geos sate,llite tracking corrections , for, refraction in ...€¦ · geos satellite tracking...
TRANSCRIPT
\, "
\.
'/
..X-514-70-467"
~REPRINT -
, ,-
&~UiU=- '/;;.:5 gX1
GEOS SATE,LLITE TRACKING CORRECTIONS, ,
, FOR,"REFRACTION IN THE ,IONOSPHERE
/
\
/
/ r
./
...... '
, ,
/
..
JOHN, H. BERBERT, HO~ACE C. PARKER'
I,
-. ' ..
DECEMBER 1970 '
" ,
/
https://ntrs.nasa.gov/search.jsp?R=19720011202 2020-06-05T16:26:35+00:00Z
GEOS SATELLITE TRACKING CORRECTIONSFOR RE FRACTION IN THE IONOSPHERE
John H. BerbertGoddard Space Flight Center
Horace C. ParkerRCA Service Company
Some of the material in this document was presented at theApril 1968 National meeting of the American GeophysicalUnion in Washington, D. C.
GODDARD SPACE FLIGHT CENTERGRE~NBELT, MARYLAND
GEOS Satellite Tracking Correctionsfor Refraction in the Ionosphere
John H. Berbert
Horace C. Parker
ABSTRACT
This document compares the analytic formulations at different elevation angles
and at a frequency of 2-GHz for the ionospheric refraction corrections used on the
GEOS satellite tracking data. The formulas and ray-trace results for elevations•greater than 100 , where most satellite tracking is done, differ in .1E, .1R, and ~R by
less than 0.4 millidegrees (1.4 arc-seconds), 12 meters, and 12 em/sec, respective
ly. In comparison to most operational requirements, this is insignificant. However,
for the GEOS Observation Systems Intercomparison Investigation, these differences
are equivalent in size to observed differences in system biases for some of the best
electronic geodetic tracking systems and are probably contributing to the ooserved
biases.
For all the GEOS ionospheric refraction correction formulas examined in this•report, the range rate refraction correction for l\R is the time derivative of the
•formula for 6R, as might be expected. It is much easier to derive.1R from.1R in
this manner, rather than independently from basic principles as was usually done in
the earlier documents.
The ray-trace results and most of the more detailed analytic correction formulas
show that the ionospheric refraction correction for range rate on an overhead pass is
a maximum for elevation angles between 150 and 300 and falls off rapidly for both
higher and lower elevation angles, contrary to the effect of the troposphere and to
some reports in the literature.
E
Ec
R
R.cR•R.cE
f
ht
hm
H
Hd
NeNTNemN.
1
N imp.
1e OJ' e
Mo
~E
.6R
~R
ReH
g
k
GLOSSARY
measured elevation
elevation corrected for refraction
measured range
range corrected for refraction
measured range rate
range rate corrected for refraction
rate of change of elevation (refer to Appendix B)
radio frequency
height of the satellite = i x 106
meters
height of maximum electron density
ionospheric scale height for the Chapman model ionosphere
ionospheric scale height for the DC program
electron density in electrons/m3
total electron content in a vertical column of area 1 m2
, NT = SNe dh
electro~ density in electrons/m3 at hm
refractivity at height h.1
refractivity at height h ,(N. < 0 for the ionosphere)m . 1mrefractive index
Napierian base = 2.718281831
N. He1m
elevation correction =E - Ec
range correction = R - R.c •
range rate correction =R - Rc
earth radius = 6,378,166 meters
ionospheric scale height factor for GPRO arising from parabolic modelionosphere. Nominally GPRO uses H = 50 km
gthickness parameter for F layer biparabolic profile in Bent modelionosphere
k = l/Hh =decay constant for topside exponential profile in Bent modelionosphElre
R =h - y in Bent model ionosphere--h m m
GEOS SATELLITE TRACKING CORRECTIONSFOR REFRACTION IN THE IONOSPHERE
INTRODUCTION
The refraction correction formulas for correcting the elevation (E), range•(R), and range rate (R) tracking data from the GEOS geodetic tracking systems are not
standardized. Different refraction correction formulas are applied to the same data by
different users. The different formulas arise from the use of different mathematical
models of the atmosphere or from different approximations to the same model.
This document presents some of the different sets of ionospheric re
fraction correction formulas now being used on GEOS data and compares the magni-«
tude, at a frequency of 2 GHz, of the corrections to E, R, andR for different ele-
vation angles. The 2-GHz frequency is close to the frequencies used by the GRARR,
the Apollo USB, and various other S-band radar tracking systems. This is a continu
ation of a study of the comparison of atmospheric refraction corr~ctionformulas for
GEOS data, in support of the GEOS Observation Systems Intercomparison Investigation.
The tropospheric refraction correction comparisons are given in reference 1. Not
all of the tracking systems represented in reference 1 are included here because the
ionospheric refraction correction is not always applied as an analytic correction
formula. In particular, the Army Secor ranging system and the Navy TRANET Dl'p
pIer system use multiple frequency refraction correction techniques. For C-band
radar tracking data, at a frequency of 5.7 GHz, the ionospheric correction is very
small and is therefore often neglected or else lumped together with tropospheric pa-o •rameters. For example, for C-band at 10 , 6R = 0.99 meters, 6R =0.13 em/sec,.
and l\E =0.076 millidegrees (0.27 seconds of arc).
•The sets of refraction correction equations for t£ I DoB" and D.R being
used by the Goddard Space Flight Center (GSFC) on GEOS data are compared with each
other in this document. The formulas for DoE and L.R are also compared with the re
sults of ray-tracings through a modified Chapman ionosphere using the Eastern Test
Range (ETR) REEK program. In addition, the AR refraction correction formulas are
compared with values of d ~R) obtained by differentiating the polynomial fitted to the
curve of the REEK ~R vs time values. Results of ray-tracing through the Bent model
ionosphere (reference 23) are plotted for comparison.
In Appendix A, the ionospheric refraction correction equations are first
given as they appear in the referenced source documents and then are converted to a
common notation and functional form to simplify the comparison of these equations.
1
•CORRECTION FORMULAS FOR AE, .6.R, AND AR
Table 1 lists the GSFC GEOS-2 tracking data processing programs and in-•dicates which programs apply AE, .6.R, and.6.R ionospheric refraction corrections.
Table 1GSFC GEOS-2 Tracking Data Processing Programs
Processing Programs AE . AR ARDifferential Correction, DC, used with GSFC operational Yes Yes YesOrbit Determination Programs.(reference 2)
Dr. J. J. Freeman Formulations (reference 5) Yes Yes Yes
GEOS Preprocessor Program, GPRO, used with the No Yes YesGeodetic Data Adjustment Program, GDAP, (ref-erence 4) and with the Network Adjustment Programs,NAP-1 and NAP-2 (reference 19)
NONA ME , all versions. prior to MONSTER (8/20/70) which No No Nocorrects for AR and AR using a table look-up for twoGRARR VHF stations
GE08-2 GRARR Validation Program, GEOVAP (ref- No Yes Yeserence 26)
Definitive Orbit Determination System, DODS (ref- Yes Yes Yeserence 24). Same formulation as DC Program, ref-erence 2, but implementation differs (reference 25)
NAP-3 (reference 2::;) Yes Yes Yes
Goddard Trajectory Determination System, GTDS, Yes Yes YesTechniques not yet finalized
The analytic corrections in these programs assume that the proper•correction to the observations E, R, and R for the net effect of the ionosphere is to
decrea.se E by the positive quaotity AE, to decrease R by the positive quantity~;
• ••• •and to decrease R by the quantity.6.R. Both Rand AR may be positive or negative,
"but both always have the same sign. Thus, the correction AR has the effect of al-•ways reducing the magnitude of R. To be compatible with these conventions for the
analytic corrections. the group option in the REEK ray-trace program was always
used. A discussion of the meaning of the group option in REEK is given in reference
28, and an analysis of the ionospheric correction for the GRARR system defining
when to use group and when to use phase corrections, is given in reference 29, The•.6.E, AR., and AR ionospheric refraction correction formulas used with GEOS-1 and
GEOS-2 data are listed in Tables 2, 3, and 4, respectively.
2
Tab
le2
Ele
vat
ion
Ref
ract
ion
Co
rrec
tio
nF
orm
ula
s
Ref
ract
ion
Eq
uat
ion
Ele
vat
ion
Ang
leR
efra
ctio
nC
orr
ecti
on
l\E
Fo
rmu
lati
on
AE
(rad
ian
s)=
E-
Ec
.G
SF
CD
C&
DO
DS
Al
4A
Eio
5
GS
FC
Fre
eman
A4
AE
. 10
.-1
[X
lco
s<X
-X
.J
NA
P-3
A26
cos
2I
20\
+X
2-
2X1
X2
CO
Se
()
7
GT
DS
un
der
dev
elo
pm
ent
-N.
HeI
ME
1m
0
io=
h tct
nE
=-h
cm
Et
.
Tab
le3-
Ran
ge
Ref
ract
ion
Co
rrec
tio
nF
orm
ula
s.
Ref
ract
ion
Eq
uat
ion
Ran
ge
Ref
ract
ion
Co
rrec
tio
nd
RF
orm
ula
tio
n:6
R(m
eter
s)=
R-
Rc
1am
ER
E2
1/2
GS
FC
DC
&D
aDS
A2
a:~R
iot-(R
::o:~)
J(E
<10
0)
-G
SF
CD
C&
DaD
SA
2b4
----(£
-.>10~
---.
"5d
Rio
.
GS
FC
Fre
eman
A5
~Rio
[1-C
;:m)ct
n2 EJ-8
HN
.csc
El
GS
FC
GP
RO
(GD
AP
)A
7g
1m
_
[I+
.25
Hgctn2 El
]1
/21
+3
(Re+~-3
Hg)
Y-
-N[_
9_
(1_
-(h
s-h
-:-
)K
)+
45
9y.]
NA
P-3
A2
8im
16K
em
480
m
R2
1/2
[1-(R
e:Rh)
cohJ
GE
OV
AP
2d
R. 1
0
GT
DS
un
der
dev
elo
pm
ent
1--
dR
.:;:
-Ni
He
csc
E=
-M
ocs
cE
10
m
Tab
le4
Ran
geR
ate
Ref
ract
ion
Co
rrec
tio
nF
orm
ula
s
Ref
ract
ion
•E
qu
atio
nR
ang AR
ate
Ref
ract
ion
Co
rrec
tio
nc1
.RF
orm
ula
tio
nt1
(met
ers/
sec)
=R
-Rc -3
/2
2[(
)JG
SF
CD
C&
DaD
SA
3a4
•R
e.
3R
eco
sE
(E<
100
)'5
l\R
.(R
/hm
)sm
E1
-R
e+
hm
10
GS
FC
DC
&D
aDS
A3b
~l\
R.
(E>
100
)1
0
•t~(1-
Sin;J
(H:>~
GS
FC
Fre
eman
A6
l\R
. 10
[2
J•
1-
4f3 2
-8N
imH
gE
1ct
nE
1esc
E1
1+
22
1/2
(1+
4f3 2
ctn
E1)
GS
FC
GP
RO
(GD
AP
)A
82
[1
+(1
+4
;,2
ctn
2E
1)1/2
.]
•(B
eY
NA
P-3
A29
E(l
\R)
Re
+R
hsi
nE
cos
E
-_(R
oco
sE)
21
R+
Re
h•
GE
OV
AP
3l\
R. 1
0
GT
DS
un
der
dev
elo
pm
ent
l\R
io=
Nim
He
1E
ctn
Eesc
E=
Ma
(esc
E)
0d
t
The DC, DODS, FREEMAN, and GEOVAP formulas listed in Table 1•employ as a factor nominal or zero-order correction functions for AE, AR, and AR,
which are defined as
E - E = AE. =c 10
1-N. He
1m'h
t
-Mctn E =.-.£ ctn E
ht
U)
R - R = AR. = -Ni H e1
csc E = -M csc Ec 10 m 0
and the derivative of ~R.10
(2)
••• 1 • ' •R - R = AR. =+N. He E ctn E csc E =M E ctn E csc E (3)c 101m 0
where M = N. H e1
• Note that M is a multiplicative factor in all these zero-ordero 1m 0 h
correction functions. It can be shown that M = N. H e1 ~ (' t N. dh, when usingo 1m j 1
the Chapman model ionosphere. 0 ,
These AE. and AR. corrections result from the following assumptions:10 10 ,
(1) Modified Chapman ionosphere
(2) Flat layered, horizontally homogeneous ionosphere
(3) Satellite height, ht, large compared to scale height, H
(4) Negligible ray path bending
The GPRO and NAP-3 formulas are different in appearance from the DC,
DODS, Freeman, and GEOVAP formulas due to the choice of a parabolic ionospheric
model in GPRO by Willmann (references 4 and 22) and a joined topside exponential,
and bottomside biparabolic, ionospheric model in NAP-3 by Bent (reference 23) com
pared to the Chapman model used by DC, DODS, Freeman, and GEOVAP. The GPRO
and NAP-3 models enable the correction formulas, which involve integration along the
ray path, to be integrated in closed form, but at the resulting expense of more com
plicated formulas.
6
ill order to compare the analytic refraction correction equations, we need
typical values for Nim, hm , and H. These are obtained as follows:
The refraction index, to first order approximation, is given by (reference 11, eq 2.78)
(4)
therefore, the refractivity at any height is
40.25 N• eNi = t-Li - 1 = - f2 •
and the refractivity at the height of maximum electron density is
40.25 Nem . (5)
which provides one of the required multiplicative factors if Nem and f are known. The
scale height iIi kilometers is determined in this study from
H= ~ [30 + 0.2 (hm - 200)]
which provides the other required multiplicative factor if h is known.m
(6a)
The origin and validity of this type of linear relationship between Hand
h is unknown to us, but the first mention we found of it was by Charnow (1959), refmerence 27, who quotes the scale height as
(6b)
This formula for scale height was also quoted in reference 2 (DC program),
reference 24 (DODS program), and refer.ence 25 (DODS program).
A similar linear relationship was proposed by Wright (1960, 1961), ref
erences 30 and 31, based on ionograms from 11 stations collected during the period
January 1959 to February 1960.
Freeman (1965), reference 5, originally quotes the scale height as
H = ~ [30 + 0.2 (hm - 200)J
but, based on 292 ionograms at Pennsylvania State University, he reduces this by a
factor of 1.61 to give equation (6a).
7
It is not known how closely other ionospheres, removed in time or dis
tance from Freeman's sample set, adhere to equation (6a). However, since equation
(6a) does represent real ionospheres for this region, and according to Wright, a
similar relationship is valid for other regions, it was used to generate values of
scale height, H, for these comparisons. The factor of i in the DC and DODS cor-•rections for ~E, ~R, and ~R in Tables 2, 3, and 4 is due to our use of equation (6a)
rather than (6b) in the definition of scale height, H.
Values of Nand h were obtained from each of the midnight and noonem m
mean N (h) profiles given in reference 11, pages 128 and 129, for Newfoundland,e
Grand Bahama, and Huancayo over a period of 1 year (1959 to 1960) near a solar
cycle maximum. The lowest, average, and highest of the Nem and hm values ob
tained in this manner were then used with equations (5) and (6a) , with a frequency of
2 GHz, to obtain the low, average, and high values of N. and H which are listed1min Table 5.
The refraction correction is proportional to N. in all of the formulas. 1m(except NAP-3 for ~ E), and hence is inversely proportional to the square of the
operating frequency. By scaling, the results in this document can be applied at any
operating frequency; in particular, the N. scaling factors given in Table 6 should be1m
used for the GEOS 1 and 2 tracking frequencies.
The refraction correction i~ also proportional to H in the DC, DODS,
Freeman, and GEOVAP formulas. The scale height, H, which arises from the use of
the Chapman model in the derivation of these formulas, is determined by means of a
semiempirical function of the maximum refractivity height,· hm • The range of H from
Table 5 is from 76.667 meters to 150.000 meters; that is, the maximum is about
twice the minimum value. N. varies by a factor of 10, which means that the product1m
NimH can vary by a factor of 20, as shown in Table 5. It follows that for a given ele-. ..vation angle, the values of ~E, ~R, and aR can vary by a factor of about 20 due to
possible variations in the ionospheric parameters •
•CORRECTIONS FOR ~E, ~R, AND ~R FROM REEK RAY-TRACES
The REEK ray-trace program, for a given elevation angle, requires as
input the true range to the target and a refractivity profile. The true range to the
satellite was obtained for each of the specified true elevation angles from the simpli
fied orbit program described in Appendix B. The refractivity profiles used as input to
8
the REEK ray-trace program were generated from all nine combinations of the speci
fied values of the Nim and H listed in Table 5 using the Chapman model and a compati
ble Bent model.
The Chapman profile is defined as follows:
_ . (1 - Z _ e -Z)N. -N. e
1 1m
h - hZ = i m
H
(7)
(8)
hi = height of each profile point going from 112.5 km to 1325 km in12. 5 km steps
The refractivity was set to zero from h. = 0 to h. = 112 km.1 1
The three Chapman and compatible Bent refractivity profiles generated
from the low, average, and high pairs of N. and H, listed in Table 5 and shown in1m
Figure 1, illustrate the shape and possible variation in these profiles •
•The ray-trace values for AE, till, and till from the Chapman model are
listed in Table 7 for all nine of the N. ,H combinations given in Table 5. These1m
values are plotted in Figures 2, 3, and 4. Several features shown in these graphs are
of interest and are as follows:
a.
b.
They verify the prior conclusion that the observed variation in N.• 1m
has more effect on AE, till, alJ.d AR than the observed variation in H,
as inferred from equation (6a) and the observed variation in h .mBecause of this, only the low, average, and high combinations of N.
1m
and H given in Table 5 are used in the subsequent analytic and ray-
trace refraction correction graphs in Figures 5 through 13.
The ratio of maximum to minimum refraction correction due to a
change in elevation angle for a gi'{en ionosphere is about 100:1 for•
~, only about 3:1 for AR, and a1)out 10:1 for AR. These generali-
zations exclude the low probability elevation angles within a degree. . .or so of zenith where AE and Ali both approach zero. Normally AR
•approaches zero at the point of closest approach where E = 0, but for
an overhead pass (used here and described in appendix B) this occurs
at zenith. For the chosen 2-GHz frequency and for our average iono
sphere, the refraction corrections fall between O. 7 and 0.007 milli
degrees (2.5 and 0.025 arc-seconds) for AE, between 9 and 3 meters•for AR, and between 1. 3 and 0.13 em/sec for AR.
9
c. At an elevation of about 100, for both ~E and AR, the correction
curves cross over so that the thicker ionospheres with larger values
of scale height, H, require a larger refraction correction for AE
and AR above 100
and less correction below 100 than do thinner
ionospheres with the same N. but smaller H.1m
Table 5Low, Average, and High N. , h , and H Values (Near Solar Cycle Maximum)1m m
N -N. h H -N. Hem 1m m 1m
(1017
el/m3
) (10-6) (km) (km) (meters)
Low (3,3) 2.19 2.21 280 76.667 0.169
Average (2,2) 10.60 10.67 364 104.667 1.117
High (1,1) 23.99 24.14 500 150.000 3.621
Table 6Correction Factors for Other Track~ng Systems
System FrequencyMultiplication
Factor for N.1m
Up (MHz) Down (MHz) Up/DownEquivalent (MHz)
C-band 5690 5765 5727 0.122
TRANET 108 342.936(NSSDC data)
162 I 152.416324 * 38.104972 4.234
Secor 420.9 224.5 280.18 50.972420.9 449.0 434.27 21. 210
420.9 '224.5 434.27 21. 210'l.. 449.0 J1. GRARR*
Channel A 2271. 9328 1705.000 1928.6 1. 075Channel C 2270.1328 1705.000 1928.0 1. 076
2. GRARR**Channel A 1799.2 2253 1988.3 1. 012Channel C 1801. 0 2253 1989.5 1.011
3. GRARR(GEOS-C) 2074.6375 2253 2158.3 0.859Proposed
*For GEOS-1 and GEOS-2**Between GEOS-2 and GEOS-C
10
Tab
le7
RE
EK
Ra
y-T
race
Val
ues
Fro
mC
hap
man
Mod
el
H7
6.6
67
76
.66
77
6.6
67
10
4.6
67
10
4.6
67
10
4.6
67
150
150
150
(km
)N
im2
.21
10
.67
24
.14
2.2
11
0.6
72
4.1
42
.21
10
.67
24
.i4
(10_
6 )
Ele
vati
onA
E~E
~E
~E
~E
~E
~E
~E
~E
(deg
)(m
lll!
deg)
(mll
lide
g)(m
ilU
deg)
(miU
ide
g)(m
illi
deg)
(mil
l!de
g)(m
illl
deg)
(mil
l!do
g)(m
llll
deg)
.0.1
50
.15
03
0.7
28
41
.64
79
0.1
39
80
.67
73
1.5
324
0.1
25
70
.60
89
1.3
77
4
1.5
00
.15
38
0.7
45
41
.68
64
0.1
43
50
.69
50
1.5
72
30
.12
93
0.6
26
3I
1.4
16
8
15
.00
0.0
96
60
.46
78
1.0
580
0.1
04
50
.50
60
1.1
44
60
.10
97
0.5
31
1i
1.2
013
,3
0.0
00
.04
41
0.2
13
60
.48
33
0.0
53
80
.26
07
0.5
89
80
.06
53
0.3
16
70
.71
64
45
.00
0.0
23
40
.11
34
0.2
56
50
.03
00
0.1
45
50
.32
91
0.0
39
00
.18
91
0.4
27
8
60
.00
0.0
12
80
.06
23
0.1
40
90
.01
68
0.0
81
60
.18
47
0.0
22
60
.10
95
0.2
47
7
90
.00
0.0
00
00
.00
00
0.0
00
00
.00
00
0.0
00
00
.00
00
0.0
00
00
.00
00
0.0
00
0
taev
atlO
n~R
(m)
~R(m)
~R(m)
t\R
(m)
~R(m)
AR
(m)
t\R
(m)
t\R
(m)
"R
(m)
(deg
)
0.1
51
.538
7.4
59
16
.87
81
.86
09
.02
12
0.4
11
2.3
05
11
.18
12
5.2
99
1.5
01
.53
27
.43
21
6.8
16
1.8
55
8.9
95
20
.35
32
.30
11
1.1
58
25
.24
7
15
.00
1.1
76
5.7
05
12
.90
81
.49
67
.25
71
6.4
19
1.9
519
.46
42
1.4
12
30
.00
0.8
11
3.9
33
8.8
99
1.0
76
5.2
17
11
.803
1.4
757
.15
41
6.1
85
45
.00
0.6
20
3.0
07
6.8
03
0.8
37
4.0
58
9.1
81
1.1
75
5.6
97
1'l
.88
8
60
.00
0.5
21
2.5
28
5.7
20
0.7
09
3.4
37
7.7
76
1.0
064
.87
71
1.0
34
90
.00
0.4
58
2.2
23
5.0
30
0.6
26
3.0
35
6.8
67
0.8
94
4.3
34
9.8
05
Ele
vati
ont\
R(e
m/s
ec)
~R
(em
/sec)
t\R
(em
/sec)
t\R
(em
/sec)
aR
(em
/sec)
L\R
(em
/sec)
t\R
(em
/sec)
t\R
(em
/sec)
~R
(em
/sec)
(deg
)
0.1
5-
0.0
32
20
.00
80
0.0
15
40
.23
98
0.0
08
00
.01
45
0.1
36
0-
1.5
00
.04
76
0.2
46
60
.52
75
0.0
42
00
.22
47
0.4
85
00
.03
95
0.1
95
40
.42
07
15
.00
0.2
55
21
.23
72
2.8
00
20
.27
60
1.3
392
3.0
30
00
.29
05
1.4
098
3.1
88
4
30
.00
0.2
28
71
.10
98
2.5
10
70
.27
86
1.3
51
23
.05
70
0.3
37
81
.639
03
.70
78
45
.00
0.1
71
90
.83
41
1.8
868
0.2
20
31
.17
14
2.4
18
60
.28
59
1.3
867
3.1
37
8
60
.00
0.1
15
60
.56
06
1.2
685
0.1
51
40
.73
43
1.6
61
20
.20
27
0.9
83
42
.22
58
90
.00
0.0
00
10
.00
03
0.0
01
10
.00
01
0.0
00
60
.00
19
0.0
00
10
.00
11
0.0
01
7
•d. The AR corrections for the ionosphere are maximum for an over-
head pass at elevation angles between 150
, for the thinner iono
spheres (low H), and 300 for the thicker ionospheres (high H). The. .AR values decrease for both lower and higher elevation angles in
sharp contrast to their behavior in the troposphere. The decrease•in A R for lower elevation angles contradicts the results by Millman
(reference 32). Weisbrod (reference 33) predicts that the AE iono
spheric refraction correction also decreases for lower elevation
angles. This is also supported by our curves.
DERIVATION OF COMPATIBLE BENT PROFILES FROM THE CHAPMAN PROFILES
The Bent profile (reference 23) is given in Appendix A, page A-I0 as
an exponential function of the parameters N. and k for the topSide ionosphere and as1ma biparabolic function of the parameters N. and y for the F-Iayer and bottomside1m mionosphere. The NAP-3 formulas "assume the Bent model and therefore incorporate
the Bent model parameters. The NAP-3 formulas were compared with REEK ray
traces through some selected Bent ionospheric profiles as well as the Chapman pro
files to determine whether the use of the different models has a significant effect.
The different model profiles were made compatible by constraining the total electron
content or total refractivity integral or zenith A R corrections to be the same.
S~ S\ 1Thus N. (Bent) dh = N. (Chapman) dh ~ N. Heo 1 0 1 1m
Since the total electron content is the dominant feature of the ionosphere
with regard to refraction corrections, this constraint reduces the differences between
the models to second order effects due to shape differences.
It is also assumed that the parameters N. and h are the same for both1m m .models.
The Bent model topside profile is represented by an exponential decay
function with scaling coefficient k=Hb
• The Chapman model profile rapidly
approaches an exponential decay function with scaling coefficient H at heights of
several H above hm
• Therefore, we somewhat arbitrarily chose Hb
=H.
Due to the above assumptions, we have defined the parameters
Bent ~im' hm , Hb] =Chapman [Nim , hm , HJ.
12,
The only remaining undefined parameter is y ,the thickness parameter for the Bentm
F-Iayer and bottomside model. Values of ym were cho~en to satisfy the total electron
content equality constraint above. The effects of this method for defining the compati
ble Bent refractivity profiles are shown in Figure I, where a slight downward shift of
the compatible Bent profile, relative to the reference Chapman profile, is observed•
•The values for 6E, ~R, and ~R obtained by REEK ray-traces through the
Bent profiles are plotted in Figures 5 through 13 for comparison with the NAP-3 for
mulas and with the ray-trace through the Chapman profiles.
COMPARISON OF ANALYTIC FORMULAS AND REEK RAY-TRACES
•Values AE, 6R, and An (calculated from the analytic formulas given in
Tables 2, 3, and 4) are plotted in Figures 5 through 13. The values of the Chapman
ionospheric parameters N. ,h ,and H given in Table 5 are used in the DC, DODS,1m mFreeman, and GEOVAP formulas. For GPRO the same N. and h parameter values
1m mare used, but the thickness parameter for the GPRO parabolic ionosphere is kept con-
stant at H = 50 km, as in the normal usage of GPRO. For NAP-3, the same N. ,. g 1mh ,and H = H
bparameter values are used and, in addition, y is determined as de-m m
scribed above so that ~R (E = 900), calculated from Bent's analytic formula (A27 or
A28), is the same .6R (E = 900) determined from the REEK ray-t~aces. The same is
• 0true for .6R (E = 90 ). The parameter values used are given below:
-N.h H ~1m
H Ym(10-6)
m g(km) (km) (km) (km) (km)
Low 2.21 280 76.667 50 76.667 172.833
Average 10.67 364 104.667 50 104.667 235.944
. High 24.14 500 150.000 50 150.000 337.603
These same parameters are used in generating the Chapman and Bent pro
files for input into the REEK ray-tracing program. The results of these ray-traces
are also shown in Figures 5 th.rough 13. There is only a slight difference between
the REEK ray-trace through the Chapman and compatible Bent profiles at low ele
vation angles, as can be seen from Figures 5 through 13.
The DC/DODS correction values are all 20% below the zero order cor
rection Do E. , Do R. , Do R. for elevation angles above 100, and about 20% below the
ill ill ill 4ray-trace corrections for angles below 100
• This is due to the ;::' factor explained. u
13
earlier in connection with the semiempirlcal equation for H. If equation (6b) rather
than (6a) had been used to define H for the Chapman profiles in the REEK ray-traces,
the DC/DaDS corrections for E <100 would have closely agreed with these ray-traces.o 0
Furthermore, if the DC/DaDS formula for E < 10 had also been used for E > 10 ,
the DC/DaDS corrections for E >100 would have better agreed with the ray-traces.
This can be seen from the dotted extensions of the DC (E <100) graphs. Multiplication
of the DC/DaDS curves by the factor ~ is equivalent to llssuming the same total
electron content for the DC/DaDS and REEK ray-trace ionospheres. This was done
for the selected elevation angles E = 1.50, 60
, 150, and 900 to show the resulting
good agreement between the DC/DaDS and the ray-trace results in Figure 5-13.
The Freeman correction for A R deteriorates rapidly for E < 300, as
. .Freeman acknowledges in reference 5. His correction for A R deteriorates for
E < 500
•
The GPRO results are quite good for our low values of the ionospheric
parameters N. and h ,but, for our average and high values of N. and h ,the1m m 1m mGPRO results are consistently too low. This is most likely due to the use of a con-
stant value of H = 50 km for the parabolic model thickness parameter. If the Hg gparameter were allowed to adjust, to keep the total electron content in the Willmann
GPRO model equivalent to. that in the Chapman profile used in RE'EK, the GPRO
results would match the ray-trace results' more closely for all cases. This can be
accomplished by setting:
thene
H = (-) Hg 4 rather than 50 km
For the chosen values of H, the new values of H are calculated belowg •
along with the proper multiplicative factors to adjust AR and AR.
Multiplicative Factor for
H = (e/4) H •H 6R and ~R(km) g (km) H
m=J50
Low 76.667 52.101 1. 0420
Average 104.667 71. 129 1. 4226
High 150.000 101. 936 2.0387
14
Applying these multiplicative factors to the GPRO graphs for A R and A:R
for the selected values of E = 1. 50, 60, 15
0, "and 900 yields the improved values indi
cated in Figures 8 through 13.
The NAP-3 values for.AE 'are consistently lower than the REEK ray-traces•for all elevations. The NAP-3 values for AR and .6R were forced to agree with ray-
trace values at E = 900, but are consistently higher than the ray-traces for all other
elevation angles. Although the magnitudes of the NAP-3 corrections do not agree with
the REEK ray-traces, the NAP-3 correction curves, in general, match the shapes of
the ray-trace curves. A modified version of the NAP-3 ionospheric refr,action for
mulas will soon be available and may improve these results.
Assuming the same total electron content, it can be shown that N. ' H e1
=1m
40.3 NT/r, and the only difference between NAP-3 (equation A27) and DC (E < 100
)
(equation A2a) formulas for 6R is in the use of Rh
= h - y in NAP-3 compared tom m
h alone in DC. In Figure A-3, if the line from the center of the earth were drawn tom
intersect the ray-line at h within the ionosphere rather than at h' - y at the bottomm m mof the ionosphere, the resulting NAP-3 formula for aR would be identical to the
DC (E < 100) formula for AR and would match the REEK ray-trace better, assuming
the NAP-3, DC/DODS, and REEK ionosphere all have the same total electron content••The same conclusions apply to the NAP-3 form.ulas for AR••
We have also tested the preliminary versions of the GTDS formulas.
However, since these formulas are still under development, the results are not
reported here.
DEVIATION AMONG REFRACTION CORRECTIONS
For elevation angles above about 100, where most satellite tracking is
done, the refraction error curves in Figures 5 through 13, deviate in general, from
each other less than at lower angles.
The maximum deviations among the corrections for the average iono
spheres in Figures 6, 9, and 12 are given in Table 8 for selected angles. The Freeman
corrections are not included below E = 300, since Freeman specifically warns 'against
it in reference 5. As explained earlier, some of the spread among the computed re
fraction corrections could be reduced by using different values of the thickness param
eters H, Hg, ~, Ym or by a different choice of effective refraction height of the iono
sphere.
, 15
Table 8Maximum Deviation Among Refraction Corrections
OAR •E OL\E OAR(degrees) (millidegrees) (meters) (cm/sec)
6 0.828 22.34 29.79
10 0.390 11.30 11.29
15 0.223 6.27 5.16
20 0.168 4.09 2.94
30 0.128 2.30 1.95
40 0.103 1.60 0.79
60 0.058 1".07 0.35
90 0.000 0.90 0.00
•RELATIONSHIPS BETWEEN REFRACTION CORRECTIONS FOR AE, AR, AND AR
NOMINAL ZERO ORDER CORRECTIONS
Several relations connecting the nominal refra•..;tion correction formulas•for AE, AR, and AR are apparent by inspection of these formulas. they are
-N. H e 1A E. =__lm-:--__
10 ht
A Rioctn E =-- cos Eh
t
1~R. =-N. He csc E = h
tDo E. sec E
10 1m 10
G 1 • . •AR. .= +N. He E csc E ctn E =- AR. E ctn E =
10 ~ 10
o .- ht E AE. sec E ctn E =
10 .
HIGHER, ORDER CORRECTIONS
d (A Rio) • d (A Rio)dt = E dE
It is of interest to determine whether relationships similar to the above
also hold for the higher order GEOS ionospheric refraction correction formuals for AE,•.o.R, and AR in Tables 2, 3, and 4. The only formula for AKwhich is more com-
plicated than the nominal formula is the NAP-3 fonnula. There seems to be no simp.le
relation betweenAE and AR for NAP-3.
16
It can easily be shown that for all the GEOS ionospheric refraction correc•tion formulas in Tables 3 and 4, the range rate refraction correction formula for AR
is simply the time derivative of the formula for AR, as might be expected for consis-•tent levels of approximation for .6.R and AR.
Th AR• = d ( AR) = E d (AR)us, dt dE
holds for all formulas. This is not generally stated in the earlier source documents•where the formulas for AR are derived more laboriously from basic principles.
The GEOS range rate measurements are all derived by determining the
number of doppler cycles in a short time interval. The number of doppler cycles in
this time interval correspopds to the range shift in satellite position relative, to
the station during the same time interval, where range shift is measured in wave
lengths of the transmitter frequency. The measured range shift includes the dif
ferential effect of the atmosphere on the doppler cycles at the beginning and end
points of spacecraft position corresponding to the beginning and end points of the
shift measurement time interval. Therefore, it is reasonable that the range rate re
fraction correction should be expressed as the difference in range refraction cor
rection divided by the corresponding differen'ce in time, or as a derivative, with
respect to time, of the range refraction correction formula.
SUMMARY
The purpose of this study was to compare the different ionospheric re- .•fraction correction techniques used with GEOS E, R, and R data to determine whether
significant deviations exist among these techniques. If deviations exist, it is difficult
to know which technique gives the best results, since no perfect model of the iono
sphere is available. However, REEK ray-trace results through typical Chapman and
Bent model ionosphere profiles were included as references for comparison of the
results from the various formulas and should indicate which formulas are best.
The following statements summarize the results of this study:
1. Some of the GEOS ionospheric refraction correction formulas and the•REEK ray-trace results for AE, AR, and AR are in poor agreement
for E <300• The zero order corrections ~R. and AR. tend to over-
10 10
correct at all elevation angles, becoming worse at the lower elevation
angles. At E = 100, AR. is twice the REEK ray-trace value and
• 10AR. is ten times the REEK value, so that no correction at all is
10
17
preferable to the zero order corrections. A E. also tends to over-o ill
correct for E <10 , but under-corrects for E> 100 and matches REEK
at about 100
•
2. For E >100, where most satellite tracking is done, the differences in
•AE, AR, and AR are insignificant in comparison to most operational
requirements. However, for the GEOS Observation Systems Inter
comparison Investigation (GOSII), these differences (refer to Table 7)
are larger in size than observed differences in system biases for
some of the best electronic geodetic tracking systems. Therefore,
the systematic differences in refraction correction techniques are
affecting the GOSII results.
3. For all the GEOS ionospheric refraction correction formulas ex-•amined in this report, the range rate refraction correction for AR
is the time derivative of the formula for AR. In the earlier source•documents, the AR and AR formulas are derived independently.
4. Assuming the same total electron content, the NAP-3 and DC/DODS•corrections for AR and AR are identical except for the use·of an
effective ionosphere height of Rh = hm - Ym in NAP-~ compared to
h alone in DC/DODS. This may account for the differences betweenm •
NAP-3 and REEK.
•5. The REEK ray-traces show that the ionospheric correction for AR
has a strong maximum between 150 and 300 elevation, contrary to the
tropospheric correction and contrary to some reports in the literature•
•6. All of the ionospheric correction formulas for AE, AR, and AR
depend on the integral of the refractivity or electron content. In·
all the ionospheric models discussed here (Chapman, Willmann,
and Bent) this integral depends on the product of N. and some. 1m
thickness parameter. Reasonably accurate values of N. and h1m m
based on bottomside ionograms, are available. The thickness param-
eter H used in the formulas based on the Chapman model is not well
known. In fact, the DC/DODS semiempiricallinear relationship used
to derive H from hm differs from the Freeman version by a factor
of 20%. The uncertainties in specifying the thickness parameters can
theoretically be reduced by means of topside ionograms or Faraday
rotation measurements, or multiple frequency techniques such as those
employed by the Secor and TRANET systems.
18
7. The DC/DaDS (E < 100
) corrections programmed for AR and ~Roshould also be used for E >10 . The currently programmed correc-
tions for E >100 are flat earth approximations which lead to sig
nificant errors above 100•
8. The DC/DODS elevation angle correction has the proper magnitude
for a single path angle correction, not the dual path interferometer
correction intended. The interferometer correction is smaller in
magnitude, and opposite in sign, and may be obtained from any of•the range rate (Ll R) correction equations, as explained in ref-
erence 1.
9. All of the formulas compared in this report assume the satellite is
well above the bulk of the ionosphere. This is a good assumption for
GEOS-1 and 2 whose perigees are both above 1080 km. However,
many satellites operate within the ionosphere with electrons above
as well as below the orbit. In these cases, the ionospheric refrac':'
tion correction formulas based on the total electron content will in
general be in error to the extent that it may be preferable not to make
corrections. Some programs, such as NAP-3, prOVide a separate
formulation for this case.
19
10
0L
.._.
....
JI..
..._
...J
...L_
_..L.~_..I..
.L..._
_.L
-__
L..
.._
_..
..I._
_-L
.--:
"_
..J-
...L...::~_...
...':"
'""--
.I
109
20
0
10
00
"N
.-6
=2
4.1
4x
10
,1m
N.
-6,
h m=
36
4kI
n=
2.2
1x
10h
m=
50
0km
1m
,9
00
.....h
=28
0km
,H
=1
04
.7km
H=
15
0.0
km
m.....
H=
76
.7kI
n,
.....,
",
.....8
00
......
,.....
,.....
",
"C
HA
PM
AN
.....7
00
......
,,
,en
/',
I',
0:: W ~ W :::;;
60
0B
EN
T....
..,
BE
NT
"0 ~ ~
......
,~
,,
:I:
,~
50
0,
"w
~:I
:.....
. ....... ,
I:\:
),
04
00
...... .....
. ~
30
0
REF
RA
CTI
VIT
Y
52
9-1
Fig
ure
1.
Ch
apm
anan
dC
om
pat
ible
Ben
tR
efra
ctiv
ity
Pro
file
s
Figure 2, Elevation Error Due to Ionospheric Refraction
21
40
f = 2 GHz50
(1.1)
(2,1)20~-...;...;;...;.--~--------
(3,1)
-6( .1) = 24.14 x 10
( .2) =10.67 x 10-6
( .3) -2.21 x 10-6(3. ) - 76.667 km
(1. ) - 150.000 km
(2. ) - 104.667 km
10
0,7
1.0
0.9
0.6
O.G
0.5
0.4"1-----7"---t--~_!~+~__:__:_!::-------l:----+.:---~~l:---:b..J~~~.I 678910
ELEVATION ANGLE (DEGREES)
(1.2)
(2,2)91--:-:~-- _
~ 81 (3,2)
~ 71:'"----------------III2;- 6II:oII: 5II:
11I11Io:~ 4
'j"i=~~II: e: 5
~~
ll! ~ 1_-:(1:,_3:)~---------------~: 2 (2,3)
~. r--~~---:------~-Cl: (3,3)II:
(H. Nim)
Figure 3. Range Error Due to Ionospheric Refraction
22
h =1,333,333 METERSTO TARGET
4 f= 2 GHz
(1. ) .. 150.000 km
(2, ) .. 104,667 km
(3. ).. 76.667 km
( ,I) = 24.14 x 10-6
(,2) = 10.67 x 10-6
( .3)"2.21 x 10-6
~7'---:'-.(3, 2)
.r~~---- (2, 2)
,,-----(1,2)
(3,3) .,/'/
(2, 3)----..:;~
0.05
0.04
1.0
0.9
0.8
0.7
0,06
0.02
0.6
u.... 0.5
i~-!0::"00
<I ~"0 ....
W~~-0::1-I-~10::~ ...o::l:!~ ....l::~0::0::...~ 0.10
~ 0.09
0.08
0.07
0.01 ~,------:-----:-----:!:--~-=__;__:__:__:::_----_;:;;__--_;;;_-~-7;;_i;;"""'~;;';;_:~
Figure 4. Range Rate Error Due to Ionospheric Refraction
23
REEK (CHAPMAN)
NAP-3
w:J!..~ .280 km
-Ntm '" 2.21 x 10-6
0.004
0.003
0.001~ """'_-....I_....L.--J~.L....L...I....L...l.... ....L._-....I_....L._L-"-L.....I....L...l.... ~_--:":-~:--:~~~~
0.1 0.3 0.4 0.6 0.6 0.70.60.91.0
1.00.90.90.7
o.e0.5
0.4
0.3 REEK (BENT)
0.2
~IIIII:
III 0.10C 0.09::::i 0.09....I
0.07i0.06
0.002
15 0.06
~III 0.04
Z
o 0.03
~e:III 0.02II:
~1=~.~. 0.010III 0.D09
0.0090.007
0.006
0.006
Figure 5. Elevation Error Due to Ionospheric Refraction
24
10987
6
4
;;;WW0::C)W0...J...J
i 1.09
0:: 800:: 70::W 6Z0i=0 4<l:0::lLW0::
ZQI-
~W...JW
0.1098
7
6
5
4
REEK (BENT)
REEK (CHAPMAN)
NAP-3
AVERAGE
h = 364.000 kmm
-N. = 10.67 x 10-61m
FREEMAN/6Eio
Figure 6. Elevation Error Due to Ionospheric Refraction
25
10987
8
4
REEK (BENT)
;;;l4Jl4JII:(!) REEK (CHAPMAN)l4J 1.00 0.9..I 0.8::!~
0.7
0.8 NAP-3II:0 0.5II:II:l4J 0.4Z0~ 0.3UC(II:....IIJ 0.2II:
Z0
S>IIJ..I 0.10 HIGHIIJ 0.09
0.080.07 hm = 500 km0.08
-60.05 -Nim = 24.14 x 10
0.04
0.03
0,02
Figure 7. Elevation Error Due to Ionospheric Refraction
26
0.4
FREEMAN
GEOVAP/4R.10
NAP-3
LOW
h = 280 kinm
-N. = 2.21 x 10-61m
DC/DaDS «10°)./
REEK (CHAPMAN)
3.0
4.0
7.0
6.0
5.0
0.3
0.2
10.0
9.0
8.0
0.25
Cil 2.S!---------...:l...__II:~ REEK (BENT)
~ 2.0 ~DC \
! I.SE=:~~~===;;;;;;;;;;;;;;;;~7;;;;i~~;;;;;;;;;;;~:~oi=o<lII: '.0IL.'" 0.9II:
'" 0.8l!>
~ 0.7II:o 0.6<5<lII: O.S
O.IS
0·11~.0---'.l...5--2.l...0-~2~.5~3.0~--4.0.L--5D'---6.0.L--l7.-0-8.l..D-9..l..0....L.,0---.l..IS--.l..20--2.l..S-...L30--...L40---...LSO--l6L..0--...L70-80L.-90J.....J,OO
ELEVATION ANGLE (DEGREES)
Figure 8. Range Error Due to Ionospheric Refraction
27
AVERAGE
h = 364.000 Ionm
-N. =10.67xlo-61m
GEOVAP/AR10
NAP-3
REEK (BENT)
so
40
80
80
100
90
80
10
iii 20Ill:ILl~ILl:::E 18r------~--- __~Ill:oIll:Ill:ILlZ 10
~ 9~==;;;=~~~~~=======1F~!t::::::::::U~~ eL~ 7p~t:.±:====:;;::=;;Z=~~~~:_-ilLI e .2-DC GPRO/!i 4~ 5 DC/DaDS « 10~Qc~ 4
Figure 9. Range Error Due to Ionospheric Refraction
28
100
90
80
70
60
50
40
30
25
'iiia: 20l&J....l&J~
15a:0a:a:l&J
Z '00f= 90cl: 8a:u.l&J 7a:l&J 6
'"Zcl:a: 500cl: 4a:
2.5
REEK (CPAPMAN)
m (GPRO)
DC/DaDS «10°)
FREEMAN
HIGH
h =500 kInm
-N. = 24.14 x 10-6un
Figure 10. Range Error Due to Ionospheric Refraction
29
10981
4
LOW
h =280 kmm
-6-Nim = 2.21 x 10
u..., 1.0Ul 0.9......2 0.8~ 0.7II: 0.8iII: O.S...,Z 0.42t-U 0.3
~II!..., 0.2
~II:...,(!)ZetII: .0.10
0,D90.08
0.07
0.06
0.08
0.04
0.03
0.02
1'\...00,_- DC/DODS (>10°)
!4.~5 io
O.OIL-_____JL_~..L..._L_.L..._L...JL...l...L.J .....L_ ___JL_...L.__1__1_L....L...I..I. ....L.__L_....L._.I......J~...............
0.1 0.2 2.0 3.0 4.0 tID 60 1.0 8D!IO 10 20 30 40 SO 60 70 8090 100ELEVATION ANGLE (DEGREES)
Figure 11. Range Rate Error Due to Ionospheric Refraction
30
GEOVAP/t.Rio
DCjDODS( > 10°)
~tb. Rio
AVERAGE
h = 364.000 kInm
-6-N. =10.67xlO1m
'098
7
6
•
E 1.00.9
~ 0.8:E~
0.7
a:: 0.80a:: 0.5a::WZ 0••0i=<.l 0.3
d(!\R)REEK (BENT)<[a::II. dtwa::w 0.2 2.DC!;( 4a::
d(!\R)REEK(CHAPMAmw<:>z dt<[
0.10a::0.090.08
0.07
0.08
0.08
0.0.
0.03
0.02
0.0~.J"I---......L"-----,J....~"""",,J....,.L-,~I,,+~---....I--......L_-1-......L--l.....l....I....J.,.J.,------I2L..O-.....J~--40l....-.J80.....J60W70'-L80.J90..J'OO
ELEVATION ANGLE (DEGREES)
Figure 12. Range Rate Error Due to Ionospheric Refraction
31
10090
8070
.60
50
40
30
20
~(/) 10..... 9~ 8~ 7II:~ 6II:IAl
~ 4i=~ 3
e:IAlII:IAl 2
~II:
IAl(l)ZC(II: I,D
0.9
".80.7
0.6
0.5
0.3
0.2
HIGH
h =500 kmm-6-N
im= 24.14 x 10
•GEOVAP/6Rio
Figure 13. Range Rate Error Due to Ionospheric Refraction
32
REFERENCES
1. Berbert, J. H., and Parker, H. C., "GEOS Satellite Tracking Corrections for
Refraction in the Troposphere, " GSFC Document X-514-70-55, February 1970.
2. Cole, Isabella J., "F116, 1, m, p. pCorrector for Ionosphere Refraction, "
AOPB Systems Manual Program Description, November, 1965.
3. Norton, Kenneth A., "Effects of Tropospheric Refraction in Earth-Space Links,"
Invited Paper by Kenneth A. Norton for the XIVth General Assembly of U.R.S.I.
held in Japan, September, 1963.
4. Lynn, Joe J., "A Users Guide for GEOS Data Adjustment Program, " September
1, 1971.
5. Freeman, J. J., "Final Report on Ionospheric Correction to Tracking Param
eters, " Submitted to Goddard Space Flight Center in fulfillment of Contract No.
NAS 5-9782, November 3, 1965.
6. Berbert, John H., "Interim GRARR Corrections at GSFC," A memorandum to
the Working Committee on the Statistical Combination of Satellite Observational
Data, May 15, 1967.
7. Schmid, P. E., "Atmospheric Tracking Errors at S-and C-Band Frequencies, "
NASA TN D-3470, August 1966.
8. "Interim Status Report on Program Development and GEOS-A Data Analysis, "
Contract No. NAS 5-99756-44A, 55, 71. Prepared by the Applied Sciences
Department, Wolf Research and Development Corporation.
9. Phister, W., and Keneshea, T. J., "Ionospheric Effects on Positioning of
Vehicles at High Altitudes," AFCRC-TN-56-203, Bedford, Mass.: Air Force
Cambridge Research Center, Ionospheric Physics Laboratory, March, 1956.
10. Bean, B. R., Horn, J. D., and Ozanich, A. M., Jr., "Climatic Charts and
Data of the Radio Refractive Index for the United States and the World, " National
Bureau of Standards Monograph No. 22, November 25, 1960.
11. Davies, K., "Ionospheric Radio Propagation, " NBS Monograph #80, April 1,
1965.
12. "Analysis of Ionograms for Electron Density Profile," Volume 2, #10 of Radio
Science. Publ. by ESSA, October 1967.
33
13. "Ray Tracing," Volume 3, #1 of Radio Science. Publ. by ESSA, January 1968.
14. Moss, S. J., and Wells, W. T., "Analysis of the GSFC Laser Range Data."
Contract No. NAS 5-9756-42, June 16, 1967.
15. Lehr, C. G., Maestre, L. A., and Anderson, P. H., "A Ruby-Laser System
for Satellite Ranging, " October 16, 1967.
16. "NONAME, An Orbit and Geodetic Parameter Estimation System," Volume II,
Contract No. NAS 5-9756-710, August 1968. Prepared by the Applied Science
Department, Wolf Research and Development Corporation.
17. Gross III, J. E., "Preprocessing Electronic Satellite Observations," Contract
No. NGR 36-008-093, Ohio State University, Dept. of Geod. Sci. Rpt. #100, March
1968. .
18.· Nelson, W. C., and Loft, E. E., "Space Mechanics" Prentice-Hall, Inc., 196~.
19. "Network Analysis Program Phase II, Mathematical Analysis Documentation,"
page 102, Contract No. NAS 5-10588, dated May 15, 1969.
20. "Documentation of ASTRO Mathematical Processes," NWL Technical Report
TR-2159, May 1968.
21. Willmann, J. B., "Refraction Correction," unpublished report. DBA Systems,
Inc., May 1969.
22. Loveless, F. M., "Intercomparison of 10 Selected GEOS-A Passes," Contract
No. NAS 5-10618, October 1970.
23. "D~scriptionof the Equations and Formulas used in the Development of the
NAP-3 Ionospheric Model and the Related Range, Range-Rate and Elevation
Angle Corrections" Contract No. NAS 5-17730, dated 28 October 1970. DBA
Systems, Inc.
24. Schmid, P. E., "Description of the 1965-1971 Ionospheric Model used in the
Definitive Orbit Determination System (DODS), "GSFC Memo, November 1970.
25. Bent, R. B., and Llewellyn, S. K., "A Criticism of the 1965-1971 DODS
Ionospheric Model, " Contract No. NAS 5-11730, dated January 1971.
26. "GEOS-B GRARR Data Validation Procedure," GSFC Document prepared by
RCA for Code 514, November 1968.
27. Charnow, M., "Comparison of Ionospheric Refraction Corrections Computed by
Various Formulas, "U.S.N~W.L. Tech. Memo No. K-22!29, November 1959.
34
28.
29.
30.
31.
32.
33.
Mallinckrodt, A. J., "Group and Phase Phenomena in an Inhomogeneous Iono
sphere," GSFC Document X-552-71-171, April 1971.
Mallinckrodt, A. J., Berbert, J. H., and Parker, H. C" "Analysis of Iono
spheric Refraction Error Corrections for GRARR Systems ," GSFC Document
X-552-71-170, April 1971.
Wright, J. W., "Comment on Models of the Ionosphere above h F2 ,"maxJ. Geophys. Research, 65, 2595-2596, 1960.
Wright, J. W. , "Diurnal and Seasonal Changes in Structure of the Mid
Latitude Quiet Ionosphere," J. Research of the NBS, Vol. 66D, No.3, May
June 1962.
Millman, G. H., "Atmosphere and Extraterrestrial Effects on Radio Wave .
Propagation," General Electric Technical Information Series No. R61EMH29,
June 1961.
Weisbrod, S., and Colin, L., "Refraction of VHF Signals at Ionospheric
Heights," Professional Group on Antennas and Propagation, August 25, 1959.
35/36
APPENDIX A
REFRACTION CORRECTION FORMULATIONS
APPENDIX A
REFRACTION CORRECTION FORMUIA TIONS
A.1 Different groups using data from the GEOS geodetic tracking systems em
ploy different refraction correction equations. This appendix lists some of the differ
ent refraction correction equations, and describes how they are being used. For com
parison of the equations, the original forms and notations are converted into a common
form and notation.
A.2 GSFC ORBIT DIFFERENTIAL CORRECTION PROGRAM (DC)
The GSFC Orbit Differential Correction Program (DC) was developed by the
Advanced Orbit Programming Branch at GSFC. It is used for operational orbit up
dating and contains optional refraction correction formulas for interferometer direction
cosines and for range and range rate. It is documented in reference 2, program
No. F1l6.
A.2.1 DC MINITRACK DIRECTION COSINE CORRECTION OR ELEVATIONANGLE CORRECTION
The direction cosine corrections are
where
i =..1- andm =mc q c q
i, m = measured direction cosines
R, ,m = refraction corrected direction cosinesc c
1 + Nq = --:-::"0o..._-:;---:-:-
N1 (h, t)1 + A - ---:---
S ht .
equation (2), F1l6
equation (4), F1l6
A = N = (It - 1) = tropospheric refractivity at the station; a constants sfor a given station for a given month. For example,
a typical N = 0.000313.s
1 + N = 1o
A-1
Keeping the io~ospheric part
+iN 1 (h, t)d£=1-i = h
c t
+mNl (h, t)d m = m - m = ----..,;;----c ht
are the cha~es in observed direction cosines due to refraction in the DC program.
As shown below, these equations are equivalent to a single refraction cor
rection equation for elevation a~le.
By definition i= cos E sin A m =cos E cos A
where E = apparent elevation angle of incident ray
A =' apparent azimuth angle of incident ray
Refraction causes small changes in elevation angle dE, but in azimuth,
AA= O.
Therefore, taking the derivatives
d1 = -dE sin E sin A dm = - dE sin E cos A,
.o.E- iNl (h, t)
- d1- n:-= sin E sinA = sin E stntA
-mN1~
dE = -dm = htsin E cos A sin E cos A
•
Nl
(h, t)- - h ctn E
t
N1
(h, t) = Nd
Hd
eV
Hd = kH =; [30 +0. 2 (hm
- 200)]
k =.!..5
h -hu =...;m~_
Hd
N =-N >0d im
A-2
N1
(h, t)- - h ctn E
t
equation (19), F116
equation (8), F116
equation (6a) and (6b)of this document
equation (5), F1l6
equation (6), F1l6 I
Average GEOS-2 height, h = i x 106 meters, and hm
= 364 km; then
Hd
= 83.73 km, and u ~ -11.58
V = 1- 9 x 10-6 = 0.99999-1
eV
= 2.718255 =-e
N1
(h, t)~E = E - E =c h
t
~E=-..!. ~E.5 10
ctn E = +(:)N. He1lr;: ctn E
t
(AI)
Thus, both of the equations (2, F116) lead to a single refraction correction
.equation for the elevation angle. Note that the sign is incorrect, that the correction is
for a single ray and not an interferometer ray pair as intended, and that the: factor
arises from assuming the equation (6b) rather than equation (6a) relation between scale
height, H, and the F-Iayer height, hm •
A.2.2
where
DC RANGE CORRECTION
From equation (12), F1l6, for E t: 100,
From equation (14), F1l6, for E > 100,
R = measured range, and R = refraction corrected rangec
For the ionosphere alone with E t: 100
where
~R1 = BsN1
(h,' t) csc cP 2
~2 = 008-1(:: ~o~~
Re
= earth radius = 6,378,166 meters
A-3
equation (17). F116
B = 1s
csc¢. =r-(:: ~o:~Ti/'1 r (Re C~SE)2]-1/2
AR1 = -Nim Hd e ll- \Re
+ hm
Since Hd ==: Hand ARio=-Nim H'e1
csc E
For E f: 100
[ ~Re cos E)2J-i/~ _ ~R'0~ sin E
1 R+ h [(R )2]1/2e m cos E1 - :e + hm (A2a)
A.2.3
For E >10~
AR2
= -N H e1
csc E = + i .6.R1,o
im d 5
DC RANGE RATE CORRECTION
From equation (13) F1l6, for E ~ 100
· .. jr,. JdNI (h, t)AR1 = R - Rc1 = LBs !. (t) • !.* (t~ dh csc 1'2
2 •-C1A s csc CPl ctn 1'1 cos ~im E sin E
(A2b)
2 ~ Re J.' 1-BsN1 (h, t) csc ¢2 ctn1'2 R + h (t) E smE
e ms
From equation (15) F1l6, for E >100
. .. 
where •R = measured range rate
•R = refraction corrected range ratec
The ionospheric part for E ~100 is
dNl
(h, t)
~R1 = Bs[l (t) • !.* (t)] dh csc<P2
equation (20), :FU6u
dN. (h, t) e v u
- (N (h t»)= 1m =-N. e edh ~ l' Hd 1m
for GEOS height; N. ~u == 01m
For E ~100
f. R)2 r, L R 2 J-3/2~R1 = +Nim Hd e
1,He +ehm Ecos E sin E [- ~e + ~n:) cos
2EJ
For E >100
•• 1 •~R2 = -N
l(h, t) E ctn E csc E = Nim Hd e E. ctn E csc E
= +.!.. ~R5 io
A.3 FREEMAN CORRECTION PROGRAMS
(A3b) .
The following formulas were formulated by J. J. Freeman Associates,
Inc. , under Contract NAS5-9782 for the Operations Evaluation Branch (OEB) at
Goddard and are documented in reference 5. The purpose of the contract was to de
velop better refraction correction equations for use in the calibration of angle, range,
and range rate systems.
A-5
· A.3.l FREEMAN ELEVATION ANGLE CORRECTION
The Freeman elevation angle correction (flat-earth approximation) is
where
&E = E - E = ctn E [N (1 _..L) -_1 fh N.dhJc s dht ht 1
o
E = observed elevation angle
E = refraction correction elevation anglec
N =(Il-l) = tropospheric refractivity at the stationsKeeping the ionospheric part
(A13)ref. 5
A.3.2
&E = -N'm H el
ctn E = &E."1 - 10ht
FREEMAN RANGE CORRECTION
The Freeman range correction (quasi -flat-earth approximation) is
(A4)
(A25)
ref. 5where
<1R =R - Re = eO~<l [/ INI dh - tA~:OC ! INI h'dh]
R = electromagnetically determined range
R = true refraction corrected range (meters)c .h =satellite heighf :::::103 km for GEOS
h' = distance from earth surface to ray path
R = distance from earth center to surfaceea: = (" /2 - E) = apparent zenith angle
IN I = absolute value of tropospheric or ionospheric refractivity =
<1R = Sl~ rJINIdh - et~2E I,NIhdh]l'o e 0
lu-11
In the previous equation, if INI assumes the value -Ni
then the ionospheric formula is
If (H+h ) 2J=-Nim HeLl -. Re
m . ctn EJ esc E
A-6
(A5)
A.3.3 FREEMAN RANGE RATE CORRECTION
From equation (A51), reference 5
V(J tan ex:
R cos ex:
where •v. =RE()
As h' approaches the GEOS satellite height of approximately 103 km, the
refractivity at the target, N , approaches zero. Therefore the range rate correctionp .
equation reduces to
•E cosE.2 ESIn
• 1 Ecos E[~ 3~R =+N. He 2 1 + 1 - . 21m sin E sm E
In the previous equation, if IN I assumes the value -N.1
) (" ::m )]= +&Rto [1 + (1 -sJ E) (H ::m )]
A.5 GSFC GPRO PROGRAM
(A6)
The GEOS Preprocessor ;program (GPRO) was developed for the OEB at
GSFC by DBA Systems, Inc. , under contr~.ct NAS 5-9860, and is documented in ref
erences 4 and 22. It is used with the Geodetic Data Adjustment Program (GDAP), and
with the Network Adjustment Programs (NAP-1 and NAP-2). The GPRO program
utilizes a parabolic model for the ionosphere.
A-7
A.5.l GPRO RANGE REFRACTION CORRECTION
The GPRO range refraction correction, from reference 4, appendix C t
section la, is
AR == 2f3l == 2f3l.. 1 ""D:'""" r.. 2 2 2 J1721 sin E1 + Lsm E1 + 4f3 2 cos E1
2f3l
csc E l
1 + [l + 4f3; ctn 2 E1]1/2
where
where
I1Rl == correction for ionospheric refraction
El
== local elevation angle of ray upon entry to ionosphere
The angle E l can be computed from
R cos EE - ecos 1 - R + h - 3 H
e m g
E == apparent elevation angle at observer
h - 3H == height of the base of the ionospherem g
{3l' {32 are given by the following functions of ionospheric parameters
where
~f ~2 80.5.N
{3 == -2H m == -2H ml. g f g f2
25 H== g
3 (R + h - 3H )e m g
-4 H N.g 1m
f == ranging frequency (or modulation frequencies)
f == critical frequency (f 2 == 80.5 N )m m m
N == maximum electron densitym
[ 1(f )2 ·-40.25 Ni ]Nim== maximum refractivity. Nim == -"2 -r- == f2 m
hm == height corresponding to maximum electron density
H == thickness factor of parabolic ionosphere == 50 kmg
R == radius of the earthe '
A-8
-8 H N. csc E1g 1m (A7)
A.5.2 GPRO RANGE RATE CORRECTION
The GPRO range rate refraction correction, from reference 22 is.
where
R cos Ee
Cos E 1 = R + h - 3 Hem' g
•• (R sin E) EE = e
1 (R + h - 3 H ) sin E 1e m g
•~R =
[
sin E 1 cos E 1 - 4/2 cos E 1 sin E 1 ] •cos E + I E11 . ( . 2 E + 4f32 2 E ) 1 2sm 1 2 cos 1
A-9
•dR
.. "-"" . ~,.... ---'-: '":'. -
. [ 2 j• 1-4~
-8 N. H E1 ctn E1 csc E1 1 + 2 2 1/2. 1m g (1 + 4{32 ctn E1)
(A8)
A.6 GSFC NAP-3 PROGRAM
The Network Adjustment Program (NAP-3) formulas were formulated by
DBA System, Inc; (Contract No. NAS5-17730), and are documented ill reference 23.
Further explanation of the Bent profile, NAP-3 Elevation Angle Correction, NAP-3
Range Correction, and NAP-3 Range Rate CQrrection is presented in the following
paragraphs (reference 23 and supplementary notes by Dr. R. Bent).
A.6.1 BENT PROFILE
The Bent profile was obtained from the analysis of many tens-of-thousands
of topside and bottomside profiles and it fits closely to actual profiles. It contains a
biparabolic lower portion and an exponential upper portion and is used in all NAP-3 .
computations on satellite refraction effects. The Bent profile is illustrated in :fig
ure A-I.
h
9 -kaN=i6 Nm 8
hm +r----:-------Jr--
N=N (1_.t)2m Y~
"--- .l.- N
52~-A-1
Figure A-I. Bent Profilewherek =exponential decay constant
N =electron density at height h
A-10
N =maximum electron densitym
y =thickness factor for biparabolicm model ionosphere
A.6.2 NAP-3 ELEVATION ANGLE CORRECTION
The passage of a ray path through the ionosphere is illustrated in Fig
ure A-2 from whence it can he shown that the deviation angle a1
=a2
=a3
=, say, a,.
The main angles under consideration are 'the angle of elevation E) the de
viation angle a 1 (or a) and the error in the elevation angle {3. The angle at the sat
ellite between the line of sight and actual path is ({3 - a).
First, obtain the deviation angle a and from this compute (3.
The deviation angle a 1 = 8T - 81' but as a 1 = a 2 = a 3' let them aR be
called a:,
(A9)
where
8T = the angle subtended at the center of the earth by the limits of
the true ray path in the ionosphere
81 = the angle subtended from the same height limits by the ap
parent ray path.
Snell's law gives:
where
P IJ. sin i = P sin I{) = constant (A10)
i and I{) =the angles of incidence of the true and apparent ray paths
respectively at height p , the following equation is derived:
d 8 = tan i d p. p (All)
which, when combined with equation (A10) yields the following equation:
(A12)
Simplifying the Appleton-Hartree equation and neglecting magnetic field effects
(which can safely be done at very high frequencies) the equation for the index of re
fraction is given by
where f = the frequency of transmission
A-ll
(A13)
fN= the plasma frequency at the heightunder investigation.
RA
Y
\
SA
TE
LL
ITE
.........
..........
.........
........
........
" " "
K=
E
J I I Jo..
-i2
I I I I I I I I I I. II
/I
I/
II
1/
II
-1I
II
I
II
II
II
/I
II
Ip.
II
\Po
l\I
ilI
\I
rI
Po\.~J/
......-1~:1.
,,\
Ii..:..
~Po
........'?
IiI
1/
'"....,
....B
.'\
1\1
II",/\l
',I
\\I
IIII
/'"
--
,,\flll
'"--
-_
o-"'
/','\
~(/
'"F
ico
SI(---l
',\\
/--
--",
o
........
"IO
NO
SP
HE
RE
........
•5
29
-A
-2
Fig
ure
A-2
.D
eriv
atio
no
fE
leva
tion
Co
rrec
tio
n
However, as the electron density N is proportional to :r2, equation (A13) can be re
written including the critical frequency as:
where
(rfC )211 2 =1- N
Nm(A14)
fc = the critical frequency of the F layer corresponding to the
electron density ~
N = the electron density at the height under investigation
Rewrite equation (A14) as follows:
11 2 = 1 - x
Combining equations (A12) and (A15) yields:
(A15)
~
where
fb .
() = sm I{) dp
TapJcos 2 I{) _ x
a and b = the lower and upper heights of the ionosphere.
(A16)
To obtain (}1 of equation (A9), assume that the ray travels in a straight line with
11 = 1 and x = O. Equation (A16) then yields:
b() -J tan I{) d1- -p- p.a
(A17)
Now the frequencies of vertical and oblique incidence radio waves are related by the
expression:
f = f sec I{)v
where
f = the frequency of the oblique incidence wave,
f = the equivalent vertical incidence frequency, andv
I{) =the angle of incidence of the apparent oblique incidence
ray path.
A-I3
fNow at the point of reflection i-sec 'P =1 and if this function i~ less than unity the
ray proceeds through the ionosphere and therefore the value of r sec 'P is a measure
of the deviation of the penetrating ray. Its maximum value will occur at or just below
the height of maximum electron density. An increase in this deviation factor indicates
an increase in the deviation angle of the penetrating and apparent.ray path. The maxi
mum deviation occurring when the vaiue of f sec 'P is just below unity . Let this de
viation factor be u where:
(A18)
which, from equations (A13) and (A15) gives
u = x sec2 'P.
Therefore equation (A16) can be rewritten as:
hence
tan 'P dp1
• '2P (l-u)
b [ 1 ]tan ~ --a = 6T - 61 = ~ -p- (l-u) 2 -1 dp. (A19)
At the height of refraction u = 1 and at the height of maximum electron density u is
simply the square of the deviation factor and for all our purposes u < 1.
For all r~ys that penetrate the ionosphere we can apply the binomial theorem to equa
tion (A19), to give:
a = 1 Jb u tan 'Pr1 + :!. + 2. 2 + 35 3 +2 P 4 u 8 u 64 u .•a
hence
a =t s: x tan 'Ppsec2
'P [ 1 + ~ u + ~ u2 + ~~ u3 + • . .] d P .
Applying the second law of the mean for integrals:
x tan 'P sec2
'P dpp
A-14
(A20)
where
~ = a number between the greatest and least values of the
series over the interval (a, b).
u = 0 at the limits (a, b) and has a maximum value, less than unity, near the height of
maximum electron density.
The value of ~ must therefore lie between unity :,lnd the value of the series at maxi
mum electron density and is defined as the mean of these two values.
Applying the second law of the mean for integrals once more to equation (A20) yields:
yp = P (1 + 0.5333 --l!L) = some radius between limits ao m Pm edb
P = R + hm e m
'Po = the value of 'P at the radius Po
where
From Figure A2
sin 'P =.J!L cos Eo Po
x dp (A21)
(A22)
then
(A23)
~ . is obtained from a table of u vs ~ values. ~me"'''' ~ 1 for f = 2 GHz.c"mean max mean a.u
Now, f 2
x= (T-) . :m (A24)
Therefore, the deviation angle is given by
(A25)
where NT is given by ~b N d P and is the total electron content of a vertical column
of unit cross-section of the ionosphere.
A-15
wher~
. A.6.3
It can now be shown from Figure A-2 that
-It Xl cosec - X2 j{3 = L\E = cos. (X~ + X ~ - 2X1 X2 cos ec ) 1/2
2 2 2 1/2 . . ( ecXl = (Rs - Re cos E) + Re cos E tan -2-)
R =R +hs e
R = radius of the earthe
ht
= height of the satellite
NAP-3 RANGE CORRECTION (REFERENCE 23)
(A26)
Assuming the total electron content (NT) has been computed in the vertical
direction through the ionosphere along AB (Figure A-3), then the one-way value ofN
L\R =4~ 3 • --'L. for the ray along AC.r. cos tp
RAY
IONOSPHERE
529- A-3
Figure A-3. Range Correction Diagram
A-16
If E is the satellite elevation angle:
where Re is the radius of the earth and Rh is the height of the bottom of the ionosphere
above the surface, (~ = hm - ym),
o o. ~R =
(A27)
YWhen the satellite height, hs ' is above the top of the biparabolic layer (hm + ;:), the
equation NT = (h Ndh gives:. J o
Ym(hs - hm -'2)
N =/ L N e-ka da +T 16 m
o
From reference 23, equations 1, 3, and 4:
[Ym )_9_ -(hs - hm ----z-' K 459 Y .
)
2 40.3xL24x10-2 16K 1-e + 480 m]
~R =(!op- [ (Re 2 2] 1 21 - R + R ) cos E
e h
. (electrons) -2 2usmg Nim . 3 = 1. 24 x 10 (fo
F2 cps)m
and N. =-40.3 Nm1m f2
A-17
we have
A.6.4 NAP-3 RANGE-RATE CORRECTION
(A28)
Equation (A27), when differ«;lntiated with respect to time, provides the
range-rate correction.
• cHdR) • . o(l\R) • __ d (l\R) E·.1R = dEE + d- h
s- hs d - E -
since•hs = 0 for our circular orbit,
• (R)2• E (dR) R : R sin E cos E
.1R =. _(eRe c~s E) 21 R +R
e h
A-I8
(A29)
APPENDIX B
ELEVATION RATE DETERMINATION
APPENDIX B
ELEVATION RATE DETERMINATION
The required expression for elevation rate is derived from the geometry
shown in Figure B-l. In this figure a circular overhead orbit around a spherical, sta
tionary earth is assumed.
POINT OF CLOSEST APPROACH·
I R Cos Er---I E
wI
~IQ:I
I
TRACKING STATION
CENTER OF EARTH
529-8-1
Figure B-l. Plane Containing Satellite, Tracking Station, and Center of Earth
R cos E RT sin cutsin cut = R ; cos E = R
T
(R + R sin E) (RT
cos cut - Rs
)cos cut == s . sin E =
R' RT
2 2 1/2R = (R
s+ RT - 2RT Rs cos cut)
substitute (B3) into (Bl).
B-1
(B1)
(B2)
(B3)
2 2 -1/2cos E = R
Tsin wt (R
s+ R
T- 2RT R
scos wt)
then by differentiation,
dE -1 -3 2- sin E dt = w RT R cos wt - w RT R (RT Rs sin wt)
substitute (B2) into (B4)
(B4)
,
-1 dE -1 2 2-2-R (R
Tcos wt - Rs)df =wRT R (R cos wt - RT Rs sin wt) (R )
~~ = [- wRT(R~ coswt + R; coswt - 2RT Rs cos 2 wt
[- wRT
(RT
- Rs
cos wt) (RT cos wt - Rs)J
R2 (RT
cos wt - Rs )
RT = distance from the center of earth to the satellite (7,711,499 meters)
Rs = distance from the center of earth to the station (6,378,166 meters)
R = distance from the station to the satellite
(B5)
w = angular velocity of the satellite = 2T"
R 3/2
( T) (page 96 of reference 18)T = period of the satellite = Tc R
s
Tc = 84.347 minutes, the period of a circular'orbit at the earth's surface
B-2