HEIGHTS, POTENTIALS AND 
GEOPOTENTIALNUMBERS 
Asst.ProfDr. Božo Ko/er 
Faculty forCivi/Engineeringand Geodesy- Department of 
Geodesy, Ljubljana 
Received for publication: 21 January 1998 
Prepared forpublication: 20 March 1998 
Abstract 
The paperpresents the connection between measured height 
diff erence and the acceleration of gravity which defines the 
difference in potentials andgeopotentialheight numbers which 
serve asa basis far the calculation of point heights in varioits 
heightsystems. 
Keywords: acceleration of gravity, dijference in potential, 
geopotential nwnbers, height difference, levelsurface, potential 
1 INTRODUCTION 
Heights and height differences have both a geometrical and a physical meaning. 
One usually envisualises height to be the vertical distance of a certain point, 
located above a certain reference surface, from that surface. In this case, height is 
defined asa geometrical quantity. Practical experience has also taught us that two 
points have the same height when water between them does not move, which means 
that they lie on the same level surface and that the height difference between them is 
proportional to the difference in their potentials. This example illustrates the physical 
meaning of heights and height differences. Which of the two meanings of height or 
height difference is more important, the geometrical or the physical one, depends above 
all on the purpose of use of the heights or height differences. The physical 
interpretation of heights and height differences proves to be more suitable for the 
majority of natura! and artificial dynamic processes which take place on Earth (the 
movement ofwater and vehicles and the dynamics of constructed buildings). For 
geodesy and determination of the position of points in three-dimensional space, on the 
other hand, increasingly greater significance is ascribed to the geometrical 
interpretation ofheights and height differences. Geometrical interpretation has gained 
a special significance with the introduction of GPS technology into geodetic surveys. 
2 POTEN'fIAlLAND DIFFERENCE INPOTENTIAJL 
he vector of gravity acts on each point on the Earth's surface and is perpendicular 
to the level surface on which this point lies. All points therefore have a specific 
potential. All points with the same potential lie on the same level surface. Geodesy is 
concerned with the connection between the acceleration of gravity g and measured 
height difference, dh. For easier analysis, a scalar field is assigned to the vector field 
W(x, y, z) of the acceleration of gravity, such that 
g=gradW. 
Geodetski vestnik 42 (1998) 1 
The magnitude of the vector of gravity is the acceleration of gravity g 
dW 
g = dh. 2.1 
he height difference between two points is determined by the length of the vertical 
line between their leve! surfaces, for whichW(x, y, z) = constant (see Figure 2.2). 
The negative sign in the equation means that g and the height difference ( change in 
height) dh are inverscly proportional. There is a constant potential difference 
(WP, - WPJ. WP, and WP, between the two leve! surfaces which run through points P t 
and P 2 with constant potentials and . TI1ese two leve! surfaces are at a distance of dh. 
Due to the irregular distribution of mass in the Earth's interior, the acceleration of 
gravity gon leve! surfaces is a variable. Changes in the acceleration of gravity ( t.g) can 
be measured with great accuracy with the use of gravimeters. The unit for L'i.g used in 
geodesy is 1 gal = 10-2 m s-2 and is not included in the intemational system of units, SI. If 
equation 2.1 is written in the following form 
dW = - g dh = constant 2.2 
and taking into account the variation of g, it is clear that the difference between the 
level surfaces dh also varies in inverse proportion. It follows from this that the 
neighbouring leve! surfaces are not parallel. At higher levels of g, the distances between 
leve! surfaces dh is smaller (Bretterbauer, 1986). 
If one desires to determine unambiguous heights, independent of the levelling route, 
the definition of heights must be bound to potentials. We are interested in the 
potential difference, and this is independent of the route. A question arises of whether 
it is possible to determine the potential difference frorn data on the measured 
acceleration of gravity and levelled height difference. The first person to give an answer 
to this question was Helmert in 1884. The problem of determining the potential 
difference from measured height differences and the acceleration of gravity was first 
analysed for a single instrument station point. 
The measured height difference between points P2 and P,is obtained asa difference 
between readings (Z, S) ona vertically placed levelling staff at points P z and P" with 
a horizontal line of sight. In this case, the line of sight is a tangent to the leve! surface 
which runs through the optical centre of the objective. TI1e potential difference between 
points Pz and P, is (Leismann et al., 1992): 
P, 
Ws Wz= f gdh= - (lzgz lsgs ), 
pt'. 
where: 
lz, 15 ••• length of the vertical line through points Pz, P s between leve! surfaces 
(Wz = constant, W5 = constant and W = constant) 
2.3 
gz, g5 ••• corresponding mean value of the acceleration of gravity in the corresponding 
part of the vertical line. 
Geodetski vestnik42 ( 1998) 1 
Figure 2.1 
It can be seen in Figure 2.1 that the slightly curved parts of the vertical line can be 
replaced with the readings on the levelling staff Z and S, which are reduced by the 
values dz and d5• According to Helmert, the error in the levelling line is negligible. The 
values (Leismann et al., 1992) 
- 1 - 1 
gz =- f gdh and g5 =- f gdh 2.4 
lz ls 
can therefore be approximated with the value obtained on one half of readings on· the 
levelling staff, since at small height differences it can be assumed that the acceleration 
of gravity falls linearly with height. Equation 2.3 can therefore be written in the 
following form 
= - (Z - dz) gz + d5) gs. 2.5 
If equation 2.5 is transformed such that individual terms are expressed as a sum or 
difference of the reading§_ on the levelling staffs (Z and S), the mean values of the 
acceleration of gravity (gz, g5 ) and the values dz and d5, the following expression is 
obtained 
(Z-S) - - (Z+S) - -
-Wz = --2-(gz+&)--2-(gz-gs)+ 
(d - d ) - - (d + d ) -
+ z 2 s (gz + gs) + z 2 s (gz - gs), 
2.6 
If levelling is performed from the middle, as prescribed for precise levelling, the 
difference ( d2 - d5) is negligible because the curvatures of the level surface between 
the instrument station point and the stati on points of the levelling staffs ( at the front, at 
the back) are almost.equal. A similar consideration applies to the levelling line, 
therefore the third term in equation 2.6 can be omitted. It was established on the basis 
Geodetski vestnik42 (1998) 1 
of research performed by Baeschlin, Ramsayer and Zeger that the second and fourth 
terms in_!he above-mentioned equation are also negligible, and therefore also the value 
of (gz - g5). The following expression is obtained 
(Z-S) -
W5 -Wz= ~(g2 +&)· 2.7 
In addition, the following relation applies to the potential difference W5 - W z 
2.8 
where 
dhz, dh5 ..• distance between level surfaces Wz = constant and W5 = constant in point 
Pz or 
g~ , g~ ... mean value of the acceleration of gravity in the intervals P z - P {' and P s P t 
(see Figure 2.1 ). 
If g is taken to be 
- l -
g =-(gz + gs) 
2 
2.9 
equations 2.7 and 2.8 are equalised, and the equation showing the difference in readings 
on the levelling staff is solved, the following equations are obtained 
-(Z-S)=dhz g{' and -(Z-S)=dh5 g~'. 2.10 
g g 
The scale factors, g{' and gt, need not be taken into account because their influence is 
g g 
smaller than lxl0-8 Equations 2.10 can therefore be written with sufficient accuracy as 
follows 
dhz =-(Z-S)= dh5• 2.11 
This means that the difference in readings on levelling staffs (Z - S), at one station point 
of the instrument can be approximated with sufficient accuracy using the distance 
between leve! surfaces which run through the stati on points of the levelling staffs. The 
difference between the station points of levelling staffs is obtained by multiplying the 
measured height difference with the acceleration of gravity at the station point (see 
equation 2.9). For each individual station point, this value equals the acceleration of 
gravity at the height of Z+ S above the station point of the instrument (Leismann et 
4 
al., 1992). Since the measurement of the acceleration of gravity at this point is 
impractical, the value of the acceleration of gravity in equation 2.9 can according to 
Helmert be taken to be the arithmetic mean of the accelerations of gravity which were 
measured on the station points of the levelling staffs. On the basis of the 
above-mentioned equations it can be seen that the potential difference between points 
Pz and P5 can be determined on the basis of data on the measured acceleration of 
gravity and the levelled height difference. 
The potential difference in the levelling line between points P I and P 2 can be 
determined in a similar manner. If the height difference on one station point of the 
Geodetski vestnik42 (1998) 1 
instrument (reading at the back of the staff - reading at the front of the staff) is 
designated as 8hi, then thc measured height difference betwecn points P I and P 2 equals 
p? 
cth;; ~ [8 h;, 
i =Pl 
Since leve! surfaces of thc gravitational ficld are not parallel to each other, and because the calibration of thc leve! vial and the position of the spirit leve! 
compensator are closely connected with the gravitational field, point heights cannot be 
determined independently of the levelling route. It can be seen in Figure 2.2 that the 
levelled height difference depends on the route. If levelling is performed from P 1 
through P 2 ··to P 2 or from P 1 through P 1 ··to P 2, different results are obtained, because 
the levelling value along the leve! surfaces P 1P 2 „ and P 1 "P 2 equals zero. Only the 
potential difference (W P, - W r,), which is obtained by integrating equation 2.2, is 
independent of the route. In practice, the integral is approximated using a sum and the 
following expression is obtained (Bretterbauer, 1986): 
P2 P2 
Wr,-WP, = - J g dh;; = I g;8h;. 
P, 
Figure2.2 
Figure 2.2 shows: 
2.12 
8h, ... height difference between the two station points of the levelling staffs ( difference 
in readings at the back and at the front of the staff) 
dh~, ... height difference between points P 1 and P 2. 
3 GEOPOTENTIALNUMBERS 
It was established in the previous section that the potential difference between points P I and P 2 can be determined on the basis of data on the measured values of 
acceleration of gravity and levelled height differences. This type of levelling can be 
named geopotential levelling. It is defined as levelling which connects direct levclling 
and the measured acceleration of gravity. Potential differences at individual points with 
regard to the reference level surface, i.e. the geoid, were named geopotential numbers 
(C) by a French geodesist P. Tardi. The following expression applies to point Pi: 
P, 
Cp = Wpn -Wp = fg; dhpP/, 
1 J I l ' 
3.1 
pjl 
Geodetski vestnik42 ( 1998) 1 
where 
W PP ... potential of the reference level surface, the geoid 
W P, ... potential of the level surface through point Pi 
... point on the reference level surface - the geoid, assigned to point Pi. 
In practice, the integral is approximatcd by a sum to yield 
P, 
= Z: g;cSh;, 3.2 
where 
ohi ... height difference on the i-th station point ofthe instrument 
g; ... mean.value ofthe acceleration of gravitybetween station points i and i-1, 
therefore 
thc height of the reference level surface or the geoid is taken to be O, then the 
difference is the natura! measure of the heights of points on the Earth's 
surface. The unit of geopotential numbers is Nm/kg, i.e. work per mass unit. At their 
conference in Rome in 1954, the International Association of Geodesists adopted a 
geopotential number unit of 1 kgal m = 1 gpu (geopotential unit) = 10 Nrn/kg = 10 
m2/s2• The differences in geopotential numbers between benchmarks P I and P 2 can be 
calculated as follows (Bilajbegovic et al., 1989): 
ACP1 = gP1cthP1 
P1 P1 P1 ' 
g ;: is calculated using the following equation 
where 
g r, ... acceleration of gravity on the P 1 benchmark 
gP, ... acceleration of gravity on the P2 benchmark 
dh ;, ... measured height difference between benchmarks P1 and P 2• 
1 
4 CONCLUSION 
though geopotential numbers unambiguously define heights and point 
,,~,,,., ••• u are determined independently oflevelling route, are unsuitable for the 
majority of users because represent point heights defined entirely physically. The 
main shortcoming of geopotential nurnbers is that they cannot be interpreted 
geometrically and are not ex-pressed in metres, which is vi tal for many users. These two 
main shortcomings of geopotential numbers can be removed by dividing geopotential 
numbers with the acceleration of gravity at a certain point. Geopotential numbers thus 
represent the basis for the determination of point heights in different elevation systems 
( orthornetric heights and normal heights ), naturally with the exception of the 
Geodetski vestnik42 (1998) 1 
determination of the ellipsoid heights of points, which are point heights defined entirely 
geometrically. 
Literature: 
Bilajbegovic, A. ctal., Jstraživanja o izborusustava visinazaNVT SFRJ s obzirom na točnost ubizanja 
sile teže. Geodetski list, Zagreb, 1989, letnik43 (66), št. 4-6, str. 97-106 
Bretterbauer, K., Das Hoehenproblem clerGeoclaesie, Oesterreichische Zeitschriftfuer 
Vermessungswesen unclPhotogrammetrie, Dunaj, 1986, letnik 74, št. 4, str. 205-215 
Leismann, M. etal., Untersuchungen verschieclener Hoehensysteme, clargestellt an einerTestschleife in 
Rheinlancl-Pfalz. Muenchen,DGK-ReiheB, 1992, str. 3-30 
Review: Dr. Miran Kuhar 
Asst.ProfDr. Bojan Stopar 
Geodetskivestnik42 (1998) 1