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02


APPENDIX A (Continued)
BIBLIOGRAPHY

Titheridge, J. E. (1959a), Ray paths in the ionosphere; approximate calculations
in the presence of the earth's magnetic field, J. Atmosph. Terr. Phys. 14,
50-62.

Titheridge, J. E. (1959b), The calculation of real and virtual heights of reflec-
tion in the ionosphere, J. Atmosph. Terr. Phys. 17, 96-109.

Titheridge, J. E. (1959c), The use of the extraordinary ray in the analysis of
jonospheric records, J. Atmosph. Terr. Phys. 17, 110-125.

Titheridge, J. E. (1959d), Ionisation below the night-time F layer, J. Atmosph.
Terr. Phys. 17, 126-133.

Titheridge, J. E. (1959e), The determination of the electron density in the
ionosphere, PhD. Thesis, University of Cambridge.

Titheridge, J. E. (1960), The accuracy of the approximate method for determining
valley size, Report for URSI Working Group.

Titheridge, J. E. (196la), The analysis of night-time h'(f) records, J. Atmosph.
Terr. Phys. 20, 209-211.

Titheridge, J. E. (1961b), A new method for the analysis of ionospheric h'(f)
records, J. Atmosph. Terr. Phys. 21, 1-12.

Titheridge, J. E. (1961c), The effect of collisions on the propagation of radio
waves in the ionosphere, J. Atmosph. Terr. Phys. 22, 200-217.

Titheridge, J. E. (1963), The analysis of ionospheric h'(f) records using the
phase refractive index, J. Atmosph. Terr. Phys. 25, 43-47.

Titheridge, J. E. (1965), The relation between the group and phase paths in a
non-homogeneous ionosphere, J. Atmosph. Terr. Phys. 27, 1115-1116.

Titheridge, J. E. (1966), The calculation of the heights of the peaks of the
ionospheric layers, J. Atmosph. Terr. Phys. 28, 267-269.

Titheridge, J. E. (1967a), Calculation of the virtual height and absorption of
radio waves in the ionosphere, Radio Science 2, 133-138.

Titheridge, J. E. (1967b), Direct manual calculations of ionospheric parameters
using a single-polynomial analysis, Radio Science 2, 1237-1253.

Titheridge, J. E. (1967c), The overlapping-polynomial analysis of ionograms,
Radio Science 2, 1169-1175.

Titheridge, J. E. (1969), The single polynomial analysis of jionograms, Radio
Science 3, 41-51.

35

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}
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APPENDIX A (Continued)

ponding to the lowest 0 ray frequency f, , the value of hj-h} indicates

the amount of underlying ionisation. A simple correction procedure is there-
fore to analyse the 0 ray trace beginning from a height hy at a frequency

f. of about 0.5 MHz . ‘Taking hy = hy -C(ht-h}) a value of C can be

found such that the resulting profile agrees exactly with the virtual height
hy . It also agrees exactly with all 0 ray virtual heights. A general

expression for C as a function of dip, gyrofrequency and f, is derived
in Titheridge (196la).

For distributions of underlying ionisation with fy < 0.7£; over most
of the height range, the precise value of f,, is not important. When the
gradient dh/dfy, is large near f, , however, the correction becomes
inaccurate. The error can be reduced by offsetting the value of f, » by
an amount Af, say (Titheridge, 1967c, appendix 2). With appropriate
choice of the two parameters C and Af results can be made exact for

any two model profiles. They will also be exact for any linear combination
of these profiles. Choosing reasonable models with and without large
gradients near fj, , the calculations can therefore be made insensitive to
such gradients. Values of C and 4£, have been determined numerically

on this basis for a range of dip angies (Titheridge, 1975b). Approximating
expressions are also given for computer analysis, and are included in
the program LAPOL.

For topside ionograms a somewhat similar single-point measurement can
be used to obtain directly information relating to the initial (top) section
of the observed profile. The virtual depth of the 0 ray is measured at the
local X ray frequency. Multiplication by a suitable constant then gives,
with no further effort, the logarithmic slope (i.e. the scale height) just
below the satellite (Titheridge, 1976).

6.3 Least-squares calculations

The program POLAN uses two parameters to define the underlying ionisa-
tion. These represent basically the total amount of ionisation with fy < £

and the mean gradient in the range f, to f, . The first calculated poly-
nomial begins at f, and gives in effect some additional information about

the unseen ionisation, by assuming continuity of all derivatives across f, .
Best results are then obtained with a rather lower value of £5 » equal to

0.6f1 . Below this point POLAN assumes a linear variation from 0.3f, to
0.6f,; . The thickness of this linear slab, and the constants of the first
polynomial, are determined by a one-step least-squares fit to the initial 0
and X ray data points.

Typically (in mode 4) 7 parameters are obtained using 5 0 rays and 5
X rays. The first S parameters are the normal polynomial terms. The last °
two represent a constant term A (giving the amount by which the starting
point hy is lower than the conservative 0 ray ex trapolation described

in 6.1), and the thickness D of the underlying linear slab. The subroutine

31



M4.

15.

16.

17.

18,

APPENDIX B (Continued)

. Gulyaeva, T.L., and V.S. Knyazyuk (1975), An analytical representation of the

virtual heights in magneto-active ionosphere, Geomagnetism and Aeronomy,
15(2), 362-364.

HicCue, C.G., and L.F. McWamara (1975), URSI Working Subgroup 3/6/2 on ‘Electron
density profiles through the ionosphere', Report of activities 1972-1975,
IPS, Australia.

Gulyaeva, T.L., and A.G. Shlionsky (1976), Localizing the maximum of the ioniza-
tion vertical gradient by using the derivatives of the trace in the iono-
gram, Geomagnetism and Aeronomy, 16(4), 698-701.

Gulyaeva, T.L. (1978), FORTRAN program ITHRAN for rapid iterative N(h) analysis
of ionogram, Moscow, IZMIRAN, in press.

Chasovitin, Yu.K., T.L. Gulyaeva, and V.G. Khryukin (197&), Change of electron
concentration during the total solar eclipse of June 30, 1973, Physica
Solariterrestris, Potsdam, DDR (in press).

Gulyaeva, T.L. (1976), Key point in the F layer for N(h) analysis of ionograms,

Radio Sci. (in press).

40

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1



BOSSY, L.

GULYAEVA, T.L.,
W. BECKER,
L.F. McNAMARA,
A.K. PAUL,
J.E. TITHERIDGE
and J.W. WRIGHT

HOWE, H.H. and
D.E. McKINNIS

McNAMARA, L.F.

McNAMARA, L.F. and
J.E. TITHERIDGE

PAUL, A.K.5

T.L. GULYAEVA,

L.F. McNAMARA,

J.E. TITHERIDGE and
J.W. WRIGHT

SCHMERLING, E.R.
TITHERIDGE, J.E.

TITHERIDGE, J.E.

TITHERIDGE, J.E.
TITHERIDGE, J.E.
TITHERIDGE, J.E.

TITHERIDGE, J.E.

TITHERIDGE, J.E.,
W. BECKER,

L. BOSSY,

T.L. GULVAEVA,
L.F. MCNAMARA and
ALK. PAUL

WRIGHT, J.W. and
G.H. SMITH, eds.

1978
1978

1967

1976

1977

1978

1967

1967a

1967b

1969

1975

1978a

1978b

1978

1967

7. REFERENCES
To be published.

"Analysis of numerical ionograms. 2. The starting problem,"
Ionospherie Prediction Service Series X Reports, IPS-X7,
Sydney.

“Ionospheric electron density profiles with continuous gradi-
ents and underlying ionization corrections. II. Formulation
for a digital computer," Radio Science, 2 (New Series), No.
10, 1135.

"The accuracy of the single polynomial method of ionogram
analysis," Ionospheric Prediction Service Series R Reports,
IPS-R32, Sydney.

"Numerical ionograms for comparing methods of N(h) analysis,"
Ionospheric Prediction Service Series X Reports, 1PS-X5,
Sydney.

"Analysis of numerical ionograms. 3. The valley problem,”
Ionospheric Prediction Service Series X Reports, IPS-X8,
Sydney.

"Ten point method of jonogram reduction," Radio Science, 2
(New Series), No. 10, 1233-1236.

"The overlapping polynomial analysis of ionograms," Radio
Setence, 2 (New Series), No. 10, 1169-1175.

"Direct manual calculations of ionospheric parameters using
a single polynomial analysis," Radio Setence, 2 (New Series),
No. 10, 1237-1254.

"Single polynomial analysis of jonograms," Radio Science, 4
(New Series), No. 1, 41-51.

"The relative accuracy of ionogram analysis techniques,"
Radio Setence, 10, 589-599.

“Increased accuracy with simple methods of ionogram analysis,"
submitted to the J. Atmosph. Terr. Physics.

"The generalised polynomial analysis of jonograms," Technical
Report 78/1, Radio Research Centre, Univ. of Auckland, New
Zealand.

“Analysis of numerical ionograms. 1. Complete monotonic
profiles," Ionospheric Prediction Service Series X Reports,
IPS-X6, Sydney.

Special issue on the analysis of ionograms for electron den-
sity profiles, Radio Setence, 2 (New Series), No. 10, 163 pp.



APPENDIX E (Continued)

Wright, J. W. and A. K. Paul (1974), Electron density profiles from
ionograms: Comparisons with rocket profiles, in Methods of Mea-
surements and Results of Lower Ionosphere Structure, ed. K. Rawer,
Akademie-Verlag (Berlin), 39-50.

Wright, J. W., A. K. Paul and BE. A. Mechtly (1975), Electron density
profiles from ionograms: Underlying ionization corrections and
their comparison with rocket results, Radio Sci., 10, 255-270.

Wright, J. W., A. R. Laird, D. Obitts, E. J. Violette and D. McKinnis

(1972), Automatic N(h,t) profiles of the ionosphere with a digital
ionosonde, Radio Sci., 7, 1103-103.

50



APPENDIX E

Historical Summary

J. W. Wright
Space Environment Laboratory
NOAA Environmental Research Laboratories
Boulder, Colorado 80303, U.S.A.

Electron density profile inversion work at the Boulder Laboratories
began on a relatively large scale shortly after orbiting of the first
earth satellite in 1957. We selected Budden's matrix method for this
purpose, obtaining copies of the relevant programs from Pennsylvania
State University. Our goal was to establish a means by which profiles
could be computed for a selection of ionosonde stations in the U.S. and
cooperating networks, with the onerous part of the data scaling performed
at the stations. By mid 1959, hourly profiles were being computed for
current data from eight U.S. stations and a total of 11 were active in
1960. (Wright, 1962) In 1960, the work of Paul came to my attention;
three advantages were expected from a reprogramming of our inversion
methods according to these ideas: a better approximation to the profile
shape with less input data, the prospect of improved corrections for
underlying ionization and valley errors, and more accurate treatment of
the refractive index at high magnetic dip angles. A new system was in
usable form by 1963 (Paul and Wright, 1963, 1964). It is described
rather completely in the October 1967 special issue of Radio Science
(Paul, 1967; Howe and McKinnis, 1967; Wright, 1967). Some efforts have
been made to compare the accuracy of the inversion, and particularly the
corrections for underlying D-region ionization and valley effects, by
comparison with rocket profiles (Wright and Paul, 1974; Wright et al.,
1975). Computation of individual hourly profiles was abandoned in 1962
following the development of a "composite" method of obtaining monthly
median profiles (Wright, 1960; Laird and Wright, 1967; Laird, Wright and
Gautier, 1974).

More recently, attention has turned to the development of methods

by which profiles could be obtained automatically within a digital
ionosonde. (Wright et al., 1972).

48



APPENDIX F (Continued)

MALTSEVA 0.A. (1979) A MULTILAYER APPROXIMATION FOR IONIZATION
DISTRIBUTION IN THE NONOBSERVED REGIONS. G.A., VOL.16, 6, 1105-1107.

MALTSEVA O.A. (1971) THE ACCOUNT OF THE UNDERLYING LONIZATION IN THE
ABSENCE OF X TRACE OF E REGION. G.A., VOL.11, 3, 542-544.

MALTSEVA O.A. (1973) THE CORRECTING METHOD FOR COMPUTATION OF N(H)
PROFILES. ARTICLES /34/, 169-146.

MALTSEVA O.A. (1974) CONTROL OF THE CORRECTING METHOD OF CALCULATION OF
N(H) PROFILES BY TEST MODELS. G.A., VOL.14, 1, 53-56.

MCNAMARA,L.F. (1976) THE ACCURACY OF THE SINGLE POLYNOMIAL METHOD OF
IONOGRAM ANALYSIS. IONOSPHERIC PREDICTION SERVICE SERIES R REPORTS, IPS-R32,
SYDNEY, AUSTRALIA.

MCNAMARA,L.F. AND J.E.TITHERIDGE (1977) NUMERICAL IONOGRAMS FOR COMPAR-
ING METHODS OF N(H) ANALYSIS. IONOSPHERIC PREDICTION SERVICE SERIES X REPORTS
IPS-X5, SYDNEY, AUSTRALIA.

PAUL A.K. (1972) THE MATHEMATICAL-PHYSICAL PROBLEM OF P'(F) TO N(H)
INVERSION FOR ANISOTROPIC MEDIA. IN "MATHEMATICS OF PROFILE INVERSION",
NASA TECHNICAL MEMORANDUM NASA TM X-62, 158. EDITED LAWRENCE COLIN

PAUL A.K. (1974) CORRECTIONS AND ERROR ESTIMATES OF ELECTRON DENSITY
PROFILES OBTAINED FROM IONOGRAMS. RADIO SCI., VOL. 9(7), 669.

PAUL,A.K.,T.L.GULYAEVA,L.F.MCNAMARA,J.E.TITHERIDGE AND J.W.WRIGHT(1978)
ANALYSIS OF NUMERICAL IONOGRAMS. 3. THE VALLEY PROBLEM. IONOSPHERIC PRED-
ICTION SERVICE SERIES X REPORTS, IPS-X8, SYDNEY, AUSTRALIA.

PAUL A.K., SMITH G.H. (1968) GENERALIZATION OF ABEL'S SOLUTION FOR BOTH
MAGNETOIONIC COMPONENTS IN THE REAL HEIGHT PROBLEM. RADIO SCI., VOL. 3(2), 163

PAUL,A.K. (1977) A SIMPLIFIED INVERSION PROCEDURE FOR CALCULATING
ELECTRON DENSITY PROFILES FROM IONOGRAMS FOR USE WITH MINICOMPUTERS. RADIO
SCI. VOL.12(1),

SAHA A.K. (1976) ON THE LOW IONIZATION PROBLEM IN IONOSPHERIC TRUE HEIGHT
ANALYSIS J. ATMOS. TERR. PHYS. VOL. 32(12), 1967.

SAHA A.K., JAIN V.C. (1972) TECHNIQUES OF REDUCTION OF IONOGRAMS TO
TRUE HEIGHT-ELECTRON DENSITY PROFILES. INDIAN J. RAD. SPACE PHYS. VOL. 1, 52.

SERAFIMOV K.B. (1978) SOME POSSIBILITIES OF MORE ACCURATE DETERMINATION
OF STRUCTURE OF THE IONOSPHERE. ARTICLES “IONOSPHERIC RESEARCHES" NO.19,
MOSCOW, "NAUKA", 114-123.

SERAFIMOV K.B. (1971) METHOD FOR DETERMINING NON-SOUNDING ELECTRONIC
CONCENTRATION BY MEANS OF PARAMETER FMIN. COMPT. RENDUS L'ACAD. BULG.
SCI., VOL.24, 12, 1633-1636.

SERAFIMOV K.B. (1973) METHOD FOR DETERMINATION OF UNSOUNDING ELECTRON
DENSITY BY MEANS OF PARAMETER FMIN. ARTICLES IN REF. 1 OF THIS APPENDIX, 147-156.

SHAPIRO B.S. (1970) DISTRIBUTION OF IONIZATION WITH HEIGHT AND VARIATIONS
OF ITS PARAMETERS FROM THE DATA OF GROUND VERTICAL SOUNDING.
ARTICLES "“IONOSPHERIC RESEARCHES" NO.19, MOSCOW,"NAUKA", 26-46.

SHAPIRO B.S. (1979) RELATION BETWEEN THE PARAMETERS OF F2 AND Fl LAYERS
AND FORECASTING OF DAYTIME N(H) PROFILES AND H'F CHARACTERISTICS.
G.A., VOL.18, 2, 344-346.

SHAPIRO B.S., VINNIKOVA T.L., FREIZON A.A. (1972) IONOSPHERIC N(H)

PROFILE CALCULATIONS USING TWO MAGNETO-IONIC COMPONENTS. ARTICLES "IONOSPHERIC
RESEARCHES" NO.26, MOSCOW, "NAUKA", 148-153.

54

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UAG-48A

UAG~49

UAG-50

UAG-51

UAG-52

UAG-53

UAG-54

UAG-55

UAG-56

UAG-57

UAG-58

UAG-59

UAG-60

UAG-61

UAG-62

UAG-63

UAG-64

“Synoptic Observations of the Solar Corona during Carrington Rotations 1580-1596 (11 October 1971 - 15 January 1973)",
[Reissue with quality images] by R. A. Howard, M. J. Koomen, D. J. Michels, R. Tousey, C. R. Detwiler, D, E.
Roberts, R. T. Seal and J. D. Whitney, E. 0. Hulbert Center for Space Research, NRL, Washington, D. C. 20375
and R. T. and S. F. Hansen, C. J. Garcia and E. Yasukawa, High Altitude Observatory, NCAR, Boulder, Colorado
80303, February 1976, 200 pages, price $4.27.

“Catalog of Standard Geomagnetic Variation Data", prepared by Environmental Data Service, NOAA, Boulder, Colorado,
August 1975, 125 pages, price $1.85.

"High-Latitude Supplement to the URSI Handbook on Ionogram Interpretation and Reduction", by W. R. Piggott, British
Antarctic Survey, c/o SRC, Appleton Laboratory, Ditton Park, Slough, England, October 1975, 292 pages, price $4.00.

"Synoptic Maps of Solar Coronal Hole Boundaries Derived from He II 304A Spectroheliograms from the Manned Skylab
Missions", by J. D. Bohlin and D. M. Rubenstein, E. 0. Hulbert Center for Space Research, Naval Research Lab-
oratory, Washington, D. C. 20375 U.S.A., November 1975, 30 pages, price 54 cents.

"Experimental Comprehensive Solar Flare Indices for Certain Flares, 1970-1974", compiled by Helen W. Dodson and
E. Ruth Hedeman, McMath-Hulbert Observatory, The University of Michigan, 895 Lake Angelus Road North, Pontiac,
Michigan 48055 U.S.A., November 1975, 27 pages, price 60 cents.

"Description and Catalog of Ionospheric F-Region Data, Jicamarca Radar Observatory (November 1966 - April 1969)", by
WoL. Clark and T. E. Van Zandt, Aeronomy Laboratory, NOAA, Boulder, Colorado 80302 and J. P. McClure, University
of Texas at Dallas, Dallas, Texas 75230, April 1976, 10 pages, price 33 cents.

"Catalog of Ionosphere Vertical Soundings Data", prepared by Environmental Data Service, NOAA, Boulder, Colorado
80302, April 1976, 130 pages, price $2.10.

"Equivalent Ionospheric Current Representations by a New Method, Illustrated for 8-9 November 1969 Magnetic Disturb-
ances", by Y. Kamide, Cooperative Institute for Research in Environmental Sciences, University of Colorado,
Boulder, Colorado 80302 and Geophysical Institute, University of Alaska, Fairbanks, Alaska 99701, H. W. Kroehl,
Data Studies Division, NOAA/EDS/NGSDC, Boulder, Colorado 80302, M. Kanamitsu, Advanced Study Program, National
Center for Atmospheric Research, Boulder, Colorado 80303, J. H. Allen, Data Studies Division, NOAA/EDS/NGSDC,
Boulder, Colorado 80302, and S.-I. Akasofu, Geophysical Institute, University of Alaska, Fairbanks, Alaska
99701, April 1976, 91 pages, price $1.60.

“Iso-intensity Contours of Ground Magnetic H Perturbations for the December 16-18, 1971 Geomagnetic Storm", by
Y. Kamide, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder,
Colorado 80302 and Geophysical Institute, University of Alaska, Fairbanks, Alaska 99701 (currently Guest worker
at Data Studies Division, NOAA/EDS/NGSDC, Boulder, Colorado 80302), April 1976, 37 pages, price $1.39.

“Manual on Ionospheric Absorption Measurements", edited by K. Rawer, Institut fiir Physikalische Weltraumforschung,
Freiburg, G.F.R., June 1976, 202 pages, price $4.27.

"ATS6 Radio Beacon Electron Content Measurements at Boulder, July 1974 - May 1975", by R. B. Fritz, Space Environment
Laboratory (currently with Wave Propagation Laboratory), NOAA, Boulder, Colorado 80302 USA, September 1976,
61 pages, price $1.04.

“Auroral Electrojet Magnetic Activity Indices AE(11) for 1974", by Joe Haskell Allen, Carl C. Abston and Leslie D.
Morris, National Geophysical and Solar-Terrestrial Data Center, Environmental Data Service, December 1976,
144 pages, price $2.16.

"Geomagnetic Data for January 1976 (AE(7) Indices and Stacked Magnetograms)" by J. H. Allen, C. A. Abston and
L. R. Morris, NGSDC/EDS/NOAA, July 1977, 57 pages, price $1.07.
“Collected Data Reports for STIP Interval II 20 March ~ 5 May 1976", edited by Helen E. Coffey and John A.

NeKinnon, National Geophysical and Solar-Terrestrial Data Center, Environmental Data Service, August 1977,
313 pages, price $2.95.

"Geomagnetic Data for February 1976 (AE(7) Indices and Stacked Magnetograms)" by J. H. Allen, C. A. Abston and
L. R. Morris, NGSDC/EDS/NOAA, September 1977, 55 pages, price $1.11. Y

“Geomagnetic Data for March 1976 (AE(7) Indices and Stacked Magnetograms)" by J. H. Allen, C. A. Abston and
L. R. Morris, NGSDC/EDS/NOAA, September 1977, 57 pages, price inte y onan

“Geomagnetic Data for April 1976 (AE(8) Indices and Stacked Magneto: ri " by d
L.-R. Morris, NGSDC/EDS/NOAA, February 1978. Snetograns)" by J. Ws AVTens C. A. Abston and



APPENDIX D

Historical Summary

Adolf K. Paul
Space Environment Laboratory
NOAA Environmental Research Laboratories
Boulder, Colorado 80303, U.S.A.

Two problems aroused my interest in electron density profile calcu-
lations, the ambiguity as described by Manning (1949) and the sensitivity
of the virtual height to details at the reflection level caused by the
singularity of the group refractive index at this point. The latter led
immediately to the conclusion that a numerical solution based on an
approximation of the profile by a number of linear segments will be
slowly convergent (Budden, 1955). This situation was significantly
improved by introducing a quadratic variation within each interval with
continuous slopes at the interval limits (Paul, 1960a). A more detailed
justification for a second (or higher) order approximation together with
a study of the effects of the singularity of the group refractive index
on the virtual heights was published later (Paul, 1967).

Some theoretical considerations indicated a solution to the ambi-
guity problem by the joint use of ordinary and extraordinary components
(Paul, 1960b). A very general numerical investigation about the rela-
tion between ordinary and extraordinary group refractive index showed
that in practice the chances for removing the ambiguity depend strongly
on latitude (dip angle) and are very low in the vicinity of a dip angle
of 39° (Paul and Smith, 1968).

In more recent development a simplified version of profile calcula-
tion was derived based on the same principles mentioned above (Paul,
1977). The number of computation steps has been drastically reduced
without significant loss of accuracy. This program will be available on
the computer of the new HF sounding system for on-line processing.

References
Budden, K. G. (1955), A method for determining the variation of electron
density with height (N(z) curves) from curves of equivalent height
against frequency ((h',f) curves), Physics of the Ionosphere, Rept.
Phys. Soc. Conference, 332-339 (Phys. Soc. London).

Manning, L. A. (1949), The reliability of jonospheric height determina-
tions, Proc. IRE, 37, 599-603.

46



T T T T T T
PROFILE 6

260

szor

/ 75° s———"*
230 b7” +1250
if
15 16 Lr 18
h
(km)
270
260+
¢
—
—— :
oo a
250h- .— ee
= —_ =>

fn (MHz)

Fig. 9. Calculated and correct real heights for the lower parts
of profile 6 used for the underlying ionization tests

18



APPENDIX A (Continued)

Titheridge, J. E. (1974a), FORTRAN program LAPOL for the 5-term overlapping
polynomial analysis of ionospheric virtual height records, Technical
Report 74/2, Radio Research Centre, University of Auckland, New Zealand.

Titheridge, J. E. (1974b), Direct analysis of ionograms at magnetic dip angles
of 26 to 30 degrees, J. Atmosph. Terr. Phys. 36, 575-582.

Titheridge, J. E. (1975a), The relative accuracy of jionogram analysis techniques,
Radio Science 10, 589-599.

Titheridge, J. E. (1975b), The analysis of night-time ionograms, J. Atmosph.
Terr. Phys. 37, 1571-1574.

Titheridge, J. E. (1976), Direct estimates of scale height from topside iono-
grams, J. Atmosph. Terr. Phys. 38, 623-626.

Titheridge, J. E. and Lobb, R. J. (1977), A least-squares polynomial analysis
and its application to topside ionograms, Radio Science 12, 451-459.

Lobb, R. J. and Titheridge, J. E. (1977a), The valley problem in bottomside
jonogram analysis, J. Atmosph. Terr. Phys. 39, 35-42.

Lobb, R. J. and Titheridge, J. E. (1977b), The effects of travelling ionospheric
disturbances on ionograms, J. Atmosph. Terr. Phys. 39, 129-138.

Titheridge, J. E. (1978a), Increased accuracy with simple methods of ionogram
analysis, submitted to J. Atmosph. Terr. Phys.

Titheridge, J. E. (1978b), The relative stability of ionogram analysis techniques,
submitted to J. Atmosph. Terr. Phys.

Titheridge, J. E. (1978c), The generalised polynomial analysis of ionograms,

Technical Report 78/1, Radio Research Centre, University of Auckland, New
Zealand.

36



TABLE OF CONTENTS

Page
ABSTRACT © we ee ee 1
1. INTRODUCTION. 2. ee ee 1
2. THE PROBLEMS IN N(h) ANALYSIS 2... ee 2
2.1 The Starting Problem 2. 2... 2
2.2 The Valley Problem 2... 2
3. ANALYSIS OF NUMERICAL IONOGRAMS 2... ee 3
3.1 Complete Monotonic Profiles... 2... ee 3
3.2 The Starting Problem... ee 4
3.3 The Valley Problem 2... ee ee 4
4. SIMPLE METHODS OF ANALYSIS . 2 1 1. eee 5
5. CONCLUSION. ©. ee 6
6. TOPICS FOR FURTHER STUDY... ee ee 7
7. REFERENCES © 6 ee ee 8
TABLE
1 The Real Height Errors (Calculated Value - Correct Value), in Meters, Obtained
Using 10 Different Methods of Analysis 2... ee ee ee ee ee 9
FIGURES
1 General Overview of the Aspects of N(h) Analysis of Ionograms ............00- 10
2 Calculated and Correct Real Heights for the Lower Parts of Profile 1 Used for
the Underlying Ionization Tests 1
3 Calculated and Correct Real Heights
the Underlying Ionization Tests 12
4 Calculated and Correct Real Heights
the Underlying Ionization Tests 13
5 Calculated and Correct Real Heights
the Underlying Ionization Tests 4
6 Calculated and Correct Real Heights
the Underlying Ionization Tests 15
7 Calculated and Correct Real Heights
the Underlying Ionization Tests 16
8 Calculated and Correct Real Heights
the Underlying Ionization Tests V7
9 Calculated and Correct Real Heights
the Underlying Ionization Tests 18
10 Calculated and Correct Real Heights
Used for the Valley Tests 2... ee 19
11 Calculated and Correct Real Heights
Used for the Valley Tests... 1 ee 20
12 Calculated and Correct Real Heights
Used for the Valley Tests»... ee 21
13. Calculated and Correct Real Heights
Used for the Valley Tests 2. 2. ee ee 22
14 Calculated and Correct Real Heights
Used for the Valley Tests. 2... 23
15 Calculated and Correct Real Heights
Used for the Valley Tests ©. 1. ee ee 24
APPENDICES
A N(h) Analysis - Historical Survey and Summary of Results by J. E. Titheridge. . 2... ee 25
B Summary of N(h) Analysis Approach by T. L. Gulyaeva . 2... 2... epee 37
C Conversion Methods for h'(f) to N(h) Used at Lindau by W. Becker. 2... 2. ee 41
D Historical Summary by Adolf K. Paul 2 1. ee 46
— Historical Summary by J. W. Wright. 2... 48
F Bibliography of N(h) Profile Analysis by L. F. McNamara... ee ee 51
G Address List of Subgroup G/6/2 Members... 56



APPENDIX A (Continued)

the width of the valley or the scale height of a peak fitting section) can
be obtained.

When a large number of data points are available, there is no need to
use a separate polynomial for each interval between scaled frequencies.
Expressions fitted over any desired range (including r real and v
virtual heights) can be used to calculate a further NH real heights. NH
is an additional free parameter in POLAN. For all current lamination pro-
cedures NH=1 . With digital ionosondes or automatic scaling procedures,
however, larger values may be employed with advantage. Thus if the amount
of available data is increased by a factor of 3, polynomials may still be
fitted over the same frequency range as before (which will now involve 3
times as many data points) with each successive step in the analysis calcu-
lating 3 new real heights.

The parameters r,v,n and NH are set by 9-term data arrays IR, IV, IT
and IH at the beginning of POLAN. A gradient fit at the last known real
height (s=1) is used with MODE 2 (parabolic Jaminations) only, since it
appears in general to have little advantage over matching virtual heights.

9 standard modes of analysis are then available by specifying a single
parameter MODE when calling POLAN with the height, frequency arrays to be
analysed. A wide range of other types of analysis can of course be obtained
by changing the constants in the data arrays. The current arrays give the
modes of analysis listed below. Starting and valley calculation procedures
are incorporated with all modes of analysis when model start/valley
constants are specified, or when extraordinary ray data is provided (as
discussed in following sections). A detailed description of POLAN is given
in Titheridge (1978c). This includes a listing of the computer program
(which is also available on paper tape or punched cards); calculations of
the accuracy and stability of the different modes; derivation, test results
and evaluation of different start, peak and valley calculations under a
wide range of conditions; and simple procedures for the accurate calculation
of group refractive index and group index integrals at latitudes up to 89°,

Standard modes of analysis in POLAN:

Linear laminations (with a parabolic peak at critical frequencies).
Parabolic laminations, matching gradients (as described by Paul).
The 5-term overlapping polynomial anaiysis, fitting 2 real and 3
virtual heights at each step (Titheridge, 1961b, 1967c).

A least-squares fit of 6 terms to 7 data points (3 real + 4 virtual).
A least-squares fit of 6 terms to 8 data points (3 real + 5 virtual).
7 terms fitting 3 real + 6 virtual heights, advancing 2 frequency
intervals at each step.

8 terms fitting 4 real + 8 virtual heights, advancing 2 points.

9 terms fitting 4 real + 9 virtual heights, advancing 3 points.
MODE=9 A single polynomial, of n terms, fitting all available virtual
heights (0 or X ray, giving a total of v data points) with

n = 0.73(v+2) . Current dimensions limit n to 16 and v_ to

30, This is normaliy adequate since the analysis splits auto-
matically into separate sections (with parabolic peaks) for each
ionospheric layer. If more than 30 points occur in one layer,

two (or more) polynomials are used, overlapped 4 points.

29



T T T T T T
PROFILE 3 et —
ee oo
260 + aot ¢—o
aan ee ae
ee 6 =e
uw —
a
270 ¢ a
—- aa
6 a
ro aa . {280
w 6 L co
250 wa 6 ‘ 280
——
h |
(km) 15 8 7 8
240 | |

240 i L — L L 1 J

fn (MHz)

Fig. 6. Calculated and correct real heights for the lower parts
of profile 3 used for the underlying ionization tests

15



190

160

150

190

180

(km)

Io

160

150

T T T T T

PROFILE 3

__

—s
.—

6

4
ga aa

——— eo

r
—— DIP = 20°

1 L 1 1

Fig. 12.

1
4.2 43 44 45 4.6
fn(MHz)

Calculated and correct real heights for the base of
F layer for the profile 3 used for the valley tests.

21

the


180

(km)

210

200

T T T T T T

PROFILE 5

‘
75° —

15 16 IT 18
i i i i

L 1 L

Fig. 8.

13 14 1S

9 1.0 Ul 12
fn (MHz)

Calculated and correct real heights for the lower parts
of profile 5 used for the underlying ionization tests

|
|
|
|
|
|



6.

6.1

APPENDIX A (Continued)

THE STARTING PROBLEM
Using ordinary (0) rays only

The major problom in ionogram analysis is the presence of ‘unseen!
regions, at low heights (where the plasma frequency is less than the lowest
observing frequency f,) or ina valley between two layers. Assuming a
reasonably continuous profile, a non-zero value of dh'/df near fy
indicates the presence of group retardation. Extrapolation of |dh'/dt|
to lower frequencies is therefore employed in POLAN, in the absence of any
other starting information. The extrapolation range is kept small to
ensure that there is no over correction. This procedure is considered a
last resort, since reasonable profiles with large amounts of underlying
ionisation can in fact produce zero values of dh'/df .

A preferred system is the provision of a starting height, at 0.5 MHz,
based on past experience or synoptic models. When this is done POLAN uses
the lower of the given height and the height extrapolated from the 0 vay
data, since the latter has been designed to give a minimum correction.

6.2 Using extraordinary (X) ray data

With an unlimited number of exact measurements on both components, the
distribution of unseen ionisation (or its monotonic equivalent) can in
theory be fully determined. In practise, however, only a limited amount of
information is obtainable. Thus when fy < Ey Hg and ty increase

almost linearly with the electron density N . The retardations of the 0
and X components then depend only on the total electron content T of the
underlying ionisation. Coefficients have been calculated which-give T ,
and the real height correction hj-h, , directly in terms of observed 0
and X ray virtual heights (Titheridge, 1959c). At high latitudes results
are accurate to about 20% even if the condition fy < 4f , for most of the
unseen ionisation, is not fulfilled. ,

At dip angles near 29° the relative retardation of the 0 and X rays is
approximately independent of f, , at frequencies below a few MHz. It is
then impossible to determine anything ebout the distribution of the unseen
ionisation. Only a single parameter can be obtained. This is however
sufficient to define the retardation of the observed rays, so that the real
height h at any frequency can be obtained accurately and directly from
corresponding values of hg and h{ (Titheridge, 1974b).

At higher latitudes a relation of the type pl = 1+aN+bNTM gives group
retardations accurate to 1% for al] fy < £-0.2 MHz , Measuvements on both

components at frequencies near f, therefore define the values of nian (=T)
and finan for the ionisation with fy < f (where £, = £,-0.2 MHz) , and
the mean gradient G in the region fy < fy < f, (Titheridge, 1959c).

With typical records only T and G can be usefully determined. These
parameters are sufficient to give accurate heights for the reflecting region,
and provide useful physical information on changes in the night E region
(Titheridge, 1959d).

In many situations only a single parameter is meaningful. This can
be found from a single value of hy . Using an X ray frequency £, corres~

30



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APPENDIX G

Address List of Subgroup G/6/2 Members

Dr. L. F. McNamara (Chairman)
Ionospheric Prediction Service
P. 0. Box 702

Darlinghurst, N.S.W. 2010
Australia

Dr. W. Becker
Max-Planck-Institut fur Aeronomie
Institut fur Tonospharenphysik
3411 Lindau/Harz

Postfach 20

Federal Republic of Germany

Dr. J. S. Belrose
Communications Research Centre
Box 490 Station A

Ottawa KIN 8T5

Canada

Prof. L. Bossy

174 Avenue W. Churchill
1180 Bruxelles

Belgium

Dr. P. Edenhofer

Deutche Forschungs-und
Vorsuchsanstalt fur Luft-und
Raumfahrt E.B.

8031 Oberfafferhofen

Post Wesseling/OBB

Federal Republic of Germany

Dr. T. L. Gulyaeva

Institute of Terrestrial Magnetism

Ionospheric and Radio Wave Propaga-
tion (IZMIRAN)

U.S.S.R. Academy of Science

142092 Troitsk

Moscow Region

U.S.S.R.

56

Prof. J. S. Nisbet

Ionospheric Research Laboratory
318 Electrical Engineering E.
University Park

Pennsylvania 16802

U.S.A.

Dr. A. K. Paul

Space Environment Laboratory
NOAA

Boulder, Colorado 80303
U.S.A.

Dr. A. K. Saha

Radio Science Division
National Physical Laboratory
Hillside Road

New Delhi - 12

India

Dr. C. Taieb

Groupe de Recherches
Tonospheriques

38-40 Rue du General Leclerc

92 Issy Les Moulineaux

France

Dr. J. E. Titheridge
Radio Research Centre
University of Auckland
Auckland

New Zealand

Dr. J. W. Wright

Space Environment Laboratory
NOAA

Boulder, Colorado 80303
U.S.A.

i



Titheridge et al. [1978] also discuss the following topics which are important facets of the N(h)
analysis problem:

Relative strengths of parabolic and polynomial methods;

Choice of scaling frequencies;

Peak extrapolation procedures;

Height-dependent gyrofrequency;

The single polynomial method;

Errors in the input data, including random scaling errors, zero-height error and receiver
delay.

OnReNe

3,2 The Starting Problem

The jonograms analyzed have been described in detail by McNamara and Titheridge [1977, Section 3].
Because no guarantee exists that the success of any analysis method that relies on O-ray data only is
not just fortuitous, X-ray data were provided in all cases.

Numerical ionograms, corresponding to six incoherent scatter profiles obtained at Arecibo, have
been analyzed. Profiles 1 and 2 are analytic approximations to typical observed profiles for the mag~
netically quiet night of 12 April 1972. They correspond approximately to mean profiles between 0200 -
0500 LT and between 2100 - 2400 LT, respectively, but both shapes appear at different times before
2100 LT. Profile 2 has a considerably greater amount of underlying ionization and a more sharply
bounded reflection layer than does profile 1. Profiles 3-6 are individual, April and July observed
profiles (averaged over 5 min) in which the assumed density variation between data points is linear up
to some height H in the base of the F layer. Above this height, the F-layer density varies linearly
with 2£n(N), closely matching the observed F-layer profile near its base.

As well as including six different underlying ionization distributions, this set of profiles con-
tains two kinds of F-layer density distributions, namely hyperbolic ones (1 and 2) and ones linear in
2n(N) (3 - 6). Thus, although a few profiles favored some methods of analysis, no technique was
favored consistently. It should be noted, however, that Titheridge's method is the only one which al-
lows for a varying curvature in the reflecting region, which is likely to be the case in practice. The
other methods are based on the assumption that there are no variations higher than parabolic.

A total of 24 analyses was requested, corresponding to six models, two dip angles and two values
for fin. The frequency spacing of the data was 0.02 MHz, and values of hy greater than 500 km were
not provided (for profile 2) since such values would not be expected in practice. Data were given for
dip angles of 30° (with fy = 1.0 MHz) and 75° (with fy = 1.5 MHz) in order to test for any dependence
of the errors on these parameters, with fy being an inverse-cube function of height.

Authors were requested to perform the analyses for two values of fine either 0.8 or 0.9 MHz and
1.5 MHz, the latter being a more likely observed value.

Details of the results are given in Gulyaeva et al. [1978]. Figures 2-9 show the correct and cal-
culated real heights for the six models. In general, the largest errors were obtained for profile 2 at
a dip angle of 30° (see Figure 4).

The right-hand axes or insets to figures 2-9 give the results for an fin of 1.5 MHz, a typical
value for that parameter. There does not seem to be any marked change in the accuracy of the results
of the different methods, although Titheridge's program POLAN did yield smaller errors for profile 2 at
a dip angle of 75° (see Figure 5).

There does not seem to be any consistent dependence of the uncertainty in the calculated real
heights on the dip angle. The effect of the height-variation of the gyrofrequency was found to be
negligible.

A figure of #5 to 10 km seems an appropriate estimate of the maximum error that should be antici-
pated when accurate 0- and X-ray data are available. Experimental errors will tend to increase the
uncertainty in the calculated profile, but their effects will be somewhat mitigated by use of least-
squares techniques in the analyses. On the average, the error can be expected to be about a factor of
four lower than the error introduced by failing to correct for the underlying ionization, but in some
cases the decrease will be much larger. It would be difficult to estimate the uncertainty of results
based on only O-ray data, but in general, the errors would be expected to exceed those obtained using
X-ray data as well.

3.3 The Valley Problem

The jonograms analyzed have been described in detail by McNamara and Titheridge [1977, Section 4].
X-ray data have been provided in all cases.



APPENDIX A (Continued)

polynomial used to represent the real height curve between successive data
points.

Successive polynomials should ideally be joined by equating veal heights
and derivatives at onc or more points on either side of the interval consid-
ered. Matching end points only gives a profile in which all derivatives
up to n-2 are continuous. This corresponds to a spline function represent-
ation of the real height curve. Tests with different procedures showed,
however, that fitting polynomials over a wider range gives slightly better
accuracy and stability.

If fitting is done symetrically, in terms of real heights only, then
a 30-point analysis with 5-term polynomials would require inversion of a
matrix dimensioned 150 x 30 . This is scarcely desirable. Gradient
matching at frequencies above the central interval of each polynomial is
therefore replaced by matching of virtual heights. There seems little if
any disadvantage in this approach, which enables a simple step-by-step
analysis (Titheridge, 1961b, 1967c, 1974a). In general a 5 x 5 matrix
must be inverted at each step. This matrix can become ill-conditioned,
giving problems on computers with limited word length. By shifting the
origin at each step to the last calculated real height, however, we obtain
well conditioned 4x 4 matrices giving errors (in the matrix inversion)
of less than 1 in 108 on 16-bit machines.

In tests using a number of different real height profiles with various
frequency intervals and dip angles, the S-term overlapping-polynomial
analysis gives results which are 100 to 1000 times’more accurate than the
use of lincar laminations, and 3 to 50 times more accurate than using para-
bolic laminations (Titheridge, 1978c). The stability of the analysis,
measured by the amplitude of spurious real height oscillations following a
cusp or discontinuity in the virtual height curve, is 20% to 50% better
than for the parabolic analysis; if a gradient discontinuity is permitted
at the cusp frequency, the oscillations (at or between scaled frequencies)
are less than half as large as for the parabolic analysis (Titheridge,
1978b) .

The ability to interpolate a point of inflection between successive
frequencies means that selection of reading frequencies is not critical,
and a fixed grid can be used for scaling ionograms. This considerably
speeds up the scaling and, since coefficients need be calculated only once
(for a given station) calculation of the real height curve takes typically
jess than one second. Both fixed and variable frequency modes, with an
optional one-point extraordinary-ray starting correction and insertion of
a model E-F valley, are provided in the program LAPOL (Titheridge, 1974a).

LEAST SQUARES SOLUTIONS

Further improvement in real height calculations requires the incorpor-
ation of more data points, which are suitably smoothed to remove the jitter
caused by randow errors. Optimum smoothing is obtained with polynomial
methods of analysis using a least-squares solution, in which the number of
terns in the polynomial (n) is less than the number of fitted data points
(m) . There is then no limit on the value of m (apart from computer
storage), while the detail required in the profile is set independently by
the value of n As a result of the least-squares procedure, fitted poly-
nomials are completely stable for values of n up to at least 16.

27



200 t T - T T
203.6 4 WZ w
7
90+ 6 : J
h Lo
(km) 5 ; 1
180 ° 4
Ys .
he
YY DIP = 75°
170 - bg 4
.
CPN
v7 /
160 - / 4
w
x
200+ 4 . , 4
7 7
7
Le Yo w
G T
L 4 LP yO 4
190 s 7
7 7
J. W, 4
vA Ww
6 x JS
180 + y 4
h /
(km) y Y w
7
170 - E oa +
A /
Lf w DIP =30°
160; / 4
T Yo
yhmE= 109.0
150 1 Wo L L Ll 1
3.5 3.6 3.7 3.8 39 4.0 4.1 42
fn (MHz)
Fig. 15. Calculated and correct real heights for the base of the

F layer for the profile 6 used for the valley tests.



A COMPARATIVE STUDY OF METHODS OF ELECTRON DENSITY PROFILE ANALYSIS
Chairman's Report, URSI Subgroup G/6/2 on N(h) Analysis

L.F. McNamara*
Tonospheric Prediction Service**
P.O. Box 702, Darlinghurst, N.S.W. 2010, Australia

ABSTRACT. URSI Subgroup G/6/2 was charged with comparing available methods
of real-height analysis of jonograms in order to determine the best method.
Experts in the field who composed the subgroup attempted to compare all known
available methods. The comparisons were made on the basis of the accuracy of
the results obtained by each method in the analyses of a series of accurate
numerical ionograms, with the correct results being known only to the chair-
man. Of those methods for which sufficient results were presented to allow
definitive conclusions to be drawn, those of A. K. Paul (Boulder, Colorado,
U.S.A.) and J. E. Titheridge (Auckland, N.Z.) proved to be the most promising.

1. INTRODUCTION

Subgroup G/6/2 of Working Group 6, Commission G of the International Union of Radio Science (URSI),
was formed after the 17th URSI General Assembly in Warsaw, 1972. Subgroup G/6/2 was asked to compare
extant methods of real-height analysis of ionograms in order to recommend the "best" technique(s).

Mr. C. G. McCue (Director of the Australian Ionospheric Prediction Service) was appointed chair-
man of the subgroup in early 1973 and immediately invited prominent workers in the field to partici-
pate in the endeavor. The nominal list of members prior to its reorganization at Helsinki is given in
Appendix G, although, as will be seen later, the effective list was much smaller. The author and pres-
ent chairman, L. F. McNamara, acted as secretary under McCue's chairmanship.

Between its inception and the 18th URSI General Assembly in Lima, 1975, the subgroup's main activ-
ity was the analysis of representative numerical ionograms in order to identify those methods that
coped best with the special problems encountered in this field. During a meeting in Lima, however,
several subgroup members expressed their dissatisfaction with the comparative tests that had been used.
They recommended that a new set be conducted with different guidelines and be organized by the new
chairman, L. F. McNamara.

This report describes the new set of tests -- those dating from 1975 -~- and the results obtained.
It also shows how the field has developed over the past 10 or so years since the Radio Science Special
Issue on at sis of lonograms for Electron Density Profiles was published in October 1967 [Wright and
Smith, 1967].

The different methods of analysis considered are discussed here mainly in terms of their accuracy
as indicated by their results for the test ionograms. Other important aspects of the problem, as far
as the use of the methods by persons other than the originators is concerned, are the availability of
the computer program, the ease with which the program may be applied routinely, and the computation
time for each complete analysis. These topics were discussed at a meeting of the subgroup in Helsinki,
preceding the 19th URSI General Assembly. A general overview of the problem is given in Figure 1.

Methods of analysis discussed in this report are those used by L. Bossy [1978]; W. Becker, J. E.
Titheridge, A. K. Paul, T. L. Gulyaeva (Appendices A-D of this report); and H. H. Howe and D. E.
McKinnis [1967] as used by J. W. Wright (Appendix E of this report). Although the set of tests con-
ducted before 1975 also included results for the methods of 0. A. Mal'tseva, A. K. Saha, and J. E.
Jackson as used by J. E. Bennett and J. S. Nisbet, these three techniques are not considered here be-
cause (1) the Mal'tseva and Jackson methods were found to be significantly less accurate for the pre-
1975 tests than the former set of methods, (2) Saha did not provide results for the second set of tests
(partly because he felt they were unrealistic), and (3) Saha provided results for only the monotonic
profile in the first set. Bossy provided results for only the monotonic profile of the second set of
tests. Becker submitted results for this profile and partial results for the "starting" problem test.

Section 2 of this report gives a brief introduction to the problems encountered in N(h) analysis,
addressing them by the analysis of test ionograms described in Section 3.

*Chairman of URSI Subgroup G/6/2
**Guest worker at World Data Center A for Solar-Terrestrial Physics, Boulder, Colorado, U.S.A.

|
|
|
{
|
|
|
|
|





250 T T T T T T
PROFILE 2
30° DIP
240 F 4
——t
—
oo et
°. [eee od 1
[ a =
230 — i |
lO “a, ce 4
Va i,
h w
(km) / “ oA
oO fof 4
/ “ /
/
) 7
te / A
or / ¥ 4
/ /
,
Va /
v /
/
iF  / 4
/
va
/
/
/
/
ua
i
1807 4
170 1 1 L 1 L 1
09 10 1 12 13 i4 15 16
fn (MHz)

Fig. 4.

Calculated and correct real heights for the lower parts of
profile 2 used for the underlying ionization tests

13



3. ANALYSIS OF NUMERICAL IONOGRAMS

This section describes the results obtained by group members for the analysis of accurate numeri-
cal ionograms. Only a summary is given here, because the test ionograms and the results have been
described in detail elsewhere [McNamara and Titheridge, 1977; Titheridge et al., 1978; Gulyaeva et al.,
1978; Paul et al., 1978].

3.1 Complete Monotonic Profiles

The ionograms analyzed corresponded to a monotonic profile composed of two overlapping Chapman
layers. Virtual heights accurate to 0.001 km were given for a dip angle of 65°. Ionograms were given
for a constant gyrofrequency of 1.2 MHz, and for a gyrofrequency of 1.2 MHz at the ground and decreas-
ing as the inverse cube of the height. The latter corresponds more closely with the real situation.
The model profile was cut off at fin? with hy = ns to avoid any starting problem.

The results of the analyses are given in detail in Titheridge et al. [1978]. Table 1 displays the
real height errors (calculated value - correct value) obtained at selected points for the constant
gyrofrequency case using the different methods. The values of frequency chosen correspond to neither a
given data point nor to the midpoint of any lamina, in order to identify methods yielding large errors
between data points. In fact none of the methods considered did so.

Titheridge provided results for his linear lamination method, LINAN [1978a], his overlapping poly-
nomial method, LAPOL [1967a + improvements], and his generalized polynomial method, POLAN [1978b]. For
POLAN, results were provided for modes 3, 4, and 5, corresponding to the use of somewhat different poly-
nomial techniques. Other members used the methods described in the appendices. Gulyaeva also provided
results for the case in which she permits the first derivative to be discontinuous at the cusp (Column
9 in Table 1).

Table 1 also lists the numbers of data points used by each member along with the smallest fre-
quency interval near the cusp of foE. The number of data points necessary to restrict the analysis
errors to below an acceptable level is an important practical consideration. Clearly, the smaller the
number the better, since a smaller number leads to less time spent scaling the ionogram and less com-
puter time spent in analyzing the scaled data. This problem will not be examined at length here.

The "Method" row of Table 1 indicates whether the calculated heights were obtained using the
assumed profile variation between two adjacent data points (F) or by some form of interpolation in the
array of calculated heights (I). For arrays fj, hj, i = 1 to n with the peak at i = n, Titheridge
used second difference interpolation to get h(f) when f < f, 2 and inverse interpolation (taking f
as a function of h) when f > f, 5. nm

In general, it would be expected that the errors would be largest near foE, depending on the
sharpness of the cusp, and near foF2, depending on the amount of curvature of the F-layer peak.

In this particular case the cusp is not very sharp, as with real ionograms, so the errors remain
relatively small, i.e., < 0.1 km. For a much sharper cusp and using a frequency interval of 0.1 MHz,
errors of 0.5-1.0 km are typical. The problem is then essentially one of under-sampling, however, and
can be overcome by the use of a smaller frequency interval (¢ 0.02 MHz), although such small intervals
are not normally used in practice.

The expected increase in error toward foF2 is not well illustrated by Table 1, apart from Becker's
results. In general, the error will be greatest for those methods using the simplest form for the
variation between scaled frequencies (excluding Titheridge's LINAN, which uses off-set frequencies) .
The error again depends on the interval between scaled frequencies and can be reduced to a negligible
size by decreasing the interval to x 0.1 MHz.

The error at the highest frequency in Table 1, 6.985 MHz, also depends on the accuracy of the peak
extrapolation procedure, if this extrapolation was commenced at a frequency below 6.985 MHz. This is
the case for Paul, Gulyaeva and Titheridge, who used 6.950, 6.980 and 6.980 MHz, respectively, as their
highest frequency. Becker used a highest frequency of 6.810 MHz, so his errors at the highest two
frequencies are affected.

It can be seen from Table 1 that the analysis errors are very small for all methods considered,
although they tend to become larger near the F2 peak. The smallest RMS errors were obtained by
Titheridge (POLAN, mode 4), but in terms of an acceptable level of error, all methods are quite ade-
quate. The errors for all methods can be made negligibly small simply by increasing the number of
data points.

In a practical situation these analysis errors would be compounded with and largely out-weighed
by uncertainties due to possible and largely irreducible errors in the input data.



T T T T T T
PROFILE 1
1
150} aa 4
a
—— G
AA _—
Ss We
wl: a a |
: Lo a —_
(km) 8 “
130} va — ao J
d Ven “ DIP = 70°
ow
120- “ |
a The key to Figures 10-15 is as follows:
1
: Correct results
G Gulyaeva's results
- P Paul's results 4
T Titheridge's results for POLAN, mode 4
W,X Wright's results for the Howe-McKinnis program
150 — 4
6
140 F rag 4
aoe w
h ae a
(km)
130 4
DIP=20°
120 08 4
4
a
—. 1 1 1 1 1
40 41 42 43 4.4 4.5 4.6 47
fn (MHz)
Fig. 10. Calculated and correct real heights for the base of the

F layer for the profile 1 used for the valley tests



REPORT UAG-68

WORLD DATA CENTER A

for
Solar-Terrestrial Physics

A COMPARATIVE STUDY OF METHODS OF
ELECTRON DENSITY PROFILE ANALYSIS

Chairman's Report, URSI Subgroup G/6/2 on N(h) Analysis

SEPTEMBER 1978



APPENDIX C (Continued)

Fig. 1
Semi-automatic scaling facility

This machine allows us to digitize film records between 8 and 35 mm,
with film images of any length and any coordinate systems. A versatile opti-
cal system (Zeiss, Oberkochen, Germany) projects the records onto the table of
a D-MAC Pencil-Follower (Edinburgh, Scotland). The enlargement can be varied
not only by replaceable optical systems but also by shifting the projector
back and forth mechanically. In order to adjust the projected ionograms to
calibrated markers on the table of the pencil-follower, the projector can
also be shifted along the other two coordinate axes whereas the film is kept
by two glass plates pressed together; they release the film as soon as the
respective push button for film transport (low or high speed, unrolling or
rewinding) is pressed. For data input such as name of the station, name of
the satellite, date and time, etc., a second disk is installed. The mechani-
cal system on the pencil-follower facilitates the digitization of records
with non-rectangular virtual-height/frequency coordinate systems. The second
projector can be used for review purposes, since very long jonograms must be
scaled section by section. Several computer programs have been written for
this data handling process. They allow the input data to be transformed into
pairs of frequencies and virtual-heights, to check their formats and finally
to get them corrected by the computer according to punch card instructions.

42



As explained by Titheridge [1967b], his single polynomial method has certain practical advantages
over Schmerling's [1967] 10-point method, as well as providing the extra information of the height of
the maximum density, the scale height of the layer peak, and the sub-peak total electron content. The
accuracy of the single polynomial method has accordingly been investigated by McNamara [1976], using
aeeaey numerical ionograms (in fact the first set of numerical ionograms distributed to members in
1972-75).

For daytime monotonic profiles, it was found that (1) around foE the calculated heights were too
low by about 10-15 km, (2) around foF2, the calculated heights were about 5-10 km too low, and (3) the
error in the scale height was around +4 km.

For nighttime ionograms, it was found that when allowance was made for underlying ionization the
errors at the lowest heights were usually of the order of 10-20 km, depending on the dip angle and the
value of foE.

For daytime non-monotonic profiles with an E-F valley, the errors yielded by the single polynomial
method are of the order of the valley width, since the method assumes that the profile is monotonic.
Similar errors would be expected for any simple method of analysis relying on the monotonic assumption.

The single polynomial method and the Schmerling 10-point method should be used only when suitable
computer facilities are not available. If possible, attempts should be made to implement at least
Titheridge's program, LAPOL, or Paul's earlier simplified method, both of which introduce simple under-
lying ionization and valley corrections and are smaller and faster than the more sophisticated methods.
These methods are discussed further in Section 5.

5. CONCLUSION

For those members of the subgroup who completed all or most of the tests, the effort has been a
learning process, the methods and computer programs being continually modified in order to perform bet-
ter in following tests. In fact, neither of the most successful programs, Titheridge's POLAN and Paul's
complicated method program, has yet reached its final stage. It is thus not possible to recommend a
"best" method of N(h) analysis, although it is obvious that these two methods are the most promising.

If Paul's or Titheridge's method is used to analyze accurate nighttime ionograms with suitable
X-ray traces, the height of the base of the F layer can generally be determined to within about 5-10 km.
If either of these techniques is used to analyze an accurate daytime ionogram with suitable X-ray
traces, the height of the F-layer base can be determined to within a few kilometers. Larger errors
must be expected in the analysis of real ionograms, which introduce experimental errors. In fact, meth-
ods of N(h) analysis seem to have reached the stage where the accuracy of the results is limited by the
quality of the input data, at least for analysis of conventional film ionograms.

For a method of N(h) analysis to be suitable for recommendation to users other than the originator,
it must satisfy at least the following three conditions:

1. It should have been found reliable in all the tests conducted, these being assumed representa-
tive of typical ionospheric conditions.

2. The corresponding computer program should be

. readily available in standard FORTRAN,

well documented,

well commented throughout,

easy to use,

fast,

as small as possible,

g. provided with a complete set of test data and results.

>waoce

3. The method should have well-defined and automatic default procedures for use when there is in-
sufficient information in the ionogram (such as no suitable X traces for use in starting and valley
corrections).

Since only Paul and Titheridge obtained good results for the valley tests, the Gulyaeva and Howe-
McKinnis programs cannot be recommended for general use. However, the latter programs gave good re-
sults for the starting tests, and in some cases were more accurate than the Paul or Titheridge (POLAN)
method. In fact, there were times when the Paul or Titheridge results were the worst, so both of these
methods have their limitations. This point will not be discussed further because Paul did not perform
all tests and Titheridge's POLAN has already been modified in the light of its relative difficulties
with some of the tests.



Profiles 1, 2, and 3 are composed of two overlapping Chapman layers as described in McNamara and
Titheridge [1977]. Profiles 1 and 2 correspond to midday conditions and have small valleys with widths
of »S and 16 km, respectively. Profile 3 corresponds to sunset conditions and has a valley width of
60 km.

Because of the method of construction, profiles 1 - 3 suffer from the limitation that wide valleys
are also deep valleys and so would favor any analysis method that made this assumption. Observed pro-
files, on the other hand, tend to be wide, shallow and fairly irregular in contrast to the smooth
variations in profiles 1 - 3, and might be expected to be less amenable to accurate analysis.

Profiles 4 - 6 have therefore been constructed using Arecibo incoherent scatter observations.

(4) Arecibo 26 duly 1972 062501~-063000 LT
(5) Arecibo 26 July 1972 073957-074456 LT
(6) Arecibo 11 April 1972 170629-171105 LT

Since the main concern is with the valley, the density distributions in the E and F layers have
been assigned convenient analytic forms that agree well with the observations. A parabolic layer,
which matches the observations closely near the layer peak, replaces the observed E layers. The ob-

served F layer is replaced by a linear in £n(N) layer which closely matches the base of the observed
F layer.

The valley distribution is taken to be linear in £n(N) between observed data points except near
the E-layer peak, where it is assumed that the peak is parabolic for densities > 0.9 NnaxE> and the
scale height is 10 percent greater than that used for the lower half of the peak. (In retrospect, it
would seem that a figure of 40 percent agrees better with observations.)

Several F-layer distributions (linear in 2n(N) for profiles 4 - 6) favor some analysis methods;
the remaining distributions (Chapman for profiles 1 - 3) favor none.

Twelve sets of analyses were requested, corresponding to two pairs of dip angle and gyrofrequency
and to six profiles. The dip angle, gyrofrequency pairs were (20°, 1.2 MHz) and (70°, 1.5 MHz) for
profiles 1 - 3 and (30°, 1.0 MHz) and (75°, 1.5 MHz) for profiles 4-6. The gyrofrequency was height-
dependent, varying inversely as the cube of the height from the given value at 6371.2 km.

Some of the largest errors were obtained for profile 3, as illustrated in Figure 12, The results
plotted in Figure 12 are representative of the general result that the methods of Paul and Titheridge
are clearly the most accurate, the errors generally being less than a few kilometers and often much
less. For the 20° dip cases, for example, the RMS errors immediately above the valley are 3.9 km
(Paul) and 3.4 km (Titheridge). At foE + 0.3 MHz, these figures have dropped to 1.8 and 1.0 km,
respectively. The error introduced by the use of the monotonic assumption would be equal to the valley
width just above foE, decreasing rapidly towards foF2. If an assumed valley width is introduced, the
error will depend on the agreement between the assumed and actual valley widths.

The Howe-McKinnis method used by Wright leads to underestimates of the height of the base of the
F layer by about 10-20 km. The results indicated by "W" in Figures 10-15 were obtained using a con-
stant gyrofrequency appropriate to an altitude of 200 km, the usual procedure followed with the use of
this program, which does not allow for a height-dependent gyrofrequency. Wright has found that more
accurate results can often be obtained by using instead the gyrofrequency at the height of the E-layer
peak. These results are indicated by the dashed line in each of the figures. However, the errors
obtained are usually still much greater than those obtained by Paul and Titheridge and do not seem to
have arisen only from the use of a constant gyrofrequency.

The need for the height variation of the gyrofrequency to be taken into account in the analysis of
the valley problem has not yet been investigated fully, but indications are that it is essential. Fail-

ure to do so results in the assignment of incorrect plasma frequencies to the X-ray frequencies used
in the analysis.

Depending on the profile shape, Gulyaeva's method yields errors of 5-10 km, but in some cases
overestimates of up to 30 km can occur. Both Gulyaeva and Wright intend modifying their methods in the
light of experience gained in performing the tests.

4. SIMPLE METHODS OF ANALYSIS

Because of their proven accuracy, the methods of Titheridge or Paul should be used wherever pos-
sible. However, if suitable computing facilities are not available, recourse may be made to one of the
simpler, but less accurate, methods of analysis, such as given by Schmerling [1967] and Titheridge
[1967b, 1969].

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APPENDIX F (Continued)

SLUTSKER S.M. (1971) CALCULATION OF UPPER IONOSPHERE N(Z) PROFILES
USING DATA OF VERTICAL SOUNDING FROM THE SURFACE OF THE EARTH.
G.A., VOL.11, 1, 157-159.

SMITH D.H. (1979) THE COMPARISON OF ELECTRON DENSITY PROFILES OBTAINED
FROM BACKSCATTER OBSERVATIONS AND IONOGRAM ANALYSIS. RADIO SCI., VOL. 5(4),685

SUTYRINA G.E. (1973) EXPRESS-METHOD OF IONOGRAMS TREATING. ARTICLES
"RESEARCHES IN G.A.S.P.", NO.28, MOSCOW, "NAUKA", 142-146.

TITHERIDGE J.E. (1969) SINGLE-POLYNOMINAL ANALYSIS OF IONOGRAMS.
RADIO SCI., VOL. 4(1), 41.

TITHERIDGE J.E. (1974) DIRECT ANALYSIS OF IONOGRAMS AT MAGNETIC DIP
ANGLES OF 26 - 39 DEGREES. J. ATMOS. TERR. PHYS. VOL. 36(4), 575.

TITHERIDGE,J.E. (1975) THE RELATIVE ACCURACY OF IONOGRAM ANALYSIS
TECHNIQUES. RADIO SCI., VOL.10(6), 589.

TITHERIDGE,J.E. (1975) THE ANALYSIS OF NIGHT-TIME IONOGRAMS. Je
ATMOS. TERR. PHYS. VOL. 37(12), 1571.

TITHERIDGE,J.E. (1976) DIRECT ESTIMATES OF SCALE HEIGHTS FROM TOP-
SIDE IONOGRAMS. J. ATMOS. TERR. PHYS. VOL.38(6), 623.

TITHERIDGE,J.E.,W.BECKER,L.BOSSY,T.L.GULYAEVA,L.F.MCNAMARA AND A.K.PAUL
(1978) ANALYSIS OF NUMERICAL IONOGRAMS. 1. COMPLETE MONOTONIC PROFILES AND
PROBLEMS ASSOCIATED WITH THE ANALYSIS OF THEIR IONOGRAMS. IONOSPHERIC
PREDICTION SERVICE SERIES X REPORTS, IPS-X6, SYDNEY, AUSTRALIA.

TITHERIDGE,J.E.,LOBB,R.J. (1977) A LEAST-SQUARES POLYNOMIAL ANALYSIS
AND ITS APPLICATION TO TOPSIDE IONOGRAMS. RADIO SCI., VOL.12(3), 451.

VASSEUR G., FELSTEIN R. (1968) CALCUL DES PROFILS D'IONISATION A PARTIR

DE SONDAGES IONOSPHERIQUES EN CONTRE-HAUT ET EN CONTRE-BAS. ANN. DES TELECOMM.
VOL. 23(7,8), 183.

VINNIKOVA T.L., FREIZON A.A., SHAPIRO B.S. (1968) CALCULATIONS OF N(H)
PROFILES OF THE IONOSPHERE WITH THE INTERLAYER AND UNDERLYING IONIZATION
TAKEN INTO ACCOUNT. GEOMAGNETISM AND AERONOMY, VOL.8, NO.5, 954-956.

VINNIKOVA T.L.-GULYAEVA (1979) THE INFLUENCE OF THE DEPTH OF AN
INTERLAYER VALLEY OF THE IONIZATION ON THE F-REGION REAL ALTITUDES.
G.A., VOL.19, 2, 346-347.

WHITEHEAD J.D., KANTARIZAS E. (1967) ERRORS IN THE MEASUREMENT OF VIRTUAL
HEIGHT USING A PHASE IONOSONDE. J. ATMOS. TERR. PHYS. VOL. 29(12), 1483.

WRIGHT J.W., LAIRD A.R., OBITTS D., VIOLETTE E.J., MCKINNIS D. (1972)

AUTOMATIC N(H,T) PROFILES OF THE IONOSPHERE WITH A DIGITAL IONOSONDE.
RADIO SCI., VOL. 7(11), 1633.

WRIGHT J.W., PAUL A.K. (1974) ELECTRON DENSITY PROFILES FROM IONOGRAMS AND
COMPARISON WITH ROCKET PROFILES. SPACE RESEARCH VOL. 14.

WRIGHT,J.W.,PAUL,A.K.,MECHTLY,E.A. (1975) UNDERLYING IONIZATION
CORRECTIONS AND THEIR COMPARISON WITH ROCKET RESULTS. RADIO SCI. VOL.
19(1), 255.

55

7
F
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APPENDIX B (Continued)

4) Full-automatic (free of a human intervention) search for the
N(h) solution among the set of the possible ones using the
rigorously-established criteria;

5) Minimum of information required in addition to the scaled
h'(f) data points (the most attractive being a version when
only geomagnetic parameters DIP and fy are required); and

6) Minimal computation times for obtaining each N(h) profile.

Also, the program must be sufficiently flexible to handle any number of
layers (and valleys) in the ionograms, with either combination of 0 and/or
X echoes in each layer. It is desirable to provide for the alternative when
the given ionogram is accompanied by some given part of the real N(h) distri-
bution obtained, for example, from the rocket experiments or incoherent
scatter observations, or from ionospheric models. All obtainable ionospheric
parameters along with the more obvious h(f) results should be provided in the
computer output. The program must be written in one of the international
algorithmic languages such as ALGOL or FORTRAN.

The first published ALGOL program /5/ and its modification for the
second order method /10/ complied with many of these requirements. Continued
employment of both programs in practice and comparison of those with other
well-known programs for N(h) analysis /14/ indicated that the present method
must and can be improved in some rather important areas.

To make it more comparable to and interchangeable with the other pro-
grams for N(h) analysis, it was decided to develop this improved version of
the program in FORTRAN /16/. This work was supported by the management of
WDC-B where the large archives of ionograms and FORTRAN-oriented computer are
available. The program ITERAN has been tested with Tests 5-9 of URSI Subgroup

G/6/2.

38



APPENDIX A (Continued)
virtual heights.

This procedure was used initially with a simple polynomial expression
for the real height profile, the number of terms in the polynomial (i.e.
the number of unknowns) being equal to the number of (h',f) data points
(Titheridge, 1961b). Considerable improvement was later obtained by inclu-
ding a term which gives a parabolic peak at the observed critical frequency.
The polynomial is constrained vo become horizontal at this frequency, so
that the peak is accurately parabolic (Titheridge, 1967b).

With the single-polynomial approach, representing the entire real
height profile by a single analytic expression, coefficients can be deter-
mined which give any required parameters of the real height profile directly.
Thus the height HM at the critical frequency, the scale height at the peak
and the sub-peak electron content can be obtained directiy from the measured
virtual heights without the need for calculating any other aspects of
the profile. For single-layer ionograms the results are of quite acceptable
accuracy using only a small number of data points. Thus with a 5-point
analysis calculated values of HM are an order of magnitude more accurate
than those obtained using Kelso-Schmerling coefficients (Titheridge, 1966).

Improved stability in the single-polynomial analysis is obtained by
assuming that at the lowest measured frequency the real height is equal to
the virtual height. Coefficients have been calculated for the analysis of
ionograms taken anywhere in the world, using 5 or 6 measured virtual heights
(Titheridge, 1969). With this number of points the results are completely
stable, and by choosing either the 5 or the 6 point frequency grid large
cusps on the ionogram can be avoided. A third set of coefficients is pro-
vided for the analysis of night-time ionograms, in which the real height
expression is fitted to 5 ordinary and 1 extraordinary ray measurement;
the resulting profile is then approximately corrected for the effects of
E region ionisation,

THE USE OF OVERLAPPING POLYNOMIALS

The single-poiynomial analysis can be usefully employed with up to 8
measured virtual heights. With more than 10 data points the results become
unstable. The problem is basically one of interpolation between given data
points. The Kelso method applies a Gaussian interpolation to the virtual
heights. In all other methods interpolation is done in the real height
domain, since the real height curve is considerably smoother. The linear
lamination analysis uses a linear interpolation between given frequencies
to derive the relation between real and virtual heights. As in any similar
problem of fitting discrete data points, accuracy is improved by an increase
in the order of the interpolating polynomial. A limit is reached, however,
beyond which the results become unstable. There is therefore an optimum
number of terms (n) for the polynomial. When the number of data points
to be fitted is greater than n the interpolation is done by using a
different polynomial for each interval between successive data points.

In many problems in numerical analysis interpolating polynomials with
4 to 6 terms are about optimum. To determine the optimum number for iono-
gram analysis, coefficients were calculated in 1959 for polynomials of up
to 8 terms (published as Table 2 of Tithoridge, 1975a). These show that
oscillatory tendencies begin with 7 terms at large magnetic dip angles,
and with 6 terms near the equator. 5 terms were therefore adopted for the

26



APPENDIX C (Continued)

Also, the h'(f)/N(h)-conversion method had to be extended for this new
project.

Now, our scaling personnel need only to punch the name of the satellite
under consideration and the date and time of observation. The computer then
determines the other data wanted, such as position of the satellite, the
intensity and inclination of the earth's magnetic field at any height, etc.
The IBM-360/91 computer allows us to make use of all available information
given by the ordinary, extraordinary, and z traces. Our present program also
allows us to use just sections of each trace, since for quality reasons a
mode change might improve the quality of the results.

For timesaving and practical reasons, no restrictions have been made as
far as the total number of readings per ionogram or the frequency spacing
is concerned. The scaler is just urged to scale more data the steeper the
gradient of the trace under consideration. Special data needed by the analy-
sis are derived by interpolation without smoothing out any real peaks.

Beyond 800 km topside profiles are approximated by exponential layer
steps and at lower heights by second order polynomial fittings, just as for
bottomside profiles. The underlying ionization problem in topside analysis
work has been solved in the same formal way as in bottomside studies. An

exponential electron density/height dependence replaces the cosine layer
step there.

The information given by Es-traces well above the peak height of the
normal E-layer has also been taken into account. Thus, the semi-thickness
of a parabolic approximation of the topside E-layer and the plasma frequency
where the Es-layer emerges can be determined. A short review of that chapter
of our h'(f) to N(h) conversion efforts was published by the author in 1969.

3 Methods Under Design

Our recent project deals with a detailed study of the valley problem
and with full-day electron density/real-height profile variations. The neces-
sary subroutines are now being submitted to their final tests.

In this project, we have been confronted with two key problems: the
underlying ionization problem in its most general form and the valley problem.
We have tried to solve them as shown in the next section.

3.1 Underlying Ionization Problem

Seven layer steps have been provided for that part of unseen ionization
distribution. The lowest one is a parabolic and the next one an inverse

43

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APPENDIX A

NCh)_ ANALYSIS by J.B. Titheridge - Historical survey and summary of results.

. LINEAR LAMINATION METHODS

In the 1950's conversion of ionospheric h'(f) records to obtain the
corresponding N(h) profiles employed the linear lamination approach.
Analysis was done by hand using integrated group index coefficients inter-
polated from graphs or tables; consequently the simplest analysis took over
2 hours, It was shown (Titheridge, 1959b) that by calculating the changes
in height from one frequency to the next a greatly simplified table of
coefficients could be used. This speeded up the analysis by a factor of
at least 10. The use of laminations in which h varied linearly with 1/f,
eliminated the need for calculation and subsequent integration of the group
refractive index in deriving the necessary coefficients.

Using a different approach - replacing individual virtual heights by
the average over all lower frequencies - the entire N(h) inversion problem
can be formulated in terms of the phase refractive index p (Titheridge,
1963). This appreciably speeds the analysis with, in general, a slight
increase in accuracy. The relation between the phase and group paths, on
which this analysis depends, is valid for any type of raypath (Titheridge,
1965).

The assumption of linear laminations produces large errors in the cal-
culated real heights in regions where the gradient is changing rapidly (as
at the peak). A simple rule was derived for determining, from the curva-
ture of the calculated profile, the approximate size of the error at each
point (Titheridge, 1961b, Appendix 1) so that a second order correction
could be applied. A more direct method of countering errors in-the lamin-
ation procedure is to calculate real heights’at frequencies between the
given data points. Thus an observed virtual height, which gives informa-
tion primarily on the electron density gradient, is used to define the
slope of a linear segment which extends either side of the observing fre-
quency (Titheridge, 1978a). This simple modification to the normal linear
lamination procedure gives in effect a second order analysis, with an
accuracy as good or better than current (parabolic) methods.

HIGHER ORDER (POLYNOMIAL) METHODS

Further increases in the efficiency of N(h) analysis, to give greater
accuracy from a given amount of data, require abandoning the ‘linear lamina-
tion' assumption in which only the real heights are contimious across lamin-
ation boundaries. The logical next step is to use quadratic Jaminations,
maintaining a continuity of gradient at all points. Would this, however,
be the final best approach or might it in turn be replaced later by an even
more sophisticated model? Thought was given as tc what might be
the end of such a process of successive refinement (Titheridge, 1959e).

The ultimate model, at least for single-layer calculations, would seem to
be one which maintained the continuity of all derivatives at all points.
This implies the use of a single mathematical expression (e.g. a polynomial)
to represent the entire profile. The mathematics implicit in this idea are
tractable provided that the adopted expression is differentiable. For any
set of scaling frequencies a matrix of coefficients can then be obtained
giving the real height at each frequency directly in terms of the observed

25



Conclusions are given in Section 5. This section and Section 6 were prepared following discus-
sions of the subgroup at Helsinki.

Appendices A, B, C, D and E give historical surveys of the development of the methods used by mem-
bers who have completed all tests, or who intend to do so. These surveys are designed to help the non-
expert through the literature, showing how each member's ideas have evolved, and the context of each
publication. They have been written by the individual members , without criticisms by other members
and may possibly contain personal opinions that are not generally accepted and that may become subjects
of further study by the subgroup. Many important aspects of N(h) analysis are also discussed in the
appendices, which can therefore be used to complement the main text.

A bibliography for the period 1969 (approximately) onwards is given in Appendix F.

2. THE PROBLEMS IN N(h) ANALYSIS

Even when very accurate virtual heights are available, there still exist two fundamental problems
in real-height or N(h) analysis. These are referred to as the "starting" problem and the "valley"
problem, and both are manifestations of incomplete data. A third difficulty, arising from a lack of
data, is the "peak extrapolation" problem. This is discussed by Titheridge et al. [1978].

Errors in the ionograms themselves and in the extraction of virtual height data from the icnograms
compound the problems mentioned above, in the analysis of practical jonograms. If individual profiles
are being studied, it is desirable to have some estimate of the uncertainty in the calculated real
heights, since if this is too large the results may be worthless. These topics are not discussed here,
since the attentions of the subgroup were directed to comparative analyses of accurate virtual height
data. This can be done only by excluding other factors which cause errors in the calculated heights.

2.1 The Starting Problem

With practical ionograms, the accuracy of a calculated N(h) profile may be severely restricted be-
cause reflections are not obtained below some limiting frequency f.... This limit is set by choice,
by broadcast station interference, by the output power of the jonoldfide, by the antenna system, or by
the rapid increase of absorption toward lower frequencies. Each of these effects leads to what is
known as the "starting" problem in N(h) analysis.

For a given ordinary ray virtual height curve, extending down to the frequency fins there exists
an infinite number of possible real height curves corresponding to different ionization distributions
below fpi,- If only ordinary ray data are available, the choice among alternatives must be made on
physical grounds. Methods of analysis must therefore incorporate some assumptions about the profile
in the "unseen" regions. These possible assumptions may be classified into six different groups, which
have been critically discussed by Titheridge [1975]. Extraordinary ray measurements are essential for
a mathematical solution to the problem. These measurements are not always available, however, so
assumptions based on expediency, physical considerations, or experience must often be employed.

2.2 The Valley Problem

The "valley" or decrease of electron density between the E- and F-layers of the ionosphere has
long been a problem in the analysis of ground-based jonograms. The simpler jonogram reduction tech-
niques have generally ignored the valley, assuming that the electron density increases monotonically
with height. This assumption, or the use of some assumed valley size, is necessary if only 0-ray
virtual data are available, since there is an infinite number of valleys that can satisfy the data.
More sophisticated techniques involve analyzing the ionogram by normal monotonic methods up to the
E-layer peak, followed by a special restart procedure.

The valley problem is essentially a daytime problem, since at night foE < fi, for most ionograms,
and any valley problem is subsumed by the starting problem. The techniques used to overcome the valley
problem are usually quite similar to those used for the starting problem and depend on the use of X-ray
data above fxE. In the absence of such data, it is necessary to assume that there is no valley (i.e.,
that the profile is monotonic) or to assume some synoptic value of the valley shape.

If in the absence of X-ray data the monotonic assumption is made, the calculated heights will al-
ways be too low if there is in fact a valley. In general, it seems preferable to introduce a valley
of some simple shape with parameters in accordance with rocket and incoherent scatter observations.
Tests of analysis methods in this case are rather pointless, because the errors depend on the fortui-
tous agreement between the actual and assumed valley profiles. It should be noted, however, that many
methods include the option of using a synoptic valley in the absence of X-ray data.



—]
PROFILE 4 —_—— =

a0 F - 5
. en _
et <= ° 4
we _
in ae
We ne
/
Lo /
260 / = 300
/
/
/ =}
/ nn eee
, — —
250 - / 75° ” a ‘ a0
/ ‘ oo
/ 15 \6 Wr 18
;
240+ d
h
(km)

” fn(MHz)

Fig. 7. Calculated and correct real heights for the lower parts
of profile 4 used for the underlying ionization tests

16



APPENDIX F (Continued)

JACKSON J.E. (1969) COMPARISONS BETWEEN TOPSIDE AND GROUND-BASED
SOUNDINGS. PROCEEDINGS IEEE, VOL.57, NO.6, 976-985.

JACKSON J.E. (1971) THE P'(F) TO N(H) INVERSION PROBLEM IN IONOSPHERIC
SOUNDINGS G.S.F.C. REPORT X-625-71-186.

KADUKHIN G.F., SHLIONSKY SH.G. (1976) A DEDUCTION METHOD OF N(H) PROFILE
BY MEANS OF ANALOGUE CALCULATION TECHNIQUE. G.A., VOL.19, 2, 268-272.

KAK A.C. (1979) ON OBTAINING ELECTRON DENSITY PROFILES OF PLANETARY
IONOSPHERES J. ATMOS. TERR. PHYS. VOL.32(2), 233.

KANTARIZAS E. (1971) MODEL PHASE IONOGRAMS FOR TESTS OF TRUE-HEIGHT
REDUCTION TECHNIQUES. J. ATMOS. TERR. PHYS. VOL. 33(1), 31.

KANTARIZAS E., WHITEHEAD J.D. (1971) AN IMPROVED’ TRUE~HEIGHT REDUCTION
TECHNIQUE RADIO SCI., VOL. 7(11), 1045.

KOEHLER,J.A. AND L.F.MCNAMARA(1975) FREQUENCY AND VIRTUAL HEIGHT ERRORS
IN I.P.S. IONOGRAMS. IONOSPHERIC PREDICTION SERVICE SERIES R REPORTS,
IPS-R29, SYDNEY, AUSTRALIA.

KUTIEV I.S. (1979) A METHOD OF THE ELECTRON CONCENTRATION DISTRIBUTION
DEFINITION IN THE IONOSPHERE. G.A., VOL.19, 3, 549-551.

KUTIEV I.S. (1978) METHOD FOR CALCULATING N(H) PROFILES ON INCOMPLETE
IONOSPHERIC TRACES BY USING BOTH MAGNETO-IONIC COMPONENTS.
COMPT.RENDUS L'ACAD. BULG. SCI., VOL.23, 6, 675-678.

KUTIEV I.S. (1972) DETERMINING THE EQUIVALENT DELAY FROM THE REGIONS
BELOW FMIN UPON CALCULATING N(H) PROFILES. COMPT. RENDUS L'ACAD. BULG. SCI.,
VOL.25, 2, 193-196.

KUTIEV I.S. (1973) COMPUTATION OF N(H) PROFILES BY MEANS OF ITERATIVE
FINDING OF RETARDATION IN UNSOUNDING PORTIONS OF THE IONOSPHERE.
ARTICLES IN REF. 1 OF THIS APPENDIX, 12-36.

KUTIEV I.S. (1974) SEMI-AUTOMATIC REDUCTION OF THE IONOGRAMS IN
COMPUTING N(H) PROFILES. REPORTS OF GEOPHYS. INST., L'ACAD. BULG. SCI.,
VOL.19, 97-163.

LOBB,R.J.,TITHERIDGE,J.E. (1977) THE VALLEY PROBLEM IN BOTTOMSIDE
IONOGRAM ANALYSIS. J. ATMOS. TERR. PHYS. VOL. 39(4), 35.

LOCKWOOD G.E.K. (1969) A COMPUTER-AIDED SYSTEM FOR SCALING TOPSIDE
IONOGRAMS. PROC. IEEE. VOL.57, NO.6, 986-989.

LOCKWOOD G.E.K. (1976) A MODIFIED ITERATION TECHNIQUE FOR USE IN
COMPUTING ELECTRON DENSITY PROFILES FROM TOPSIDE IONOGRAMS. RADIO SCI.,
VOL.5, NO.3, 575-577.

LYANNOJ B.E., SHOYA L.D. (1972) THE ESTIMATION OF THE EFFICIENCY OF THE
METHOD OF THE ADDITIVE NOISE MINIMIZATION IN THE PROBLEM OF THE N(H) PROFILE
CALCULATION FROM THE VERTICAL SOUNDING IONOGRAMS. ARTICLES “RESEARCHES IN
G.A.S.P.", NO.25, IRKUTSK, 154-159.

MALTSEVA O.A. (1969) METHODS OF COMPUTING IONOGRAMS FOR A MODEL OF THE
IONOSPHERE WITH A NON-MONOTONIC N(H) DISTRIBUTION. G.A., VOL.9, 2, 280-281.

MALTSEVA O.A. (1969) CORRECTION METHOD OF F-REGION N(H) PROFILE WITH THE
INTERLAYER E-F IONIZATION TAKEN INTO ACCOUNT. G.A., VOL.9, 5, 936-938.

MALTSEVA 0.A. (1976) SOME PROBLEMS OF N(H) PROFILE CORRECTION WITH
CONSIDERATION OF THE UNOBSERVED IONOSPHERIC IONIZATION. G.A., VOL.16, 5,
915-918.

MALTSEVA 0.A. (1979) ON ANALYTICAL EXPRESSIONS FOR COEFFICIENTS OF THE
CORRECTING METHOD. G.A., VOL.16, 5, 918-921.

53

|
|
|


APPENDIX A (Continued)

With X ray data above a layer peak, the width of the valley is calculated
{along with the constants of the first polynomial) by a least-squares fit
to 0 and X ray data. The shape of the valley is maintained approximately
constant, by setting D«W* . This requires an iterative procedure, 3
cycles are sufficient in all cases since large changes in the asswned depth
D have comparatively little effect on the calculated width W.

With accurate virtual height data the program can determine W and D
independently. This requires an expanded iteration cycle, to find the value
of D which gives the smallest RMS deviation in the fit to the given virtual
heights. As in the starting calculation, physical constraints are applied
at each stage to ensure positive values for the widths V and W, and the
initial gradient of the polynomial. With accurate data this provides a
good estimate of valley depth at most dip angles. Near 29° dip the depth D
may be far too great, but this has littie effect on the calculated width and
the F region heights.

. BASIC ERRORS IN THE CALCULATED PROFILES

Apart from the starting and valley problems, errors arise due to
inaccuracies in the data and to possible limitations in the theory on which
the inversion process is based. It has been shown that, for a single mag-
neto-ionic component, data errors must have a similar effect on all methods
of analysis (Titheridge, 1975a). Further calculations show that random
errors Ah' in the virtual height data produce real height errors of the
order of 0.5(A£)9-6,h! » where Af is the interval between scaled fre-
quencies (Titheridge, 1978c).

Thereare several possible sources of error in the basic relation
h' = | w'dh (Titheridge, 1959e). Even in the absence of collisions, the

ray theory assumed becomes inapplicable near the peak of a layer; the errors
are however appreciable only within about 0.01% of the critical frequency,
and so are of little concern.

Electron collisions in the ionospherereduce ' in the region near
reflection, and can considerably decrease the height h' given by the above
integral. Under these conditions, however, 4 is not zero at any real height
and ray theory fails. In the absence of a magnetic field, and replacing the
true profile by a linear approximation near reflection, integration in the
complex plane shows that collisions have zero effect on the true value of ht!
(Titheridge, 1961c). Numerical calculations show that this result is gener-
ally applicable. An exact virtual height is obtained by integrating the
complex value of «' , including the collision terms, up to the complex height
at which w =0 . At all dip angles this gives, to within 0.1%, the same
result as integrating the no-collision expression for ' up to the classical
reflection height (Titheridge, 19672). ‘The effect of collisions should there-
fore be ignored at all stages in the analysis.

A further assumption inherent in the expression for h' , that of a
vertically propagating ray, can cause considerable errors. With an inclined
magnetic field the 0 and X rays are deviated horizontally in opposite
directions by a distance of typically 20km (Titheridge, 1959a). Detailed
comparisons of hy and hg to obtain information about the unseen regions

will therefore become unreliable in the presence of horizontal gradients in
the ionosphere.

33

|
i



APPENDIX A (Continued)

The presence of large irregularities in the ionosphere can cause
horizontal deviations of over 10Ckm (Lobb and Titheridge, 1977b). The
rays avoid regions of decreased density. Thus a decrease travelling over
a station may pass unseen on the jonograms. A similar increase will appear
perhaps 200km larger (horizontally) than its true dimensions. In either
case, when there is an irregularity within a few hundred km of the station,
ionograms do not give a reliable measure of even the overhead critical fre-
quency. Analysis of topside ionograms, taken in the presence of wavelike
disturbances, gives most nearly the profile of the undisturbed ionosphere.

34



APPENDIX E (Continued)

Bibliography

Howe, H. H. and D. E. McKinnis (1967), Ionospheric electron-density
profiles with continuous gradients and underlying ionization cor-
rections. II. Formulation for a digital computer, Radio Sci., 2
(New Series), 1135-1158.

Laird, A. R. and J. W. Wright (1967), Computation and application of
synoptic ionospheric profiles, Radio Sci., 2 (New Series), 1255-
1262.

Laird, A. R., J. W. Wright and T. N. Gautier (1974), Remarks on achieving
representative electron density profiles of the ionosphere: Per-
formance and advantages of the "composite virtual height method",
Radio Sei., 9, 825-835.

Paul, A. K. (1967), Ionospheric electron-density profiles with con-
tinuous gradients and underlying ionization corrections. I. The
mathematical-physical problem of real-height determination from
ionograms, Radio Sci., 2 (New Series), 1127-1134.

Paul, A. K. and J. W. Wright (1963), Some results of a new method for
obtaining ionospheric N(h) profiles with a bearing on the structure
of the lower F region, J. Geophys. Res., 68, 5%13-5420.

Paul, A. K. and J. W. Wright (1964), Electron density profile analysis
of topside sounder ionograms, J. Geophys. Res., 69, 1431-1435.

Wright, J. W. (1959), Note on quiet-day vertical cross sections of the
ionosphere along 75°W geographic meridian, J. Geophy. Res., 64,
1631-163).

Wright, J. W. (1960), A model of the F region above h_ F2, J. Geophys.
Res., 65, 185-191. max

Wright, J. W. (1962), Dependence of the ionospheric F region on the
solar cycle, Nature, 194, 461-62.

Wright, J. W. (1962), Diurnal and seasonal changes in structure of the
mid-latitude quiet ionosphere, J. Res. NBS, 66D, 297-312.

Wright, J. W. (1967), Ionospheric electron-density profiles with con-
tinuous gradients and underlying ionization corrections. III.
Practical procedures and some instructive examples, Radio Sci., 2
(New Series), 1159-1168.

49



180

170

(km)

160

150

140

180

170

(km)

160

150

140

T
PROFILE 4

i

ine =114.8 km Lo

i wal 1 it 1

Fig.

2

13.

2.2 2.3 2.4 2.5 2.6
fn (MHz )

Calculated and correct real heights for the base of
F layer for the profile 4 used for the valley tests.

22

the



APPENDIX F (Continued)

DANILKIN N.P. (1974) ON GETTING AND POSSIBLE USE OF TRANS-IONOGRAMS.
G.A., VOL.14, 2, 369-371.

DANILKIN N.P., MALTSEVA 0.A., FAER YU.N. (1974) POLARIZED IONOGRAMS AND
CALCULATION OF N(H) PROFILES OF THE IONOSPHERE. G.A., VOL.14, 4, 636-639.6

DVINSKIKH N.I., STARKOVA I.P. (1972) THE N(H) PROFILES CALCULATION USING
THE DATA OF THE FIXED FREQUENCY TOPSIDE SOUNDING. MOSCOW, VINITI,
NO.5947-72 DEP.

DYSON P.L. (1972) A METHOD FOR DETERMINING THE ELECTRON DENSITY

DISTRIBUTION ABOUT THE F2 PEAK OF THE IONOSPHERE. AUSTRAL. J. PHYS., VOL.25,
NO.3, 293-297.

DYSON P.L. (1974) DETERMINING A STARTING POINT FOR THE ANALYSIS OF
GROUND-BASED IONOGRAMS J. ATMOS. TERR. PHYS. VOL.36(3), 553.

FLEURY L., TAIEB C. (1971) SIMULTANEOUS MEASUREMENTS OF ELECTRON DENSITY
USING TOPSIDE SOUNDINGS AND INCOHERENT SCATTER SOUNDINGS. Jd. ATMOS. TERR. PHYS
VOL. 33(6), 999.

FREIZON A.A., SHAPIRO B.S. (1968) VARIATIONS OF THE E-F INTERLAYER REGION
THICKNESS, GEOMAGN. AND AERONOMY, VOL.8, NO.5, 943-945.

GERSHENGORN G.I. (1972) AN IONOGRAM TREATMENT METHOD. ARTICLES
“RESEARCHES IN GEOMAGNETISM, AERONOMY AND SOLAR PHYSICS", ISSUE NO.21,
IRKUTSK, 291-292.

GULYAEVA T.L.-VINNIKOVA (1971) ON THE POSSIBILITY OF DETERMINATION OF
THE TOPSIDE IONOSPHERE N(H) PROFILES FROM THE SATELLITE SOUNDING DATA AT
FIXED FREQUENCIES, PROCEEDINGS OF THE 3D SCIENTIFIC CONFERENCE OF YOUNG
SPECIALISTS OF IZMIRAN. MOSCOW, IZMIRAN, 44-55,

GULYAEVA T.L. (1971) SET OF THE ALGORITHMS FOR COMPUTING OF IONOSPHERIC
N(H) PROFILES IN ALGOL~60. MOSCOW, VINITI, NO.3341-71 DEP.

GULYAEVA T.L. (1972) ON A NON-AMBIGUOUS STATEMENT OF THE PROBLEM OF
COMPUTING THE N(H) PROFILES OF THE BOTTOMSIDE IONOSPHERE. G.A., VOL.12, 3,
551-553.

GULYAEVA T.L., MALTSEVA O.A., FREIZON A.A., CHASOVITIN YU.K. (1973) A

COMPARATIVE ANALYSIS OF ROCKET MEASUREMENTS NE(H) AND THE DATA OF GROUND
VERTICAL SOUNDING. G.A., VOL.13, 4, 659-663.

GULYAEVA T.L., KUTIEV I.S., SERAFIMOV K.B., SHAPIRO B.S. (1973) ON
UNIFICATION OF METHODS OF IONOSPHERIC N(H) PROFILES COMPUTATION.
G.A., VOL.13, 4, 752-753.

GULYAEVA T.L. (1973) COMPUTATION OF N(H) PROFILES USING THE FIRST AND
SECOND ORDER METHOD WITH A VARIABLE PARAMETER IN UNOBSERVED REGIONS.
ARTICLES IN REF. 1 OF THIS APPENDIX, 76-99.

GULYAEVA T.L. (1974) ON THE CHOICE OF THE BEST APPROXIMATION OF N(R)
PROFILE AND ITS DERIVATIVE FOR CALCULATING VIRTUAL HEIGHTS OF THE IONOSPHERE.
G.A., VOL.14, 1, 48-52.

GULYAEVA T.L. (1974) THE EFFECT OF THE IONIZATION VERTICAL GRADIENT ON
AN IONOGRAM. G.A., VOL.14, 4, 741-743.

GULYAEVA,T.L. ,W.BECKER,L.F.MCNAMARA,A.K.PAUL,J.E.TITHERIDGE AND J.W.
WRIGHT (1978) ANALYSIS OF NUMERICAL IONOGRAMS. 2. THE STARTING PROBLEM.
IONOSPHERIC PREDICTION SERVICE SERIES X REPORTS IPS~X7, SYDNEY, AUSTRALIA.

GULYAEVA T.L., KNYAZYUK V.S. (1974) AN ANALYTICAL REPRESENTATION OF THE
VIRTUAL REFLECTION HEIGHTS. GEOM. AERON. (USA) VOL.15(2) ,299.

JACKSON J.-E. (1969) THE REDUCTION OF TOPSIDE IONOGRAMS TO ELECTRON-
DENSITY PROFILES. PROCEEDINGS IEEE, VOL.57, NO.6, 9608-976.

52





APPENDIX C

Conversion Methods for h'(f) to N(h) Used at Lindau
W. Becker

Max-Planck-Institut fur Aeronomie
Lindau/Harz, Germany

1 Methods Used Before 1967

Real-height analysis of multifrequency ordinary and extraordinary
virtual-height traces has been carried out at Lindau since 1956 with methods
of our own design. All of them take account of the influence of the earth's
magnetic field, even the model method we started with in 1956. It merged
into a more general manual method allowing the conversion of any monotonic
ordinary or extraordinary h'(f)-trace. In 1961, an IBM-650 computer became
available to us. Just as the manual methods had done, here also a height-
independent geomagnetic field was assumed. Underlying ionization was taken
into account in the same way as the manual methods had done. But the criti-
cal frequencies were no longer estimated by eye; they were analytically
deduced. By doing so, a better accuracy of the layer peak parameters could
be attained. The method itself was a laminar method with linear electron
density increases within each layer step. This program was mainly intended
to continue our studies on the solar control of the F-region and its dynamics
throughout half a sunspot cycle.

All the methods just mentioned and the physical results deduced with
them were reviewed by the author in 1967. Through an oversight, all the
references of this paper had been put together with those of my former assis-
tant and attached to his paper (Herbert, 1967).

2 Methods Used Since 1967

In 1964, an IBM-7040 computer replaced the IBM-650 machine and allowed
us a further improvement of the accuracy of our results, to store them on
magnetic tapes and to provide many other options. Concerning higher accuracy,
the former linear electron density increases within each layer step were
replaced by second order polynomial approximations with equal gradients at
the common boundary of adjacent layer steps. Now the height dependence of
the geomagnetic field could be taken into account also.

Tn 1969, we began to use an IBM-360/91 computer in addition. Thus, we
could start extensive studies of the topside ionosphere by converting a large
number of Alouette I and ISIS2 ionograms. For that Purpose a new semi-
automatic scaling facility had to be designed first. It is shown in Figure 1.

41

|
|
|


APPENDIX B (Continued)

References

1. Shapiro, B.S., and T.L. Vinnikova (1963), The programming of computations of
ionospheric true heights on electronic computers, Geomagnetism and Aeronomy,
3(2), 218-225,

2. Special Issue on Analysis of Tonograms for Electron Density Profiles (1967),
Radio Sei., 2(10).

3. Vimnikova, T.L., A.A. Freizon, and B.S. Shapiro (1968), Calculations of N(h) pro-
files of the ionosphere with the interlayer and underlying ionization taken
into account, Geomagnetism and Aeronony, 8(5), 954-956.

4. Vinnikova-Gulyaeva, T.L. (1970), The effect of the depth of interlayer valley of
the ionization on the F region real heights, Geomagnetism and Aeronomy,
10(2), 346-347.

5. Gulyaeva, T.L. (1971), Set of the algorithms for the computation of ionospheric
N(h) profiles in ALGOL-60, Moscow, VINITI, No.3341-71 Dep.

6. Gulyaeva, T.L. (1972), On a non-ambiguous statement of the problem of computing
the N(h) profiles of the bottomside ionosphere, Geomagnetism and Aeronony,
12(3), 551-553.

7. Gulyaeva, T.L., O.A, Maltzeva, A.A. Freizon, and Yu.k. Chasovitin (1973), A com-
parative analysis of rocket measurements of n, (a) and the data of ground
vertical sounding, Geomagnetism and Aeronomy, 13(4), 659-663.

&. Gulyaeva, T.L., 1.8. Kutiev, K.B. Serafimov, and B.S. Shapiro (1973), On unifica~
tion of methods of ionospheric N(h) profiles computation, Geomagnetism and
Aeronomy, 13(4), 752-753.

9. Methods of computation and investigations of ionospheric N(h) profiles (1973),

edited by B.S. Shapiro, T.L. Gulyaeva, and N.I. Potapova, Moscow, IZMIRAN.

10. Gulyaeva, 1.1. (1973), Computation of N(h) profiles using the first and second
order methods with a variable parameter in unobserved regions, in 'Methods
of computation and investigations of ionospheric N(h) profiles', edited by
B.S. Shapiro, T.L. Gulyaeva, and N.I. Potapova, Moscow, IZMIRAN, p.76-99.

11. Gulyaeva, T.L. (1974), Selection of the best approximation of N(h) profile and
its derivative for computing the virtual heights of the ionosphere,
Geomagnetism and Aeronomy, 14(1), 39-41.

12. Gulyaeva, T.L. (1974), The effect of the ionization vertical gradient on the
donogram, Geomagnetism and Aeronomy, 14(4), 741-743.

39



Table 1. The real height errors (calculated value - correct value), in meters,
obtained using 10 different methods of analysis

FN REAL 1 2 3 4 5 6 7 8 9 10
HEIGHT

0.905 93.133 20 -7 13 -6 11 3-130 5 5 -1
1.505 99.305 -40 4 3 2 7 61 3 20 20 2
2.002 102.386 -48 8 0 0 4 35 15 19 19 2
2.503 105.269 -44 8 -3 -2 2 -4 -88 «15 15 1
3.004 108.397 -38 8 -4 -3 0 -65 -39° «14 14 1
3.505 112.406 -28 4 -9 -6 -5 -118 -11 15 15 2
4.005 120.144 10 8 -12 -8 -13 -64 67 40 40 4
4.056 121.860 8 10-13 -9 = -16 6 86 39 39 6
4.105 124.165 -4 41 12 10 -5 9 85 49 49 8
4.156 127.814 54 49 17 11 28-18 5732 32 8
4.207 132.270 70-18 -60 ~-47 -47  -35 62 30 47 0
4.256 135.725 80 67 12 19 -22 -46 57-31 46 4
4.303 138.306 58 47 -5 -14 0 -35 48 32 45 5
4.355 140.667 38 33 3 6 -8 10 61 36 48 7
4.501 145.906 26 21 5 5 -3 25 47-34 44 8
5.002 159.030 -5 8 -1 -1 -5 -3 54-37 44 10
5.503 170.685 -29 5 -2 -1 -3 -45 40 39 44 12
6.004 183.298 -19 5 -4 -2 -4— -87 30 38 43 14
6.505 199.363 14 4 -7 -5 -7 -116 25 48 52 20
6.606 203.545 10 6 ~-10 -7 -9 -129 24 «49 53 22
6.707 208.358 21 6 -13 -9 -13  -105 25 54 58 25
6.808 214.194 61 15 -10 -10 -11 -74 26-71 74 31
6.909 222.116 22-21 -15 -7 30 9-23 25 82 86 76
6.985 232.938 57 35 35 30 66 -338 37, 109 113-23
RMS Errors 40 25 17 14 20 92 56 45 50 20
No. of Points 51 51 49 49 49 38 85 56 56 142
Min. Delta F -05  .05 .05 05) 605.04 -02 .02 .02 «.02
Method I I I I I F F F F F
Maximum Freq. 6.98 6.98 6.98 6.98 6.98 6.95 7.00 6.98 6.98 6.81

NOTE: The first two columns give the corresponding plasma frequencies and
real heights; the columns marked 1-5 are Titheridge's results for LINAN,
LAPOL, and POLAN, modes 3, 4, and 53; the remaining columns are the results
due to Paul, Bossy, Gulyaeva (2 sets) and Becker. Also given for each
method are the number of data points, the minimum frequency interval used
near foE, the method of obtaining real heights at frequencies between the
data points (see text), and the frequency above which the profile is fitted
by an extrapolated peak.



APPENDIX F

BIBLIOGRAPHY OF N(H) PROFILE ANALYSIS URSI SUBGROUP G/6/2

L.F.MCNAMARA JUNE 1978

ARTICLES “METHODS FOR COMPUTING AND RESEARCHES OF N(H) PROFILES OF THE
IONOSPHERE. PART 1. ALGORITHMS AND PROGRAMS OF N(H) COMPUTATION". EDITED BY
SHAPIRO,B.S.,GULYAEVA,T.L.,POTAPOVA,N.I. MOSCOW,IZMIRAN, 1973.

BARASHKOV A.S., DMITRIEV V.I., KOSTOMAROV D.P. (1972) ON THE INVERSE
PROBLEM OF THE VERTICAL SOUNDING. ARTICLES "RESEARCHES IN G.A.S.P.", NO.25,
IRKUTSK, 37-49.

BARASHKOV A.S., DMITRIEV V.I. (1973) THE INVERSE PROBLEMS OF THE
IONOSPHERE SOUNDING. "COMPUTATIONAL METHODS AND PROGRAMMING", NO.20,
MOSCOW STATE UNIVERSITY, 315-322.

BENCZE P., SHAIKO J., SHATORY G. (1973) COMPUTATION OF N(H) PROFILES
FROM IONOGRAMS AND SOME RESULTS OF ITS INVESTIGATION IN HUNGARY.
ARTICLES IN REF. 1 ABOVE, 37-52.

BENNETT P.E. (1974) THE MACHINE REDUCTION OF IONOGRAMS, TESTING OF AND
IMPROVEMENTS TO THE PROCESS. PENN. STATE UNIV. IONOSPHERE RESEARCH LAB.
SCIENTIFIC REPORT 427,

BENNETT P.E. (1974) TESTS OF UNDERLYING IONIZATION CORRECTION METHODS
IN ELECTRON DENSITY CALCULATION PROGRAMS. PENN. STATE UNIV. IONOSPHERE
RESEARCH LAB. INTERNAL REPORT NO.41

BENNETT P.E. (1974) GENERAL ACCURACY AND PEAK EXTRAPOLATION TEST OF
JACKSON ELECTRON DENSITY PROGRAM. PENN. STATE UNIV. IONOSPHERE RESEARCH LAB.
INTERNAL REPORT NO. 42.

BOHM S. WEIB J. (1973) METHOD OF COMPUTING N(H) PROFILES FROM DATA OF
GROUND VERTICAL SOUNDING OF IONOSPHERE IN USE AT CENTRAL INSTITUTE OF THE
SOLAR-TERRESTRIAL PHYSICS OF GERMAN ACADEMY OF SCIENCES IN BERLIN.

ARTICLES IN REF. 1 ABOVE, 53-75.

CHORNY F.B. (1973) ON REDUCTION OF HEIGHT-FREQUENTIAL CHARACTERISTICS
INTO THE ELECTRON DENSITY PROFILES. G.A., VOL.13, 2, 242-247.

CHUANG,S.L., YEH,;K.C. (1977) A METHOD FOR INVERTING OBLIQUE-
SOUNDING DATA IN THE IONOSPHERE. RADIO SCI. VOL.12(1), 135.

COLIN L., CHAN K.L., LEE J.G.K. (1968) PHASE AND GROUP REFRACTIVE
INDICES FROM THE COLLISIONLESS MAGNETOIONIC THEORY. NASA TM X-1553; ALSO
RADIO SCI., VOL. 4, NO.6, 573, 1969.

COLIN L., CHAN K.L., LEE J.G.K. (1969) COMPUTATION OF TOPSIDE IONOGRAMS
FROM N(H) PROFILES. NASA TN D-5952; ALSO RADIO SCI., VOL.4, NO.6, 573, 1969.

DANILKIN,N.P. ,DENISENKO,P.F.,SOTSKIY,V.V. (1975) COMPUTATION OF
IONOSPHERIC N(H) PROFILES BY QUADRATIC PROGRAMMING. GEOM. AERON. (USA)
VOL. 15(2), 292.

DANILKIN N.P., MALTSEVA 0.A. (1972) N(H) PROFILES WITH CONSIDERATION FOR
IONOSPHERIC HORIZONTAL IRREGULARITIES. G.A., VOL.12, 4, 625-630.

DANILKIN N.P., MALTSEVA 0.A. (1973) THE CALCULATION OF N(Z) PROFILES OF
THE IONOSPHERE WITH THE USE OF DATA ON RADIOWAVE ABSORPTION.
G.A., VOL.13, 2, 256-269.

DANILKIN N.P., MALTSEVA 0.A. (1973) THE USE OF DATA ON ABSORPTION FOR
COMPUTING OF IONOSPHERIC N(Z) PROFILES. ARTICLES IN REF, 1 ABOVE, 157-162.

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WORLD DATA CENTER A
for
Solar-Terrestrial Physics

REPORT UAG-68

A COMPARATIVE STUDY OF METHODS OF
ELECTRON DENSITY PROFILE ANALYSIS

Chairman’s Report, URSI Subgroup G/6/2 on N(h) Analysis

by

LF. McNAMARA

lonospheric Prediction Service
Sydney, Australia

SEPTEMBER 1978 jl
Published by World Data Center A for ~ WW,
Solar-Terrestrial Physics, NOAA, Boulder, Colorado z
BZ i

and printed by

U.S. DEPARTMENT OF COMMERCE WS
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION, iy 7
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PRICE THIS ISSUE $1.41



APPENDIX A (Continued)

SOLVE is designed so that these final calculated parameters can if necessary
be reset to zero, and corresponding adjustments made directly to the other
calculated parameters and to the fitting error. If the calculated vaiue of
D is negative then it is set equal to zero and a new result obtained, corres-
ponding to a fit of 6 parameters to the 10 virtual heights. Similarly if A
comes out negative it also is eliminated from the solution. If, even after
these adjustments, the initial gradient of the polynomial should be negative
this (first) parameter is set to zeru, the weight given to the X ray data is
reduced, and the equations resolved. Thus the number of parameters defining
the underlying ionisation is reduced as required to ensure that, even with
poor data, the final profile is always physically acceptable,

VALLEY CALCULATIONS

Because of the higher frequencies involved, and the resulting decrease
in Bom hy » even less can be determined about the valley region. The one

parameter of importance is the width of the valley. With practical ionograms,
different assumptions about the valley shape give calculated widths varying
by about 15% (Titheridge, 1959c). All these results fit the 0 ray data
exactly and fit the X ray data to well within experimental error. Exact
model calculations using square and triangular valleys of different depths
gave a similar result, with errors of up to 20% in the widths calculated
assuming a fixed shape and depth (Titheridge, 1960). Errors in the calculated
F region heights are a similar percentage of the uncorrected error. Thus
only a one-parameter valley calculation is normally feasible, and this will
reduce the errors in calculated F-Layer heights by a factor of about 5.

At dip angles near 29° the same considerations apply as for the starting
problem. No information can be obtained about the valley shape, but the width
(ard F region heights) can be obtained directly from corresponding 0 and X
ray measurements (Titheridge, 1974b).

Applying a least-squares analysis to ionograms calculatcd from realistic
model profiles, at a dip angle of 60°, showed that the virtual height data
could be satisfied to within normal oxperimental error over a considerable
range of assumed valley depths (Lobb and Titheridge, 1977a). In particular,
the value of fcE must be known to better than 1% before meaningful estimates
of valley depth can be obtained. Because of the increased effect of errors in
foE , and of the assumed valley shape and depth, the reliability of the calcu-

lated widths is decreased if data points are used within about 0.1 MHz of
foE .

In the computer program POLAN a model valley is normally inserted above
the E layer. To provide a reasonably realistic
mode! the valley consists of three parts: (a) a parabolic section extending
from the peak down to a plasma frequency of foE-D , using a scale hei ght 25%
greater than that calculated for the E layer; (b) a constant plasma frequency
for a height range 0.6V; and (c) a linear increase between plasma frequencies
foE-D and foE , over a height range 0.4V .

. The total width W of the valley is V plus the height of the parabolic
section. In the absence of X ray data we assume W = 20km and D = 0.05 MHz.

These constants can however be changed as required, or the valley can be
omitted,

32



T T T T T
PROFILE 2
160/- 4
150- 7
140 - 4
h
(km)
130}- 4
wor
a
I290<7" 4
160 - 4
150}- 4
h
a -- —
‘«n) a ceo a
140}- . a a 4
— ——
P - ia —
rae aon
ae
130} wo DIP = 20° a
wo
12005-— L 1 1 1 1 L
4.0 4) 4.2 4.3 44 4.5 46 47
fn(MHz)
Fig. 11. Calculated and correct real heights for the base of the

F layer for the profile 2 used for the valley tests

20



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APPENDIX C (Continued)

parabolic layer step. Their peak densities and the parameter values of the
second order polynomial approximations of the other five steps are determined
by iteration. The final solution is supposed to result in a least-squares
deviation of the extraordinary virtual heights of the profile just derived
from the observed extraordinary virtual heights, if ordinary data were used
primarily. The number of readings for this selection process is optional.
Also, no restrictions have been made as far as the frequency of the first
extraordinary reading is concerned. Any formal solution inhibiting a negative
gradient is generally rejected.

3.2 Valley Problem

3.2.1 No Es-Layer Observed

This problem is being solved similarly to the underlying ionization
problem just described. From former studies we know that the semi-thickness
of a parabolic approximation of the topside E-layer is about 20 percent
thicker than the bottomside semi-thickness. Therefore, we start with such
a value for the topside E-layer and apply the underlying ionization proce-
dure after having rejected its lowest (parabolic) step. The lowest plasma
density of the valley fin is considered a free parameter between 0.0 and

foog:, the ordinary critical frequency of the E-layer.

3.2.2 Es-Layer Observed

3.2.2.1 Es-Layer Within the Topside E-Layer

This problem differs from the above one only slightly. The semi-
thickness of the topside E-layer is no longer an unknown parameter and fiji
needs only to be varied between 0.0 and the plasma frequency at the height of
the emerging Es-layer.

3.2.2.2 Es-Layer Within the Valley

In such a case we proceed in two steps. The first one is the determina-
tion of the real height of this Es-layer; the second one is similar to the
underlying ionization procedure described under 3.2.2.1.

The analysis starts with an unknown semi-thickness of the topside E-
layer. It assumes an inverse parabolic layer shape of the electron density
distribution within the valley from fm, up to the Es-layer level. By itera-
tion, that solution is selected which results in practically equal real
heights for the observed Es-layer.

44

|



APPENDIX A (Continued)

For single-layer ionograms, all available data can be fitted in a
single step. This is particularly valuable for the analysis of topside
ionograms where full use can be made of fragmentary ordinary and extra-
ordinary ray traces. These are incorporated into a single analysis which
interpolates smoothly across any unobserved regions, and can give any de-
sired number of points on the real height profile (Titheridge and Lobb,
1977).

With daytime bottomside ionograms single-polynomial solutions are
obtained first for the E region and then for the F region, By incor-
porating a model valley into the analytic expression for the upper section,
and using all available ordinary and extraordinary ray data for the F
region, the best-fitting valley width and F region real height profile
are obtained directly (Lobb and Titheridge, 1977a).

A GENERALISED FORMULATION OF THE N(h) INVERSION PROBLEM

All published methods of N(h) analysis (apart from the Kelso procedure)
can be included in the following scheme. Virtual height data in the form of
a series of points (h!,f.) is to be converted into a corresponding series

Pa

of real heights thy £3) . Suppose real heights have been calculated up to
the point k . (corresponding to hy £0 . The next real height section from

k to k+l is defined by an expression which is a linear function of n

variables. This is normally an n-term polynomial. The final 1 or 2 terms
may however be replaced by a parabolic-peak expression, by terms which re-
present unseen ionisation with fy < f£; , or by a term which specifies the

width of a model valley (as in 7 below). The n-term real height expression
is fitted to the last r calculated real heights, to the gradient dh/dN
at the last s calculated points, and to v_ successive virtual heights.
The virtual heights will include the point Oh Fear » and may also

include further points at both higher and lower frequencies.

These conditions specify a set of m= r+s+v simultaneous equations
in n unknowns. In the generalised polynomial analysis program POLAN the
parameters r,s,v and n can be specified independently. A subroutine
COEFIC is called at each step to calculate the mxn array of coefficients.
For the real height and gradient conditions the coefficients are simple
functions of the relevant frequencies. The virtual height equations involve
group index integrals, which are evaluated using 5 or 12 point Gaussian inte-
gration depending on the desired accuracy. At high latitudes both 5 and 12
point integrals are employed, over different sections of the integration
range, in a way which reduces errors by a factor of 10 to 100 at dip angles
of 78 to 88°. At a given frequency the n polynomial integrals are obtained
from the same 5,12 or 17 calculated values of wu! . Calculation time
therefore depends primarily on the number of virtual heights (v) fitted at
each step, and not on the number of terms in the fitted polynomial.

Solution of the m simultaneous equations is carried out by a sub-~-
routine SOLVE. This gives an exact solution.if n=m and a least-squares
solution if n<m. The calculation uses Householder transformations
which give accurate results up to large values of n . From the n_ calcu-
lated real height coefficients any required features of the profile (e.g.
the real heights at any desired frequencies, the total electron content,

28



APPENDIX B

Summary of N(h) Analysis Approach

T. L. Gulyaeva
IZMIRAN
Troitsk, Moscow Region, U.S.S.R.

This work has developed out of the other well-known methods for N(h)
analysis of ionograms. Since the implementation of the linear lamination
analysis /1/ for the O-ray data, the method used at IZMIRAN has been con-
siderably improved. The first attempt at combining the 0 and X data for
the valley correction /3/ turned out to be unsuccessful due to the assump-
tion of a many-parameters model in the E-F region that produced ambiguous
N(h) results in many cases. Fairly good criteria for conforming the N(h)
results to the 0 and X echoes were established when a one-parameter model
of starting and valley ionization was adopted /4,5,6,10,16/.

Because we intended to develop a program for the users (rather than for
the originator), we prefer an iterative approach /6,8,16/ to the least-
squares strategy when both 0 and X data are available, since the former gives
less breakdowns in practice though at the cost of some additional computation
times required for performing the iterations. The modification of the itera-
tive strategy /16/ leads to large savings of computation time.

Since the Special Issue of Radio Science /2/, efforts have been contin-
ually made to test the analysis methods /4,6,9,14/, to compare N(h) profiles
deduced from the ionograms with the data of rocket measurements /7,17/, and
to unify the different methods of N(h) analysis /8,18/. While developing
the method of combined use of 0 and X data, the urgent need was discovered
for recommendations concerning parameters for the start and valley in the
most common cases when only O-ray data were available. The conclusion was
drawn that further progress in the field must be founded on study of the
relationship of realistic N(h) profiles to the corresponding numerical iono-
grams, and several studies were carried out in this direction /11,12,13,15,18/.
Gulyaeva /18/ suggests a procedure for starting and valley corrections using
the observable parameters on each ionogram when only one magnetoionic com-
ponent is available.

Progress of the methods for computing electron density profiles from the
vertical-incidence ionograms permits us to lay down the key principles, which
must be satisfied by a good program for N(h) analysis:

1) High accuracy of N(h) approximation over a frequency interval;

2) The best possible accuracy of allowance for the underlying and
valley ionization;

3) Simplicity of the algorithm through a little number of unknown
parameters of "unseen" ionization;

37

|
|
|
|


T T T

T
PROFILE 5

160 4

150 - s toon 4 |
h _ a |
S 1-7 |

(km) Boe w |
_— a _— |

140- —_— Te w | |
oa

coe |

ae “
130+ a DIP = 75° 4

youn
-

wo
120+ 4
160+ 4

150}
h
(km)
140 4
wo
130 4
“ DIP = 30°
{hme =114.8 km a |
120 1 ra | \ 1 |
27 26 29 30 a 32 a5 |
|

fn(MHz)

Fig. 14. Calculated and correct real heights for the base of the
F layer for the profile 5 used for the valley tests.


At the time of writing, Titheridge's POLAN is more readily available and better documented than
Paul's program. They require about the same computer storage and have very similar computation times.
For some users, POLAN's "one-pass" solution may be an important consideration - at present Paul's pro-
gram still requires visual inspection of the results to determine whether further iteration is neces-
sary.

Both Paul's program and POLAN are fully supplied with default conditions, for use when there is
insufficient information in the ionogram. This is an important consideration since these default con-
ditions may be invoked in the majority of analyses. The relative merits of different default condi-
tions is a topic for further study by the subgroup.

Computer programs and documentation will be available both from the originators and WDC-A in the
near future. Some descriptions of POLAN are given by Titheridge in Appendix A.

For organizations with only small computer facilities, POLAN or Paul's complicated method (as used
in the present tests) may be too large, in which case recourse must be made to programs such as
Titheridge's simpler program LAPOL or Paul's earlier "simplified method for use with mini-computers."
For these particular programs, accuracy of the results has been traded off in favor of simplicity, the
starting and valley corrections being rather unsophisticated.

At the present time LAPOL requires slight modifications to prevent negative slopes at fin, which
arise occasionally when the calculated starting heights obtained using one X-ray point are too large.
(This problem does not arise if the fixed frequency mode of LAPOL is used, as recommended by Tithe-
ridge.) Paul's simple method does not use X-ray data and is essentially the default version of his
complicated method for use when no X-ray data are available. Both methods are available from their
originators.

6. TOPICS FOR FURTHER STUDY
The following topics will be given the group's attention over the next 3 years:
1. Members will intercompare the default procedures they have adopted for the underlying ioniza-
tion and valley problems. These methods are largely based on experience and are probably not capable

of comparative testing, but a reasonable consensus view should be obtainable.

2. Members will make a determined attack on the problem of error bars, since the value of a re-
sult is greatly enhanced when its range of uncertainty is known.

3. A Handbook of N(h) Analysis will be prepared, illustrating with real ionograms typical prob-
lems encountered and their suggested solutions (with error bars).

4. Some discussion is necessary to determine whether it is feasible to develop a "best method of
N(h) analysis," which would combine techniques used by different members. To this end, members will
interchange programs and documentation.

5. A study will be made of methods for including angle of arrival information in the N(h) analy-
sis, thus permitting allowance for horizontal gradients. Present methods of analysis assume that
propagation is vertical, and this can cause errors, particularly in combined 0-X solutions.

6. Procedures will be developed for including Z-trace information in the analysis of high-lati-
tude ionograms.

7. Comparisons will be made of different methods of analysis of topside ionograms.
8. The "overlap" problem will be studied, using topside and bottomside ionograms, and TEC data.

9. A study will be made of problems of analysis which have been encountered at different dip
angles.

10. Attempts will be made to develop an automatic (adaptive) method of analysis for sequences of
jonograms.

11. A study will be made of the use of Es layers to provide information about the E-F valley.

Some of these topics are already under consideration by some members.



NGSDC DATA SETS (SOLAR-TERRESTRIAL PHYSICS) April 1978

Listed below are "fliers" that describe many data sets available from the National Geophysical and
Solar-Terrestrial Data Center, NOAA/EDS, Boulder, Colo. 80303. Place a check mark in the box at the
left of each flier that you would like to receive; print your name and mailing address on the reverse
side, fold, and mail to NGSDC, or give to NGSDC representative. Please allow 4 weeks for receipt of

e fliers.

|
|

fy 1974 (A) Special Ionospheric Data Services j
Tonospheric Electron Density Profiles--N(h) '
lam] 1974 (B) Special Geomagnetic Data Services
Magnetograms Reduced to Common Scale -- Analog or Digital
£7 1974 (C) Speciat Geomagnetic Data Services t
Digital Geomagnetic Data |
am 1974 (D) Special Geomagnetic Data Services
AE Indices and AH for Auroral Zone Stations
LT 1974 (G) Special Auroral Data Services
Auroral Imagery and Precipitating Electon Data from
DAPP Satellites
Ly 1974 (H) New Data Available on Solar and Interplanetary Phenomena
lam 1975 (Cc) Special Ionosphere Data Services
Riometer Data on Microfilm
lami 1975 (E) Special Ionospheric Data Services
Incoherent Scatter Radar Data
fy 1975 (G) Special Geomagnetic Data Services
Magnetic Pulsation Data
lam] 1976 (A) Special Ionospheric Data Services
Monthly Composite Data -- h'f and N(h) I
fT 1976 (B) Special Ionospheric Data Services |
Whistler Data -- 1956-1966 I
am) 1976 (D) Special Geomagnetic Data Services f
New Acquisitions of Magnetograms (South America) i]
|
Oo 1977 (B) Special Geomagnetic Data Services |
New Acquisitions of Magnetograms (East Africa) i
L7 1977 (C) Special Solar Activity Services i]
Historical Data on Microfilm/Microfiche |
C7 1977 (D) Special Cosmic Ray Services |
Neutron Monitor Data on Magnetic Tape -- 1955-76 |
fT 1977 (E) Special Solar Activity Services |
Solar Activity Data on Magnetic Tape |
rm 1978 (A) Special Ionospheric Data Services |
Detailed Ionospheric Vertical Soundings Data on ‘i
Magnetic Tape
k
Ly 1978 (B) Price List for Solar-Terrestrial Physics Data Services |
Lf] 1978 (C) Precipitating Electron Data from DMSP Satellites on Magnetic Tape

1
|

USCOMM-NOAA-ASHEVILLE, N.C. 9-78-600


APPENDIX C (Continued)

The second part of the analysis allows for a slight discontinuity of
the inverse parabolic electron density distribution at the Es-layer level
and proceeds in the same way as under 3.2.2.1.

4 Conclusions
The present preliminary test results are very promising. In spite of
that fact, the author does not want to publish them beforehand; he just
wants to mention that the above analysis needs a big computer and a remark-
able amount of computer time.
5 References
Becker, W. (1967), On the manual and digital computer methods used at Lindau
for the conversion of multifrequency ionograms to electron density-
height profiles. Radio Science 2, 1205-1232.

Becker, W. (1969), Die Bestimmung der Feinstruktur der Ionosphare aus Iono-
grammen. Kleinheubacher Ber. 13, 37-44.

Herbert, Th. (1967), Tables of virtual heights for models of monotonic and
nonmonotonic ionospheric layers. Radio Science 2, 1269-1278.

45

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1
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if


APPENDIX D (Continued)

Paul, A. K. (1960a), Bestimmung der wahren aus der scheinbaren Reflexionshthe,
Arch. Elek. Ubertragung 14, 468-476.

Paul, A. K. (1960b), Kritische Bemerkungen zur Berechnung der "wahren |
Héhen", Geofisica Pura e Applicata 47, 69-78. |

Paul, A. K. (1967), Ionospheric electron-density profiles with contin- |
uous gradients and underlying ionization corrections. I. The |
mathematical-physical problem of real-height determination from
jonograms, Radio Sci. 2 (New Series), 1127-1133.

Paul, A. K. and G. H. Smith (1968), Generalization of Abel's solution
for both magnetoionic components in the real-height problem, Radio
Sci., 3 (New Series), 163-170.

Paul, A. K. (1977), A simplified inversion procedure for calculating
electron density profiles from ionograms for use with minicomputers,
Radio Sci., 12 119-122.

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