

The large particles in this range are the furthest from Rayleigh in

their scattering properties, i.e., in their cross-section wavelength

dependence and in the shape of their phase function. Because of their largeo

cross section the number density is low, running from 23.4 /cm at the

ground to 7.7x10 /cm at 100 km.

The great difference between these particles and other atmospheric

constituents makes them the most readily inverted of all size ranges. It is

possible to invert all four aerosol parameters from a given horizon scan

when the phase function is not minimized by an unfavorable scattering angle

14

i.e. sun angle. Although the four sensitivities are reasonably close, results

from the previous size ranges indicate that all four physical characteristics

are separable. Variable scattering angles are readily obtainable for nearly

all conceivable orbits from equatorial to polar orbits with the exception of

the 600 and 1800 local hour angle near polar-orbits.

There is, however, a problem with this size range that does not appear

in the other ranges. That is, when the scattering angle approaches 110°,

the phase function is so small that the aerosol energy contribution and

respective sensitivities drop drastically and only number desity is highly

invertible from the data (Figure 2.1-12). The variations in sensitivity with

changing solar angle can be clearly seen in Figures 2.1-9 through 2.1-12.

Covariance propagation results for the large particles are shown in

Figure 2.1-15 and the corresponding numerical values are listed in Table

2.1-II. In the case of the large particles all the parameter uncertainties

are substantially reduced including the real part of the index of refraction

which was essentially unaffected for the other particle size ranges.

2.1.5 Modeling Errors

With the standard filter approach to the inversion problem it is

theoretically possible to drive the variances on all the parameters as close

to zero as desired by taking enough measurements. However, since there

are inherent errors in the radiative transfer modeling a near perfect variance

is unrealistic. The purpose of this section is to determine the effects of

aerosol modeling errors on the final accuracy obtainable in an inversion.

This is done by computing the vectors (k, Ax) and the scalar (AI) in the

equation Ax = k (AI) where Ax is the state error, (k) is the filter gain, and

AI is the intensity error.

The results are presented in sets of tables. The first table of each

set (Table 2.1-IV) illustrates the gain (k) which is used to compute errors

in the state given errors in intensity. This is done with a geometry having

15

a zenith angle = 30 and an azimuth angle = 0° for all size ranges, with

standard initial errors (see section 2.1.2), and using a 300% initial uncertainty

in aerosol number density. The additional tables of each set are the computed

state error Ax for errors in the albedo and aerosol models and for an

instrument Mas. ^The state errors are based upon the gain (k) of the first

table.

It is important to realize that the initial uncertainties are important

in the propagation of error. For example, a parameter which has both a

large partial derivative and a large initial uncertainty will receive a large

update and become more in error by inaccurate models than a parameter

with a small initial variance and partial derivative. The numbers given

represent the maximum error that will occur in an inversion using 3000,

4000, 5500, and 7000 A as wavelength channels. It should be noted that the

(k) values for 3000 A at 20 and 40 km, and for 4000 A at 20 km are set

equal to zero. This is because these wavelengths are saturated at the

designated altitudes and therefore yield no information about the densities.

Tables 2.1 -V, VI, IX, X, XV, and XVI list the errors in the state

vector elements produced by errors in the aerosol extinction and phase

function models. The model errors used here are representative of the

errors associated with the aerosol models presented in Section 3.3 of this

report. Tables2.1 -V, VI list the state errors for aerosols in the (0.01-0. IA/)

size range. The modeling errors for this size range are small («2% in

extinction and «5% in phase function) and consequently the effects on the

state are also small. The largest resulting error is due to the phase function

error, and is a 12.3% error in the particle number density at 60 km as

shown in the fourth column of Table 2^1 -VI.

The (O.l-l.O/^) particle size range has larger associated errors and

consequently larger effects on the state. Tables 2.1 -IX, X list the state

errors associated with this size range. As with the smaller size range,

the phase function error (10%) produces larger state errors than the

extinction error (5%). The largest error induced is in the aerosol number

16

density at 60 km, and is 26%. This error is still small compared to the

assumed initial number density uncertainty of 300%, and is less than the

60% uncertainty obtained after a coarse covariance propagation.

For the large particle range the state errors are presented in Tables

2.1 -XV, XVI. Again the number density estimates .are primarily affected,

and the errors are comparable to the final state estimates obtained via a

covariance propagation.

Table 2.1 -XII shows the effects of the assumption of the wrong size

range for aerosols, i.e., the table shows what errors result if the size

range is assumed to be (0.1-1.0/u) whereas the actual size range is (.01-0.1/u).

The resulting errors are: large as expected since the extinctions differ by

two orders of magnitude for. the two size ranges. Aerosol number density

is shown to be in error, by one order of magnitude or « 1000% at 40 km.

Other state elements are also strongly affected. The neutral density estimate

is in error by 52% at 80 km, and the ozone density by 242% at 60 km.

Errors in the other aerosol parameters are significant though not large

(these are given in absolute units). At 40 km, the error in (a) approaches

one which represents a significant alteration of the size distribution. The

results of this size range shift emphasize the importance of either knowing

rather accurately the particle size, limits, or expanding the state vector to

include these limits. .

Table 2.1 -XIII shows the effects of the introduction of a 10% instrument

bias error in each wavelength channel. The error induced in the aerosol

number density is moderately large because the signal contribution by the

aerosol to the total signal is small. Since the aerosol number density

uncertainty is large, the filter attempts to correct the signal error by

adjusting mainly the aerosol number density.

Finally, Tables 2.1 -VII, XI, and XVII show the effects on the state

of the introduction of anunestimated effective surface albedo deviation from

the expected value. The effect of an albedo uncertainty is similar to the

17

effect of an instrument bias, i.e., there is a broad-band shift in the measured

intensity away from, the expected value. Since aerosol density is assumed

to have a relatively large initial uncertainty, it is this quantity which is

mainly adjusted to account for the intensity shift. The amount of adjustment

done to the aerosol number density is inversely proportional to the

contribution that the aerosol makes to the total intensity. Error values in

Table 2.1 -VII, XI, and XVII are based upon a scattering angle that coincides

with the minimum in the angular scattering functions for the medium and

large-sized particles.

Thus these aerosols are making a minimal contribution to the total

intensity at the receiver, and therefore the errors related to an uncertain

albedo are maximized. Tables 2.1 -VII, XI, and XVII illustrate this effect

dramatically. Each aerosol size rangeis defined to have the same extinction.

As the particle size increases, the scattering phase function decreases (at

the chosen sun angle) and the error increases rapidly for the same extinction

in each size range. There are two implications of these results. One is

that the albedo should be added to the state vector and estimated to minimize

its uncertainty. The other implication is that the solar scattering angle

for primary radiation should be chosen to minimize the effect of albedo

uncertainties. At more desirable scattering angles the large particle errors

(Table 2.1 -XVII), are substantially reduced, and are comparable to the

small particle errors (Table 2.1 -VII). For example, the 1160% error in

number density for large particles at a scattering angle of 80 (Table 2.1

-XVII, 40 km) is reduced to 60% when the scattering angle is changed to

30°.

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ALTITUDE (KM)

Fig. 2.1-1 Scattered Sunlight Constituent Sensitivities

for Aerosol Size.Range (0,1-l.Oy..) , Zenith Angle = 60°,

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29-

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ALTITUDE (KM)


for Aerosol Size Ranqe (0.1-1.On), Zenith Angle = 60°,

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ALTITUDE (KM)


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ALTITUDE (KM)


for Aerosol Size Range (0.1-l.Oy), Zenith Anale = 0°,

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32

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ALTITUDE (KM)

'70 80 9O 100

Fig. 2.1-5 Scattered Sunlight Constituent Serisitivities ":

for Aerosol Size Range (0.01-0.ly), "Zenith Angle = 60 ,

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ALTITUDE (KM,)


for Aerosol Size Ranae (0.01-0.ly), Zenith Angle = 60°,

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ALTITUDE (KM)

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ALTITUDE (KM)


for Aerosol Size Range (0.01-O.ly), Zenith Angle = 60°,


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ALTITUDE (KM)


for Aerosol Size Range (1.0-lOy), Zenith Angle = 60°,

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ALTITUDE (KM)


for Aerosol Size Range (1.0-lOy), Zenith Angle = 90 ,


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ALTITUDE (KM)


for Aerosol Size Range (1.0-lOy), Zenith Angle = 50°,

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ALTITUDE (KM)

90 10O


for Aerosol Size Range (1.0-lOy), Zenith Angle .= 0°,

, Azimuth Angle = 0°, and la values (p=300%, a=l, n=.05,

n'=.005), Wavelength in Angstroms.4 0 , . . . - . - • '

o

CO

UJQ

DiUJCO

40

- *

i-t-

-\—

T

60 80 0 20

ALTITUDE ( KM )

54

3

2

1

<cn:Q_

+1.6

1.5

1.4

•0.02

-0.01

040 60 80 100

Fig. 2.1-13 Covariance Propagation for Aerosol Size Range

(0.1-l.Oy), Zenith Angle = 60°, Azimuth Angle = 0°,

and la values (p=166%, a=l, n=.05,n'=.005), Wave-

lengths of 3000, 4000, 5000, 6000, and 7000 8,

and Tangent Heights Spaced at 4 km.

41

CO

O

on~2.LUO

CQ

•* +• +

20 40 60T t t T t

80 0 20 . 40ALTITUDE (KM) '

6

5

4

32

1

••1.6

-1.5

60 80 100

1.4•0.02

-0.01

0


(0,01-0.ly) Zenith Angle = 60°, Azimuth Angle = 0°,

and lo values (p=166%, a=l, n=.05, n'=.005), Wave-

lengths of 3000, 4000, 5000, 6000, and 7000 &,


42

o

CO

UJo

O3-£EIT)'

1 1 1-

H 1-

0 20 40 60 80t t t T T

5

4

3

2 :

1

jl.6

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20 40 60 80 100

ALTITUDE (KM)


(1.0-lOy), Zenith Angle =' 60°, Azimuth Angle = 0°,

and la values (p=166%, a=l, n=.05, h'=.005), Wave-

lengths of 3000, 4000, 5000, 6000, and 7000 S,


43

2.2 STELLAR OCCULTATION SIMULATION

2.2.1 Introduction

-4" One of the contract- subtasks was concerned with the application of

the Kalman-Bucy filter to the inversion of stellar occultation data. The

data, however, was unobtainable during the contract period. The inversionscheme was nevertheless developed and a sensitivity analysis was

performed. Several computer runs were made using simulated data which

illustrates the invertibility of high altitude aerosol layers with a star

occultation experiment. :

2.2.2 Sensitivity

The sensitivity function for stellar occultation is identical to that

used for scattered light. The partial derivatives are, however, applied to a

different radiative transfer model. Figure 2.2-1 shows the envelope of

peak sensitivities for a complete stellar occultation scan at several

wavelengths. It can be seen from this figure that both ozone and neutral

density will be invertible down to an altitude of 20 km. The curves indicate

ozone cannot be inverted below this level. This is because the intensity

attenuation of ozone sensitive wavelengths below 20 km is too great to permit

measurement of the signal with the assumed instrument signal to noise

ratio of 100:1.

Figure 2.2-2 shows the effect of a high altitude aerosol layer (50

km, one order of magnitude anomalous increase) on the stellar occultation

sensitivity curves. It can be seen that standard levels of aerosols do generate

a sufficient signal increase to be. detected, however, an anomalous increase

of this nature will be measured and inverted for the assumed 100:1 signal

to noise ratio. . . . . .

44

2.2.3 Selected Simulation Results

For the purpose of simulation, fictitious, but realistic, density

distributions other than standard were used to generate simulated data e.g.

anomalous layers were included based on measurements of Rossler, 1968

and Elliot, 1971. The data was then processed by the inversion routine.

In Figure 2.2-3 the crosses represent the estimate of the density with two

error bars given for each-point". The horizontal bar represents the un-

certainty in the altitude at which the average density occurs while the

vertical bar represents the density uncertainty. The density error

bars are derived from the filter while the altitude error bars are derived

a priori in simulations. The solid line represents the fictitious density or

"right" answer. If the densities fall within the RMS error bars 67% of the

time the simulation is considered a success.

Figure 2.2-3 is a simulation in which an order of magnitude anomalous

aerosol layer at 50 km was used to generate fictitious data. The inversion

results demonstrate the ability of the inversion technique to determine such

layers if they exist with a 100:1 signal to noise ratio and wavelengths of

3000, 4000, 5000, and 7000 A.

2.2.4 Conclusion

The Kalman-Bucy filter is easily adapted to the inversion of stellar

occultation data and produces excellent results. Although a simplified

radiative transfer model was used in these simulations there is no reason

why a more sophisticated model incorporating refraction and dispersion

effects could not be used. This would extend the range of invertibility well

below 30 km for both neutral density and aerosols in addition to providing

more accurate results above 30 km!.

45

V<IdQ-

enZLJQ

CDZLJCO

1 1 1 1 1 1

1O 2O 30 SO 60 70 8O 90 1OO

ALTITUDE (KM)

Fig 2.2-1 Stellar Occultation Constituent Peak SensitivityProfiles with 100:1 signal to noise and standarddensities.

46

LJQ_

tnzuo

enZuen

80 9O 1OO20 30 -tO SO 6O ?O

ALTITUDE (KM)

Fig. 2.2-2 Stellar Occultation Constituent Peak SensitivityProfiles with 100:1 signal to noise and an (orderof magnitude increase) anomalous aerosol layer at50 km.

47

CW>I/T> siosoyav

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2.3 HORIZON PROFILE PARAMETER SIMULATION

2.3.1 Introduction

In this section we discuss the results of a horizon profile parameter

study aimed at determining how horizon profiles are altered by changes in

the various measurement parameters. These include season, latitude,

satellite altitude, solar zenith and azimuth angles, and either the earth's

albedo or the altitude and albedo of a cloud layer. The newly developed

mulitple scattering code (C.K.W.) was used to produce 324 radiance profiles

representing various combinations of the measurement parameters. The

horizon profiles are plotted in multiples of from three to five profiles per

graph and are compiled in Appendix 5.1. Figure 2.3-1 is a sample profile

set showing zenith angle as a variable. The measurement parameter being

varied in each graph is identified at the top of the graph and values of a

given variable corresponding to a given profile are listed to the right of

each graph. The fixed parameters for each profile set are listed in the

legend. The right hand graph in each figure displays the fractional change

of each radiance profile relative to one member of the set.

Figure 2.3-2 shows the limb scan geometry, defining the tangent height

(H), the satellite altitude (S), and the line of. sight (L). The direction of

the sun's rays are specified in relation to a Cartesian coordinate frameA ' A '

attached to the satellite, with (Z) along the local vertical and with (Y) inA A

the plane defined by (Z) and (L). The solar zenith angle (9) is measuredA A A

relative to (Z) and the solar azimuth angle (0) is measured in the ( X , Y )A ' A A

plane relative to (Y). The angle which (L) makes with (Y), i.e., the scan

angle, is defined as (6).

All of the horizon profiles given in this report pertain to a satellite

altitude of 500 km. The scan angle (5) therefore varies from about 19.5

to 22 as the tangent height varies from 100 km to zero. The single scattering

angle is determined by the angles (0, $, and 6 ) through the expression

= (cos(f>sinOcos6 - cos0sin<5). Thus, for the coplanar cases (<£=0° and ^ = 180 )

0=90 +6-e and 0=90°+S+G, and for the ^=90° case cos^ is simply given by

(-cos0sin&).

49

2.3.2 Discussion of Results

For reasons of organizational convenience the 84 figures for this

parameter study are located in Section 5.1. In each case the atmosphere

was considered to have two-kilometer layers (DZ = 2) that extended to an

altitude of one hundred and twenty two kilometers (ATM = 122). The seasonal

and latitudinal distributions for the mean neutral and ozone densities used

in determining these profiles are shown in Figures 5.1-73 to 5.1-84. Figures

5.1-1 to 5.1-12 show the variations in horizon profiles resulting from

seasonal variations in the neutral and ozone densities for each of twelve

latitide-wavelength conditions (latitude = 0, 40, and 70°N and X= 3000,

4000, 5500 and 7000 Angstroms). At 0°N latitude the same neutral density

distribution was used for all seasons. The changes in the profiles with

season at 0 N latitude reflect, therefore, only the changes in the ozone

density distribution. This profile set represents the only case where the

differential horizon profiles can be interpreted simply in terms of the

changes'occurring in just one of the atmospheric constituents . These changes,

which are referenced to the winter profile, are greatest at 3000 A and

smallest at 4000 A in agreement with the known spectral characteristics

of the ozone absorption cross section (Wu, 1970). The remaining horizon

profiles in this parameter study are arbitrarily based upon the latitude

and seasonal choices of 40°N latitude, winter.

Figures 5.1-13 to 5.1-24 show the variations in horizon profiles with

wavelength for twelve cloud-zenith angle conditions (a zenith angle of 30

and 80° and a cloud at 4, 1, and 10 km with an albedo of 0.4 and 0.8). The

solar irradiances at 3000, 4000, 5500, and 7000 A are 5.14, 14.92, 17.25,2 :

and 13.25 u watts /cm -A respectively (Thekaekara and Drummond, 1970).


wavelength for four zenith angles (0, 30, 50, and 80 degrees) and a ground

albedo of 0.3. These figures show that clouds are only observable at the

longer wavelengths like 5500 and 7000 A which penetrate deepest into the

atmosphere. In addition, these figures show that the enhancement of radiance

50

due to an increase in cloud albedo is strongly dependent upon the zenith

and azimuth angles, which determine the scattering angle (0) and hence

the fraction (sin0) of the solar irradiance that is actually incident upon the

cloud surface.


zenith angle for 36 combinations of azimuth angle, wavelength, and ground

albedo (azimuth angle = 0, 90, and 180 degrees; X= 3000, 4000, 5500, and

7000 Angstroms; ground albedo = 0.2, 0.5 and 0.8). The differential horizon

profiles are all referenced to the zero zenith angle profile. These figures

show that the shape of a profile is not affected greatly by changes in the

albedo and in the solar azimuth and zenith angles. For the brightest profile

(80° zenith and 0° azimuth) there is little albedo effect because sin (0) is

relatively small and therefore most of the signal is produced by strong

forward scattering high in the atmosphere. Albedo effects are most

prominent for azimuth and zenith angle combinations that make the scattering

angle (0) fa i r 12.


ground albedo for eight combinations of zenith angle and wavelength (zenith

angle = 30 and 80 degrees; X= 3000, 4000, 5500, and 7000 Angstroms).

The differential horizon profiles are referenced to the albedo = 0 profile.

These figures show that the enhancement of the line of sight radiance due

to the earth's albedo is strongly dependent upon wavelength and scattering

angle.

51

l-Ll l/>

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52

Fig. 2.3-2 Limb Scan Geometry

53

REFERENCES

SECTION 2

1) Elliott, D.D., "Effect of a High Altitude (50Km) Aerosol Layer on

Topside Ozone Sounding", Space Res., 11,857-861, 1971.

2) Elterman, L., "Vertical-Attenuation Model with Eight Surfaces

Meteorological Ranges 2 to 13 Kilometers", AFCRL Report 70-0200,

March, 1970.

3) Elterman, L., "An Atlas of Aerosol Attenuation and Extinction Profiles

for the Troposphere and Stratosphere", AFCRL Report 66-828,

December, 1966.

4) Malchow, H.L., "Standard Models of Atmospheric Constituents and

Radiative Phenomena for Inversion Simulation", MIT Aeronomy

Program Internal Report No. AER 7-1, January, 1971.

5) Newell, R.E., and C.R. Gray, "Meteorological and Ecological


Report, NASA CR-2094, August, 1972.

6) Rossler, F., "The Aerosol Layer in the Stratosphere", Space Res.,

8, 633-636, 1968.

7) Thekaekara, M.P., and A.J. Drummond, "Standard Values for the Solar

Constant and its Spectral Components", Nature, Phy. Sci., 229, 6-9,

January 4, 1971.

8) Volz, F.E., "Infrared Optical Constants of Ammonium Sulfate, Sahara

Dust, Volcanic Pumice, and Flyash", Appl. Opt., 12, 564-598, March,

1973.

54

3.0 MODELING

3.1 FILTER DEVELOPMENT

3.1.1 Introduction

There are a number of approaches to the problem of inverting aerosol

parameters, neutral atmospheric density, and ozone density from horizon

profile data. Ideally the method chosen should provide a best estimate (in

some statistical sense) of the desired parameters incorporating not only

the data, but the instrument characteristics arid previous knowledge of the

parameters as well. What follows is a discussion of several techniques

which have been considered as well as the reasoning behind the choice of

the Kalman-Bucy filter.

3.1.2 Comparison of Techniques

The first and most obvious approach to the inversion problem is to

attempt an analytic solution of the equations of radiative transfer, that is

solving for density as a function of intensity. This technique, has been

used (Anderson, 1969) for atmospheric probing; however, even for simple

geometries with single scattering and only one unknown constituent several

simplifying assumptions must be made in order to reach an analytic solution.

Also, once the solution is reached it is difficult to perform a meaningful

error analysis on the final answer. Thus given the stated goals of the

method to be chosen and the complexity of the radiative transfer equations

for horizon scan geometry, it is clear that this approach to the inversion

problem is limited.

Since it is not practically feasible to solve the equations analytically

the next possibility to consider is an indirect solution to the equations using

partial derivatives and some iterative scheme to arrive at an approximate

solution to a set of simultaneous equations. This would probably be possible

and would give reasonable answers, however, there would be no statistical

55

error analysis. Also if there were more data points than needed to solve

the simultaneous equations only the first data taken would be used. Thus

this technique involves incomplete data utilization and does not provide

for the desired error analysis.

Next, a regression technique might be considered. A regression would

provide not only a best estimate in a least squares sense but error statistics

as well. However there are some problems with the application of a

regression and these are: 1) Previous knowledge of the parameters is not

incorporated in the estimate or the statistics, 2) Instrument noise is not

incorporated, and 3) The large amount of data must be processed

simultaneously rather than sequentially. However this would be a good

technique if there was not a better approach to the problem.

The use of Bayesian statistics makes possible the incorporation of

previous knowledge and instrument noise in the inversion scheme. It also

allows data to be processed one point at a time thus incorporating each

new datum into the estimate of the statistics of the estimate. There are a

number of Bayesian approaches to the problem such as maximum likelihood

estimation, Bayesian regression, weighted least squares, and the Kalman-

Bucy filter.

For the case in which the equations relating the measurement and

the state are linear, and noises are Gaussian, all these techniques are

equivalent. However in real physical situations these assumptions are often

invalid and the techniques differ. The Kalman-Bucy filter was chosen

because: 1) It has all the advantages of Bayesian statistics, 2) The filter

equations are simple and easy to understand and code, and 3) Because of

the simplicity of the equations the filter readily lends itself to iterative

techniques which ensure convergence in spite of the nonlinearity of the

equations.

56

3.1.3 Filter Review

The Kalman-Bucy filter is a recursive technique for estimating the

state (in this case aerosol parameters, neutral atmospheric density, and

ozone) of a system (the atmosphere) from measurements of that system

(horizon profile data). At each recursion the previous estimate of the state

is updated. The degree and distribution among parameters of the update

is determined by the measurement; the previous estimate of the state, the

instrument noise, and the statistics (covariance matrix) associated with the

estimate of the state. After the state is updated the covariance matrix is

also updated so that it reflects the current knowledge of the state.

The updating procedure is done in three steps. First is the calculation

of the filter gain which determines the degree and distribution of the update

among the elements of the state vector.

K(m+l) = P(m) BT(m+l) [B(m+l) P(m) BT(m+l) + R]"1 (3.1-1)

where

(m) is an index of measurement

(K) is the filter gain

(P) is the covariance matrix

(B) is the measurement vector whose elements are

• b. = 3h(x, m+l)/3x.

h(x, m+1.) is the predicted value of the intensity based, on the

state estimate and measurement parameters

and

(R) is the instrument noise.

57

The second step is the updating of the state estimate

x(m+l) = x(m) + K(m+l) [l (m+1) - h(x, m+1)] (3.1-2)

where (x) is the state estimate and (I) is the measurement.

Thirdly the covariance matrix is updated to incorporate the knowledge

gained of the state.

P(m+l) = P(m) - K(m+l) B(m+l) P(m) (3.1-3)

. As can be seen from the equations there are three steps which must

betaken before applying a filter to the problem of limb scan data inversion.

First a computer code giving a direct solution to the equations of radiative

transfer is needed for the calculation of h(x, m) and 3h(x, m)/9x (partials

are usually calculated by the approximation Ah(x, m)/Ax). Second the state

(x) must be defined, and third some initial estimate of the covariance matrix

P(0) must be made to start the recursive process.

Two computer codes are available for the first requirement and these

are discussed in detail in Section 3.2. For all of the inversion work to

date the hybrid single scattering code has been used since the multiple

scattering code had not been completed. However, the characteristic shape

of the hybrid profile agrees well with the multiple scattering code. Also,

the hybrid codes use less computer time and it is simpler to extract the

partial derivatives from the single scattering code. It is important to note

that one of the advantages of the filter inversion technique is the capability

of incorporating any solution into the equations of radiative transfer no

matter how complex. Thus in an actual experiment the more accurate

multiple scattering code would be used.

The definition of the state has been previously discussed (Newell and

Gray, 1972) for the purposes of inverting three constituent densities.

Basically that scheme had density points defined at arbitrary altitude

58

increments. The elements of the state were the logs of the densities (or

extinction in the case of aerosols) at each altitude. By using this definition

of the state at intervals of from 3-10 km along with a finer integration

step of 1 km in the radiative transfer codes it was found that the filter

would converge (Newell and Gray, 1972) for realistic densities that would

be encountered in the atmosphere.

3.1.4 Sensitivity Review

For this current work the state definition was expanded to include

the log of aerosol number density (P), size distribution parameter (a), and

both real and complex indices of refraction (n and n1) at each chosen altitude.

This expanded state was first analyzed through sensitivity studies and then

through covariance propagation to determine the feasibility of estimating

each parameter. The results of this study are discussed in Section 2.1. .,

The sensitivity vector is derived from the filter equations and indicates

the relative information derived about each constituent parameter at each

altitude for every measurement condition (i.e. wavelength and viewing angle).

Since after each measurement the covariance matrix records the information

gained from the measurement it is natural to look at the covariance update

equation for the definition of sensitivity. The term of interest here is

K(m+l) B(m+l) P(m), which gives the decrease in the covariance after the

(m+1) measurement. Normalizing by P(m) yields the sensitivity matrix

K(m+l) B(m+l) whose diagonal elements comprise the sensitivity vector.

For practical data presentation and speed in calculation, the sensitivity

vector can be defined by assuming that the covariance matrix maintains

its initial variances and has zero cross correlation. With these assumptions+Vi

the sensitivity of the j element of the state vector is

s. + R (3.1-4)

59

It is important at this time to realize both the usefulness and drawbacks

of sensitivity analysis; The main purpose of the analysis is to show just

which constituent parameters and altitude regions are important in a

measurement. Thus sensitivity analysis will reveal for example that at

4000 A and a tangent height of 20 km with a certain set of initial uncertainties

the greatest information will be derived about aerosol number density and

aerosol absorption in the region from 20 to 23 km. The analysis will also

reveal that at 3000 A and 20 km the only information derived is from ozone

between 40 and 50 km. (This is due to the strong ozone absorption which

prohibits this wavelength from penetrating below 30 km.)

The one disadvantage of the sensitivity analysis stems from the

assumption that each update has zero cross correlation. It often happens

that a set of measurements do not yield linearly independent sensitivity

vecfers. For example consider a simple experiment designed to determine

aerosol number density and neutral density at 50 km by two measurements

at 3800 and 4000 A. The sensitivity vectors would be high for both

constituents at both wavelengths seemingly indicating that both are readily

observable. However, the actual final covariance matrix of such an

experiment would show that little has been learned of either constituent

since there is no way to distinguish between constituents from the

measurements. Therefore, the covariance matrix would indicate that the

sum of both constituents is well known.

This is not a serious problem however since visual inspection of

sensitivity vectors is often sufficient to determine whether or not there

will be a problem with ambiguous measurements. If there seems to be

some doubt then a complete propagation of the covariance matrix can be

made to determine exactly what information is recoverable from the data.

3.1.5 Stellar Occultation

The stellar occultation inversion technique is essentially the same

as the inversion of scattered sunlight (Newell and Gray, 1972). There are,

however, some aspects of the stellar occultation radiative transfer model

60

that make improvements possible in the construct of the.-state elements.

These improvements yield finer data resolution and more accurate error

statistics. The next few paragraphs describe the essential differences

between the two inversions.

. The radiative transfer model used for the stellar occultation work

i s . . . • : • • ' . • • : • • ' • . • • • • • • . . • : • . '

I-= I0e"T (3.1-5)

where (I) is the measured intensity, (I ) is the star intensity, and (r) is

the optical depth along the line of sight. The model .used assumes a linear

line of sight and neglects dispersion and multiple scattering effects., These

assumptions will not cause errors in excess of 10% above 30 km .(Hayes

and Roble, 1968). This altitude is approximately the maximum depth to

which ozone information can be retrieved. , . , "

Since the Kalman-Bucy filter assumes linearity in the .measurement

equation it is desirable to obtain a radiative transfer model which is as'

nearly linear as possible. In the case of stellar occultation it is possible

to create an exact linear equation by measuring the log of the intensity

rather than the intensity thus giving the new measurement equation

ln(I) = ln(Io) - r ,. . .. (3.1-6)'

In the scattered light case discussed previously it is necessary, to

estimate densities at a resolution approximately two to three times as coarse

as the tangent height data. This was necessary along with a gain damping

term .to insure convergence of the filter. For the linearized stellar,

occultation case it is-possible to estimate density at each tangent height

and also to dispense with the gain damping factor. The final density estimates

are therefore average densities of a layer between two tangent heights.

61

3.2 RADIATIVE TRANSFER MODELING ;

3.2.1 Introduction

This section discusses the extension of computer algorithms for

simulating the horizon profile caused by radiative transfer within the earth's

atmosphere (previously reported in Newell and Gray, 1972). Here, we begin

by reviewing the computational problems to which these codes are addressed

and introduce briefly computer codes applicable to related problems. Their

applicability to present work objectives is discussed, with emphasis on

the unsolved problems that required the development of new codes. These

new codes, a simple, modified single scattering code and a more

sophisticated, full multiple scattering code, are compared in terms of

accuracy obtainable versus computational burden required. The various

applications to which they have been put are presented.

3.2.2 The Computational Problem

The problem of horizon profile simulation is a difficult one because

in the earth's atmosphere there are important polarization effects in the

presence of multiple scattering involving not only Rayleigh but also aerosol

scatterers, as well as absorption. Furthermore, there is the

nonhomogeneous structure of the atmosphere, and its slight curvature in

conformity with the earth. Many of these factors contribute substantial

difficulty. Polarization requires a matrix rather than a scalar treatment,

thus increasing the complexity of operations and the amount of storage

space required. Multiple scattering is a well-known primary source of

difficulty in all radiative transfer calculations. Rayleigh' scattering and

absorption are straightforward, but aerosol scattering is characterized by

an exceedingly complicated ill-behaved phase function, or angular pattern,

that is difficult to incorporate in a radiative transfer model. The

nonhomogeneous structure of the atmosphere imposes an altitude dependence

of the scattering into any given solid angle. Finally, the curvature of the

earth entails formidable geometry problems.

62

3.2.3 Codes Applicable to Related Problems

Basic computational techniques for radiative transfer modeling that

existed at the outset of our work are reviewed by Hunt (1971). We shall

refer to the two basic approaches he discusses as the matrix operator

method and the Monte Carlo method. The matrix operator method represents

the atmosphere (or any sub-layer of it) as a matrix operator that transforms

an input column vector of stream irradiances to an output column vector

of stream irradiances. This matrix formulation is not to be confused with

the matrix formulation required by the inclusion of polarization in the

problem; the matrix operator method has matrices even if polarization is

ignored. The Monte CarlOrmethod represents the atmosphere with a random

number generator that chooses scattering locations, angles etc. for individual

photons, very large numbers of which are followed to accumulate a horizon

profile.

The matrix operator method has been used mainly in homogeneous

plane parallel atmospheres, where application of the doubling technique

makes it impressively efficient. It has not, however, yet been developed

to handle inhomogeneous curved atmospheres. Thus, while it was available

in principle at the outset of our work, in practice, its use would have required

extensive development. By contrast, the Monte Carlo method has actually

been applied to essentially our problem in a program called FLASH written

by Collins and Wells, 1970. The FLASH program will indeed accommodate

every one of the sources of difficulty mentioned in the preceeding section.

Furthermore, the program existed at the outset of our activities. It would

have been an attractive solution to our problems except for one factor: it

requires a large amount of computing time (approximately 2000 seconds

of CDC 6600 cpu time per horizon scan to achieve an accuracy of

approximately four percent).

63

3.2.4 The Radiative Transfer Codes

Within the Draper Laboratory, there are now two radiative transfer

codes that simulate horizon profiles. One is a refinement and completion

of the code REV.HYBRID which is described briefly in the earlier report

by Newell and Gray, 1972 and in greater detail in an internal report by

Var, 1971. It is based essentially on single scattering plus an effective

planetary or cloud albedo as a secondary source. The second code (C.K.W.)

is a more recently developed full multiple scattering model with polarization.

It is based on mathematical techniques described in Newell and Gray, 1972

and in more detail in a recent journal article by Whitney, 1972. Some of

the computational procedures involved in implementing the model are

described in Appendix 5.3 of this document.

Considerable effort has been expended to provide a basis for judging

which of the two radiative transfer codes should be used for any particular

application. The remainder of this section reports on these activities.

There are three main areas of comparison: accuracy achievable, run time

requirements, and suitability for specific applications.

In attempting to establish the absolute accuracy of the C.K.W. code

we compared its results to FLASH results through the courtesy of the Air

Force Cambridge Research Laboratory. Differences between FLASH and

C.K.W. were found to be small and related primarily to fluctuations in the

Monte Carlo results or differences between the details of layer definitions

in the two codes.

The run times for both the C.K.W and the REV.HYBRID codes appear

to be significantly less than those required for Monte Carlo simulations,

in spite of the fact that specific comparisons are complicated by a number

of operational and statistical factors. Our comparisons have been based

rather arbitrarily on the times required to produce horizon profiles which

are in apparent agreement with each other. Typically, the C.K.W code

produces a horizon profile in approximately twenty seconds, and

64

REV.HYBRID in.approxirnately seven.seconds,:both on an IBM 360/75; while

the Monte Carlo, code of .FLASH uses approximately two thousand /seconds

of CDC 6600 time to produce comparable results; .The significaritcomparison

here, is between, the :C.K.W. code and the Monte Carlo code, since-only

they are really addressed to the full multiple scattering problem. A tentative

estimate of, the .time saving with C.K.W-..code is a factor of 'several hundred.

,-. ....:.. . In general, one would anticipate that multiple scattering would increase

.observed radiances'to .those that have-been scattered directly into the

.receiver field of:view. . The magnitude of that contribution as determined

by the C.K.W.; code ,is- illustrated in Figures .3:2-1 and 3.2-2. The

enhancement of the radiance occurs without substantially altering the profile

shape. Such an effect can be simulated also with the REV.HYBRID code

by introducing a fictitious planetary albedo. The ficticious albedo required

to match multiple scattering results depends upon sun angle and wavelength.

Figures 3.2-3 and 3.2-4 show the effects of an albedo range of zero to

unity upon REV.HYBRID horizon profile at 4000 and 5500 A. At 5500 A

the vertical optical depth of the atmosphere is smaller than at 4000 A,

thus more sunlight reaches the surface and is reflected back into the

atmosphere. The reflected radiation is scattered into the receiver field

of view adding substantially more radiation than when the albedo is zero

causing a significant variation in the enhancement for the two wavelengths.

There is no way that the enhancement required to make REV.HYBRID

conform to the C.K.W. code can be derived from basic principles. Thus

we feel it is wise to use the full multiple scattering code (C.K.W.) for

applications requiring absolute radiances, rather than simply horizon profile

shape. Where only shape is required, REV.HYBRID is sufficiently accurate.

We come now to the question of suitability for different applications.

There are two important applications to consider here: the generation of

horizon profiles for the inversion simulations used to produce the aerosol

invertibility results of Section 2.1, and the generation of .the 324 sample

horizon profiles in Appendix 5.1 illustrating the effects of varying wavelength,

65

latitude, season, cloud height, zenith and azimuth angles. For the inversion

simulations, the single scattering REV. HYBRID was judged suitable because

the simulation does not require absolute radiances but rather fractional

changes in radiances induced by small perturbations in each of the quantities

defining elements of the state vector. Furthermore, the calculation of these

differential changes would have required the calculation of many complete

horizon profiles, and hence a considerable amount of computer run time

with C.K.W. This is not the case with REV.HYBRID; differential changes

can be obtained without recalculating complete profiles. By contrast, for

the parametric horizon profile study, the computer code (C.K.W.) was used

because the effects of the parameters to be illustrated could not have been

accurately obtained from REV.HYBRID.

66

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70

3.3 AEROSOL MODELING

3.3.1 Introduction

.The application of recursive filter algorithms to limb profile inversion

of aerosols requires that at each recursion the aerosol optical properties

be recomputed given an updated physical characteristic estimate. Aerosol

optical and physical properties are related theoretically through the Mie

equations which express the angular scattering contributions and cross

section as a function of the physical parameters such as size and index of

refraction. At each recursion the filter requires recomputed values of

these quantities and also the partial derivatives of radiant intensity with

respect to those physical characteristics that are included in the state vector.

For example the partial derivative of intensity with respect to index of

refraction is of the general form 3I/3n = ( d l / d r ) (d r ldn) + (d l ldP (6>))

OP (0)/3n). Evaluation of drldn and 3P (0 ) /3n involves the use of detailed

aerosol models. Since computational times on the order of a minute are

involved in determining these quantities using the Mie series, the total

computer time required for inverting a single scan would be on the order

of hours per scan just for aerosol calculations. Such an amount of time

would be out of proportion to other calculatipnal requirements posed by

the filter algorithm, and inconsistent with the goal of near-real-time

inversion capability.

The implications of this calculational complexity when matched with

the desire for computational economy are that multicalculations with the

Mie series are to be avoided, and some alternate procedure must be provided.

3.3.2 Choice of Models

Several alternate procedures for the Mie computations are suggested.

For the simple one parameter size distribution with which we are concerned,

there are four parameters to consider within each size range. These are

wavelength (A), size distribution parameter (a) (used in the distribution

71

law n(r) = Ar ), and the two part index of refraction m = (n) - i (n1).

Thus each scattering phase function and associated cross section is a

function in four dimensions P(0; a,\, n, n1) and a(a, \, n, n1). The empirical

fit problem is thus to find a set of four dimensional functions to fit computed

data to within some accuracy criterion. In addition the (0) dependence of

(P) must be described for any chosen set of parameters.

One approach would be to establish a four dimensional array of

computed points and to interpolate between them. It was felt however that

the search times involved in this procedure would be longer than desirable,

and that the four dimensional interpolation procedure would be excessively

complex.

Another approach, and the one taken here, is to fit, in a least squares

sense, the computed data with continuous functions of the four variables.

This has been done by applying a multivariate regression analysis to the

.computed Mie data for the extinction, scattering, and absorption cross

sections (a,-,-., a and a), and to the computed Mie phase function data at

six selected angles, viz., 0°, 20°, 60°, 110°, 164°, 180°. The Mie data for

each function (P,a) was obtained for 81 points corresponding to all possible

combinations of the three values assigned to each parameter, viz., a = 2,

4, 7; X = 4000, 6000, 8000A; n = 1.3, 1.4, 1.5; and n1 = 0, 0.02, 0.04. The

continuous functions thus obtained for the phase functions and cross sections

are listed in Appendix 5.2.

There are several options for the choice of a "goodness of fit"

criterion. In general one can force fit with a least path moment weight

called the best Lp approximation (Kahng, 1972). When p = *) the result is

the Chebyshev "maximum residual" fit. When p = 2 we have the usual

least squares fit. For this problem some (p) in between would perhaps be

desirable; however since a multivariate computer code was readily available

only for p = 2, this fit criterion was used and the resulting functions are

least squares fits in four dimensions.

72

Phase Function Interpolation

To obtain P(0; a, A, n, n1,) for (0) values other than 0^= 0°, 20°, 60°,

110°, 164°, 180°, an interpolation formula is required. A least squares

polynomial fit could be made to the selected angle points, however,

comparable accuracy was found to be obtainable by fitting log-linear straight

lines between the function values P'j at the selected angles 0.. Thus between

the selected 0., values of'P(0) are given by

In P ( 6 . , , ) - ' In P ( 6 . )P ( 6 . ) e x p 1+1 x

(3.3-1)

Normalization

The interpolation formula has the advantage that it can be easily

normalized to insure proper conservation of scattering by shifting each

modeled (P) up or down by an amount determined by a closed form integral.

The constant of normalization is just

5 {exp(m.. 6 ...., ) Y •. -exp (m• .6 • ) . 3 • }c.= 1/2TT £ P ( e . ) e x p ( - m . e . ) ——i , ^^—i (3.3-2)

i=l 1 + m.

where '

. - Yi = misinei+1-cos0i+1. (3.3-3)

•Bi- = misin6i - C0s8i . (3.3-4)

and

ii^ = {In P(8 i+1) - 1" p(ei)

}/(9 i+1-e i) (3.3-5)

73

3.3.3 Error Sources

It is possible to have considerable error in the phase function models

for a particular set of parameters and still gain useful information from a

model. The important thing is that the model represent an average over

an ensemble of computed points that are representative of the phase function

over the range of the parameters. For example if a modeled P(0) has an

RMS deviation of 25% at some angle, but variations of a parameter (say

n') over its range cause P(#) to change by say 200% then reliable partial

derivatives and the concomitant inversion sensitivities can be obtained.

The error sources and related limitations are described in the following

subsections.

Several strong trends in the data that should outweigh the RMS

residuals are evident. These are noted below.

1.0-lOA/Size Range

1. There is a strong (n1) dependence at both 164° and 180° as well as

other angles in this region. The phase function is generally an order

of magnitude lower for n1 = 0.02 than for the n' = 0 case. The n1 =

0.04 case drops P(0) another 50%.

2. The P(0°) values are twice as high for the n1 = 0.02, 0.04 cases as

for the n1 = 0 case.

3. (n1) effects are also strong at 20° and 60° (200% to 500% variation),

as well as at other intermediate angles.

4. (X) effects are strong (300%) at 0° and 180°. As expected, the (X)

dependence changes from P(0°) <* 1/X to P(180°) or X.

5. (a) effects are as much as 400% at P(0°) and diminish with increasing

(0)/ Beyond 110° complicated forms result.

6. (n) effects are most noticeable and consistent at 164° (up to 300%

variation). As (n) increases, P(0) increases.

74

0.1 -1. Q/J Size Range

1. The(n') dependence at 164° and 180° is strong; (200% to 500%) unless

a = 7, elsewhere it is very weak (10%).

2. (n) effects are up to 200% at 164° and 180°, but diminish with increasing

' Q and X.

3. (X) dependence is strongly linked to (a) over all (0).

4. (n) effects at 164° are strong (200% to 300%) unless a= 7, X = 8000A.

5. (a) effects are strong at 0 (up to an order or magnitude), but correlated

with (X), P(0°) oc I / a -A . - .

0.01-0.1/v Size Range

1. (n1) dependence is negligibly small over the range of all the other

parameters.

2. (n) dependence is at most a 10% effect at the larger angles for a = 2.

3. (a) and (X) both strongly influence the phase function, but the effects

are on a smaller (200%) scale than for the larger size ranges.

Mie Code Errors

To normalize the phase function generated by the weighted size

distribution average, the Mie code performs a trapezoidal integration

between the selected defining points, and adjusts the amplitudes by the proper

constant to achieve conservative scattering. However, since much of the

accumulated value of the normalizing integral 2w / p(0) sin e do is obtained

at small angles, the adjustment of the entire functional amplitude is very

sensitive to the fineness of the angular defining intervals. This is

particularly important for the larger particles which have rapidly decreasing

phase function values near 0 = 0 . In the (1.0-10/u) size range, a decrease

in the small angle (0) step size from 10 to 0.2 results in a factor of

twenty adjustment of the phase function upward. The upwards adjustment

is caused by the excessive integral accumulation under the trapezoidal

function between 0° and the first defined point at 10° in the coarse sample.

75

Figure 3.3-1 shows an extreme case of this with angle step increments

of 0.2 and 10 . Both curves in Figure 3. 3-1 are normalized by integration to

unity. The dashed curve is the actual curve under which the trapezoidal

integration is performed rather than the computer-drawn straight line

connecting the 0 and 10 points. In general the error due to the overall

phase function adjustment for normalization is proportional to the difference

between the scattering cross section as predicted, by the evaluation of the

0 = 0° Mie series and the value obtained by integration of the phase function.

For the (1.0-10A/) size range the maximum value of this error is about

25% forthe Ae intervals that were used in the calculations. In the (O.l-l.O//)

range the error is about 5% or less, and forthe (0.01-0.1/u) range the error

is less than 0.3%.

Size Distribution Sampling Errors

The power law distribution n(r) = Ar is sampled at discrete (r)

values within the chosen size ranges. Each (r) choice corresponds to a

particular Mie scattering result for a particle of radius (r), and the resulting

phase functions are averaged by combining the individual (r) phase values

with the (-a) weight. Since this process has the effect of averaging out the

oscillations related totheBessel and Legendre series within the Mie series,

the results depend somewhat on the fineness of the Ar integration steps.

Table 3.3-1 shows an example for the size range (1.0-10A/) with complex

index of refraction. Differences between the phase function values for the

range of 10 to 200 radius sample points are generally of the order of a

few percent except for the 180° values which differ by 20%. In Table 3.3-II

similar values are shown for non-absorptive aerosols (n1 = 0). Thirty

percent differences in the phase function values occur here fora few angles,

and indicate that the number of radius steps required for a few percent

accuracy is nearer 100 than 10. However the required computer time is

directly proportional to the number of integration steps.

76

CD

CD

TABLE 3.3-1

PHASE FUNCTION VALUES

NOMINAL CASE, a=2, A=.4, 11=1.4, n' = .02, l-10y

PHASE ANGLE

O.TLU

s 10^ 50

g 100LU

£ 200u_CD

0

__P°372368368368

20°,0367,0373

,0371,0372

60°,00406

,00411,00418

,00418

110°,00120,00122,00119

,00120

164°,00126,00118,00124

,00123

180°,00122,00164

,00152,00154

TABLE 3.3-II

PHASE FUNCTION VALUES

I>ex:LU|

£ 105 50

£100i0/3 200

0°256249

253253

a=2,

20°,205,236

,203,204

A=.4, n=1.4, n

PHASE ANGLE

60°

,0151,0190

,0196,0196

'=0, l-10y

110°

,00264,00285

,00245

,00276

164°, 0114,0134

,0141,0138

1.80°,0578

,0676

,0728

,0731

77

The derived aerosol model is based upon a 10 step integration, thus

30% errors may be expected in the (1..0-10A/) size range from this error

source. Figure 3.3-2 shows a plot of the differences between the phase

function values resulting from 10 step and 200 step size distribution

sampling. The differences are expressed, as a percentage of the phase

function value for the 200 sample case. It shows that the effects of changing

the sample step size are essentially random with regard to angular position.

Errors Related to Piecewise Function Fitting

The errors just, discussed were related . to the computational

uncertainty of the phase function at a particular angle. The modeling

procedure we have chosen connects the points f(P(0.) , 6.) with exponential

curves of the form P(f>) = Aexp(B0); thus even if the parameter model P(0-;

n, n ' . o r . X ) were perfectly accurate there would be errors caused by the

fluctuations of P(0) between the select model points P(0.). This would be

true, whether the points were connected by exponential curves or by, e.g. a

least squares polynomial fit. This error source would decrease in magnitude

if the P(0.) sample points were.more closely spaced, however the six chosen

B- represent a compromise between accuracy and the need to limit model

complexity. Each new 9. requires a new Lp fit to find the modeled P(6.',

n, n1, a, X), however the data shows that the Q. choices should be optimized

for each size range separately. For example, Figure 3.3-3 shows the

log-linear fit to the (0. 01-0. In) case havingthe most curvature. The fit is in

error by a maximum of 14%. The RMS error for this case is 7%. For

some of the better fits in this size range the RMS error is about 4%. The

case illustrated in Figure 3.3-3 could be improved by the addition of points

defined around 30°, 90°, and 110°. However the choice of defining angles

for a given size range is not a simple matter to be solved by examination

of a single curve. The noticeable structure in the curves in the backscatter

region moves over a range of (0) when the various parameters are changed.

Figure 3.3-4 shows the six point fit to one of the worst-fit cases in

the (0.1-1.0/y) range. In this size range the points of fit correspond quite

78

closely with the major points where curvature changes, however it can be

seen from the figure that additional defining points would be helpful here

at 130° and 176°. The maximum error for the (0.1-1.0/y) case is 56%,

with an RMS error of 21.5%, l<r. '

In the (1.0-10/y) size range the standard points of definition do not fit

the Mie curves well, and three or four additional points should be used for

a better fit. A particularly bad-fitting case is shown in Figure 3.3-5.

The minimum for the phase function is at 130° and is missed by the

approximate fit point at 110° causing large errors at 130°. The 6-point

function also ignores the secondary peak at 150°, and fits badly at 10°.

Over the range of parameters considered, the position of the phase function

minimum for the (1.0-10A/) size range varies over the interval 110°-130°.

Thus, improved modeling of this size range would require inclusion of extra

defining points within this interval. The maximum error for this case is

630% at 130°. The RMS error is 400%.

The RMS errors listed in this section are based upon the sequence

of discrete angles used in the Mie computations. Since these angles are

defined with smaller increments at small and large angles, there is a built-in

weighting which may adjust the RMS error value somewhat either upward

or downward (depending on the particular fit) from the RMS error which

would result from a uniform disbursement of sample points.

In summary, the errors due to the piecewise log-linear fit at six

preselected angles are moderate but within useable limitsforthe (0.01-0.1/u)

and (0.1-1.0/u) size ranges. The errors are too large for the large particle

range (l.O-10/i/) which implies that an optimization procedure should be

constructed for choosing the angular points of definition.

Parametric Fluctuations

It is of interest to examine the effects of small changes in the defining

parameters (a ,A,n , n1) to see whether or not the phase function is undergoing

rapid fluctuations near the selected nominal parameter values. If this were

the case there would be no guarantee that the value of P(9) at the nominal

79

parameter value was actually representative of P(0) in the region around

the selected value.

This problem was investigated by means of a set of Mie calculations

with small perturbations from the standard points. Deviations from the

case a = 2, X = 4000A, n = 1.4, n1 = 0.02 were considered by running cases

with a= 2.1, X.= 4100A, n = 1.41 and n' = 0.021. The (1.0-10/u) size range

was considered because it exhibits the most erratic behavior and should

represent a worst case. The results are tabulated in Table 3.3-III.

TABLE 3.3-III

LINEARITY PARAMETER 5 FOR A l-10y CASE

(a) (n) (n') (X)

5=2.3

9.0

0.36

0.91

1.2

0.39

2.4

1.6

5.0

0.18

1.6

0.63

0.12

0.2

2.7

0.27

0.12

0.027

4.2

7.7

18.0

1.0

2.1

6.7

0°

20°

60°

110°

164°

180°

AP Ax- The number in the table is J- = =- -^— where AP and Ax aretherangeso tr o x

of the phase function and the parameter expressed as a percent of thenominal

value. The quantities 6P and 6x are the perturbed phase function and

parameter values also expressed as a percent or fraction of the nominal

values. If all the functional behavior were linear these values would be

unity. Typical changes in the phase function indicate the £= 4 corresponds

80

to quadratic dependence. The results show that at X = 4000A one has a

rapidly varying phase function at 60°, however the excursion from the value

at 4000A to the value at 4100A is a variation of 21% which is comparable

to the scale of other errors for this size range. Otherwise the fluctuations

are relatively calm. The small values for (n1) are related to the fact that

P changes rapidly in the forward and backward directions for very small

deviations of (n1) from zero. Thus the range of P is large between values

corresponding to n1 •= 0 and n1 = 0.02, hence the small values of (£) for

(n1). Since the modeling functions used were primarily quadratic, one expects

to find relatively smooth behavior of the phase functions with respect to

parametric variations over the parameter range.

Summary of Phase Function Modeling Errors

The models listed in Appendix 5.2 exhibit a complex error structure

in general. However one can estimate the maximum RMS errors by assuming

that the various sources of error are independent and thus that the errors

can be added. The error sources are, in approximately decreasing order

of importance, regression residuals, piecewise fitting errors, . size

distribution sampling errors, Mie Code normalizing errors, and higher

order parametric fluxuations.

The regression residuals are listed in Table 3.3-IV with both maximum

and RMS values given.

At the defined angles the total error consists of the regression error,

the size distribution sampling error, and the normalizing error. If we

take the typical size distribution sampling errors of Table 3.3-1 as la values,

and the normalizing errors discussed above and combine them by means

of a root mean sum of squares, the errors in Table 3.3-V are obtained at

the defined phase function angles.

81

TABLE 3.3.-IV

RMS AND MAXIMUM RESIDUALS FOR PHASE FUNCTION PARAMETER MODEL

Phase 0.01-O.ly 0.1-i.Oy 1.0-10y

Angle a(%) Max.(%) a(%) Max.(%) o(%) Max.(%)

0° 0.3 0.9 38 3 10

20° 0.2 0.5 1.8 5 11 45

60° 0.1 o.3 4 9.5 8 23

110° 0.4 2.7 4.5 17 18.9 97

164° 0.25 0.6 6 15 31 120

180° 0.2 0.5 10 25 32 127

TABLE 3.3.-V

RMS ERRORS AT DEFINED ANGLES

INCLUDING NORMALIZATION AND SAMPLING INTERVAL ERRORS

Phase 0.01-O.ly o.l—l.O/i 1.0-10 yAngle

0°

20°

60°

110°

164°

180°

2%

2%

1%

3%

4%

4%

6%

5%

9%

15%

9%

16%

25%

27%

35%

32%

43%

46%

82

In .between these "defined" values the piecewise fitting errors must

be added in proportion to the distance from the defined angle. These errors

have been discussed in one of the previous paragraphs, and since no attempt

has been made to assess them, statistically they will not be combined with

the above table to produce a grand model error table. For the (0.01-0.1/u)

size range the "in-between" errors are of the order of 10% at maximum.

For the (0.1-l.Oju) range the maximum fitting error is roughly 50%, and

for the (1.0- 10A/) range it is approximately 100%. A graphical display of

the error sources is shown in Figure 3.3-6.

3.3.4 Models of the Extinction, Scattering and Absorption Cross Sections

The cross sections were modeled by applying the regression

procedure described above for the phase functions. Since the cross sections

for absorption and scattering sum to the extinction cross section, only two

of the three cross sections need be modeled in each case. Thus the two

with the smallest variances were chosen from each set of three models

and used to produce the third by algebraic combination. The errors for

these quantities are listed in Table 3. 3- VI.

TABLE 3. 3 -VI

CROSS-SECTION ERRORS

0.01-O.ly 0.1-T.Oyi 1.0-lOy

cr(%) Max(%) a(%) Max(%) a(%) Max(%)Error Error Error

a (absorption) 3 1 0 2 5 2 9cl

a (scattering) 0 . 3 1 - -s

a (extinction) - - 10 34 18 52e

The standard errors for the unlisted cross sections can be determined

if the correlations are known, however, since these errors were not used

explicitly in the inversion procedure the correlations were not computed.

83

H 1

A = 1.1110 +00•a = -2.0X = .100 Mn = 1.400 n1 = -0.02EXT = 6.6575 -O7

Fig. 3.3-1 Dependence of Normalization Adjustment

Upon A6 Sample Size.84

w.H(0

Q)-PC

-H

QJ

g(0

OJN

•HU)

O41

OJ

Q)

(NI

ro

Cn•H

Q)tnC(0

OJN

-HCO

C -HO O

-H W-P O

-H (0

-P '-tn P.

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tna<u-PU5

ooCM

O 'OH CLt (0Q)

c•HW

°°zdaNV°ld

102t

A. = i.mo -02o = -2.0X = .400n = 1.500 n1

EXT = 1.033* -11

PIECEWISE MODEL

1 II 1 • I I I 1 1 1 h 1 1 1 1

30 60 90 120THETA

150 180

Fig. 3.3-3 Worst Case Fitting Errors,

0.01-0.ly Range

86

10*;

zg

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-H

REFERENCES

SECTION 3

1) .Anderson, G.P., "The Vertical Distribution of Ozone between 35 and

55 Km as Determined from Satellite Ultraviolet Measurements",

Thesis for Master of Science, Department of Astro Geophysics,

University of Colorado, 1969.

2) Collins, D., and M. Wells, "Flash, A Monte Carlo Procedure for Use

in Calculating Light Scattering in a Spherical Shell Atmosphere",

AFCRL Report No. 70-0206, 1970. ;

3) Hayes, P.B., and R.D. Roble, "Stellar Spectra and Atmospheric

Composition", J. Atm. Sci., 25, 1141-1153, November, 1968.

4) Hunt, G.E., "A Review of Computational Techniques for Analyzing

the Transfer of Radiation Through a Model Cloudy Atmosphere", J.

Quant. Spectrosc. Radiat. Transfer, 11, 655-690, 1971.

5) Kahng, S.W., "Best Lp Approximation", Mathematics of Computation,

26, 505-508, 1972.

6) Newell, R.E., and C.R. Gray, "Meteorological and Ecological


Report, NASA-CR-2094, August, 1972.

7) Var, R.E., "A Hybrid Algorithm for Computing Scattered Sunlight

Horizon Profiles", MIT Aeronomy Program Internal Report No. AER

7-3, June, 1971.

8) Whitney, C.K., "implications of a Quadratic Stream Definition in

Radiative Transfer Theory", J. Atm. Sci., 29, 1520-1530, November,

1972.

90

4.0 SUMMARY

4.1 CONCLUSIONS — AEROSOL PHYSICAL PROPERTIES

The vertical distribution of aerosols from approximately 10 km

upwards to 100 km can be determined globally by a satellite horizon inversion

technique. Additionally, information such as the number density, the size

distribution, and the complex parts of the index of refraction can be obtained.

Three particle size ranges (0.01-0.1/v, 0.1-1.0/w, 1.0-10/c/) were

considered in the simulations, and the results show that the quality of the

physical characteristic information is size range dependent, i.e., that the

relative observability of the physical characteristics changes from one size

range to the next. For the smallest size range (0.01-0.1/u) the number

density (f t) and the size distribution parameter (Q-) are observable, but the

indices of refraction are not. However this result rests partly upon the

assumption that the initial uncertainties of the four modeled physical

parameters are as follows: a = ±300%, o• • = ±1, a , . = ±0.005,

CT = ±0.05. Reductions in a and cr have the effect of increasing the

observability of (n1). A simple covariance propagation case in this size

range, using four kilometer horizon sample intervals and five wavelength

channels, reduces the initial variances as follows: a (±300%)->0p = ±150%,

= ±0.1. cr - + c r , a , -»• a , .•' n no' n1 n'o

For the medium sized particles (0.1-l.OA/), using the same initial

covariances, the simulations show that the complex part of the index of

refraction becomes generally observable. The peak sensitivity of (n1) jumps

from a value of «0.01 to«0.4 which is in the highly observable range. A

simple covariance propagationcase, using four kilometer horizon sample

intervals and three wavelengths showed initial covariance reductions as

follows: or (±300%)-*o>>= ±60%, crao(±l)-«ra = ±0.2,

an,o(±0.005)-*an, = ±0.001, anQ(±0-05^an = ±0.05.

91

All four physical parameters are found to be observable for the large

particle (1.0-10yu) size range. However the observability of the index of

refraction is strongly dependent upon the scattering angle, i.e. the sun

azimuth and zenith angles. For a scattering angle in the forward region

(«50 ), (n) is found to be unobservable. In the side-scattering region of

the phase function(n') is unobservable and the observabilities of (n) and (a)

are considerably reduced. However in the backscattering directions all

four aerosol parameters are observable.

An estimate of the state errors resulting from errors in the models

was made based upon the filter gain and computed horizon intensity

deviations. The calculations show that the aerosol modeling errors induce

physical parameter estimation errors that are generally smaller than the

final variances resulting from a covariance matrix propagation. However

the error analysis also shows that a deviation of the albedo of 0.5 from

the assumed value can cause large uncertainties in the state vector elements.

Thus the albedo should be included as an element of the state vector.

4.2 RESEARCH EXTENSION

The results obtained in study show how well one can determine the

parameters of a specific aerosol model. For example the inchoate model

developed for this research uses a single parameter, power law size

distribution. If the real aerosol size distribution is of some other functional

form, then the results of applying the chosen model in an inversion would

be to obtain a best power law fit to the real function. If the best power

law fit to the real function can be closely related to the real distribution,

then the results are useful for aerosol research. To this end, then, it is

important to attempt to generate the most realistic aerosol models possible.

A first step in this direction is to insure that the ranges of the chosen

variables cover the real physical range. This has been done in the model

used here in every respect except the aerosol size range. Hence one of

the next steps in a continuation of aerosol model development would be to

reconstruct the model with variable particle radius limits. This would

92

increase the size of the state vector by two elements, however, the present

model could be used to optimize the selection of wavelength channels. By

fixing wavelength, the dimension of the model could be reduced by one

element for a net increase of one dimension in the model.

To insure that fitting errors are minimized in extended aerosol models,

a procedure should be constructed which determines a minimum set of

phase function defining angles consistent with accuracy goals. The models

should also make maximum use of the analytical relationships derived from

ray optics (Van de Hulst, 1957) and information theory (Whitney, 1972).

The design of the filter algorithm should be further extended to include

the additional states resulting from improvements in the aerosol model

and from the inclusion of polarization measurements. The effects of model

type error sources such as approximation errors in the radiative transfer

simulation, fitting errors in the aerosol models, and relative fluctuations

in the solar power output should be explicitly included in the filter

formulation. As the aerosol models become more complex, additional

sources of information will be required besides the measurements at

different wavelengths. Two possible sources are measurements at different

solar angles and measurements of polarization. The present radiative

transfer simulation for a curved atmosphere includes the necessary

polarization calculations, however, these simulations would have to be

checked against other results to insure their accuracy. Accurate inclusion

of cloud scattering effects at low sun angles (<10°) would require

modifications of the existing code as well as simulations of the transfer

problem within a cloud.

93

REFERENCES

SECTION 4

1) Van de Hulst, H. C., Light Scattering by Small Particles, John Wiley

& Sons, Inc;, New York, 1957.

2) Whitney, C. K., "Stream Transformation and Phase-Function Integrals

in Radiative Transfer Theory", MIT Aeronomy Program Internal

Report No. AER 22-3, January, 1972.

94

5.0 APPENDICES

5.1 THEORETICAL HORIZON PROFILES

This appendix contains the theoretical multiple scattering horizon

profiles discussed in Section 2.3. The profiles illustrating the effects of

changes in various measurement parameters are arranged as follows.

Figure Numbers Variable Measurement Parameter(s)

5.1-1 to 5.1-12 Season; for 12 combinations of latitude

and wavelength.

5.1-13 to 5.1-28 Wavelength; for 16 combinations of cloud

cover, ground cover and zenith angle.

5.1-29 to 5.1-64 Zenith angle; for 36 combinations of azimuth

angle, wavelength, arid ground albedo.

5.1-65 to 5.1-72 Ground albedo; for 8 combinations of

zenith angle and wavelength.2The units of solar irradiance (SUN PWR) are in /uwatts/cm A.

Figures 5.1-73 to 5.1-84 show the vertical distributions of neutral

and ozone densities for 12 combinations of season and latitude. These

figures illustrate the mean and one sigma standard deviation values about

the mean.

Neutral density data was obtained from Valley, 1965 and from the

data compiled by Groves, 1970. The designations winter, spring, summer,

and fall refer to the months January, April, July, and October. The ozone

density data was generated from the modified Fermi distribution function

developed by Wu, 1970 and the peculiar appearance of the ozone density

distribution for the minus one sigma curve is due to the fact that the one

sigma values are equal to or greater than 100 per cent of the mean value

below approximately 10 km and again above approximately 50 km. Aerosol

extinction data was compiled from a variety of sources by Malchow, 1971

and the composite model illustrated in Figure 5.1-85 was constructed.

95

Latitudinal variations in the aerosol extinction data were accounted for on

the basis of data gathered by Salah, 1971 as illustrated in Figure 5.1-86.

Due to a lack of sufficient seasonal distribution, this aerosol extinction

data was used to represent the four seasons.

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~2LgHo-2LH-X

10 "•::

10 -"::

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10 -7.

10 -6.

0-5 km Elterman (1970)»

5-35 km Elterman (1966) '

35-80 km Exponential (0 .57 x Rayleigh @5500 A)

10 20 30 60 70 80

ALTITUDE (KM)

Figure 5.1-85 Aerosol Extinction Profile181

0.1 2.50 10 20 30 40 50 60 70 80 90

v LATITUDE l°l •

Figure 5.1-86 Stratospheric Aerosol Latitude Factors

182

5.2 MODELED AEROSOL FUNCTIONS

This appendix gives the analytic expressions obtained (by a

multivariate regression procedure discussed in Section 3.3 for scattering

phase-functions P(0.; a , X , n, n1) at 6L = 0°, 20°, 60°, .110°, 164° and.180°,

the total cross-section v,(a,\, n, n1) and either the absorption cross-section

v ( c t , X, n, n1) or the scattering cross-section <r (a, X, n, n'). Thesea sexpressions are grouped according to the particle size ranges (Q.Ql-O. l / j ,

O . I - I . O A / , and 1.0-10/v).

In the 0. 01 to 0. 1/y range we have:

P(0°) = .exp(-2.937E-2 + 5.234E-2 aX - 4 . 7 4 7 A - 5.812E-2 aA2

- 3.388E-2 a2 + 3.259 X2 + 6.963E-2 a2A + 1.195E-1 n2

- 3.961E-2 a 2 A 2 - 3.744E-1 nA)

n

P ( 2 0 ° ) = exp(-2.396E-l + 4 .447E-2 aA - 4 . 6 8 7 , A - 4.803E-2 aA

- 3.229E-2 a2 + 3 .069 A2 + 6.728E-2 a 2 A . + 1.295E-1 n2

- 3.887E-2 a 2 A 2 - 1.451E-1 n 2 A )

n

P ( 6 0 ° ) = exp(-1.776 + 6 .964E-2 aA - 1.906 A - 9.225E-2 aA

-1.402E-2 a2 + 1.388 A2 + 2.051E-2 a2A + 9.914E-2 n2

- 2.343E-1 nA - 4.050E-3 n a 2 A 2 ) .

0 ' OP(110°)=exp(-4.482 + 4 . 9 2 8 A + 3.946E-2 a - 3 .747 A - 1.013E-1

a2A - 2.372E-1 n2 + 6.485E-2 a 2 A 2 + 2.946E-1 nA .

+ 1.534 E-l n2A + 1.100E-2 n 2 A 2 a)

183

P(164°)=exp(-6.612 - 1.685 aX + 1.227E+1 X + 1.166 aX2

- 8.577 X2 - 3.798E-1 n2 - 7.140E-3 a2X2 + 4.126E-1 n2X

+ 5.990E-1 a + 5.850E-3 na2)

P(180°)=exp(-6.686 -1.711 aX + 1.249E+1 X + 1.179 aX2

'-'8.693 X2 - 3.715E-1 n2 - 6.800E-3 a2X2 + 4.027E-1 n2X

+ 6.117E-1 a + 5.800E-3 na2) ,

a = exp(1.51lE+l - 1.779E-1 aX +1.189E-2 naX2 +3.959 X2s

- 1.604 a - 9.541 n"1 - 1.043E+1 X + 2.689E-2 a2)

a = .007259 + 77.895 n1{(2.58(a/2)2'35 - .79a) (X/.4)-1* 3} ~1

{1 + .0208 X(a-2) (7-a){| (n-1.234) |/.066}

- {.1667(a-2) (a-4) (n/1.3)5'75(.06-n1)}{1+7.5(X-.4) (X-.6)}}

In the 0.1 to ly range we have:

P(0°) = exp(6.415E-l + 2.302 naX + 1.992E-1 nX2a +5.082 nX2

+ 1.841E-1 a2X2 + 9.519E-1 na - 1.404E-1 na2 + 9.720 nn1

- 1.533 nn'aX - 1.560E+1 nX + 1.679E-1 (na)2

+ 2.865E-1 nn'(aX)2 - 2.336 aX2 + 3.003 n2 - 1.569 n2a

+ 9.547. X - 2.570E-1 na2X + 3.095E-1 n2aX - 6.137 nn'X2

- 9.415E-1 nn1)

P(20°)= exp(-3.668 - 9.038E-3 (na)2 + 2.348E-1 naX - 3.470E-3

(na)2 + 2.986 n"1 - 8.749 a2X + 4.534E-2 (naX)2 - 4.293 X2

+ 1.473E-1 na - 3.111E-1 (nX)2a + 7.388 nX2 + 1.166E+1

Xn'n2 - 2.672E+1 n'X - 1.666E-1 n'(Xna)2 - 2.416 (nX)2

+ 2.292E-1 n'(Xa)2 + 8.322E-2 a + 2.412E+1 X2n' + 1.235E-1

n'n2a - 9.250(Xn)2n')

184

P(60°)= exp(-5.344 - 1.100E-2 n2a + 3.714 n3X + 1.168E-1 n

- 2.729E-2 (naX)2 - 4.697 n2X + 2.213E-1 (n)n a + 2.068E+1

n'X/a - 1.715 Xn + 2.363 naX - 1.429 n2aX - 4.949E-1 nn

+ 3.470 n2/{(l+5n')a> - 2.016E+1 n2a~3 -I- 5.328E-1 (n1)'5

4- 9.547 n2a~2 - 3.757E-1 n2(l+5n')~2 - 2.719 n1 + 1.058E-1

exp(-10 n'a) - 4.793E-3 (na)2X - 2.276E-1 n2 + 2.996E+1

(n'X/a)2 + 4.266E-2 n2(l+5n')~1)

P(110°)=exp(4.751 - 4.099 n2aX - 8.511 n + 3.866 aX + 9.015E-2

(naX)2 - 3.342E+1 n1 + 8.520 Xn'a - 3.278E-12 na12/X

+ 6.585E-1 a2X + 2.164 n2a - 5.439 aX2 - 3.228E+1 nX2

- 9.783E-2 nn'(aX)2 - 7.001E-1 (aX)2 + 4.486E+1 nX

+ 3.729E+1 X2 - 1.196E-1 na2 - 5.399E+1 X + 1.259E+1 nn1

- 3.436 nn'aX2 - 5.098 n'(na)2 + 1.143E-I-2 X(n')2 + 6.578

naX2 - 2.303 na - 2.545E+1 a(n'X)2)

P(164°)=exp(2.472E+l + 7.151E-1 (a-2) (a-4) (X-.4) (X/.6)"la°8

+ 2 .546 (n-1.35) |a-7| (n/1.4)"3 '88 - 4.058E+1 nn1 + 7.673E-1

n n ' ( a X ) 2 - 1.056 a2 - 5.101E+1 X2 - 1.843 a2X -f 7.514E-1

nn 'a 2 X - 1.471 a + 1.769E+1 aX - 7.267E-1 (1+50 n ') /ct

+ 2.921 na2 + 5.524E-1 X ( n a ) 2 + 7.434 ( n a X ) 2 - 1.854

n ( a X ) 2 - 2 .226 X/U+50 n1) + 4.555E+2 ,n (n 1 ) 2 + 3.278E+1

nX2 - 7 .245 aXn2 - 1.336E+1 na - 1.815E+1 n + 9.171 n2a

- 1.379 (na) 2 + 1.352 ( a X ) 2 + 2.562E-1 a ( n X ) 2 )

2P(180°)=exp(1.340 + 5.193E-2 ln{ (a-2/X) (2/n ) (1+50 n')

- 2.180E+1 n - 1.070 n n ' ( a X ) 2 - 1.092E-1 (na) 2 - 1.379E-1

na2 + 3.134E-1 Xa2 + 3.732 nn 'aX 2 + 5.469E-1 (naX) 2

185

- 3.927E+1 nn1 + 8.804 n'aX - 8.464 n2aX+ 1.187E+1 X2

- 3.627E+1 nX2 + 3.261 aX - 1.255 (aX)2 + 1.495E-1

n'(na)2 + 1.427 n' + 3.153 naX2 + 3.236E-1 (a-2/X)2

+ 4.157E+1 n(n')2 + 3.377 aX2 + 8.490 nX + 3.580 an2

- 5.849E+1 X)

4 ' 4 .a,,, = exp{2.3{X A.a3 - exp ( B .aD) Hl+. 036a (n-1.4) I'1}. j=0 D j=0 D

.where AQ = -8.896 E-l A, = 1.288

A2 = -5.316 E-l A3 = 6.727 E-2

A = -2.855 E-3

BQ = 1.948 Bj^ = 1.159 E-l

B2 = -2.551 E-2 B3 = 2.491 E-3

B = -8.889 E-5

a = -1.518E-3 + FX {1 +12.5U-.6) (X-.8) (.215 + DX + .5(n-1.3)3.

CX + (n-1.3)(ri-1.4) BX>. + 25(X-.4) (X-.8)(.25-.03(7-a)

- .7(n-1.3)) T- 12.5(X-.4) (X-.6) (.47 - .032(7-a) - .7(n-l.

EX - .2(n-1.3)(a-4)(a-2)/15)}(1.61(a/2)2'32 - .ei)'1

where FX = 1.372 (n'/.02)AX

AX = .585 + 2.5U-.4) (.138 - .0163(a-2)) + .0644(a-2)

CX = |(a-2)|a~a/8

BX = a2 CX/8

DX, = | (n-1.3) | (n/1.4)~10*21

. EX = I (a-7) la/12

186

In the 1 to 10y range we have:

P(0°) = exp(8.457 - 1.357 a + 6.576 X + 9.534E+1 n1 - 6.383 a2

- 2.374E+4 (n1)2 - 8.454 X2 - 1.596E+1 Xn'a2 + 2.076E+3

X(n'a)2 - 7.498E-1 aX2 - 1.161E+1 (aX)2 4- 9.258 na2

+ 8.853 X(na)2 - 3.303 (na)2 - 3.573E+1 nn1 + 1.211E+2

Xn'a + 1.375 Xn'n2 - 6.465E+1 Xa(n')2 - 3.425E+3 Xn(n')2

- 9.796E+1 (Xan1)2 - 3.656E+1 an1 + 1.262 n2 + 5.706E+1

na(Xn')2 + 4.348 n'a2 + 5.356E+2 a(Xn')2 + 2.365E+3

n(Xn')2 + 1.036E+3 X(nan')2 - 7.508 n'(Xna)2 - 6.004

(naX)2 + 1.221 aX + 1.677E+1 n(aX)2 - 2.909E+3 nX(an')2

- 1.155E+4 (nn1)2 + 3.359E+4 n(n')2 - 1.199E+2 an'X2

+ 9.740 nX2 + 1.736E-3 nn'a2 - 1.195E+1 nX - 2.378E+1

nXa2 + 1.717E+1 Xa2 + 2.148 nn'(aX)2)

P(20°)= exp(-6.551 - 5.221E+1 nn1 - 6.469E+2 naX(n')2 + 6.866E+1

n(aXn')2 + 3.327E+2 nn'(aX)2 - 8.600E+4 (nn1)2

+ 3.156E+3 aXn1 - 1.193E+2 n'(anX)2 - 2.377E+2 n'(aX)2

+ 7.901/n + 2.525 n2 - 5.764E+1 n1 - 1.585E+3 X2

+ 5.231E+2 Xn2 + 9.972E-2 (na)2 - 1.822E+1 Xa2

+ 2.823E+1 (aX)2 - 8.998 X(na)2 + 2.520E+1 nXa2

+ 2.304E+3 nX2 - 8.260E+2 (na)2 - 1.941E-2 na2 + 2.430E+5

n(n')2 - 1.689E+5 (n1)2 - 3.983E+1 n'X2 - 1.402E+1 nan1

+ 9.910 a(nX)2 + 1.379E+1 (naX)2 - 4.360 naX+ l.OOOE+3 X

- 3.913E+1 n(aX)2 - 1.560 na + 1.587E+3 Xan'n2 - 4.434E+3

nn'aX - 1.463E+3 nX - 1.903E+1 naX2 + 1.422E+1 aX + 6.747

an'n )

187

P(60°)= exp(-2.065E+2 - 1.942E+3 n' + 1.348E-1 aXn2 + 3.908

(aX)2 + 5.889E+2 an'X2 - 2.647E+1 (n1)2 + 3.968E+2 a(Xn')2

+ 1.789E+2 n"1 + 3.761E+1 n2 + 3.453 aXn1 + 2.123E+2 nX

- 3.399E+2 na(n')2 - 2.754E+1 an2 - 1.225E-2 a2 - 8.192E+1

Xn2 - 1.354E+2 X + 5.389 Xa2 - 4.490E+2 na(Xn')2

- 1.054E+3 n'n2 + 7.534E+1 na + 1.800 (naX)2 + 1.274E+3

(nn1)2 + 3.513E+2 an'(nX)2 - 2.453 n'(Xna)2 + 3.865E+1

n(aXn')2 + 1.520 an'n2 - 4.816 aX2 - 5.124E+1 a - 5.320

n(aX)2 + 2.786 E+3 nn1 - 9.080E+2 nn'aX2 - 5.976E+2

X(nn')2 + 1.106E+1 (arm1)2 - 7.938 nXa2 + 2.916 X (na) 2

+ 3.193 naX2)

P(110°)=exp(2.558E+2 + 5.608E+1 a(nX)2 + 1.412E+4 h1 - 2.236E+4 nn'

+ 8.719E+3 n'n2 + 9.753E+3 (n1)2 - 2.447E+2/n - 4.403E+1 n2

- 1.587E+2 naX2 + 2.201 a(nn')2 - 1. 782E+1, aX - 1. 075 an2

- 1.409E+4 n'X2 + 1.440E+4 X(nn')2 + 1.113E+2 aX2

- 4.937E+3 Xn'n2 - 1.838E+1 an'n2 - 2.539E+2 aXn' + 7.063

nn'X + 1.474E+4 nn'X2 - 3.422E+3 n1 (nX) 2 + 1.419 (ann1)2

-5.161 Xn' a2 - 5.923 n' (na)2 - 4.134E-3 (na)2 - 2.600E+3

an'X2 - 5.367E-1 X(ann')2 - 3.557E+3 (nn1)2 - 2.589E+4

nX(n')2 - 7.125 aXn2 + 2.336E+1 naX + 6.357E-3 a + 1.431

na + 5.565E+3 (Xn')2 - 1.618E+3 an'.(nX)2 +.4.130E+3 ann'X2

.+ 1.549E+2 aXn'n2 + 1.079 nn'a2)

P(164°)=exp(2.689E+2 + 1.618E+4 n' + 8.978E-2 a(nX)2 -2.602E+2/n

- 2.900E+4 (n1)2 -- 3.299E+3 an'X2 - 3 . 436E+4 ah (n1 X) 2

+ 6.153E+3 n'n2 - 2.036E+4 nn' - 4.332 n'(aX)2 + 1.493 aX

+ 1.669E+4 (nn')2 + 2.443E-2 na2 + 2.274E+3 nn'aX2,

188

+ 4.619E+1 cm'(n-A)2 - 5.877E+1 na - 1.269 E-l (aX)2

..- - 1.429 Xn2 - 9.151E+3 aX(nn')2 '+ 7.659E+1 X(ari')2

- 1.297E+3 aXn'n2 - 8 . 899E+4; ,n.(Xn') 2 - 5.749E+3 Xn1

- 4.313E+1 n2 - 8.996E+3 a(nn')2 + 2.629E4-3 aXn1

+ 1.798E+4 a(n')2 - 1.885E+3 an1; + 1.339E-f 3 annf

- 1.059E+5 aX(h')'2 + 5.003E+4 a(Xn',)2 + 8.643E+4 aX(n')2

+ 4.100E+1 a + 1.062E+1 an2 + 4.159E+3 Xnn' + 1.225E+5

(Xn')2)

P(l'8d°)=exp(1.759E+2 .+ 1.334E+2 n'.+ 3.142E+4 aX(nn')2 - 2.056E+3

X(an')2 -. 3.018E-fl n2 - 1..074E+3 Xn'a2 + 4.978E+3 (n1)2

+ 5.162E+2 aXn'n2 - 2.187E+1 (aX)2 - 1.068E+1 (naX)2

,. +:6.399E+2 rin'Xa2 + 2.924E+5 naX(n')2 + 4.239E+1;Xn1(na)2

+ 1.371E+2 aX' -n'6.876 aXn2 - 9.354E+4 a(n.')2 + 5.932E+2

• X(nn')2 - 6.690E+2 nn1 (aX)2 - l'.508E-f3 Xn1 — l,663E+2/n

- 1.945E+2 naX - 6.090 n'an2 + ;1.004E+3'n1(aX)2 - 9.508E+3

a(nn')2 + 8.024E+4 an(n')2 + 1.169E+3 n'X2 + 1.629E+5

na(Xn')2 - 2.267E+5 a(Xn')2 + 3.438E+5 aX(n')2 + 3.062E+1

n(aX)2 - 3.701E+2 an'(nX)2 + 1.939E+2 n'a2 - 1.748E+2 nn'a

- 8.555E+1 n'(na)2 + 1.618E+3 nX(an')2)

OT = exp(-1.498E+2 - 1.020E+1 a + 3.730 a2 + 8.622E+2 nX

- 2.967E+2 n'(aX)2 + 1.548 (na)2 - 5.009 na2 - 2.402E+2

n'(naX)2 - 4.761E+2 n'Xa2 + 1.484E+2 (nX)2 + 4.393E+1 n'a2

+ 3.614E+2 X2 - 6.377E+2 X + 2.016E+1 n2 - 4.702E+2 nX2

- 2.852 Xn2 + 5.636E+2 nn1(aX) 2 + 1.086E+1 n'(na)2

- 3.906E+1 nn'a2 + 1.545E+2/n - 1.476E+2 Xn'(na)2

1&9

+ 5.258E+.2 nn'Xa2 + 8.306 na + 3.476E+3 an1 (nX)2

- 5.662E+2 aXn'n2 - 1.181 aX2 + 3.284 a(nX)2 •+. 1.239E+1 aX

- 1.423E+1 naX + 5.827E+3 an'X2 - 9.137E+3 nn'aX2

+ 9.489E+2 nn'aX - 8.068E+1 n'a + 3.763E+1 n'n2

- 1.981E+3 ,Xn' +. 2.406E+3 nn'X - 1.159E-1 X'(na)2

- 6.879E-1 (aX)2 + 8.424E-1 nXa2 - 7.663E+2 Xn'n2

+.. 8.076E+1 (Xn1)2)

a = 1.195E-3 - 2.507E-4 a + 1.087E+3 n' + 3.008E+3 a(n')2a+ 2.789E-5 a2 - 2.547E+2 (an1)2 + 1.191E+1 n'a2 - 1.091E+4

(n1)2 - 1.940E+2 an' - 6.414E-4 Xn2 + 4.157E+3 X(n')2

- 3.911E+2 Xn1 - 6.612E^-2 nn' + 1..380 nan1 - 4.455E+1

a(Xn')2 + 2.947E-2 n' (aX)2 - 2.262 n' (na)2 - 2.715E+2

h'(nX)2 +. 2.815 an'X2 -i- 1.853E+2 n'n2 + 2.201E+1 X(an')2

+ 2.609E+2 nn'X + 5.240 nn'a2 - 3.396E+2 n'X2 - 2.056E+2

aX(n')2; - 5,251 nX(an')2 + 1.3T7E+3 h(n')2 r- 1.977E+3

nX (n-1) .

190

5.3 MULTIPLE SCATTERING CODE C.K.W.

5.3.1 Introduction

Because it has not been given more than a cursory description in

print, and because its speed makes the model concept potentially valuable

to the scientific community as a whole, and because it is the model to be

adapted for all our future work we present in this and the following

subsections a brief description of the multiple scattering code C.K.W. in

it's present state of development.

For the program C.K.W. the atmosphere is modeled as a set of

concentric spherical layers, each characterized by extinction coefficients

for Rayleigh scatterers, aerosols, and ozone, and by an index of refraction,

all 'assumed constant within a given layer. The acceptability of assuming

constancy depends on the thickness of layers. Typically, we use 50 to 60

layers of 2 km thickness, and the calculated horizon profile is insensitive

to further refinement.

The calculations naturally fall into a hierarchy of increasing difficulty.

The simplest problem is single scattering, which is handled in great detail,

with the full spherical geometry, exact integrations including refraction,

detailed phase function data with polarization, and a Feugelson law for

albedo ( Kondratyev , 1969). The next case includes some nearly forward

scattering on the path from the sun to the site of one major scatter into

the detector line of sight. This is done with almost as much refinement as

single scattering. The next, more difficult case includes some nearly forward

scattering on the path to the detector. For this case, an approximate

estimation of coupling between scan lines is introduced. The final case is

full multiple scattering, which requires some additional .approximations

for the geometry. The following sections detail the methods and

approximations used to treat each of the successively more difficult cases.

191

5.3.2 Single Scattering

Basic Geometry

The modeling of -the-horizon profile requires that we first of all

establish a coordinate frame in which all the vectors that occur in the

calculation can be expressed unambiguously. For this purpose, we affix a

coordinate frame to the detector. The (z) axis is along the local vertical

at the detector, the (x) axis is perpendicular to the plane of scan, and the

(y) axis lies in the scan plane, pointing roughly in the direction from which

the detector seeks light (Figure 5.3-1). This choice of coordinate frame

is arbitrary, but convenient on several counts. First, this coordinate frame

is independent of the detector altitude. This is useful because one of the

applications for horizon profile modeling is in spacecraft navigation, .for

which the altitude may be unknown. Second, the zenith and azimuth of the

sun can be measured directly from the craft, so the vector representing

the input sunlight can be defined directly for this frame. Third, in the

special but common case that the sun lies in the scan plane, the (x) axis is

perpendicular to the scattering plane, so there is a coincidence in regard

to the naming of polarization states: both the usual geometric conventionand the convention used in scattering theory agree that polarization along

(x) corresponds to the same direction in Poincare space.

A second geometric requirement to be met is the specification of

the earth and its atmosphere. Thus we introduce the height of the detector,

(HDET), and the earth radius (RE), shown in Figure 5.3-1. Also we fix

the radii to the bottoms of each of the layers, RT :i_j

RL = RE + L KM (5.3-1)

where (KM) is the single-layer thickness and (L) is an integer member of

set (NL) defining the number of layers. Next we precalculate geometric

quantities that will be used over and over in the horizon profile calculation.

192

These are the scan path increments through the layers, DXT for L"= 0(1 )NL.J _ j - ' . , ' . ' '

For the scan path tangent to the earth, the path increment through layer

(L) is

-C' • • ' - " • - . • . DXL £ ^2 RE KM (/L+T - ,/L) (5.3-2)

Almost these same path increments occur on all the other scan paths too,

and so the numbers are saved for use on all the scan paths.

The final item of geometric information to be handled is the location

of any cloud. We denote the altitude of a cloud by the variable SOLIDTOP

in Figure 5.3-1. If the cloud happens to fall between layers, then it occludes

aportionof some layer. This portion is called SOLIDTIP. In the subsequent

calculations, any path increment through the partially occluded layer should

be adjusted from the appropriate DX^ above. The adjustment factor is

SQ = /KM - /SOLIDTIP (5-3.3)/KM

Average Attenuation

The attenuation within a path increment (DD) is one of the factors

that determine how much radiation is scattered to the detector from (DD).

• At any point in (DD), the radiation arriving from the sun is damped by

passage through a total optical depth denoted by (TTS). Similarly, the

radiation scattering to the detector is further attenuated by passage through

' 'a total optical depth denoted by (TTD). Thus at a point, the attenuation is

-TTS - TTDe

193

A full contribution to the output from the whole increment (DD) is of the

form

J a-TTS - T T D - E a . p . W dxDD i

where a. and p.(0) are the scattering coefficient an'd phase function for the11i constituent/and (dx) is. a differential increment of length. Assuming

constancy of all extinction coefficients in the layer (or at least their local

ratios), we split the contribution into two factors:

'*-' /.-DD DD

TTS - TTD ,dx

and

DD

The first factor is the average attenuation, and its evaluation is the subject

of this subsection. .

To perform the integration in the average attenuation we introduce a

change of variables: .

dx = J d(TTS + TTD) (5.3-4)

Here (J) is the Jacobian of the transformation,

J = l/(d(TTS + TTD)/dx) (5.3-5)

194

We assume the derivative in (J) can be approximated by

A (TTS + TTD)/DD

Thus we reexpress the average attenuation by

f -TTS - TTDJ eA(TTS + TTD) DD6 ^" d-<TTS .*. TTD)

This can be evaluated immediately as

: •:.-.- .... DAMP: = -Ae~TTS TTD/A..(TT_S + TTD), , , (5.3-6)

The above trivial result.is modified ;slightly by the inclusion of the

phenomenon of refraction. The basic principle involved is that light travels

not exactly in a straight line, but rather in a slightly curved line. At a

point on the line, we let (n) denote the refractive index, (r) the distance to

(the earth center, and sine the angle of incidence on a surface parallel to

.the earth. The curved line is described by - . , . - . _ ;

(n)(r) sine = constant (5.3-7)

where the constant is the nominal tangent height RR of the scan line.

Referring to the greatly exaggerated Figure 5.3-2, we see that where the

unrefracted path would be intersecting the bottom of layer (L) at an angle

whose sine is Ru/R T , the refracted path is passing through a somewhat• - • • i-P- ' . V * ' ' • ' • , • : • • .

lower point and at a lesser angle. Let (y) be the nominal scan line, and

(z) be the perpendicular displacement of the true path from the unrefracted

oheV Then the true angle is

f l . . /M • '/dz-.-^V , . " • : : " - - . , . - . . . -9 ^ arcsmr—1- arcsmf^ — J ( 5 3_8 }• • • - . ' , ' • ' • - • • \ L I / \ T J / - • • • . • - . '

195

We also have

/RH \arcsinfjp RiNDEXLJ (5.3-9)

\ L /e

v

where RINDEXT is the refractive index in layer (L). The derivative dz/dy\-ican be isolated and simplified by appropriate series expansions. It is used

in the code to develop the true path as a function of the nominal path, and

so to provide the correct boundaries, etc., at which to evaluate (TTS) and

(TTD).

Scattering Factor

Besides the average attenuation discussed in the preceding subsection,

the contribution to the output from a path increment (DD) depends on the

previously introduced scattering factor of the form

£ a± PI(T|») DD

Since the atmosphere being modeled contains both Rayleigh and aerosol

scatterers, there are two a's and p's that have to be evaluated. In both

cases, the (CT) is the local extinction coefficient, denoted as (RDTAU) for

Rayleigh scatterers, and (ADTAU) for aerosol scatterers. Also in both

cases, the polarized output (if any exists) can be expected to be linearly

polarized in the plane perpendicular to the scattering plane, so that in both

cases two polarization parameters of the phase function constitute sufficient

input data. Finally, for both cases it is essential to specify the scattering

angle (0). Actually, this is done by specifying the cosine of that angle

(COPSI). We simply form the dot product of a vector representing the

input sunlight and a vector representing the light traveling to the detector.

In the case of Rayleigh scattering, the phase function can be calculated

analytically within the program. Ideally, the two polarization parameterssy .

are 6/1677 and 6 COPSI /16j7 . Sometimes it is appropriate to modify this

196

slightly. But for aerosol scattering, the situation is much more complicated.

At present, we read in the phase function for selected values of (COPSI),

usually 46 in number, and interpolate on the basis of (COPSI).

Albedo

Scattering from the atmosphere is only one of two possible mechanisms

whereby light reaches the detector. There is also scattering from the

surface of the earth, or some cloud layer above it. The total fraction of

the incident light scattered from such a surf ace is its albedo. The simplest,

.and most commonly used angular. dependence for surface scattering is the

well-known Lambert law:

• - - ' . yradiance out = (irradiance in) ALBEDO -2. (5.3-10)

where (A/ ) is the cosine of the incidence angle. It is known that natural

surfaces usually are not Lambertian. Nevertheless, better models are

only occasionally available. Our model has incorporated a Feugelson law

for single scattering from clouds, but otherwise uses Lambert's Law.

The Feugelson law for the approximate albedo (A ) of a cloud of opticalC*

thickness (r) is given by

2y + 1

A = 1 -

(5.3-11)uo/3 + c 2 ( 3 v - D/8 {1 - 2u (ci/3 - 1)}

where c^ = 2.48 and Cg = 3.70. : .

5.3.3 Forward Scattering on the Sun Path

After single scattering, the next less simple mechanism whereby light

reaches the detector includes one or more nearly forward scatters on the

path from the sun to the primary scatter. This subsection details the way

197

in which this contribution is handled by adapting the methods already

described in the preceding sections. That is, we discuss modifications to

the geometry, the average attenuation, the scattering fraction, and the albedo

calculation.

Geometry

The geometry used to calculate the contribution to the output that

includes some forward scattering on the sun path differs from the single

scattering geometry only in that what is assumed to propagate along the

sun path is a stream rather than a plane wave. The explicit definition foro

the stream (Whitney, 1972) is radiance integrated against a cos weighting

function. There occurs in the calculation a factor (FPRL), which is part

of the stream definition, and factors (NORM) and (ESTPRL) that relate to

the inversion of streams back to radiances at the output. This inversion

is discussed in detail in a subsequent subsection on full multiple scattering.

Attenuation

The notion of a stream rather than a plane wave propagating along

the sun path modifies the average attenuation in two ways. First, since

forward scatters do not remove anything from the stream, forward scatters

have to be excluded from the optical depths along the sun path. This is

done by modifying the optical depths by a factor of (1-PHS), where (PHS)

is a combination of streamlike phase function integrals that represents

forward scattering from a stream into itself. The construction of (PHS)

is detailed in the above mentioned paper. The modified optical depth from

the sun is (MSTTS). With that replacement, the analog of (DAMP), the

average attenuation, is

MSDAMP = -Ae~MSTTS ~ ^^/ L (MSTTS + TTD') (5.3-12)

198

The second modification required is that we exclude from (MSDAMP)

the case in which absolutely no scatters occur on the sun path, since we

have already counted single scattering. This done by decreasing (MSDAMP)

by that part of the average that represents just single scattering. Thus

we put

MSDAMP = MSDAMP - DAMP

Scattering

The scattering factor is also affected by the inclusion of forward

scattering along the sun path. Where before it was appropriate to have a

plane-wave transformation, or phase function p(0), it is now appropriate

to have a stream transformation, or (PHS), evaluated for the nominal

scattering angle (0). For both Rayleigh and aerosol scatterers, these PHS's

are entered as data, precalculated at the angles that occur in the dodecahedral

arrangement of streams introduced in (Whitney, 1972) namely 0°, 180°,

63 and 117°. An interpolation routine then obtains the (PHS) for the desired

angle (</>).

Albedo

The final modifications required to accommodate a stream on the

sun path concern the albedo calculations. For a Lambertian surface, a

discretized version of Lambert's law is required. The continuous law

involves the factor | cos(0)| In-, where

TT = / | COS (9) |dfihemisphere

199

That is, the law is formulated to say that uniform radiance input provides

uniform radiance output, just scaled down by the albedo. The discrete law

must be formulated to say the same thing, but with the integral replaced

by a sum over the dodecahedron angles, and the |cos(e) l replaced by

averages of |cos(0)| with quadratic weighting functions. The normalization

for all the averages is

(p-k)

hemisphere= 2TT/3 (5.3-13)

The actual averages are as follows: for 0 ,

< | c o s ( 9 )•j I /\. /N.

= ^ I JU-k)

hemisphere

dfi = 3/4 (5.3-14)

ofor 63 ,

<|cos(8)|> = -^ I (z-k)

hemisphere

hemisphere

hemisphere

= 9/20

A A A A O4z-k(x-k)

(5.3-15)

200

Thus the sum analogous to the integral is

5 terms =

3/4 + 9/20 = 3 (5.3-16)

Thus for the discrete case, the (77) in Lambert's law is replaced by a 3.

5.3.4 Forward Scattering on the Detector Path . •

We suppose now that the photons sustain one or more nearly forward

scatters on their path from the primary scatter to the detector. This case

is slightly more complicated than that .discussed in the preceding section,

and so entails a few more modifications, in regard to geometry, average

attenuation, and scattering factor.

Geometry

The major geometric consideration with forward scattering on the

detector path is the possibility of transfer of radiation from one line of

sight to another at a slightly different angle. To handle this, we formulate

for each line of sight an approximation for the probability that forward

scattering ultimately contributes to that particular line of sight. This

probability varies from zero for the uppermost line of sight to unity for a

scan tangent to the earth. In between, the probability is approximated as

follows: the difficult feature requiring approximation is that where forward

scatters are -concerned we must think not of contributions from a path

increment, but rather from whole three dimensional regions. That is, if

subsequent forward scatters are allowed, contributions to a given scan line

come not from the line, but from a region surrounding the line. We formulate

the above mentioned probability function as a line integral of the form

PMm = / e~ T ' dT1 (5.3-17)

201

where the path over which the integral is done traverses, roughly, the optical

region that can, with no mechanism other than subsequent forward scattering,

contribute to the m scan line. The path proceeds from the detector,

through atmospheric layers down to and including the mth one, and then

out again. The AT' to be associated with any particular layer has to be

approximated in a rather gross way, because with the curvature of the

earth, the different propagation .directions in a stream can traverse

drastically different optical depths. We begin by formulating Ar1 in such

a way that if the whole atmosphere were condensed in one uniform layer

(whose geometric thickness would therefore be the atmospheric scale

height), that layer would have the.optical depth tangent through itself:

AT' = EXTINCTION /2 RE SH (5.3-18)

Thus for a layer of thickness KM, we put

AT' = EXTINCTION /2 RE KM /KM/SH . (5.3-19)

This formulation guarantees that for very small (KM), (r1) converges to a

well-defined function of the tangent height associated" with (PM) and is

insensitive to the number and geometric thickness of layers that happen to

be used in the atmospheric model. Finally, we must consider the effect of

the scattering phase function. Clearly in the case of extreme forward

scattering, the contributing region that Ar1 represents should shrink to

zero; that is, forward scattering becomes ineffective for transfer from

one line of sight to another. We represent this volume effect by modifying

Ar1 to Ar1 (1-PHS)3. ,

Attenuation .

The attenuation for contributions to the output that will undergo one

or more forward scatters on the detector path is modified in several ways.

First, the attenuation on the detector path is modified by a factor (1-PHS)

202

to omit forward scattering just as the attenuation along the sun path was.

The modified optical depth on the detector path is (MSTTD), so that at

each point the attenuation is

-MSTTS - MSTTDe

Second, as we have already counted single scattering and forward scattering

on the sun path, we have to subtract the portion of the function representing~MSTTD~TTDthose events: e . Third, and novel to the case where there

are to be subsequent forward scatters, there is no one unique and readily

identified path over which this function should be evaluated and averaged.

Recall that instead of contributions from a path increment, we are considering

contributions from a spatial region. Instead of attempting to average the

attenuation over a region, we simply leave it as a point function, evaluated

at representative points (e.g. the tangent points of scan lines directed to

the bottom of each atmospheric layer).

Scattering

The absence of a particular path to integrate over is again evident

in the scattering factor, for no particular path length (DD) appears. Instead,

(DD) is replaced by a standard length for subsequent forward scattering,

which is just the /2 RE KM /KM/SH previously introduced in connection

with the PM's.

5.3.5 Full Multiple Scattering

The final and most complicated mechanism whereby light reaches

the horizon profile detector involves two or more scatters that cannot be

classified as nearly forward. This section discusses the application of a

method that handles these events, and also incidentally provides some

additional information concerning the other mechanisms already discussed.

The method is basically that introduced in Whitney, 1972. In that paper,

the integro-differential equation of radiative transfer is discretized as a

set of twelve coupled differential equations for streams. For symmetry

the streams are arranged pointing to the faces of a regular dodecahedron.

203

Geometry

The geometry of the horizon scan situation is modified slightly to

achieve a speedy application of the method without significant loss of

accuracy. Specifically, thicker layers are used (usually 20 km), and they

are imagined to be flat until the final step, where curvature is introduced

as a correction. Because the layers are flat, the introduction of a cloud

is simplified; the analog of

SQ = /KM - VSOLIDTIP (5.3-20)

/KM

becomes simply KM - SOLIDTIP

. . KM

Also because the layers are flat, the original continuous equation of radiative

transfer can be expressed with inverse cosines of incidence angles. In

the discrete case, these are all replaced by inverses of cosines averaged

with quadratic weighting functions; that is, inverses of the values 3/4 and

9/20 previously derived in connection with the discrete analog of Lambert's

Law.

Attenuation

The attenuation of contributions to the output is calculated in much

the same way as it was in the preceding section. All optical depths are

scaled by a factor of the form (1-PHS), in order to eliminate forward

scattering from the calculation. Then the integrals are done exactly. The

general form of the integral required is

204

where r, , r*, TO • • • are the optical depths at which the first, second, third

. . . scatters occur, and CQ, c, , c2 ... are the cosines for the stream

traveling from the sun, from the first scatter, from the second scatter, . . .

. . . The various T'S are constrained so that each of the exponentials is

less than unity. The constraints are handled by a tree structure, or set of

nested do-loops, that governs the calculations. The program first chooses

the layer for the first interaction (Ll), then the inclination for the scattered

stream (PI), then its direction (Ql), and then the layer for the second scatter

(L2), and so on. If the first inclination (PI) is upward (downward) the

second layer (L2) is required not to be below (above) the first layer (Ll).

In the code, the multiple attenuation integral is not done all at once,

but in pieces; since the TV integral is done in a loop which is within the

loop that does the r_ integral, and which is in turn inside the loop that£

does the T integral. As an example, the T-, integral is just

(I/Co - 1/Cl) -

Because it has been isolated from the functions of ?•„, it is no longer true

that the exponential is necessarily less than unity; it may as well be greater.

However, when all the required integrals are multiplied together, the result

is less than unity, as is physically required.

A small novelty is required to handle the case where two or more

scatters occur in sequence within the same layer. The problem for that

layer is then a micro version of the problem for the atmosphere as a whole.

We could handle it the same as the full atmosphere, breaking the layer

into smaller sub layers. But such a procedure would be time consuming,

so we attempt to get an approximate answer without resorting to it. For

the case where all the scatters from(i) through (i+n) occur in one layer, we

approximate the required integral with the product of the' r., T. + , , . . . T.+n

integrals, but with an important modification. Consider the case where

205

all the inclinations are the same. The problem then reduces to a familiar

one in Poisson statistics, and the required integral is known to be simply

the above product, but divided by (n.1). Thus we induce that generally a

factor of (1/n!) should be incorporated.

Scattering

We come now to the actual scattering operations for full multiple

scattering. These are constructed by a procedure whose goal is to minimize

the storage space and computational burden required. The main source of

difficulty to be overcome is the inclusion of polarization, requiring four

Stokes parameters s~, s., s2, s3 to characterize a stream and, in general,

sixteen parameters to characterize a stream transformation. The input

stream transformations are provided for streams in the Z-X plane separated

by the dodecahedral scattering angles 0°, 63°, 117°, and 180°. For these

cases, symmetry reduces the sixteen parameters to far fewer, so that storage

space is minimized. In particular, for 63° and 117°, there are only eight

parameters: one 2x2 matrix operating on ( gO j and another operating

(s )» and for 0 and 180 there are only three parameters, only two

on

which are independent. For any particular scatter that occurs in the code,

the stream transformation is synthesized by combining the minimal stored

information with appropriate rotations of the plane-polarization parameters

(s / • That is, first a 2x2 rotation is applied so that the polarization

state in the nominal scattering plane acquires the name 1. Then the

appropriate stream transformation is applied. Then another 2x2 rotation

is applied to restore the polarization state in the nominal scattering plane

to its final correct name.

Program Control

In the case of multiple scattering it is not enough simply to attenuate

and scatter. The program must be prevented from expending excessive

effort on negligible contributions to the output, and must be prevented from

reconsidering the contributions it has more accurately calculated in the

single scattering part of the code. These goals are accomplished by several

tests in the code.

206

Of primary importance is the so-called twig pruning, or (ENDTEST)

for multiple scattering. This test compares the current contribution to

the output with a numerical standard. The standard is based on the output

that was obtained from single scattering, on the total number of terms of

the particular scattering order that are left to consider, and on an input

parameter expressing the programmer's judgement of how accurate he

wants to be. If the test is failed, it is judged unprofitable to further pursue

that particular branch of the multiple scattering tree. Were it not for this

test, the program would run for outlandishly long times without appreciably

changing the output.

Inversion of Streams

The final step in creating the horizon profile is to take the streams

SA provided by the multiple scattering code, and invert them to estimates

of physical radiance, S(k). This problem is the inverse of that involved inA

forming the stream S A from the radiance S(k). Since forming SA is analogousP A P A

to performing an integral transform on S(k), it follows that estimating S(k)

is analogous to inverting the transform. Because we do the original

transform only for power n = 2, and not arbitrary (n), it is possible to

invert only approximately. This section discusses what is thought to be

an optimum procedure for approximating S(k).

A

We begin by considering a simple case where S(k) is assumed to beA

equal to S(-k), and to have the form

6 terms~ SP

(k) = 2, (5.3-21)

with this form, it can readily be verified that

5 terms

S(e> '• - - (5.3-22,

207

Referring to Whitney, 1972, we see that the inverse of this relationship

would be

5 terms

:. £ V (5.3-23)

As in Whitney, 1972, we have, to separate forward-traveling from, . • • : • ) - • : . ' • ; . 7 " , . .

backward traveling radiation by extending these simple six term

relationships to twelve terms. For the numbers used, the analogous

inverse is

. 1+ 8"

5 terms 5 terms3 V^ 1 NT20" X S+6' + 40" Z S-6'

(5.3-24)

This is the basic relationship currently used to invert the array of output

streams in the code,

208

SCAN LINES

Fig. 5.3-1 Horizon Scan Geometry

209

NOMINAL. UNREFRACTEDSCAN PATH

REFRACTED SCAN PATH

Fig. 5.3-2 Effect of Refraction

210

REFERENCES

SECTION 5

1) Groves, G.V., "Seasonal and Latitudinal Models of Atmospheric

Temperature, Pressure, and Density", Air Force Surveys in

Geophysics, No. 218, May, 1970.

2) Kondratyev. K.Ya.. "Radiation in the Atmosphere", 386-399, Academic

Press, New York and London, 1969.

3) Malchow, H.L., "Standard Models of Atmospheric Constituents-and

Radiative Phenomena for Inversion Simulation", MIT Aeronomy

Program Internal Report No. AER 7-1, January, 1971.

4) Salah, J.E., "On the Nature and Distribution of Stratospheric

Aerosols", Private Communications, February, 1971.

5) Valley, S. L., Handbook of Geophysics and Space Environment,

McGraw-Hill, New York, 1965.

6) Whitney, C.K., "implications of a Quadratic Stream Definition in

Radiative Transfer Theory", J. Atm. Sci., 29, 1520-1530, 1972.

7) Wu, M.F., "Ozone Distribution and Variability", MIT Aeronomy

Program Internal Report No. AER 1-2, August, 1970.

21-1

3-30€¦ · 2.3.1 introduction 49 2.3.2 discussion of results 50 references 54 3. 0 modeling 55...

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