The signal characteristics and the geopotential parameter recovery capability of the SST Doppler sensor flown on the geopotential research mission (GRM) are discussed. Simulation studies of the velocity profiles resulting from the perturbation produced by a 1 deg/w/1 deg, 1 mgal anomaly as sensed by two GRM spacecraft orbiting altitudes of 160 km and 200 km respectively are described. It was found that the amplitude of the gravity signal drops off by a factor of 1.5 when going from an altitude of 160 km to 200 km. By extrapolation the signal amplitude is further decreased by a factor of 3 when the orbital altitude is increased to 250 km. Thus the amplitude of the measurement drops off as the altitude is increased to the point where it is insignificant at the 1 mgal level for altitudes above 200 km. Spectral analysis results show that for a GRM mission altitude of 160 km and a system precision of 1 micrometer/sec, gravity field information can be sensed up to 230 cycles per orbital revolution - beyond that frequency the gravity signal is characterized by white noise. It follows that at the GRM mission altitude of 160 km and a satellite to satellite Doppler system precision of 1 micrometer per second, 1/1 deg gravity and geoid anomalies can be determined to an accuracy of 3.4 mgals and 8.6 cm respectively

Felsentreger, T. L.

Kahn, W. D.

English

NASA Technical Reports Server

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https://ntrs.nasa.gov/search.jsp?R=19830002404 2020-03-21T06:40:02+00:00Z
JULY 1982
National Aeronautics and
Space Administration
Goddard Space Flight Center
Greenbelt, Mary'and 20771
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NASA
^. Technical Memorandum 83970
(NA.)A-Tli- d391J )	 SlUNAL ANALYSIS AND $l+Nt-a 	 N8J -1Ub74
ANALYSIS STUUlk-S PUa A GLOPUTENTIAL IIESIAhkk H
PIA..iSIUN (Gt(t'1)	 (NASA)	 19 p Lit- A021!1f AJ 1
CSCL 08N	 U cclaS
G-s/ 146 3S-'b4
Signal Analysis and Error Analysis Studies
for a Geopotential Research Mission (GRIVI)
W.D. Kahn and T.L. Felsentreger
4a
TM 83970
M
s	 SIGNAL ANALYSIS AND ERROR ANALYSIS STUDIES
FOR A. GEOPOTENTIAL RESEARCH MISSION (GRM)
i
By
W, D, Kalin
T. L, rcisentreger
July 1982
GODDARD SPACE FLIGHT CENTER
Greenbelt, Maryland 20771
b
SIGNAL ANALYSIS AND ERROR ANALYSIS STUDIES
FOR A GEOPOTENTIAL RESEARCH MISSION (GRM)
W. D. Kahn
T. L. Felsentreger
ABSTRACT
The Geopotential Research Mission (GRM), formerly known as GRAVSATJMAGSAT, has been
proposed for the mapping of the Earth's gravity field. It is a candidate for a NASA new start in the
near future. GRM is composed of a pair of surface-force compensated satellites in identical polar
orbits, spaced approximately 3° apart at an altitude of 160km with a system precision of 1 µm/s,
The objectives of this paper are to disuuab she signal characteristics and the geopotential param-
eter recovery capability of the SST Doppler sensor flown on GRM.
Simulation studies of the velocity profiles resulting from the perturbation produced by a I° X
1°, 1 mgal anomaly as sensed by two GRM spacecraft orbiting altitudes of 160km and 200km re-
spectively are described. It has been found that the amplitude of the gravity signal drops off by a
factor of 1.5 when going from an altitude of 160km to 200km. By extrapolation the signal ampli-
tude is further decreased by a factor of 3 when the orbital altitude is increased to 250km. Thus the
amplitude of the measurement drops off as the altitude is increased to the point where it is insignifi-
cant at the 1 mgal level for altitudes above 200km.
Spectral analysis results show that for a GRM mission altitude of 160km and a system precision
of I micrometer/sec., gravity field information can be sensed up to 230 cycles per orbital revolution
beyond that frequency the gravity signal is characterized by white noise. It follows that at the GRM
mission altitude of 160 km and a satellite-to-satellite Doppler system precision of ±1 micrometer
per second, I' X 1 0
 gravity and geoid anomalies can be determined to an accuracy of 3.4 mgals and
8.6 cm respectively.
These results are compatible with the scientific requirements for GRM which are: Accuracy -
2.5 mgals; Horizontal Resolution - 100 km; Accuracy of Oceanic Geoid Undulation Difference -
10cm (over wavelengths of 100 to 3000 kilometers).
PRECEDING PAGE 1 LAl'isi, NOT FUMED
► 1 ,- ^V
	
v
SIGNAL ANALYSIS AND ERROR ANALYSIS STUDIES
FOR A GEOPOTENTIAL RESEARCH MISSION (GRM)
I. INTRODUCTION
Studies have been conducted at the Goddard Space Flight Center to investigate the accuracy
with which the Earth's gravity field can be determined from a gravity field mapping mission such as
the Geopotential Research Mission (GRM), formerly designated as GRAVSAT/MAGSAT. The
studies performed at Goddard are threefold: (1) gravity field signal analysis, (2) rapid error analysis
which can be used for mean gravity anomaly and mean undulation accuracy estimation for almost
continuous data, and (3) full-scale simulation studies. This paper reports on the results of (1) &
(2),
II. ANALYSIS
A. Signal Analysis
The perturbation by the gravity field induced on the intersatellite Doppler signal is given by
the following relationship:
S(.r^i) =	
tst
	 1 aT 
dt_ 
rtl l aTdt
r8^
J
	r 70
ft'o+St 	 to
where
T = disturbing potential
r = a + h
a = Earth's mean equatorial radius
h = satellite orbit altitude
0	 true anomaly
St = time separation interval of two satellites in identical orbits,
(1.0)
The form of the disturbing potential most useful for formulating (1,0) is given by the formula
	
T (r' ) 1 
a	
J J X g S (r, 0) du	 (1.0)
	
Orr	 a
where
(r, ^) _ 2a	 a	 3ak	 a2	 ("	 r —a cos	 Q
..Q + r — rz — r2 cos ^ 15 +3 In (	 2r	 )]
Q = [r2 + a2 — 2r a cos fl 1/2
0I = mean gravity anomaly
da = element of area.
Using (1.0) and (1.1) for an orbit configuration as depicted in Figure 1, velocity profiles resulting
from the perturbation produced by a 1 mgal gravity anomaly as sensed by the 2 satellites a,e shown
in Figure 2, The satellites are spaced 330 km (3°) apart — results are shown for 160 km and 200 km
satellite altitudes. The amplitude of the gravity signal drops off by a factor of 1.5 when going from
an altitude of 160 km to 200 km, By extrapolation, the amplitude is further decreased by about a
factor of 3 when the orbital altitude is increased to 250 km. Thus, the am plitude of the measure-
ment drops off as the altitude is increased to the point where it is insigni,',cant at the 1 mgal level for
altitudes above 200 km, By way of contrast, Figure 2A shows the SST signal generated by ORM for
one full orbit, under the influence of a full (36, 36) spherical harmonic gravity field (GEM10B).
The two satellites are at 160 kia altitude, separated by 3 degrees.
The ability to resolve the signal from a (I' x I) 1 mgal gravity anomaly for 2 spacecraft orbit-
ing at a 160 km altitude, whose orbits are displaced from the anomaly by 1° and 5°, respectively, is
shown in Figure 3, From a 1° orbit displacement, better than 60 percent of the gravity signal is
sensed. However, when the orbit is displaced by 5° from the gravity anomaly, only 10 percent of
the gravity signal,is sensed. This would imply that the gravity anomaly signal from an adjacent
block drops off drastically with increased distance. Nevertheless, it is significant that remote gravity
anomalies do contribute to the overall gravity signal by as much as the 10 percent indicated.
e
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B, Terror Analysis
A rapid error analysis procedure developed by 0. Jekcli and R. II. Rapp (Ref, 1) has been used
for the estimation of mean gravity anomalies ,-.od mean undulation accuracies that can be o btained
from SST missions such as the future Geopotential Research Mission (GRM), The basic method
used is the analysis of the gravity field mapping in the frequency domain with results obtained in
terms of mean gravity anomalies and mean geoid undulations. The measurements to be considered
are velocity differences, with the errors in the measurements assumed to be uncorrelated,
fit 	 to estimate the expected "signal" for SST Doppler we shall evaluate its ; power spec-
trum (Refs, 2, 3) derived from the Earth's anomalous gravity potential, That is,
GM (a/r)Q PRiii(sin ) (C^Rm Cos mn + S Rnn sin mM	 (2.0
r	 CO
where
r	 is the geocentric distance
,O	 is the subsatellite geocentric latitude
r
is the subsatellite east longitude
a	 is the mean radius of the Earth
PRIII(sin cp)	 are the fully normalized associated Legendre polymonials of degree R and
order in (R is also the orbital frequency of a satellite iti cycles per revolution)
and
CRnr, S Qm	 are the fully normalized coefficients of a truncated harmonic series expansion
representing the geopotential,
The power spectrum of the velocity call 	 determined by utilizing the Law of Conservation of
energy.
Y2vz — U = la	 (3.0)
where U is the total gravitational potential, v is the velocity, and E is the total (constant) energy.	 E
Decomposing U into the reference potential V and the disturbing potential T of the Earth, we have
7
orr^ ray	= Jw, u Vn
'/Zv2 « V	 T = E	 (3.1)
The variation of (3.2) then reads
v Sv
	
8T	 (4.1)	 v
where v2
 = GM/r.
From Equation (2.0), together with (4.1) we obtain the variation of the r.., iellite velocity mag-
nitude as:
Sv = v(a/r) Q Pgm(sin p) (CQm cos mx + gm sin mx)	 (4,2)
E 2 m n 0
The velocity power spectrum oaf Sv is now obtained by squaring Equation (4.2) and integrating
over the unit sphere:
VQ2 (Sv) = V2 (a/r)Z Q Z T9 (sin tp) (C l 2 + 22	 (4.3)M=O
Using Kaula's rule (Ref. 4), the summation containing 'the C, S terms c.An be estimated as follows:
m=0 (C& + S ^m ) 
_ (10-1 /R2 )2
	 (4.4)
Then
VQ2 (Sv)
	
v2 (a/r)' 2 (2Q+ 1) (10-5 /Q2 )2	 (4.5)
R>2
In order to derive the velocity variations for the low-low SST configuration, Equation (4.1) is
to be modified as follows:
5T = (T2 — Tl ) = v SvIi	(4.6)
where T i and T2 are the disturbance potentials at satellite positions 1 and 2 in the orbit with
an angular separation, X12.
8
VYGu+v,G^ ^, G 	
^	 3
^.,^	 .r,1p r R
OF Po r ,r , t-; '	 .^
The corresponding vi,, &%nce in the horizontal component of velocity is now to be derived as
follows:
aR2 (ST) = aQ2 (T l ) + ql (T2 ) -- 2ag(T 1 T2 )	 (4,7)
By assuming that ag2 (T 1 ) = aR2 (T2 ) = ue (T) and ak(T 1 T2 ) = PR(cos ^ 12) aQ2 (T) (Ref 2)
one obtains
age (6T) = 2aQ2 (T) [I — PR(cos di ll) ]	 (4,8)
and from Equation (2,0) together with (4,6) and (0),
cre (5 al, ) = 2v2 (a/r) 2 2 [ 1 _ pR(coS 012) ] (22 + 1) (10-5 /Q2 )2 	(4,9)
Equation. 1.9 represents the horizontal velocity "signal", variance,
The intersatellite velocity signal spectra is given in Figure 4, It can be seen that for the punned
GRM altitude of 160 km and the SST system precision of tl µm/s, gravity signals at the orbital
frequency of 180 cycles/revolution (R 180) can be sensed by GRM. Also shown in this figure is
the level of signal attenuation with increasing orbital altitudr;.
The frequency spectrum only provides information on the specific frequency (spectral line) at
which the SST' velocity signal will sense the gravity field. It does not sense the maximum frequency
at which a SST system with a specified signal precision and a specified orbit altitude can sense
gravity field information. Determining the total gravity signal sensitivity for a SST system having a
given level of precision necessitates integrating the velocity cpectrum. That is,
aR ' .(8v,,) _	
f 
0Q2 (6vii ) dfl]'h
	(5.0)
1
The lower integration limit can be interpreted as that orbital frequency at which the gravity
signal-to-noise ratio is unity foi a given SST system precision. In Figure 5, it can be seen that for
the system precision of 1 pm/sec and orbital altitude of 160 km, the gravity signal cut-off frequency
+M
9
U06y)
1.OE+02
1.0 E+01
E
:3.
J
z(9
1.OE+01
w
1.0E-C
OF POOR QUALITY,
50	 100	 150	 200	 zbu	 Jvv
	 a;i%i
DEGREE.Q
Figure 4. GRM Velocity Spectrum for Variable Orbit Altitudes.
10
w
-1
cn
z0
oc
:s
cc 'I.OE-03
0
UA
.j
C.9
vi
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;Awy
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SPACECRAFT SEPARATION -, 300 loil
(41 12 - 3")
h w 300 kni
	 A
	1.0c-06 1	 (h w SPACECRAFT ORBIT ALTITUDE)	
it, w 350 km
	
0
	
so	 100	 150	 200	 250	 300	 360
LOWER BOUND OF DEGREE INTE11VAL (UPPER BOUND * 3000)
Vigulk, S. 0 RM Velocity Signal Vrror.
O"RIC"It"AL ^ at bN ^taA05 POOR ^^ ^^l 1 i'Y
occurs at the orbital frequency of about 230 cycles per revolution (R = 230). All gravity hSorma-
tion above that frequency (i.e. limited by the SST sensor precision) would be characterised by
white noise. Thus, a higher "cut-off" frequency at a given orbit altitude necessitates a SST' sensor
of a higher precision.
To assess the accuracy with which geophysical parameters (such as gravity or geoid anomalies)
can be determined when using the previously selected 0 R sensor precision of±1 pm/s,, the models
developed by Jckeli and Rapp (Ref. 1) are used.
The total error budget associated with these parameters is the combination of the (-rrors of
commission and that a omission (generally the larger terms). Tile models for these error sources
read as follows;
(a) The error of commission in terms of mean gravity anomalies is given by:
R	 PR2 (R - 1)2 (^-+1) r \)
2 R+11'h
ac (
_ (7/^l)'` 3 (DA/47r)'/' v(Sv,.,)	 R 2	 2[ 1--I' cos	 )] a /	 (6.0)R(	 ►
 i2
(b) The error of omission in terms of mean gravity anomalies is given by:
C* 	 (R— 1) (2R+ 1) 'h
	
eo(Sb) = y x 10-5 
IR + 1 RR2
	
R^	
(6.1)
1
where
R	 cot 
^	 r PQ1 (cos ^o)a	 L	 (6.2)
2	 R(R+ 1)
00 = CS° x S° /7r]	 (63)
7 = (GM/a2)	 (6.4)
x
Ri = Value obtained from Figure 5 for which the signal-to-noise ratio is unity for a specified
SST system precision a (&vii ),
AA = Area of block in units of radians squared.
00 = The radius of a circular cap that has about the same area AA as a bloeic whose sides are
S° (in degrees).
12
u^^^^u^vL'.-,
05
012- Angular separation of the two GRM spacecraft (in degrees),
a = Mean equatorial radius of the Earth,
S°	 aide of anomaly block (in degrees),
and
(c)	 'rhe total error for mean gravity anomalies is as follows;
v,r(Sb) _ [ac. 2 (sg) + ao2(s6)I'h
In terms of geoid anomaly errors, the error models are as follows,
(a) The error of commission in terms of mean geoid anomalies is given by;
>	 91	 Og (2Q+ 1)	
r 2R+L 'h
	
ac (N) = (a/ ,r) h (DA/4^r) /' 0(5V H )' E 	(---L 2 2[1-1? (cos
	 \ a)Q	 ^i2>]
(b) The error of omission in terms of mean geoid anomalies is given by;
ao(N)	 ax 10-5 [ Z OR, (2Q+1/Q1)]'1214.1
(c) The total error for mean geoid anomalies is.,
a i, (N) = [(y 2(N) + 002 (N) ]'h
Figures 6 and 7 show the total expected errors in terms of the gravity and geoid anomaly
parameterization as obtained from Equations (6.5) and (7.2). The errors shown are for the GRM
mission SST sensor precision a(5%)	 ±1 µm/s, satellite separation distance X 12	 3° as a func-
tion of orbital altitude and block size.
It is shown that by using a pair of spacecraft in a 160 km polar orbit separated by 3°, one can
map the geopotential in terms of 1° x 1° blocks to about 3'h milligals or P em respectively.
CONCLUSIONS
As is evident from Figures 6 and 7, the planned GRM mission requirements can be met using
the SST low-low approach and a I Jun/s Doppler system, that is, global gravity and geoid anomalies
(6.5)
(7,0)
(7.1)
(7.2)
13
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X	 X
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OF POOR
QUAILITY
	
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si	 M	 M7	 N	 N	 r	 Gr
(W9) HOHH3 NOuvin0Nn 01039
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OW
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14
can be recovered with a resolution of 100 kill and a precision of 3 mgals and 9 cm
respectively.
AC'hNOWI,1 OMI-Wr
nic authors are indebted to Mr. G.11, Wyatt for programming and computational support.
RE"F FIRE NCI S
1. Jckeli, C"., Rapp, R. II., "Accuracy of the Determination of Mcan Anomalies and Mean C cold
Undulations from a Satellite Gravity Field Mapping Mission," The Ohio State University
Department orGeodetle Science Report 307, August 1950.
'	 IIciskauen, W„ Moritz, 11., 1967, "Physical Geodesy," W, 11. Freeman.
3,	 Umbeek, K,, " MethoLIS alld Geophysical Applications or Satellite Geodesy," Reports on
Progress in Physics 1973, Vol, 421 , pp. 547-01-8.
4. Raula, W. M„ —rlie Appropriate. Representation of the Gravity field for Satellite Gcodesy,"
Proceedings or the IV Sy ►nposium or Matlie ►uatiCal Ueodesy,'1'rieste, 1969,
5. Lerch, R J., ct al., 1975, Gravity Itlodel Improvement Wing GEOS-3 (G1 M-9 and -10)
GSUC \-921 1-77- 240.
1s


Signal analysis and error analysis studies for a Geopotential Research Mission (GRM)

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