SU3150 – PRINCIPLES OF GEODESY
Gravity
(Chapter 7)
Survey Measurements and the Gravity
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Surveyor and the gravity
Nature of the gravity field on or near earth
Equipotential surfaces
Modeling the geoid
Orthometric heights and the plumbline
3-D Positioning
Since most survey measurements are made under the
influence of gravity, they are affected by it
All direction measurements and hence the angles are affected
by the deflection of vertical
That is, instrument axis is aligned with the vertical, and
therefore, horizontal angles and directions are measured on a
plane that is perpendicular to the vertical, not the ellipsoidal
normal
Magnitude of deflection of vertical also varies due to local
variation of gravity field
This results in the unevenness of the geoid, i.e. the distance
separating the geoid and ellipsoid (geoid undulation) is not
constant
Geoid undulations affect the reduction of long lines, measured
on the topographic surface, to the surface of the ellipsoid
Geoid undulations are also needed if orthometric heights are to
be derived from geodetic heights such as those obtained from
GPS measurements
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Even though spirit leveling is used for the determination of
accurate orthometric heights, errors due to non-parallelism of
equipotential surfaces may affect the accuracy
Accurate orthometric heights are needed in the following
applications
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Topographic mapping
Construction of water systems
Monitoring of water systems
Other construction
Subsidence studies
Crustal motion studies
Gravimetry for prospecting and plate tectonic studies
Navigation
Property relationship to water
Elevations from GPS surveys
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Gravity Field of the Earth
Following properties apply to Earth’s gravity field
 combination of gravitational attraction and centrifugal
acceleration
 the combined effect is called gravity
 acts in the direction of plumbline and coincides with the
local vertical
 gravity generally increases from equator to poles
 there are also local variations due to change in mass
density of earth
 the presence of the force field of gravity creates a potential
It is more convenient to deal with potential than it is to deal with
gravity as potential is not a vector but gravity is
The total potential is the sum of gravitational potential and
centrifugal potential
W(X,Y.Z) = V(X.Y,Z) + (X,Y)
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The magnitude of gravity (g) is the gradient of potential (w) in
the direction of plumbline,
i.e.
g = - dw/dh
where h is the distance along plumbline (elevation)
Negative sign is due to the fact that gravity increases when
elevation decreases
A surface on which the potential has a constant value is called
an equi-potential surface or a level surface (of gravity)
Geoid is a particular equi-potential surface chosen to coincide
approximately with the mean ocean surface
The plumbline, i.e., direction of gravity, is perpendicular to the
surface of an equi-potential surface at every point on that
surface
Equi-potential surfaces converge towards poles, and therefore,
plumblines are curved, as a result of systematic increase of
gravity
The equi-potential surfaces are also undulated, the result of
local variations of gravity, and therefore, plumblines change
direction from point to point (deflection of vertical)
Determination of Geoid
The fundamental problem in physical (gravimetric) geodesy is
to determine the shape and size of the earth (geoid)
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The surface of the geoid is determined in relation to a
reference ellipsoid, called mean earth ellipsoid
This is done by both gravimetric methods and astro-geodetic
methods that use a combination of astronomic and geodetic
measurements
In recent years satellite altimetry and other satellite methods
have helped achieve good results
The problem of determining the geoid reduces to a problem of
determining geoid undulations (N) and deflection of vertical
components (,)
N gives the distance of any point on the geoid from the
reference ellipsoid and,  and  can determine the slope of
geoid surface relative to ellipsoidal surface
Recall that the reference ellipsoid is a mathematical surface
that has a chosen size and shape
The reference ellipsoid chosen must preferably be such that
 N2 = minimum, or
(2 + 2) = minimum
In astro-geodetic methods, geoid undulations and deflection of
vertical are relative to the coordinate origin but the reference
ellipsoid used has its minor axis parallel to the earth's rotational
axis
This type of ellipsoid when used as a datum is called a regional
datum
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In gravimetric method, a mean earth ellipsoid is defined to have
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same center as the mass center of earth
same mass (M)
same potential (wo)
same angular velocity ()
same difference in principal moments of inertia (C-A) as
that of geoid
C = moment of inertia w.r.t. rotational axis
A = moment of inertia w.r.t. an axis in the equatorial plane
The gravity field of the reference ellipsoid is defined to have
only systematic variations with latitude, and is called the normal
gravity
The difference between the normal gravity (computed) and the
actual gravity (measured on earth and reduced to geoid) is
called gravity anomalies (g)
In gravimetric method following formulas can be used to
determine the geoid undulations (N) and deflection of vertical
components (,)
These are computed by using the gravity anomalies g stated
above in Stoke’s formula and Vening Meinesz's formula shown
below
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Stoke's formula
N = ( R/4)   g. s(). d
and Vening Meinesz's formula
 
 cos  
 =(1/4)  g S()/d 
 d
 
 sin  
where
R = spherical radius of earth
 = an average normal gravity
d = surface element of gravity anomaly, g
S() = stokes' function
 = spherical distance(angle) from point where N is
computed to point where g was measured
 = azimuth of g
Above can be separately written as
 = (1/4)  g S()/d cos  d
 = (1/4)  g S()/d sin  d
If sufficient gravity anomalies are available, above equations
can be evaluated by summation of finite surface elements
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One problem in getting an accurate estimate of the geoid is
non-availability of gravity data for the whole earth
Very accurate values can be obtained by integrating over a
large, e.g.  =100  as the influence of gravity anomalies
beyond this is less than 1"
Gravity on the surface of earth can be measured directly with
the use of gravimeters
As gravity can only be measured on the topographic surface,
certain corrections are needed to reduce them to geoid
A combination of spherical harmonics and stokes' formula is
usually employed
External gravitational potential of the earth or the disturbing
potential can be expressed in terms of a spherical harmonic
function
Lower order harmonic coefficients have been evaluated using a
combination of gravity measurements and measurements from
low orbit satellites
Satellite altimetry determines the range from a satellite to the
ocean surface
Sea surface topography is determined after applying some
corrections
Data collected by all the above methods has been used in
realization of an accurate geoid for geodetic applications
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Geoid model is an algorithm that generates geoid heights for
any given point, e.g., Geoid96, Geoid99
Geoid models generally use gravity data in a grid and an
interpolation algorithm to generate geoid undulations and/or
deflection of vertical components
Effect of Anomalous Gravity in Geodetic Surveying
The effect of the deflection of vertical on horizontal directions
measured by conventional methods were discussed in chapter
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Heights referenced to the ellipsoid (geodetic heights) are also
needed in the reduction of measured distances to the datum
Since, in practice, only orthometric height may be available,
they need to be converted to geodetic heights by applying the
geoid undulation (geoid heights - N)
An accurate geoid model is needed when geoid undulations
need to be estimated for converting orthometric heights to
geodetic heights
Effect on Geodetic Leveling
Orthometric height is the linear distance along the plumbline
from the geoid to the point in question, generally on
topographic surface
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Two points having the same orthometric height may not be on
the same equi-potential surface as equi-potential surfaces are
not parallel
In differential leveling, same elevations are obtained for points
on the same equi-potential surface
This is because the level bubble that is leveled with respect to
gravity follows an equi-potential (level) surface
Non-parallelism of equi-potential surfaces are mostly prominent
in the north-south direction
For this reason, the difference between true orthometric height
and the height obtained by differential leveling is only
significant in this direction
Orthometric correction is applied to precise elevations
determined by leveling, in order to obtain accurate orthometric
heights
Elevations of a point from geoid can also be expressed in
terms of the difference in potential and are called geopotential
numbers
Unit of geopotential number is geopotential unit and 1 g.p.u.=
kgal.m = 1000 gal.m
Geopotential numbers in g.p.u. are almost equal to the height
above the geoid in meters and are sometimes called Dynamic
height
Satellite positioning provides coordinates X, Y, Z in a three
dimensional geocentric Cartesian system
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They can be converted to geodetic coordinates , , and h
(geodetic height)
If the geoid undulation N can be determined with a sufficient
accuracy, the orthometric height H can be derived from
H=h–N
The accuracy of both the geodetic height (h) determined with
current positioning technology and the geoid undulation (N)
available from current geoid models are not sufficiently
accurate for most surveying applications
However, the geodetic height differences between the end
points of baselines determined by GPS static relative
positioning method accurate within 1-2 cm.
Similarly, the current geoid models offer a better accuracy in
geoid height differences in short distances
Therefore, the geodetic height difference obtained by GPS can
be converted to an orthometric height difference by applying
the geoid height difference as follows
i.e.
h1 = H 1 + N 1
h2 = H 2 + N 2
h1- h2 = H1 - H2 + N1- N2
h =H + N
If the orthometric height of one point is accurately known then
the orthometric height of the other point can be accurately
computed
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This is achieved by including at least one good benchmark in
the survey
More accurate and reliable results can be obtained by including
a number of benchmarks in the GPS net and performing a
network adjustment (see NGS specifications)
Orthometric heights of intermediate points between
benchmarks can also be determined by interpolation
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