National Bureau of Standards

Library, N.W. Bldg

^ecUnlcai v2ote 9Zo. 3/5

A METHOD FOR OBTAINING THE PARAMETERS

OF ELECTRON-DENSITY PROFILES

FROM TOPSIDE lONOGRAMS

R. S. LAWRENCE AND MARGOT HALLENBECK

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards is a principal focal point in the Federal Government for assuring

maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. Its responsibilities include development and maintenance of the national stand- ards of measurement, and the provisions of means for making measurements consistent with those

standards; determination of physical constants and properties of materials; development of methods

for testing materials, mechanisms, and structures, and making such tests as may be necessary, particularly for government agencies; cooperation in the establishment of standard practices for incorpora^

tion in codes and specifications; advisory service to government agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; assistance

to industry, business, and consumers in the development and acceptance of commercial standards and

simplified trade practice recommendations; administration of programs in cooperation with United

States business groups and standards organizations for the development of international standards of

practice; and maintenance of a clearinghouse for the collection and dissemination of scientific, technical, and engineering information. The scope of the Bureau's activities is suggested in the following

listing of its four Institutes and their organizational units. Institute for Basic Standards. Applied Mathematics. Electricity. Metrology. Mechanics. Heat. Atomic Physics. Physical Chemistry. Laboratory Astrophysics.* Radiation Physics. Radio Standards

Laboratory:* Radio Standards Physics; Radio Standards Engineering. Office of Standard Reference

Data. Institute for Materials Research. Analytical Chemistry. Polymers. Metallurgy. Inorganic Materials. Reactor Radiations. Cryogenics.* Materials Evaluation Laboratory. Office of Standard Refer- ence Materials.

Institute for Applied Technology. Building Research. Information Technology. Performance Test Development. Electronic Instrumentation. Textile and Apparel Technology Center. Technical Analysis.

Office of Weights and Measures. Office of Engineering Standards. Office of Invention and Innovation.

Office of Technical Resources. Clearinghouse for Federal Scientific and Technical Information.**

Central Radio Propagation Laboratory.* Ionospheric Telecommunications. Tropospheric Tele- communications. Space Environment Forecasting. Aeronomy.

* Located at Boulder, Colorado 80301. ** Located at 5285 Port Royal Road, Springfield, Virginia 22171.
NATIONAL BUREAU OF STANDARDS

technical ^Vtote 3/5

ISSUED August 3, 1965

A METHOD FOR OBTAINING THE PARAMETERS OF

ELECTRON-DENSITY PROFILES FROM TOPSIDE lONOGRAMS

Robert S. Lawrence and Margot Hallenbeck

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price: 25(Jr

TABLE OF COMTEETS

Abstract 1

1. Introduction 1

2. Description of the Computer Program 5

3. Conclusions 22

k. Acknowledgments 22

5. References 23-2^

6. Figures 17-18

and 25-26

7. Tables 27-31

ux

A METHOD FOR OBTAINING THE PARAMETERS OF

ELECTRON DENSITY PROFILES FROM TOPSIDE lONOGRAMS

Robert S. Lawrence and Margot J. Hallenbeck

We present sample resiilts of a mo del- fitting procedure

which involves adjustment of physically meaningful parameters

of a model ionosphere so as to minimize the mean square

difference between the observed virtual depths and those

calculated for the model. The adjustment is made subject

to additional constraints such as (l) physically reasonable

li mi ts to the values of the parameters and (2) agreement with

any available auxiliary information such as the electron

density at the satellite. The method requires more computer

time than do conventional methods but compensates by making

extremely economical use of observational data. It has been

demonstrated that the method will provide useful information

from data which are too scanty to be usable by lamination

or polynomial methods. When, as occasionally happens, the

process fails to find a loseful answer, the reason appears

to be that the ionosphere cannot be represented by the model

being used. On the other hand, the use of physical limits on the parameters seems to insure that erroneous or multiple

solutions will not occur.

1. INTRODUCTION

Many methods are available for deriving information about the

vertical distribution of electrons from sweep-frequency ionospheric

soundings. One of the earliest of these, suggested by Appleton and

Beynon [19^0], involved the assumption that the profile of the ionosphere

is approximately parabolic. With this assumption, three points on the

ionogram were used to determine the parameters of the parabola. In fact,

the electron distribution is much more complicated than a simple parabola,

and for realistic results it is necessary to elaborate this simple

procedure.
Modern attempts at elaboration lie between two limiting types of

approach. One of these, the "polynomial" approach, involves the use of

a polynomial of high order to represent the entire ionosphere [Titheridge,

1961]. The other, the "lamination" method [Budden, I961] , involves a

large number of slabs, each of which is assumed to have a linear

variation of plasma frequency. Between these two extremes there exists

many lamination methods which involve the use of non-linear slabs.

Such methods, with various elaborations to surmount practical

difficulties, have been used quite successfuly to produce large mombers

of useful electron-density profiles below the peak of the F region [Thomas

et al,, 1958; Wright et al., I963] . With the advent of the first topside

ionograms from the satellite -borne sounder, Alouette (1962 Beta Alpha

One), it was only natural to use the same procedures to obtain topside

profiles. Neljns [I963], Thomas et al. [1963]^ Doupnik [I963],

Fitzenreiter and Blumle [19614-], and Paul and Wright [196^]-], among "others,

have prepared computer programs for this purpose. These programs produce

satisfactory topside profiles from the Alouette soundings and they have

been widely used.

-2-
The primary purpose for producing ionospheric profiles is, of

course, to learn more about the physical conditions existing in the

upper atmosphere, especially the involved processes of the formation,

loss, and movement of the ions and free electrons. For example, at

heights well above the F peak, the plasma scale height is determined by

the slope of a semi -logarithmic plot of electron density. If the ionic

composition is known, the plasma scale height gives a direct measixrement

of temperature. This type of reasoning has been carried several steps

further. Rishbeth (see Calvert et al., l^Gk) for example, has compared

topside profiles, deduced from ionograms by a lamination method,

with a nijmber of theoretical profiles based upon various assumptions

about the temperature and ionic composition. In this way, he succeeded

in determining values for several parameters of his model at heights well

above the F peak where production and loss processes could be neglected.

The success of Rishbeth' s approach using a small number of

adjustable parameters led us to consider the possibility of mechanizing

his method so that, with a computer, more data could be analyzed and

more parameters coiild be adjusted with greater ease and objectivity.

If a computer program to fit a model to a known profile is feasible, it

is, conceptually, only a small step to consider a still more complex

computer program which fits the parameters of the model directly to the

ionogram. Both of these computer programs have been written and they

operate successfully. It is the purpose of this paper to describe the

methods used by these programs and to present sample resiolts.

-3-
There is a compelling reason for taking the second, and larger, of

these two steps, thereby repleicing the conventional determination of an

electron- density profile. The conventional methods require observation

of the virtual depth at a large number of frequencies; large, that is,

compared to the number of parameters we hope to adjust. They are

generally ill adapted to the use of incomplete ionograms or to the

simxiltaneous use of bits and pieces of the ordinary and extraordinary

traces and of the ground-reflected trace. Therefore, in the case of

incomplete ionograms or data from the fixed- frequency topside sounder,

196h-'^lA (S-i|-8) [-Khecht et al., 19^2], the conventional methods are

frequently quite incapable of producing accurate results.

To recapitulate, we are presenting a method of analysis which is

applicable to cases where the ionospheric profile is reasonably smooth

and which is particularly valuable for those cases where the number of

data points is limited or some of the values are imprecisely known.

Because of the complicated structure of the lower ionosphere, the method

is not recommended for use with ground-based ionograms.

In this paper we illustrate our method with a particular model

(Rishbeth*s) for the upper ionosphere, a particular method of calculating

virtual depths (VIRDEP), and a particular minimization process (MENMTZ).

However, the reader must keep in mind that the success of the method is

not based upon these particulars but rather upon the incorporation of

non-rigid boimds on the individual variable parameters aad upon the use

of available constraints on the model as a whole.

-k-
2. DESCRIPTION OF THE COMPUTER PROGRAM

The computer program is given: (l) an ionospheric model involving

several adjustable parameters j (2) an initial set of parameters with their

individual bounds; (3) the frequencies, modes (ordinary or extraordinary),

and observed virtual depths of several pulses reflected back to a topside

sounder; (k) any known information about the ionosphere which may serve

as a constraint; and ^5y essential information at the sounder - height,

geomagnetic dip, and gyrofrequency. The program must then adjust the

parameters of the model so that the theoretical virtual depths calculated

from the model will best agree with the observed virtual depths. Implicit

in this statement 3s the assumption that there is a unique solution which

gives "best" agreement. The problem seems to the authors far too

complicated to permit a mathematical demonstration of uniqueness, but

experience sioggests that, for practical cases, if a solution exists which

gives reasonable agreement it is unique. Ve define best agreement as

that which minimizes the mean square of the residuals.

From this statement of the problem we can already see two of the

essential ingredients of the program. First, there is a subroutine,

VIRDEP, that computes the virtual depth to be expected for a pulse of

arbitrary frequency reflected from the model ionosphere. Second, there

is a subroutine, MINMSfZ, that tries to adjust a set of parameters in such

a way as to minimize the sum of the squares of an arbitrary set of

residuals which result from the use of those parameters. These subroutines

are described later in some detail.

-5-
AcLOther essential ingredient of any practical program of this

natiire is a procedure for accepting bounds on the parameters and for

accepting other constraints on the model. The parameters of the model

are intended to have at least a modicum of physical meaning. ¥e are not

interested in any narrowly empirical solution which involves such impalatable features as a negative amount of some atmospheric constituent, and

we do not want the computer to spend time searching for such a solution.

If we specify physically realistic boimds on the values which the

parameters may take, we can ensure that the computer will find only

acceptable solutions.

In addition, it is highly desirable to be able to include, as

constraints, any pertinent information which might be available from

independent sources. For example, the critical (penetration) frequency

of the ionosphere might be known from ground-based observations, or the

total electron content might be known from Faraday-rotation measiirements

of the satellite signal. Such information comprises a constraint on a

function of several parameters rather than a bo\md on the value of any

one parameter. The methods of handling boimds and constraints are

detailed in a later paragraph.
General operation The program must be giver a set of initial values

of the parameters. Its first task is to check to see whether these

initial values satisfy all the constraints. Generally, they will fall

within the specified bounds, but they may violate a constraint such

as a known critical frequency or a known electron density at the sounder.

If one of these constrains is violated, the program uses the subroutine

MINMIZ to adjust the parameters until all constraints and bounds are

satisfied. This process, which is relatively simple, reduces the amount

of adjustment which will be needed later when virtual depths will have

to be calculated laboriously for each trial model.

After a suitable set of parameters is obtained, the program uses

MIKMIZ again, this time trying to adjust the parameters so that the

virtual depths calculated by VIRDEP will nearly agree with the

observations and the parameters will continue to satisfy all boiinds and

constraints. If the actual ionosphere differs radically from any

acceptable configuration of the model, there will be no solution.

Experience suggests that when a solution is found it is unique, i.e.,

it does not depend upon the initial values assumed for the parameters. -

Failure to find a solution appears to indicate incompatibility between

the data and the model. Provision is made for printing the current

values of the parameters, the trial virtual depths, and a graph of the

trial model profile as frequently as desired to check progress.

Naturally, this information is always presented for the final result.

-7-
The virtual depth subroutine, VTRDEP The virtual , or apparent,

depth of reflection of a pulse from a topside sounder is

h

s d = J'^x'dh, (1)

h

r

where h is the height of reflection of the pulse; h is the height

± s of the sounder; and [i ', the group refractive index of the ionosphere,

is a complicated function of the soimding frequency, the electron

density and the geomagnetic field. VIRDEP uses the formulation

presented by Shinn and Whale [1952], assuming that the propagation is

vertical and that the dip angle of the magnetic field is independent

of height. VTRDEP assumes that the strength of the magnetic field

varies inversely as the cube of the distance from the center of the

earth.

Before integrating equation (l), VIRDEP must find h . Since

our model is monotonic above h » h can be determined by an

max' r

"^

ordinary binary search.

VIRDEP divides the range of integration into several slabs of

thickness averaging about 100 km, the ones near the soimder being the

thickest. In order to achieve an accuracy of one kilometer, 8-point

Gaussian integration is required for each slab. The lowest slab

presents a special problem because |j, ' becomes infinite at h . The

bottom kilometer of this slab is evaluated by an analytic integration

which assumes a linear variation of electron density. In this case p,'

is approximated by the first of two terms of a series expansion given by

Budden [I961].

8-
The minimization process, MTNMYZ MIKMIZ is a gradient-projection

subroutine developed by Slutz and Winkelman [196i^-]. Its purpose is to

adjust a set of variables so as to minimize an arbitrary function of them.

In the present application, the variables are the parameters of the model

ionosphere and the function to be minimized involves the mean square

difference between the observed virtual depths and the corresponding

depths calculated by VIRDEP.

Given m adjustable parameters P. and a procedure for computing

n (^m) errors (such as discrepancies in virtual depth) E^, MINMIZ first

J evaluates numerically the partial derivatives ^E./^P. for j ^ n and

all i ^ m. This is done by one-sided rather than central differencing

in order to reduce from 2mn to mn+1 the number of times VTRDEP must

be used.

In geometrical terms, the operation of MTNMTZ may be described as

follovs. From the array of partial derivatives, MINMIZ determines the

gradient of the error surface in (m+l) -dimensional space and from the

individual derivatives estimates the distance which that gradient shoiild

be projected to find a minimum. The step is taken and VTRDEP is used

n times to assess the result. Comparison of the actual chan.ges in the

errors with those which were expected permits MDMTZ to decide whether

it will be profitable to take additional steps in the same direction. If

so, they are taken, using VTRDEP n times to check the result of each

step. MXNMyZ continues until no further improvement can be made

locally in that direction. The program may use MINMrZ several times

to get new gradient directions.

-9-
When MIEMTZ finds a minimum there is naturally no assurance that

it is the global minimum. It is the responsibility of the user to design

the function which defines the errors so that there will be few^ if any^

.local minima in which to become trapped. Our procedure for doing this

includes the use of physical constraints and boimds on the parameters.

These are described below. In addition, we have found it necessary to

invent artificial definitions of the error for those cases where, in

the trial ionosphere, the probing frequency cannot propagate in the

vicinity of the soimder or where it penetrates all the way to the

groLind. Unless something special is done with these cases, the partial

derivatives are zero and MINMYZ receives no indication of which way

to go.

The accommodation of bounds and constraints At first glance, the

present method seems to fall in the category of problems commonly faced

in operations research. Nonlinear programming involves the minimization

(or maximization) of a given function of several variables while

simultaneously satisfying a set of linear or nonlinear constraints on

those variables. An enormous effort has been devoted to the solution of

this economically important problem [see for example, Zoutendijk, I96O;

or Rosen, I96I] , and it is tempting to try to use here the results of

that work. This seems to be a vain hope because of important differences

in emphasis. The typical problem in operations research involves, perhaps,

hundreds of variables, and the fimction to be minimized is a simple

function of those variables. Furthermore, the bounds on the variables

are absolutely rigid.

-10-
As an example^ the number of oil refineries which can be used to

produce a specific grade of fuel cannot be negative nor can it exceed

the total number available. Rigid boimds cause abrupt changes in the

direction of progress toward a minimum and so may increase the number of

calculations of the function to be minimized.

Our problem^ on the other hand^ involves a small number of variables

(the parameters of the model) _, but the function to be minimized is

extremely complicated. Its calciilation involves many lengthy numerical

integrations of the group refractive index. The refractive index is an

involved function of the plasma frequency which, in turn, is a complicated

function of the parameters. The boiuads on our parameters are not, by

nature, absolutely rigid. To be sirre, the temperatiure cannot be negative,

but we can give more information than that. We know that it probably lies

between 500°K and 1500°K and we can specify those values as bounds,

although we would be just as satisfied with an answer of 1+99° as with

501°.

-11-
Thus, "we have chosen to ignore the absolutely bounded methods of

operations research and to include our bounds and constraints in the

error function to be minimized. In this way we tend to avoid an

excessive number of calciilations of our complicated error function. Each

boimd or constraint is expressed in terms of a function which can be

treated in exactly the same way as a residual in virtual depth. In

essence all errors are "normalized" to a virtual depth residual. For

example, we have defined the "error", E , connected with temperature T

as Iooo-tV (2)

so that temperatures of 500° or 1500°would have the same effect on

MUTMIZ as would a virtual-depth residual of 30 km, and temperatures

venturing outside this raxige would rapidly become less acceptable.

Likewise, an "error" of similar form is used for any of the available

constraints. In the case of a constraint upon the critical frequency,

we might require that a computed frequency, f be within 0.1 Mc/s of

the given frequency, f . The error wo-uld then be defined as

,f - f N3 E =lo(-2 ii-,_. (3) ^ (O.lf

The inclusion of these "errors" defined upon the variable parameters,

themselves, and upon the given constraints lends much to the success of

the method. It not only permits the incorporation of any extra

information (constraints) into the final results, but it also helps to

guarantee an acceptable solution by keeping the parameters within

physically reasonable bounds.

-12-
The final function to be minimized is the mean square value of all

such errors and of all residuals between the observed and computed

virtual depths. We describe the degree of failure of the program to

make a perfect fit to the observations by evaluating the estimated rms

error,

r,Q - t i=l

= = ir^I-i^T, (M

where r. are the residuals in virtual depth, n is the number of

observed virtual depths, and k is the number of independently adjustable

parameters, i.e., the number of adjustable parameters reduced by the

number of constraints and further reduced by one to allow approximately

for the fact that all parameters are bounded.

Sample results We now illiistrate our method with several examples

which were rim on the IBM 709O computer. The model used in our examples

involves the following adjustable parameters:

h - the height at which the concentration of equals the

concentration of He (km) + + a - the ratio of concentration of H and He at height h

N - the concentration of electrons at height h (cm"®)

a a

^ ^

0, T - the electron temperature ( K)

h - the height of maximum electron density (km).

max

The electron density R at height h (km) is calciilated from the

parameters in the following way:

'-N -• 2 + a

' (5)

-13-
where

7 = K (h - h^.)

^ o a^ o

'

K(h - h )

^ ma:x: a m ~ T (R^ o + ha^}(R

o +li

max Y

R = 6370 km

K = 765.826 X 10^ °K km.

We do not intend to try to justify this particiilar model, but only to

illustrate our method with its use. The method should work as well for

any reasonable model involving a similar number of parameters.

In the sample computer runs, functions of the form

/P - P \n

^n {-\^ (^)

were used as "errors" for MENMIZ in order to establish the following

upper and lower bounds on the parameters

500° K < T < 1500° K

< a < i.25

375 km < h < 900 km

.5 X 10'^ cm"^< N < 5.5 X 10* cm"^

3. 100 km < h < i^-OO km.

max

-U-
The P is the meaxi value of the desired upper and lower bounds

m

for each parameter, P is the current value of the parameter, P is

the half-range of permitted variation for the parameter and is simply the

difference between the upper bound and P , and P is a "normalization"

m n

factor. In most computations it was I5 in order to make the effect of a

parameter at the limit of its acceptable range comparable to a virtualdepth error of I5 km. The power n is any odd positive integer;

generally it was 3 but occasionally values up to 9 were used. The

purpose of n is to control the rate of increase of the "error" as the

value of the particular parameter ventures outside the acceptable range.

The samples we shall discuss involve several computer runs,

niimbered arbitrarily. Each run took about a minute of computer time.

[These cases involve the use of the program to find an ionospheric model

consistent with various observed points from a particular Alouette

ionogram. The ionogram was made at 3 ^f ^0 UT on October 1, I962.

The sounder was at a height of IO38 km, located at 3^° '9 N. latitude

and 89 •! W longitude. The geomagnetic dip angle was 67 degrees, and

the electron gyrofrequency at the sounder was 0.9^ Mc/s. We shall see

that the results agree well with the true -height profile obtained from

the same ionogram using the lamination method of Nelms [I963]

-15-
For the run which we have arbitrarily labeled #l^-, we chose to scale

from the ionogram the virtual depths at those frequencies which would have

been observed by the fixed-frequency topside sounder_, 196ii- 5IA (S-ij-8).

Only six virtual depths with corresponding frequencies and modes were

used as input, and a mild constraint on the critical frequency was used.

No constraint on the electron density at the soimder was used. Figure 1

displays the electron density profiles of the initial trial model and the

final model. The model used in this case incorporated a "peak", i.e., it

involved h as a parameter. Since the input data contain no informamax

tion from which either h or the critical frequency could possibly

be deduced, we made another run (run #5)^ this time simplifying the

model by'' omitting^ the factor which involves h . The results of this

max

rxm. are shown in figure 2.

Agreement of the electron-density profiles is gratifying but only

of secondary interest. Our chief concern is whether or not the method

detenaines the parameters of the model xmiquely and with acceptable

accuracy. It is important that the final set of parameters be determined

solely by the observed data and not by the particular initial parameters

used for the trial model. This is the case when a satisfactory fit is

obtained. Aij example is given in table 1 which summarizes the data

illustrated in figures 1 and 2 and gives the parameter values that

resulted.

-16-
1200

No (cm-')

Figure 1. The electron-density profile (solid line) resulting from sample run ^h, is compared with the profile (thin dashed line) calculated

by a lamination method using 29 virtual depths scaled from the

extraordinary trace of the same ionogram. The initial trial

parameters produced the profile shown by the thick dashed line.

Vertical arrows show the true heights of reflection, corres- ponding to the virtual depths of the final model, for the six

pulses that were used. The final estimated rms error for run

#i^ of table 1 = 6 km.

.

-17-

J
J200

Ne (cm-')

Figure 2. The result of sample rim #5. The only difference from figure

1 is the omission of a peak in the model. Final estimated rms error for run #5 of table 1=8 km.

18-
In reference to the question of uniqueness of the final parameters

we made two runs using the same input frequencies and virtual depths but

"with different sets of initial values for the parameters. These two runs

used six virtual depths scaled from the extraordinary trace of the

ionogram. The plasma frequency at the satellite was given as a constraint,

and the plasma frequency at a probable height of the maximum electron

density obtained by the lamination method was also given as a mild

constraint. Table 2 provides a summary of the input data and the results.

The resiilting electron-density profiles for each case are shown in

figures 3 and. kj respectively. Both figiores show the initial models, the

final models and also, for comparison, the profile obtained by the

lamination method. Also shown in each figure is the result of the

preliminary adjustment of the parameters which was made in order to

satisfy all bounds and constraints.

We made several other runs to test the use of the various types of

information which would have been available from the fixed frequency

so\mder. Three of these (6, 9^ and lO) are summarized in table 3« The

same initial values for the parameters were used in all cases and no

added constraints were incorporated except mild constraints on the

critical frequency in 9 snd 10. From the particular ionogram which we

were using, it was possible to scale the virtual depths of three of the

specified fixed frequencies: 1.5^ 2.0, and 2.85 Mc/s. Both the extraordinary and the ordinary traces were present and were used for run =ff6.

We made run #9 using only the 1,5 and the 2.0 Mc/s frequencies in order

to investigate the capability of the method in dealing with very sketchy

data.

-19-
The program is designed to use delay times of ground-reflected pulses

as additional data. Unfortunately^ the particular ionogram which we had

selected for analysis showed no ground-reflected traces. Consequently,

we calculated the virtual depths to be expected for the extraordinary

and ordinary ground traces at 5»^T Mc/s and used this information together

with the data of run #9 to make run #10 in our analysis of the simulated

S-i|-8 data. Results appear in table 3» The final parameters, virtual

depths, and true heights agree well among the three different runs.

With the fixed- frequency sounder there exists a possibility of

identifying the ordinary and the extraordinaiy pulse of a particular

frequency incorrectly. In order to study the convergence behavior of

the method in such cases, we made run #17 using both the ordinary and

extraordinary modes of three S-i^-8 frequencies but with the modes

interchanged for the 2.0 Mc/s frequency. Table k summarizes the results

of this run, and the corresponding 2run (#3) using the correct data. The

results for the run using the incorrect data show differences of more

than 100 km between the observed and computed virtual depths while the

corresponding differences for the correct run do not exceed 7 km* The

estimated rms error for the incorrect run was 107 km, which indicates

the lack of a good fit as compared to 6 km for the correct run.

-20-
A summary of the final parameters and estimated rms errors of

eight different riins is given in table 5* These rims utilize different

minimization controls, different initial values of the parameters _, various

types of constraints such as the plasma frequency at the vehicle and

h , and virtual depths scaled at different frequencies. Two of the

max

rims used data from the electron-density profile determined by the

lamination method. Thus, the computation of virtual depths was

eliminated in these cases. Table 6 gives the averages and standard

deviations of the final parameters for sixteen different types of rims,

including those described in table 5^ and also includes Rishbeth's

estimate for comparison. We see that, for practical purposes, the

results are quite insensitive to initial values of the parameters.

The programs have been used subsequently for about I50 ionograms.

More than three -fourths of these produced excellent results; the

remainder failed to fit the data within the estimated error of observation.

Some of the failures exhibited residuals which indicated that more

complicated models would be needed to fit the observations; five or six

stubbornly refused to make aay progress at all.

21-
CONCLUSIONS

We have presented a method of analysis of topside ionograms which

is particiilarly useful for sketchy or incomplete traces such as those

from the fixed- frequency sounder. The method is a model-fitting method

applicable to a wide variety of models and avoids the intermediate step

of determining an explicit electron-density profile if only the parameters

of the model are desired. The success of the method stems largely from

the use of realistic bounds on the parameters and the ability to incorporate other constraints on the model. The method makes efficient use

of input data^ but requires more computing time than conventional

true -height methods.

Tests with simulated data of the kind obtained from the fixedfrequency topside sounder show that several parameters of the ionosphere

can be deduced from such records, even when it is impractical to produce

complete electron-density profiles by conventional methods.

ACKNOWLEDGMENTS

We are indebted to R. W. Knecht and T. E. VanZandt for suggesting

the problem,, to H. Rishbeth for providing his formulation of the

ionosphere model, to J. R. Winkelman for the subroutine MJNMIZ, to G.L.

Nelm.s for the lamination programs, and to E. L. Crow for helpful

discussion. Financial support was provided by the National Aeronautics

and Space Administration under Order No. S-2i|-8l-G.

-22-
REFEREJIJCES

Appleton^ EoV., and W. J. G. Beynon, "The application of ionospheric

data to radio communication problems. Part I", Proc. Phys. Soc,

52, 518-533 (19^0).

Budden, K. G. , Radio Waves in the Ionosphere , Chapter 12, Cambridge

University Press (I96I).

Calvert, W., H. Rishbeth, and T. E. VanZandt, "Studies of Alouette

Observations at the Central Radio Propagation Laboratory", IG

Bulletin 83 in Trans. Amer. Geophys. Union 1+5, 398-1+02 (196I+).

Doupnik, J. R., "A flexible method of determining the electron-density

distribution in the ionosphere", Penn. State ISaiv. lono. Res. Lab.,

Sci. Report No. I90 (August 1, I963).

Fitzenreiter, R, J., and L. J. Blurale, "Analysis of topside soimder

records", J. Geophys. Res., 69, 3, I+O7-I+I5 (196I+).

Khecht, R. W., T. E. VanZandt, and J. M. Watts, "The NASA fixed- frequency

topside sounder program", in Electron Density Profiles in the

Ionosphere and Exosphere , ed. B. Maehlum (Pergamon, Oxford) 2l+6-260

(1962).

Nelms, G. L., (Defence Research Board, Ottawa, Canada) private

communication (1963).

Paul, A. K. , and J. W. Wright, "Electijon-density profile analysis of

topside soimder ionograms", J. Geophys. Res., 69, 7? 114-31-11^-35 (l961t-).

Rosen, J. B., "The gradient projection method for nonlinear programming

Part II. Nonlinear constraints", J. Soc. Indust. Appl. Math., %

\, 51i+-532 (1961).

Shimi, D. H., and H. A. Whale, "Group velocities and group heights from

the magneto -ionic theory", J. Atmos. Terrest. Phys., 2, 85-IO5

(1952).

Slutz, R. J. and J. R. Winkelman, "Shape of the magnetospheric boundary

under solar wind pressure", J. Geophys. Res. 69^ 23, 1+93 3 -14-91^-8 (196I+). ~

Thomas, J. 0., J. Haselgrove, and A, Robbins, "The electron distribution

in the ionosphere over Slough, I. Quiet days", J. Atmos. Terrest.

Phys., 12, 1, 1+6-56 (1958).

Thomas, J. 0., A. R. Long, and D. Westover, "The calculation of electron

density profiles from topside souader records", J. Geophys. Res., 68 , 3237-321+2 (1963).

-23-
Titheridge, J. E., "A new method for the analysis of ionospheric h*(f)

records", J. Atmos. Terrest. Phys., 21, 1-12 (1961).

Wright, J. W., L. R. Wescott, and D. J. Brown, "Mean electron density

vajriations of the quiet ionosphere, 10: December 1959"^ NBS

(Boulder, Colorado) Tech. Note it-0-10 (March I963). Also Tech.

Notes UO-1 through kO-9.

Zoutendi jk, G. , Methods of Feasible Directions , Elsevier Publishing

Company, Amsterdam (I96O).

-24-
1200

Ne (cm-5)

Figure 3. The electron-density profile (solid line) is compared with the

profile (thick dashed line) produced by the initial trial

parameters and the profile (dot-dashed line) obtained by

preliminazy adjustment to the bounds and constraints. The arrows show the true heights of reflection, corresponding to

the virtual depths of the final model, for the six pulses that were used. The thin dashed line shows the profile produced by

the lamination method. Final estimated rms error for run #1

of table 2b = 3 km.

-25-
1200

i

Ne (cm"')

Figure k. Same as figure 3 except that the model used different initial

trial parameters. Final estimated rms error for run #2 of

table 2b = 6 km.

-26-

i
Table 1:

Frequency

(Mc/s)

1.5

2.0

2.85

1.5

2.0

2.85

INPUT

Mode

X

X

X

OUTPUT

Virtual

Depth (km)

IO3U

878

838

713

Ikk

825

Error in Virtual True -he ight of

Depth (km) Reflection (km)

Run k Run 5 Run h Run 5

1 -2 785 792

-2 -1 588 593

k 8 k6l 1+62

-5 -3 551 556

5 6 480 1+82

-2 -10 397 399

Table la: Frequencies, virtual depths, and true -heights of reflection

for the sample calculations illustrated in figures 1 and 2.

The estimated rms error for run k was 6 km; the estimated rms error for run 5 ^a-s 8 km. Both estimated rms errors are based

on n - k = 3 degrees of freedom.

Parameters h

rna.x (km)

Initial Model .500 250

Final Model

Run k .053 271+ Final Model

Run 5 .01+7 —

Rishbeth's

Estimated value .05

Estimated

Uncertainty .10

Table lb: The initial and final par

h N

a a

(km) (cm"^ X lO"*)

575

651

666

650

10

2.00

1.^3

1.1+0

T

(°K)

1000

878

928

81+0

50

figures 1 and 2. For comparison are given the values which

Rishbeth (1963) deduced from the same ionogram by comparing

a number of theoretical curves with the lamination profile.

-27-
Table 2:

UTPOT OUTPUT

Frequency

(Mc/s)

Mode Virtual

Itepth (km)

Error in Virtual

Depth (km)

True -height of

Reflection (km)

Run 1 Run 2 Run 1 Run 2

1.3 X 600 -2 8 989 981^

1.5 X 103i^ 3 2 795 792

1.8 X 930 3 6 6kl 61^3

2.1 X 861 -2 -1^ 573 570

2.6 X 833 1 -5 i^95 l^9i^

3.i^ X 863 2 iH3 14-12

Table 2a: Freque]2cies, virtual depths. and true -heights of reflect ion for

sample calculations illustrated in figures 3 aJid. k. Estimated

rms error for run 1 was 3 km; estimated rms error for run 2 was

6 km. Both estimated rms errors are based on n - k = ^i- degrees

of freedom.

Parameters h h N T

max a a

(km) (km) (cm'^xlO"^) (°K)

Initial Model

Run 1

,500 225 600 2.25 900

Final Model

Run 1

.029 291^ 61^8 1.5^ 872

Initial Model

Run 2

.i^oo 275 600 2.25 900

Final Model

Run 2

.032 328 635 1.62 828

Table 2b: The initial and final parameters of the models illustrated in

figures 3 and k»

-28-
Table 3:

INPUT OUPUT

Freq.uency Mods Virtual

(Mc/s) Depth (km)

103^

878

838

713

Ikh

825

11^26

1128

1.5 X

2.0 X

2.85 X

1.5

2.0

2.85

3M GX

3M GO

Table 3a: F:

Error in Virtual

Depth (km)

Run 6 Run 9 RuJi 10

- 2 - 2 - 8

3

k

- 2

8 11

- 6 -10

3 - 6

-lU —

2

- 1

Tirue -height of

Reflection (km)

Run 6 Run 9 Run 10

792 810 833

590 597 608

)+62 — —

553 560 570

1+81 1+87 U96

J+00 —

Frequencies, virtual depths and true-heights of reflection for

sample S-U8 data. Estimated rms error for run 6 was 11 Km with

n - k = 2^ run 9 was 10 km with n - k = 1^ and run 10 was 10 km

with n - k = 3 degrees of freedom.

Parameters

Run 6:

Ordinary 1.5, 2.0, 2.85

Extraordinary 1.5, 2.0, 2.85

Run 9:

Ordinary I.5, 2.0

Extraordinary 1.5, 2.0

Run 10:

Ordinary 1.5,

Extraordinary 1.5,

Ground Ord. 5.i+7

Ground Ext. 5-^7

2.0

2.0

,050

,106

.061

^max

(km)

251

261

260

ha Na T

(km) (cm-^XlO-*) (°K)

652

653

6k6

1.1+8

1.55

1.73

896

881+

883

Table 3b: Final parameters for sample S-1+8 data.

-29-
Table k:

INPUT OUTPUT

Frequency

(Mc/s)

Mode Virtual

Depth (km)

Error in

Depth

L Virtual

(km)

True -height of

Reflection (km)

Run 3 Run 17 Run 3 Run 17 Run 3 Run 17

1.5 X X 103i+ 9h 802 798

2,0 X 878 -2 -101+ 599 1+86

2o85 X X 838 7 36 1+67 1+63

1.5 713 -5 -1 562 571

2.0 X 7kk 1+ 112 1+87 611

2.85 825 -1+ 31 1+01 388

Table 1+: A test of the effect of mis -identifying ordinary and

extraordinaiy echoes. The estimated rms error for run 3 "w^as 6 km with n - k = 3^ whereas IO7 km was the estimated rms error for run I7 based on n - k = 3 degrees of freedom.

30-
Table 5:

Run

Number

a max

(km)

h

a

(km) (cm "^x\o"^)

T

(°K)

Estimated

rms error

1 .029 29i^ 61^8 1.5i^ 872 2.6 km

2 .032 328 635 1.62 828 6.3 km

3 .078 21^3 673 1.1+1 929 6.3 km

k .053 ZJk 651 l.i+3 878 6.3 km

5 ,0k7 „ 666 l.UO 928 8.U km

6 .050 251 652 1.1+8 896 11.0 km

7 .111 282 626 1.88 899 .10 Mc/s

8 .109 283 631 1.82 899 .07 Mc/s

Mean of

8 runs ,06k 279 61+8 1.57 891

Standard

Deviation .O3O 26 I5 .17 31

of 8 r\ms

Runs 1 and 2 used six extraordinary traces and the plasma frequency at

the vehicle; runs 3^1^-^5^ aJ^^ 6 used the extraordinary and ordinarytrace of three S-1+8 frequencies; and runs 7 aJ^d. 8 used nine and twenty

nine points respectively of the lamination profile. Five different

sets of initial parameters were used.

Table 6:

h

max

(km)

h

a

(km) (cm"^ X 10"^

)

T

(°K)

277 650 1.55 833

29 19 .16 1+2

650 81+0

10 50

Mean of 16 runs .09I+

Standard deviation

of 16 runs .061

Rishbeth*s

Estimated value ,05

Estimated

Uncertainty .10

The sixteen different runs include the eight described in table 5-

They use various data points, various Initial sets of parameters, and

a number of different constraints on the data.

-31-

GPO 852-692


U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DBPARTMENT OF COMMERCE

OFFICIAL BUSINESS