Vol.3, No.5, 388-396 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.35052
Copyright © 2011 SciRes. OPEN ACCESS
Determination of the geopotential and orthometric
height based on frequency shift equation
Wenbin Shen1*, Jinsheng Ning1, Jingnan Liu2, Jiancheng Li1, Dingbo Chao1
1Department of Geophysics/Key Lab. of Geospace Environment and Geodesy, School of Geodesy and Geomatics, Wuhan University,
Wuhan, China; *Corresponding Author: wbshen@sgg.whu.edu.cn
2GNSS Engineering Center, Wuhan University, Wuhan, China.
Received 15 July 2009; revised 20 August 2009; accepted 10 September 2009.
ABSTRACT
The orthometric height (OH) system plays a key
role in geodesy, and it has broad applications in
various fields and activities. Based on general
relativity theory (GRT), on an arbitrary equi-geo-
potential surface, there does not exist the grav-
ity frequency shift of an electromagnetic wave
signal. However, between arbitrary two different
equi-geopotential surfaces, there exists the gra-
vity frequency shift of the signal. The relation-
ship between the geopotential difference and
the gravity frequency shift between arbitrary
two points P and Q is referred to as the gravity
frequency shift equation. Based on this equa-
tion, one can determine the geopotential dif-
ference as well as the OH difference between
two separated points P and Q either by using
electromagnetic wave signals propagated be-
tween P and Q, or by using the Global Posi-
tioning System (GPS) satellite signals received
simultaneously by receivers at P and Q. Sup-
pose an emitter at P emits a signal with fre-
quency f towards a receiver at Q, and the re-
ceived frequency of the signal at Q is
f
, or
suppose an emitter on board a flying GPS satel-
lite emits signals with frequency f towards two
receivers at P and Q on ground, and the re-
ceived frequencies of the signals at P and Q are
f
P and
f
Q, respectively, then, the geopotential
dif- ference between these two points can be
determined based on the geopotential frequen-
cy shift equation, using either the gravity fre-
quency shift
f
f or
f
Q
f
P, and the corre-
sponding OH difference is further determined
based on the Bruns’ formula. Besides, using
this approach a unified world height datum
system might be realized, because P and Q
could be chosen quite arbitrarily, e.g., they are
located on two separated continents or islands.
Keywords: Equi-Frequency Geoid;
Gravity Frequency Shift Equation; GPS Signal;
Geopotential; Orthometric Height;
World Height Datum System Unification
1. INTRODUCTION
The orthometric height (OH), the height above the ge-
oid along the gravity plumb line, plays an important role
in geodesy, and has broad applications in various fields.
Conventionally, the OH is determined by leveling with
additional gravimetry [1], due to the fact that the level-
ing goes along the equigeopotential surface, and the
non-parallel influences of different equigeopotential
surfaces should be considered based on the measured
gravity data. The conventional approach has at least
three drawbacks: 1) the error is accumulated (becomes
larger and larger) with the increase of the length of the
measurement line; 2) it is difficult to connect two sepa-
rated points which are located on two continents or is-
lands separated by sea; 3) the leveling is a very laborious
work requiring a lot of manpower and equipments, espe-
cially in mountainous areas.
To conquer the mentioned drawbacks in conventional
approach, Bjerhammar (1985) put forward an idea to
determine the OH based on the general relativity theory
(GRT) [2]: the OH might be determined by precise
clocks. This approach is referred to as the clock ap-
proach for convenience. Since the clock approach is
based on the comparisons between precise atomic clocks
between two stations by clock transportation approach
[3], it is seriously constrained in practical applications
due to the fact that atomic clocks are very expensive for
general use and very difficult to control the normal work
condition during their transportation. Just due to this
reason, Shen et al. (1993) suggested that the OH could
be determined by gravity frequency shift, which is re-
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
389
ferred to as the frequency shift approach. Both the clock
approach and the frequency shift approach are referred
to as the relativistic approach [4]. Using the relativistic
approach, the above mentioned drawbacks existed in the
conventional approach could be overcome. Especially,
the Global Positioning System (GPS) technique provides
a good opportunity to determine the OH by using the
GPS signals based on the frequency shift approach [4-7],
which is referred to as the GPS frequency approach.
Though GPS leveling provides an approach in deter-
mining the OH [8], to determine the OH with high preci-
sion, e.g., at the centimeter-level accuracy, it requires the
condition that a global or local geoid with the corre-
sponding precision (e.g., centimeter-level accuracy) has
been a priori established. This condition can not be satis-
fied in many cases, e.g., in mountainous areas. Espe-
cially, since a precise global geoid is not yet established,
the GPS leveling approach is seriously constrained in
connecting the height datum marks located in different
continents.
In this paper, after introducing the definition of the
relativistic geoid by precise clocks in Section 2, the de-
finition of the equi-frequency geoid and the derivation of
the gravity frequency shift equation are provided in Sec-
tion 3. Then, in Section 4, based on the gravity fre-
quency shift equation, we provide the approach to de-
termine the geopotential and OH using electromagnetic
wave signals propagated between two points on ground,
especially using GPS signals received by two separated
receivers on ground. In the following section, we discuss
some problems related to the unification of the world
height system, and in the last section, we discuss the
problems related to the stability of the atomic clocks,
and conclude that the frequency shift approach for de-
termining the geopotential and OH is prospective. This
paper is an extension of Shen et al. (2008b) [7].
2. DEFINITION OF THE RELATIVISTIC
GEOID
2.1. Equi-Geopotential Surfaces
We point out that there does not exist essential differ-
ence between gravitation and gravity, and the only dif-
ference is due to the choices of different reference sys-
tems [9]. Similarly, we can say the same about the gra-
vitational potential and geopotential. In fact, the metric
tensor
g
has the character of the gravitational poten-
tial [9-11], and consequently it has the character of the
geopotential. According to GRT, a precise clock runs
quicker at the position with higher geopotential than a
precise clock at the position with lower geopotential. To
establish the relationship between the keeping time of
clocks with the geopotentials with which the clocks are
located at the positions, we investigate the proper time
interval [11,12]
22
00 0
d dd d+2dddd
iij
iij
g
xxg tgtxgxx


  (1)
in which, 2
d
is the proper time and it is an invariant
quantity,
x
are the 4-dimensional coordinates, where
0
x
is the time coordinate,

1, 2, 3
i
xi are the space
coordinates. The Einstein summation convention is ap-
plied throughout this paper: the summation will be ap-
plied if and only if there are two same indexes, one be-
ing up and another being sub. In addition, the light unit
system, 1c
, is used. In this case, the speed is a pure
quantity without unit, and the length has the same unit as
that of time. Since 00
of
g
corresponds to energy,
the geopotential could be expressed by 00
g
[10,13,14].
Hence, set
00
Cg
(2)
where C is a constant, which defines a set of equi-
geopotential surfaces. Eq.1 can be rewritten as
2
00 0
2
d +2
d
iij
iij
g
gv gvv
t
 (3)
where dd
ii
vxt denotes the particle's velocity. Since
the geopotential surface should keep the static balance
state, it holds 0
i
v
. Then, equation (3) becomes
2
00
2
d
d
t
(4)
From Eqs.2 and 4 one gets
2
2d
dC
t
(5)
Eq.5 shows that on the equi-geopotential surface pre-
cise clocks run with the same rate. Based on this equa-
tion, Bjerhammar (1985, 1986) defined the equi-geopo-
tential surface as “a closed curve surface on which all
the precise clocks run with the same rate” [2,15], which
could be properly called the equi-time-rate surface
[4,10,16].
The equigeopotential surface defined as above was
first put forward by Bjerhammar (1985, 1986) [2,15],
later redefined by Soffel et al. (1988b) in a more rigor-
ous sense [14], and it can be properly called the
equi-time-rate surface [16]. On the equigeopotential
surface, the clock’s running rate keeps the same, and
consequently the vibration frequency of the clock must
also keep the same [4,11]. That is to say, if there are two
points
A
and B on the equigeopotential surface,
there does not exist gravity frequency shift. In fact, as
the light signal propagates on the quigeopotential surface,
there does not exist the gain or loss of energy. Based on
this viewpoint, we can define the equi-geopotential sur-
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
390
face as follows [4,10,16]: the equigeopotential surface is
such a closed curve surface on which there does not exist
gravity frequency shift. The equigeopotential surface so
defined may be properly called the equi-frequency sur-
face [4,16].
2.2. Relativistic Geoid
In conventional geodesy, the geoid is defined as “the
closed equi-geopotential surface nearest to the mean sea
level” [1], which is referred to as the conventional geoid
for convenience.
In relativistic geodesy, based on the definition of the
equi-time-rate surface, Bjerhammar (1985, 1986) defined
the relativistic geoid as “the closed curve surface nearest
to the mean sea level on which precise clocks run with
the same rate” [2,15], which is properly referred to as the
equi-time-rate geoid [4,10]. In fact, based on the defini-
tion of the equi-time-rate surface, the relativistic geoid
can be simply defined as the equi-time-rate surface
nearest to the mean sea level.
According to the relativistic definition, the geoid can
be determined by using precise clocks. Combining Eqs.4
and 5 one can write down

12
12
00
1
ddd tg
C



 (6)
which gives rise to a clock’s running rate on an arbitrary
equi-time-rate surface. Suppose the equi-time-rate geoid
0
S and an arbitrary equi-time-rate surface
H
S are re-
spectively given by the following equations:
00 0
00
H
g
C
g
C
(7)
where 0
C and
H
C are the geopotential constants on
the equi-time-rate geoid 0
S and the equi-time-rate sur-
face
H
S, respectively. Then


12 12
00
dd, dd
HH
tCt C
 (8)
and consequently we have
0
0
dd
H
H
C
tt
C
(9)
where 0
dt and d
H
t denote the clocks’ running rates
(unit seconds) on the equi-time-rate geoid and the H-
equi-time-rate surface that passes the point just above
the datum point on the geoid with the OH, denoted by
H
, respectively.
It is noted that the difference between the relativistic
geoid and the conventional geoid is about 0.5 cm [10,17].
Such a difference could be neglected in general applica-
tions, but should be taken into account in high precise
geoid determination.
According to Eq.6 the geopotential value at an arbi-
trary point on the Earth’s surface can be determined
based on the clock transportation approach [3]. Though
there are other approaches for time comparison between
two separated clocks located at two stations, e.g., the
GPS common-view approach and the approach of
two-way time transfer by satellite [18], they provide the
accuracy about parts of nanoseconds, and consequently
they are too poor to determine a meaningful geopotential
difference. In non-rela- tivistic geodesy, the measure-
ments of the geopotentials are generally realized by
combining gravimetry and leveling. The measurement
procedure is very laborious, and the accumulated meas-
urement error becomes larger and larger as the length of
the measurement line increases. These drawbacks could
be overcome by clock transportation approach. We note
that the accuracy of determining the geopotentials by
using precise clocks depends on the accuracies of the
clocks. If the accuracy level of the atomic clocks is on
the order of 1016, the accuracy level of the determined
geopotentials corresponds to the height difference of 1
meter. In recent years, the time and frequency science
develop quickly. Atomic clocks with the stability level
of 1016 have been created [19-21]. It is noted that there
are several study groups investigating the “optical fre-
quency standard”, and significant results have been
achieved [22-27]. They compared the different “optical
frequency standards”, and found that all the stabilities
are in the level of 1018 to 1019 [27]. Scientists predict
that, in the near future, “optical clocks” with the stability
of 1018 could be realized. This will provide a firm
foundation for determining the geopotential or OH at the
centimeter level using clock transportation approach [3]
or frequency shift approach (see Section 3).
However, concerning the clock transportation ap-
proach, at present, the atomic clocks available are very
expensive, very heavy, and quite difficult for normal work
during the transportation. Hence, only if portable, rela-
tively cheap and precise clocks were created, one has to
pursue other approaches to determine the geopotential
differences. This is the motivation that the frequency shift
approach was proposed [4,7].
3. FREQUENCY SHIFT EQUATION OF
ELECTROMAGNETIC SIGNALS
On the equi-geopotential surface, an atomic clock’s
running rate keeps the same, and consequently the fre-
quency of an atomic clock must also keep the same
[6,10-12]. Since a clock’s running rate is controlled by
the vibration frequency, we can conclude that for arbi-
trary two points
A
and B at rest on a same equi-
geopotential surface there does not exist electromagnetic
signal’s frequency shift, which is referred to as the grav-
ity (or geopotential) frequency shift. In virtue of this
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
391
viewpoint, an equi-geopotential surface could be defined
as “a closed curve surface on which there does not exist
gravity frequency shift” [4,6,10,16], which is referred to
as the equi-frequency surface.
Then, based on the definition of the equi-frequency
surface the relativistic geoid could be defined as “the
closed curve surface nearest to the mean sea level on
which there does not exist gravity frequency shift”,
which is referred to as the equi-frequency geoid [4]. Or,
the relativistic geoid can be simply defined as the
equi-frequency surface nearest to the mean sea level
[4,6,10,16]. Based on the definition of the equi-frequ-
ency geoid one can determine the relativistic geoid by
measuring the gravity frequency shifts of electromag-
netic signals.
Since the frequency is inversely related to the period
based on which the unit second is defined (see http://en.
wikipedia.org/wiki/Second_unit), according to Eq.9 one
has [6,10,12]
0
0
H
H
C
f
f
C
(10)
where 0
C and
H
C are the geopotential constants cor-
responding to the geoid and the H-equi-frequency geo-
potential surface that passes the point just above the da-
tum point on the geoid with the OH, H, respectively, 0
f
and
H
f
are the atomic clocks’ frequencies on the
equi-frequency geoid and the H-equi-frequency geopo-
tential surface, respectively. By frequency shift observa-
tions,

00H
f
ff might be determined. Hence, based
on Eq.10, if the geopotential constant 0
C on the geoid
is determined,
H
C can be determined.
It is noted that, at least at present or in the near future,
the equi-frequency geoid is more realizable than the
equi-time-rate geoid [10]. At present, it is difficult to
generally realize the comparisons between two separated
clocks by clock transportation approach, due to the fact
that precise atomic clock are very expensive for general
usage. On the contrary, it is quite easy to generally real-
ize the frequency shift observations, e.g., the generally
used GPS observations.
Suppose a light signal with frequency
f
is emitted
from point P and it is received at point Q. Because
of the geopotential difference between these two points,
the frequency of the received signal is not
f
but
f
.
Based on Eq.4, the running rates of the atomic clocks
P and Q at arbitrary two points on ground are given
by the following equation


00
00
d
d
Q
P
PQ
g
t
tg
(11)
Based on the above equation one has


00
00
= 1
QP Q
PP
g
ff
fff
ff fg

 (12)
where
P
f
f
, Q
f
f
. In Eq.12,
P
f
and Q
f
are the
frequencies at P and Q, respectively. Accurate to the
order 2
V (V is the gravitational potential), 00
g
can
be expressed as [10,11]
22
00 12 22122
g
VV WV   (13)
where WV
 is the classical Newtonian geopoten-
tial,
is the centrifugal force potential. Throughout
this paper the definition of the geopotential in physical
geodesy is applied: it always holds that 0W, which is
different from the definition in physics. Combining Eqs.
12 and 13, accurate to W
, one has

QP
f
fffW fWW
  (14)
where
P
W and Q
W are the geopotentials at point P
and Q, respectively. Eq.14 is the gravity frequency shift
equation, which was confirmed by various physics ex-
periments [28-32].
The frequency approach has special advantages com-
pared to the clock transportation approach (Cf. Section
4). As mentioned before, concerning the OH determina-
tion, clock transportation approach is difficult for gen-
eral applications (Cf. Section 2.2). However, the gravity
frequency shift between arbitrary two points P and Q
on ground could be directly determined using GPS sig-
nals, even these two points are located far away from
each other (Cf. Section 4.2).
Suppose the geopotential at point P is given, then
from Eq.14 one can determine the geopotential at an
arbitrary point Q by measuring the gravity frequency
shift
f
between P and Q, in virtue of the following
equation
QP
f
WW
f
 (15)
If the point P is chosen on the geoid, then one has
(Shen et al., 2008a)
0Q
f
WC
f
 (16)
where 0
C is the geoid geopotential constant, the deter-
mination of which could be found in e.g., Chao et al.
(2007) [33]. Once 0
C is determined, the geopotential at
an arbitrary point Q on the Earth’s surface can be deter-
mined by using frequency shift observation method. The
basic principle of measuring the frequency shift is stated
in the sequel.
Referring to Figure 1, set at point P an emitter which
emits a signal with frequency
f
and a receiver at point
Q receives the emitted signal with frequency
f
com-
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
392
Figure 1. An emitter at point P on ground emits a light signal
with frequency f towards a receiver at point Q on ground, and
the received signal’s frequency is not f but f', the difference
between the received frequency and the emitting frequency is
the gravity frequency shift Δf = f' f.
ing from P. Then, comparing the frequency
f
of the
received signal with the standard frequency
f
itself,
the frequency shift
f
ff
 might be determined.
Consequently, according to Eq.16 one can determine the
geopotential difference
P
QQP
WWW between P
and Q. Using the same principle one can find the geo-
potential difference 00PP
WWW between the geoid
and the equi-frequency surface which contains the point
P. If 0
C is a given constant (generally it can be deter-
mined by satellite geodesy approach, see Section 5),
P
W
as well as Q
W can be found. According to Eq.16, once
the gravity frequency shift QP
f
ff between points
P and Q is determined, the geopotential difference
P
Q
W can be determined. If what it measured is the
gravitational frequency shift G
f
, it can be found the
gravitational potential difference
P
QQP
VVV be-
tween these two points by following equation:
G
PQQ P
f
f
f
VVV
f
f
  (17)
where
f
denotes the emitting frequency,
f
denotes
the received frequency due to the gravitational potential
difference between P and Q. Once
P
Q
V is deter-
mined, the geopotential difference
P
Q
W can be de-
termined, due to the fact that,
P
QPQPQ
WV,
P
QQP
 ,
P
and Q
are centrifugal force
potentials at P and Q, respectively, and they are
known quantities.
4. DIRECT ORTHOMETRIC HEIGHT
DETERMINATION
4.1. Orthometric Height Determination
between Two Points on Ground
In the sequel we consider how to determine the OH
difference
H
between two points P and Q accor-
ding to the measured gravity frequency shift
P
Q
f
be-
tween P and Q.
Without loss of generality, it is assumed that 0
PQ
f
.
In this case, from Eq.15 one gets
QP P
f
WW W
f
 (18)
This means that the geopotential value at point Q is
smaller than that at point P, and P and Q can be
taken for granted that they are located on two different
equi-frequency surfaces
P
WC and Q
WC, respec-
tively. It is noted that,
Wr is Newtonian geopotential
at the field point r, taking positive value, and the less
the value of the geopotential
Wr, the field point is
further from the center of the Earth. Let 0
WW denote
the equi-frequency geoid, then the geopotential differ-
ences between the equi-frequency geoid and the point
P as well as Q can be respectively expressed as
0
0
00
,Q
P
PQ
f
f
WW
f
f
 
(19)
where 0
P
f
and 0Q
f
express the gravity frequency
shifts between the equi-frequency geoid and the point
P as well as Q, respectively. Expanding the equi-
frequency surface
P
WC
into Tayler series with re-
spect to the OH H on the equi-frequency geoid 0
WW
,
one has
0P
P
W
WW H
H



 (20)
where
P
P
WH g
 is the gravity value on the
geoid corresponding to the point P, and
P
H
is the
OH of point P. When the height is not so large (e.g.,
less than 200 meters, i.e., the mountainous areas are not
considered), only the first two terms are kept in the
right-hand side of Eq.20, and instead of
P
g
one uses
the average normal gravity
. Hence one has
00PP
P
WWW
H


  (21)
Similarly
00QQ
Q
WWW
H


  (22)
It is noted that the condition under which Eqs .2 1 and
22 hold is that the height
H
is much smaller than the
Earth’s radius. From Eqs.21 and 22 one can find the
height difference between P and Q:
QP PQ
QP
CCW
HH H


   (23)
Substituting Eq.14 into Eq.23 one gets
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
393
1
f
H
f
 (24)
From Eq.24 one can see that the accuracy of
H
depends on that of
f
, and consequently it is related to
the stabilities of the frequencies of the emitter and re-
ceiver. In theory, if the stabilities of the frequencies of
the emitter and receiver are better than 1018 (which is
possible to be achieved because of quick development of
time science, Cf. Section 2), the accuracy in determining
the height difference between two different points could
achieve the order of centimeter. Based on the above
analysis one can see that, no matter what is determined,
the geopotential difference (14), the gravitational poten-
tial difference (17), or the height difference (24), the key
problem is how to measure the gravity frequency shift
and estimate the accuracy of
f
. We note that the gra-
vitational frequency shift can be determined if the grav-
ity frequency shift is determined and vice versa. It
should be emphasized that Eq.24 is only suitable to the
non-rough areas. In the mountainous areas, Eq.2 0 should
be kept to the second order of the height H. The details
are referred to [6,17].
4.2. OH Determination Using GPS Signals
Referring to Figure 2, suppose an emitter is set on
board a flying satellite (e.g., GPS satellite), which can
emit electromagnetic wave signals with regular intervals.
Then, by receiving the signals from the emitter simulta-
neously at two points P and Q, one could determine
the geopotential difference between P and Q, based
on the gravity frequency shift Eq.14.
Now, suppose the signal emitter E is set on board a
satellite, and two signal receivers P and Q on ground
receive the signals coming from E corresponding to an
emitting time t. Further suppose the received frequen-
cies of the signals corresponding to time t are recorded
Figure 2. Two receivers at points P and Q on the Earth’s sur-
face  receive simultaneously the light signals with fre-
quency f emitted on board a flying satellite S.
by P and Q receivers in some way, respectively, i.e.,
P
f
and Q
f
at
P
t and Q
t (P
tt and Q
tt, due to
the delay of the signal propagation) are recorded by re-
ceivers at P and Q, respectively. Note that the time
P
t at which the signal is received by P receiver is
generally different from the time Q
t at which the signal
is received by Q receiver. By comparing the received
frequencies P
f and Q
f it could be determined the
geopotential difference
P
QQP
WWW
 [4], which is
just given by Eq.14 .
One of the advantages by using the geopotential fre-
quency shift approach lies in that a unified global height
datum system could be established: two receivers lo-
cated at two height datum points
A
and B, which
belong to two separated continents or islands, could si-
multaneously receive the signals emitted by a satellite
source emitter, and consequently the frequency shift
between
A
and B is determined; then, based on the
geopotential frequency shift equation the geopotential
difference as well as the OH difference between
A
and
B is determined. By such a way, the height datum of
one continent (or island) could be connected to the
height datum of another continent (or island). Then, a
unified global height datum system might be established.
In practical applications, however, the gravity fre-
quency shift signals in GPS frequency observations are
largely contaminated by other noise frequency shifts,
which include the first-order Doppler frequency shift,
ionosphere frequency shift, troposphere frequency shift,
clock errors and random influences. To separate the
gravity frequency shift signals from other noises is not
so easy as generally imagined. Hence, to determine the
geopotential or OH of an arbitrary point on ground, the
key problem is how to draw the gravity frequency shift
signals from the GPS frequency observations. The in-
vestigations on this problem will be provided in a sepa-
rated paper.
5. UNIFICATION OF THE WORLD
ORTHOMETRIC HEIGHT DATUM
SYSTEM
Theoretically, the final determination of the geoid de-
pends on the choice of constant 00
WC on the geoid.
Given different 0
Ws, we get different equigeopotential
surfaces. In practice, to determine the geoid, we always
choose some tidal gauges’ average sea level as the da-
tum (standard) of a local geoid. Theoretically, if 0
W is
not determined properly, the real geoid will deviate from
the datum. The key problem lies in that at this situation
we still regard the datum as in consistency with the ge-
oid. As a result, there will be a systematic error in the
height system; and furthermore, since the mean sea level
does not coincides with any equigeopotential surface
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
394
[10,34-41], various datums in the world are in fact lo-
cated on different equigeopotential surfaces. This will
give rise to the inconsistency of the world height system.
If one chooses point A as the datum of the geoid, the
geopotential constant 0
W on the geoid cannot be cho-
sen arbitrarily but determined uniquely [10]. This is be-
cause of two causes. One is in that to determine the ge-
oid, a reference ellipsoid is needed. The normal potential
0
U on the surface of the ellipsoid is given a priori, or
can be uniquely determined by the given parameters of
the ellipsoid. No matter which method is chosen, 0
W is
completely determined, since one should investigate
gravity and normal gravity, geopotential and normal
geopotential, etc., in the same system (a unified coordi-
nate system). Another one lies in that to solve the boun-
dary value problem we require that the gravitational part
of W is regular at infinity. This condition (combining
with the choice of the ellipsoid) will limit the variation
of 0
W. Theoretically, we may suggest different methods
to determine 0
W. The most basic method might be
stated as follows [10]: Suppose we have chosen a refer-
ence ellipsoid r
E, e.g., WGS84 ellipsoid [42]. If a defi-
nite shape of the ellipsoid is given (i. e. , given r
E’s se-
mimajor axis a and semiminor axis b), the normal geo-
potential 0
U can be calculated theoretically, and con-
sequently the geopotential constant 0
W on the geoid is
determined (because of the condition that 00
UW
).
However, a dilemma occurs: If 0
U is given, the r
E’s
semimajor and semiminor axises a and b are determined
uniquely, but in this case we need to know 0
W a priori;
inversely, if a and b are given, 0
U is determined uni-
quely, but different a and b will introduce different 0
Us.
Without previous knowledge, it is impossible to choose
a and b so that 00
UW. The best way might be like this:
one determines 0
U so that it app- roximates 0
W grad-
ually. Hence, to precisely determine 0
W is a delicate
matter [33,43]. Any error stemmed from 0
W will give
rise to a systematic error to the geoid. In fact, How to
precisely determine 0
W is an open problem.
Once 0
Wis determined, the standard level geoid is
determined. Then, using GPS frequency shift approach,
we can unify the world height datum system. The basic
principle is stated as follows.
Suppose 0
W is precisely determined, e.g., with the
accuracy of 1 cm level. Then, with the same accuracy
level one can determine the OH of a datum point
A
located at least in a relatively plain area with small OH.
Then, the point
A
is taken as the datum point of world
height datum system. Referring to Figure 2, suppose the
OH
P
H
of a point P on ground is determined by
leveling plus gravimetry approach between
A
and P.
Then, using GPS frequency shift approach one can de-
termine the OH Q
H
of an arbitrary point Q on
ground. Since the point Q on ground is quite arbitrary,
one can unify the world height datum system using the
frequency shift approach.
6. DISCUSSIONS AND CONCLUSIONS
According to the GRT, a precise (atomic) clock lo-
cated at the position P with higher geopotential runs
quicker than a precise clock located at the position Q
with lower geopotential [12,44,45]. Equivalently, the
vibration frequency of the clock at P is larger than that
of the clock at Q. Then, the relativistic geoid can be
defined based on the clock’s running rate, which is re-
ferred to as the equi-time-rate geoid. In another aspect,
the relativistic geoid can be defined based on the clock’s
vibration frequency, or can be defined by the frequency
shift equation, and so defined geoid is referred to as the
equi-frequency geoid. The realization of the equi-freque-
ncy geoid is based on the frequency shift approach, and
the realization of the equi-time-rate geoid is based on the
clock transportation approach.
With clock transportation approach [3,46], the key
problem is to compare two clocks located at different
places by transporting portable clocks. Hence, one needs
portable clocks to complete the comparisons. At present,
though portable clocks with the stability better than 1 ×
1016 are not yet available, we may determine the geo-
potential difference at the accuracy level of 1 m2·s2
(equivalent to 0.1 m) between two separated points on
ground by using clocks with the stability around 1 ×
1014 (Shen et al., 2009). The problem is whether the
portable clocks with the stability 1 × 1014 [3] for the
aim of the transportation comparisons are available.
With frequency shift approach, especially the GPS
frequency shift approach [4,5,10,16,46], the key problem
is to draw the frequency shift information from the fre-
quency observations, which include other influences
except for the gravity frequency shift. In fact, the GPS
frequency observations include not only the gravity fre-
quency shift, but also other noise frequency shifts such
as the Doppler frequency shift, ionosphere frequency
shift, troposphere frequency shift, etc. The noise fre-
quency shifts should be removed from the observations.
Obviously, after removing the noise frequency shifts, the
accuracy in determining the OH depends on the stability
of the time-keeping system. If the stability of the
time-keeping system is better than 1 × 1018, we can de-
termine the OH with the accuracy of 1 cm.
To directly determine the geopotential and OH using
GPS signals, the GPS frequency shift approach is pro-
spective. This is due to the fact that the time and fre-
quency science develop very quickly, and clocks or
time-keeping systems and portable clocks with the sta-
bility better than 1 × 1018 might be available in the near
W. B. Shen et al. / Natural Science 3 (2011) 388-396
Copyright © 2011 SciRes. OPEN ACCESS
395
future [22,27]. Then, a new era may come that the geo-
metric position (coordinates) and the geopotential (as
well as OH) could be simultaneously determined using
GPS technique.
7. ACKNOWLEDGEMENTS
This study is supported by National 863 Program of China (grant No.
2006AA12Z211) and the National Natural Science Foundation of
China (grant No. 40637034).
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